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[1] Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems
Yong Huang Hunan university Erwin Lutwak New York University Deane Yang New York University Gaoyong Zhang New York University
Analysis of PDEs Differential Geometry Convex and Discrete Geometry mathscidoc:1702.03003
Silver Award Paper in 2017
Acta Mathematica, 216, (2), 325–388, 2016.11
[ Download ] [ 2017-02-05 23:02:26 uploaded by YHuang ] [ 1479 downloads ] [ 0 comments ]
A longstanding question in the dual Brunn–Minkowski theory is "What are the dual analogues of Federer's curvature measures for convex bodies?" The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.
[2] Optimal Sobolev norms and the Lp Minkowski problem
Erwin Lutwak New York University Deane Yang New York University Gaoyong Zhang New York University
Analysis of PDEs Convex and Discrete Geometry mathscidoc:1703.03002
International Mathematics Research Notices, 62987, 2006
[ Download ] [ 2017-03-02 06:38:57 uploaded by deaneyang ] [ 1354 downloads ] [ 0 comments ]
[3] Trace theorems for some nonlocal energy spaces with heterogeneous localization
Xiaochuan Tian Columbia University Qiang Du Columbia University
Analysis of PDEs mathscidoc:1703.03009
SIAM J. Mathematical Analysis, 2017
[ Download ] [ 2017-03-14 19:26:04 uploaded by qd2125 ] [ 1312 downloads ] [ 0 comments ]
It is a classical result of Sobolev spaces that any $H^1$ function has a well-defined $H^{1/2}$ trace on the boundary of a sufficient regular domain. In this work, we present stronger and more general versions of such a trace theorem in a new nonlocal function space $\cS(\Omega)$ satisfying $H^1(\Omega)\subset \cS(\Omega)\subset L^2(\Omega)$. The new space $\cS(\Omega)$ is associated with a nonlocal norm characterized by a nonlocal interaction kernel that is defined heterogeneously with a special localization feature on the boundary. Through the heterogeneous localization, we are able to show that the $H^{1/2}$ norm of the trace on the boundary can be controlled by the nonlocal norm that are weaker than the classical $H^1$ norm. In fact, the trace theorems can be essentially shown without imposing any extra regularity of the function in the interior of the domain other than being square integrable. Implications of the new trace theorems to the coupling of local and nonlocal equations and possible further generalizations are also discussed.
[4] Calabi-Yau manifolds with isolated conical singularities
Hans-Joachim Hein Fordham University Song Sun Stony Brook University
Analysis of PDEs Complex Variables and Complex Analysis Differential Geometry Geometric Analysis and Geometric Topology Mathematical Physics Symplectic Geometry mathscidoc:1806.01002
Distinguished Paper Award in 2018
Publications mathématiques de l'IHÉS, 126, (1), 73-130, 2017.11
[ Download ] [ 2018-06-27 17:39:48 uploaded by hhein ] [ 1246 downloads ] [ 0 comments ]
[5] Semiclassical measures on hyperbolic surfaces have full support
Semyon Dyatlov University of California, Berkeley and Massachusetts Institute of Technology Long Jin Yau Mathematical Science Center, Tsinghua University
Analysis of PDEs Dynamical Systems mathscidoc:1808.03001
Best Paper Award in 2018
Acta Mathematica, 220, (2), 297 – 339, 2018
[ Download ] [ 2018-08-06 16:02:33 uploaded by jinlong ] [ 1229 downloads ] [ 0 comments ]
We show that each limiting semiclassical measure obtained from a sequence of eigenfunctions of the Laplacian on a compact hyperbolic surface is supported on the entire cosphere bundle. The key new ingredient for the proof is the fractal uncertainty principle, first formulated by Dyatlov-Zahl and proved for porous sets in Bourgain-Dyatlov.
[6] Wave propagation speed on fractals
Sze-Man Ngai Hunan Normal University and Georgia Southern University Wei Tang Hunan Normal University Yuanyuan Xie Hunan Normal University
[ Download ] [ 2017-01-05 18:04:58 uploaded by smngai ] [ 1108 downloads ] [ 0 comments ]
We study the wave propagation speed problem on metric measure spaces, emphasizing on self-similar sets that are not postcritically finite. We prove that a sub-Gaussian lower heat kernel estimate leads to infinite propagation speed, extending a result of Y.-T. Lee to include bounded and unbounded generalized Sierpi\'nski carpets as well as some fractal blowups. We also formulate conditions under which a Gaussian upper heat kernel estimate leads to finite propagation speed, and apply this result to two classes of iterated function systems with overlaps, including those defining the classical infinite Bernoulli convolutions.
[7] Uncertain Multilevel Programming: Algorithm and Applications
Baoding Liu Tsinghua University Kai Yao University of Chinese Academy of Sciences
Computers & Industrial Engineering, 89, 235-240, 2015
[ Download ] [ 2016-11-26 13:41:09 uploaded by utlab ] [ 1087 downloads ] [ 0 comments ]
Zhou J, Yang F, Wang K, et al. Multi-objective optimization in uncertain random environments[J]. Fuzzy Optimization and Decision Making, 2014, 13(4): 397-413.
Liu B, Chen X. Uncertain Multiobjective Programming and Uncertain Goal Programming[C]., 2015, 3(1).
Ding S. Uncertain multi-product newsboy problem with chance constraint[J]. Applied Mathematics and Computation, 2013: 139-146.
Ding S. Uncertain minimum cost flow problem[J]. soft computing, 2014, 18(11): 2201-2207.
Krichen S, Chaouachi J. The Resource Constrained Project Scheduling Problem[C]., 2014: 69-82.
Liu B. Toward uncertain finance theory[C]., 2013, 1(1): 1-15.
Yongchao Hou. Uncertain Programming with Recourse. 2012.
Wang L, Huang H, Ke H, et al. Chance-Constrained Model for RCPSP with Uncertain Durations[C]., 2015, 3(1).
Zhang C, Guo C, Yi S, et al. Airline Overbooking Problem with Uncertain No-Shows[J]. Journal of Applied Mathematics, 2014: 1-7.
Li R, Liu G. An uncertain goal programming model for machine scheduling problem[J]. Journal of Intelligent Manufacturing, 2014: 1-6.
[ Cited by 28 ]
[8] A Holder estimate for non-uniform elliptic equations in a random medium
Li-Ming Yeh National Chiao Tung University Shiah-Sen Wang National Chiao Tung University
[ Download ] [ 2016-09-21 22:30:23 uploaded by liming ] [ 1074 downloads ] [ 0 comments ]
[9] Generalized characteristics and Lax-Oleinik operators: global theory
Piermarco Cannarsa Universit\`a di Roma ``Tor Vergata'', Italy Wei Cheng Nanjing University, China
[ Download ] [ 2017-01-24 22:44:01 uploaded by treatise ] [ 1066 downloads ] [ 0 comments ]
For autonomous Tonelli systems on $\R^n$, we develop an intrinsic proof of the existence of generalized characteristics using sup-convolutions. This approach, together with convexity estimates for the fundamental solution, leads to new results such as the global propagation of singularities along generalized characteristics.
[10] On improved fractional Sobolev–Poincaré inequalities
Bartłomiej Dyda Faculty of Pure and Applied Mathematics, Wrocław University of Technology Lizaveta Ihnatsyeva Department of Mathematics, Kansas State University Antti V. Vähäkangas Department of Mathematics and Statistics, University of Jyvaskyla
Analysis of PDEs Functional Analysis mathscidoc:1701.03014
Arkiv for Matematik, 1-18, 2014.5
[ Download ] [ 2017-01-08 20:34:06 uploaded by arkivadmin ] [ 954 downloads ] [ 0 comments ]
We study a certain improved fractional Sobolev–Poincaré inequality on domains, which can be considered as a fractional counterpart of the classical Sobolev–Poincaré inequality. We prove the equivalence of the corresponding weak and strong type inequalities; this leads to a simple proof of a strong type inequality on John domains. We also give necessary conditions for the validity of an improved fractional Sobolev–Poincaré inequality, in particular, we show that a domain of finite measure, satisfying this inequality and a 'separation property', is a John domain.
[11] Estimates of the gaps between consecutive eigenvalues of Laplacian
Daguang Chen Tsinghua University Hongcang Yang Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Tao Zheng School of Mathematics and Statistics, Beijing Institute of Technology,
Pacific J. Math. , 2016
[ Download ] [ 2016-12-30 15:18:59 uploaded by Daguang ] [ 950 downloads ] [ 0 comments ]
Tao Zheng. Estimates of eigenvalues of a clamped problem. 2015.
[12] Efficient Parallel Simulation of Blood Flows in Abdominal Aorta
Shanlin Qin Rongliang Chen Ngai-Ching Wong 陳界山
SIAM Journal on Scientific Computing, 34, 2019.5
[ Download ] [ 2019-09-05 14:02:06 uploaded by NgaiChing_Wong ] [ 911 downloads ] [ 0 comments ]
It is known that the maximum diameter for the rupture-risk assessment of the abdominal aortic aneurysm is a generally good method, but not sufficient. Alternative features obtained with computational modeling may provide additional useful criteria. Though computational approaches are noninvasive, they are often time-consuming because of the high computational complexity. In this paper, we present a highly parallel algorithm for the numerical simulation of unsteady blood flows in the patient-specific abdominal aorta. We model the blood flow with the unsteady incompressible Navier-Stokes equations, and solve the discretized system with a highly scalable domain decomposition method. With this approach, the complete flow field can be obtained in less than an hour, instead of days with older methods. We show experimentally that the proposed method offers accurate solutions of the pressure, the velocity and the wall shear stress, and the parallel efficiency is higher than 70% on a parallel computer with more than 1,000 processor cores.
[13] Evolution of the two dimensional Boussinesq system
Tai-Man Tang Xiangtan University Ye Xie Xiangtan University
Dynamics of Partial Differential Equations, 13, (4), 333-350, 2016
[ Download ] [ 2017-03-11 11:56:09 uploaded by taimantang ] [ 900 downloads ] [ 0 comments ]
[14] On the formation of shocks for quasilinear wave equations
Shuang Miao University of Michigan Pin Yu Tsinghua University
[ Download ] [ 2016-08-17 22:17:13 uploaded by pinyu ] [ 859 downloads ] [ 0 comments ]
Demetrios Christodoulou · Daniel Perez. On the formation of shocks of electromagnetic plane waves in non-linear crystals. 2015.
[15] L^p gradient estimate for elliptic equations with high-contrast conductivities in R^n
Li-Ming Yeh National Chiao Tung University
[ Download ] [ 2016-09-21 22:37:14 uploaded by liming ] [ 859 downloads ] [ 0 comments ]
[16] Asymptotic-Preserving schemes for kinetic-fluid modeling of disperse two-phase flows with variable fluid density
Thierry Goudon INRIA Sophia Antipolis Méditerranée and Labo. J. A. Dieudonné Shi Jin Institute of Natural Sciences, and Ministry of Education Key Laboratory of Scientific and Engineering Computing, Shanghai Jiao Tong University Jian-Guo Liu Duke University Bokai Yan University of California, Los Angeles
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, 75, (1), 81–102, 2014.2
[ Download ] [ 2017-02-08 10:40:07 uploaded by jianguo ] [ 858 downloads ] [ 0 comments ]
We consider a system coupling the incompressible Navier–Stokes equations to the Vlasov–Fokker–Planck equation. Such a problem arises in the description of particulate flows. We design a numerical scheme to simulate the behavior of the system. This scheme is asymptotic-preserving, thus efficient in both the kinetic and hydrodynamic regimes. It has a numerical stability condition controlled by the non-stiff convection operator, with an implicit treatment of the stiff drag term and the Fokker–Planck operator. Yet, consistent to a standard asymptotic-preserving Fokker–Planck solver or an incompressible Navier–Stokes solver, only the conjugate–gradient method and fast Poisson and Helmholtz solvers are needed. Numerical experiments are presented to demonstrate the accuracy and asymptotic behavior of the scheme, with several interesting applications.
[17] Generalization of Selberg's $$ \frac{3}{{16}} $$ theorem and affine sieve
Jean Bourgain School of Mathematics, Institute for Advanced Study Alex Gamburd Department of Mathematics, University of California at Santa Cruz Peter Sarnak School of Mathematics, Institute for Advanced Study
Analysis of PDEs Combinatorics Functional Analysis mathscidoc:1701.03005
Acta Mathematica, 207, (2), 255-290, 2009.12
[ Download ] [ 2017-01-08 20:33:57 uploaded by actaadmin ] [ 833 downloads ] [ 0 comments ]
An analogue of the well-known $$ \frac{3}{{16}} $$ lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non-elementary subgroup$L$of SL(2,$Z$). The proof in the case that the Hausdorff of the limit set of$L$is bigger than $$ \frac{1}{2} $$ is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting. In the case that delta is less than $$ \frac{1}{2} $$ we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function. These "spectral gaps" are then applied to sieving problems on orbits of such groups.
[18] Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime
Yuan Gao Fudan University Jian-Guo Liu Duke University Jianfeng Lu Duke University
We study in this work a continuum model derived from 1D attachment-detachment-limited (ADL) type step flow on vicinal surface, $ u_t=-u^2(u^3)_{hhhh}$, where $u$, considered as a function of step height $h$, is the step slope of the surface. We formulate a notion of weak solution to this continuum model and prove the existence of a global weak solution, which is positive almost everywhere. We also study the long time behavior of weak solution and prove it converges to a constant solution as time goes to infinity. The space-time H\"older continuity of the weak solution is also discussed as a byproduct.
[19] On convolution groups of completely monotone sequences/functions and fractional calculus
LEI LI Duke University Jian-Guo Liu Duke University
We study convolution groups generated by completely monotone sequences and completely monotone functions. Using a convolution group, we define a fractional calculus for a certain class of distributions. When acting on causal functions, this definition agrees with the traditional Riemann-Liouville definition for t>0 but includes some singularities at t=0 so that the group property holds. Using this group, we are able to extend the definition of Caputo derivatives of order in (0,1) to a certain class of locally integrable functions without using the first derivative. The group property allows us to de-convolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general Gronwall inequality (or comparison principle) with the most general conditions. This then opens the door of a priori energy estimates of fractional PDEs. Some other fundamental results for fractional ODEs are also established within this frame under very weak conditions. Besides, we also obtain some interesting results about completely monotone sequences.
[20] Spectral analysis and computation of effective diffusivities in space-time periodic incompressible flows
N. Benjamin Murphy University of California at Irvine Elena Cherkaev University of Utah Jack Xin University of California at Irvine Jingyi Zhu University of Utah Kenneth M. Golden University of Utah
Annals of Mathematical Sciences and Applications, 0, (0), 2014.1
The diffusive transport of passive tracers or particles can be enhanced by incompressible, turbulent flow fields. Analyzing the effective behavior is a challenging problem with theoretical and practical importance in many areas of science and engineering, ranging from the transport of mass, heat, and pollutants in geophysical flows to sea ice dynamics and turbulent combustion. The long time, large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity tensor D* . Two different formulations of integral representations for D* were developed for the case of time-independent fluid velocity fields, involving spectral measures of bounded self-adjoint operators acting on vector fields and scalar fields, respectively. Here, we extend both of these approaches to the case of space-time periodic velocity fields, allowing for chaotic dynamics, providing rigorous integral representations for D* involving spectral measures of unbounded self-adjoint operators.We prove the different formulations are equivalent. Their correspondence follows from a one-to-one isometry between the underlying Hilbert spaces. We also develop a Fourier method for computing D* , which captures the phenomenon of residual diffusion related to Lagrangian chaos of a model flow. This is reflected in the spectral measure by a concentration of mass near the spectral origin.
[21] Global existence of smooth solutions and convergence to Barenblatt solutions for the physical vacuum free boundary problem of compressible Euler equations with damping
TAO LUO Georgetown University Huihui Zeng Tsinghua University
Comm. Pure Appl. Math. , 69
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For the physical vacuum free boundary problem with the sound speed being C^1/2-Holder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the porous media equation with the same total mass when the initial datum is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in the supremum norm, and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher-order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher-order nonlinear energy estimates, and elliptic estimates.
[22] On the regularity of solutions to a generalization of the Minkowski problem
Erwin Lutwak New York University Vladimir Oliker Emory University
Journal of Differential Geometry, 41, (1), 227–246, 1995
[ Download ] [ 2017-03-02 06:30:27 uploaded by lutwak ] [ 818 downloads ] [ 0 comments ]
[23] The logarithmic Minkowski problem
Károly J. Böröczky Alfréd Rényi Institute of Mathematics Erwin Lutwak New York University Deane Yang New York University Gaoyong Zhang New York University
Journal of the American Mathematical Society, 26, (3), 831–852, 2013
[ Download ] [ 2017-02-28 04:45:36 uploaded by deaneyang ] [ 791 downloads ] [ 0 comments ]
In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.
[24] Spectroscopic imaging of a dilute cell suspension
Habib Ammari ETH Zürich Josselin Garnier Université Paris VII Laure Giovangigli University of California at Irvine Wenjia Jing Tsinghua University Jin-Keun Seo Yonsei University
J. Math. Pures Appl., 105, (5), 603-661, 2016
[ Download ] [ 2017-02-22 08:41:12 uploaded by wjing ] [ 781 downloads ] [ 0 comments ]
The paper aims at analytically exhibiting for the first time the fundamental mechanisms underlying the fact that effective biological tissue electrical properties and their frequency dependence reflect the tissue composition and physiology. For doing so, a homogenization theory is derived to describe the effective admittivity of cell suspensions. A new formula is reported for dilute cases that gives the frequency-dependent effective admittivity with re- spect to the membrane polarization. Different microstructures are shown to be distinguish- able via spectroscopic measurements of the overall admittivity using the spectral properties of the membrane polarization. The Debye relaxation times associated with the membrane polarization tensor are shown to be able to give the microscopic structure of the medium. A natural measure of the admittivity anisotropy is introduced and its dependence on the frequency of applied current is derived. A Maxwell-Wagner-Fricke formula is given for concentric circular cells, and the results can be extended to the random cases. A randomly deformed periodic medium is also considered and a new formula is derived for the overall admittivity of a dilute suspension of randomly deformed cells.
[25] Higher integrability for vector-valued parabolic quasi-minimizers on metric measure spaces
Jens Habermann Department Mathematik, Universität Erlangen
Arkiv for Matematik, 54, (1), 85-123, 2015.1
We establish local higher integrability estimates for upper gradients of vector-valued parabolic quasi-minimizers in metric measure spaces, satisfying a doubling property and supporting a weak Poincaré inequality.
[26] 2D Homogeneous Solutions to the Euler Equation
Xue Luo Beihang University Roman Shvydkoy University of Illinois at Chicago
Communications in Partial Differential Equations, 40, 1666-1687, 2015
[ Download ] [ 2017-01-29 10:33:49 uploaded by xluo ] [ 772 downloads ] [ 0 comments ]
In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $\bu = \n^\perp \Psi$, $\Psi(r,\th) = r^{\l} \psi(\th)$, for $\l >0$, we show that only trivial solutions exist in the range $0<\l<1/2$, i.e. parallel shear and rotational flows. In other cases many new solutions are exhibited that have hyperbolic, parabolic and elliptic structure of streamlines. In particular, for $\l>9/2$ the number of different non-trivial elliptic solutions is equal to the cardinality of the set $(2,\sqrt{2\l}) \cap \N$. The case $\l = 2/3$ is relevant to Onsager's conjecture. We underline the reasons why no anomalous dissipation of energy occurs for such solutions despite their critical Besov regularity $1/3$.
[27] Uniform bound and convergence for elliptic homogenization problems
[28] Asymptotic behavior of equilibrium states of reaction–diffusion systems with mass conservation
Jann-Long Chern Central University
Journal of Differential Equations, (264), 550–574, 2018
[ Download ] [ 2018-03-05 21:29:05 uploaded by chern ] [ 753 downloads ] [ 0 comments ]
We deal with a stationary problem of a reaction–diffusion system with a conservation law under the Neu-mann boundary condition. It is shown that the stationary problem turns to be the Euler–Lagrange equation of an energy functional with a mass constraint. When the domain is the finite interval (0, 1), we investigate the asymptotic profile of a strictly monotone minimizer of the energy as d, the ratio of the diffusion coef-ficient of the system, tends to zero. In view of a logarithmic function in the leading term of the potential, we get to a scaling parameter κsatisfying the relation ε:=√d=√logκ/κ2. The main result shows that a sequence of minimizers converges to a Dirac mass multiplied by the total mass and that by a scaling with κthe asymptotic profile exhibits a parabola in the nonvanishing region. We also prove the existence of an unstable monotone solution when the mass is small.
[29] Entire solutions with merging three fronts to the Allen-Cahn equation
Yan-Yu Chen Tamkang University Jong-Shenq Guo Tamkang University Hirokazu Ninomiya Meiji University Chih-Hong Yao Tamkang University
[ Download ] [ 2016-09-15 06:48:09 uploaded by jsguo ] [ 751 downloads ] [ 0 comments ]
In this paper, we study entire solutions of the Allen-Chan equation in one-dimensional Euclidean space. This equation is a scalar reaction-diffusion equation with a bistable nonlinearity. It is well-known that this equation admits three different types of traveling fronts connecting two of its three constant states. Under certain conditions on the wave speeds, the existence of entire solutions with merging these three traveling fronts is shown by constructing a suitable pair of super-sub-solutions.
[30] Global Well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional Compressible Navier-Stokes equations
XIANGDI HUANG University of Science and Technology of China JING LI Academy of Mathematics and Systems Science ZHOUPING XIN The Chinese University of Hong Kong
Analysis of PDEs Mathematical Physics mathscidoc:1705.03006
CPAM, 14, (4), 791-825, 2012
[ Download ] [ 2017-05-31 10:00:43 uploaded by yauawardadmin ] [ 750 downloads ] [ 0 comments ]
We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field, which could be either vacuum or nonvacuum. The initial density is allowed to vanish, and the spatial measure of the set of vacuum can be arbitrarily large; in particular, the initial density can even have compact support. These results generalize previous results on classical solutions for initial densities being strictly away from vacuum and are the first for global classical solutions that may have large oscillations and can contain vacuum states. © 2012 Wiley Periodicals, Inc
[31] Heterogeneously localized nonlocal operators, boundary traces and variational problems
Qiang Du Columbia University Xiaochuan Tian Columbia University
Proceedings of ICCM 2016, 2017
[ Download ] [ 2017-03-14 19:30:05 uploaded by qd2125 ] [ 749 downloads ] [ 0 comments ]
It is a classical result of Sobolev spaces that any $H^1$ function has a well-defined $H^{1/2}$ trace on the boundary of a sufficient regular domain. We discuss its recent extensions given in \cite{TiDu16} in some heterogeneously localized nonlocal function spaces. The new trace theorems are stronger and more general than the classical result. They can be established essentially for all functions having only square integrability away from the boundary or in any compact subset of interior domain. Yet, the heterogeneous localization offers the necessary regularity precisely at the boundary to have well-defined traces. A consequence is that we may study associated Dirichlet type boundary value problems, as well as the coupling of local and nonlocal equations through co-dimension-1 interfaces. %Their derivations not only involve various extensions of classical inequalities but also %require new techniques absent from traditional approaches.
[32] Regularity of the Schrödinger equation for the harmonic oscillator
Bruno Bongioanni Instituto de Matemática Aplicada del Litoral, CONICET Keith M. Rogers Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas
Arkiv for Matematik, 49, (2), 217-238, 2009.8
We consider the Schrödinger equation for the harmonic oscillator$i$$∂$_{$t$}$u$=$Hu$, where$H$=−Δ+|$x$|^{2}, with initial data in the Hermite–Sobolev space$H$^{−$s$/2}$L$^{2}(ℝ^{$n$}). We obtain smoothing and maximal estimates and apply these to perturbations of the equation and almost everywhere convergence problems.
[33] Elliptic and parabolic equations in fractured media
Liming Yeh. Lp gradient estimate for elliptic equations with high-contrast conductivities in RnRn. 2016.
[34] The existence and nonexistence of global solutions for a semilinear heat equation on graphs
Yong Lin Renmin University of China Yiting Wu Renmin University of China
Analysis of PDEs Combinatorics mathscidoc:1804.03003
Calculus of Variations and Partial Differential Equations, 56, 2017.3
[ Download ] [ 2018-04-03 14:46:58 uploaded by linyong01 ] [ 737 downloads ] [ 0 comments ]
[35] Pointwise estimate for elliptic equations in periodic perforated domains
[36] On the topology of the set of singularities of a solution of the Hamilton-Jacobi equation
Piermarco Cannarsa Universit\`a di Roma ``Tor Vergata'', Italy Wei Cheng Nanjing University, China Albert Fathi ENS de Lyon \& IUF, France
[ Download ] [ 2017-01-24 22:33:10 uploaded by treatise ] [ 733 downloads ] [ 0 comments ]
We address the topology of the set of singularities of a solution of a Hamilton-Jacobi equation. For this we will apply the idea of the first two authors (Generalized characteristic and Lax-Oleinik operators: global result, preprint, arXiv:1605.07581, 2016.) to use the positive Lax-Oleinik semi-group to propagate singularities. {\it To cite this article: P. Cannarsa, W. Cheng, A. Fathi, C. R. Acad. Sci. Paris, Ser. I 340 (2005).
[37] Mean Field Equation of Liouville Type with Singular Data Topological Degree
CHIUN-CHUAN CHEN National Taiwan University Chang-Shou Lin National Taiwan University
CPAM, 887-947, 2015
We consider the following mean field equation: $$\Delta_{g}v+\rho\left(\frac{h^*e^v}{\int_Mh^*e^v}-1\right)=4\pi\sum_{j=1}^N\alpha_j(\delta_{q_j}-1)$$ on M, where M is a compact Riemann surface with volume 1, $h^*$ is a positive $C^1$ function on M, and $\alpha_j$ are constants satisfying $\alpha_j>-1$. In this paper, we derive the topological degree counting formula for noncritical values of $\rho$. We also give several applications of this formula, including existence of curvature +1 metric with conic singularities, doubly periodic solutions of electroweak theory, and a special case of self-gravitating strings.
[38] Transonic Shocks for the Full Compressible Euler Systems in a general two-dimensional de Laval nozzle
Jun Li Stanford University Zhouping Xin The Chinese University of Hong Kong Huicheng Yin Nanjing University
ARMA, 207, (2), 2013
In this paper, we study the transonic shock problem for the full compressible Euler system in a general two-dimensional de Laval nozzle as proposed in Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New York, 1948): given the appropriately large exit pressure pe(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes pe(x). We solve this problem completely for a general class of de Laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible Euler system. The problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system. One of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted Hölder space with low regularities for a second order quasilinear elliptic equation with a free parameter (the position of the shock curve at one wall of the nozzle) and non-local terms involving the trace on the shock of the first order derivatives of the unknown function.
[39] Mean Field Equations, Hyperelliptic curves and modular forms II
Chang-Shou Lin National Taiwan University Chin-Lung Wang National Taiwan University
Analysis of PDEs Geometric Analysis and Geometric Topology mathscidoc:1705.03001
The Journal de Ecole polytechnique , 2017.3
[40] On the constant scalar curvature K\"ahler metrics-apriori estimates
Xiuxiong Chen Stony Brook University Jingrui Cheng Stony Brook University
Analysis of PDEs arXiv subject: Differential Geometry (math.DG) mathscidoc:2005.03003
[ Download ] [ 2020-05-13 03:38:59 uploaded by jrcheng ] [ 712 downloads ] [ 0 comments ]
In this paper, we derive apriori estimates for constant scalar curvature K\"ahler metrics on a compact K\"ahler manifold. We show that higher order derivatives can be estimated in terms of a $C^0$ bound for the K\"ahler potential. We also discuss some local versions of these estimates which can be of independent interest.
[41] A geometric approach to generalized Stokes conjectures
Eugen Varvaruca Department of Mathematics and Statistics, University of Reading Georg S. Weiss Mathematical Institute, Heinrich Heine University
Analysis of PDEs Geometric Modeling and Processing mathscidoc:1701.03003
Acta Mathematica, 206, (2), 363-403, 2009.8
We consider the Stokes conjecture concerning the shape of extreme 2-dimensional water waves. By new geometric methods including a non-linear frequency formula, we prove the Stokes conjecture in the original variables. Our results do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity. Part of our results extends to the mathematical problem in higher dimensions.
[42] Classification of nondegeneracy of Toda system with singular sources
Chang-Shou Lin National Taiwan University Juncheng Wei Chinese University of Hong Kong Dong Ye Université de Metz
Analysis of PDEs Differential Geometry mathscidoc:1705.03005
Inventions , 169-207, 2012
[43] Resonance-free region in scattering by a strictly convex obstacle
Long Jin Department of Mathematics, University of California, Berkeley
Arkiv for Matematik, 52, (2), 257-289, 2012.12
Long Jin. Scattering Resonances of Convex Obstacles for General Boundary Conditions. 2014.
We prove the existence of a resonance-free region in scattering by a strictly convex obstacle $\mathcal{O}$ with the Robin boundary condition $\partial_{\nu}u+\gamma u|_{\partial\mathcal{O}}=0$ . More precisely, we show that the scattering resonances lie below a cubic curve ℑ$ζ$=−$S$|$ζ$|^{1/3}+$C$. The constant$S$is the same as in the case of the Neumann boundary condition$γ$=0. This generalizes earlier results on cubic pole-free regions obtained for the Dirichlet boundary condition.
[44] Analytical validation of a continuum model for the evolution of a crystal surface in multiple space dimensions
Jian-Guo Liu Duke University Xiangsheng Xu Mississippi State University
In this paper we are concerned with the existence of a weak solution to the initial boundary value problem for the equation ∂ t u=Δ(Δu) −3 . This problem arises in the mathematical modeling of the evolution of a crystal surface. Existence of a weak solution u with Δu≥0 is obtained via a suitable substitution. Our investigations reveal the close connection between this problem and the equation ∂ t ρ+ρ 2 Δ 2 ρ 3 =0 , another crystal surface model first proposed by H. Al Hajj Shehadeh, R. V. Kohn and J. Weare in [1].
[45] Non-uniform elliptic equations in convex Lipschitz domains
[46] Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media
Jack Xin
Journal of statistical physics, 73, 893-926, 1993.12
[ Download ] [ 2019-12-24 21:01:26 uploaded by Jack_Xin ] [ 658 downloads ] [ 0 comments ]
We review the existence results of traveling wave solutions to the reaction-diffusion equations with periodic diffusion (convection) coefficients and combustion (bistable) nonlinearities. We prove that whenever traveling waves exist, the solutions of the initial value problem with either frontlike or pulselike data propagate with the constant effective speeds of traveling waves in all suitable directions. In the case of bistable nonlinearity and one space dimension, we give an example of nonexistence of traveling waves which causes quenching (localization) of wavefront propagation. Quenching (localization) only occurs when the variations of the media from their constant mean values are large enough. Our related numerical results also provide evidence for this phenomenon in the parameter regimes not covered by the analytical example. Finally, we comment on the role of the effective wave speeds in
[47] An Efficient Semi-Implicit Immersed Boundary Method for the Navier-Stokes Equations
Thomas Y. Hou Caltech Zuoqiang Shi Tsinghua University
Analysis of PDEs Numerical Analysis and Scientific Computing mathscidoc:1709.03002
Journal of Computational Physics, 227, 8968-8991, 2008
[ Download ] [ 2017-09-27 16:07:30 uploaded by shizqi ] [ 657 downloads ] [ 0 comments ]
The immersed boundary method is one of the most useful computational methods in studying fluid structure interaction. On the other hand, the Immersed Boundary method is also known to require small time steps to maintain stability when solved with an explicit method. Many implicit or approximately implicit methods have been proposed in the literature to remove this severe time step stability constraint, but none of them give satisfactory performance. In this paper, we propose an efficient semi-implicit scheme to remove this stiffness from the immersed boundary method for the Navier–Stokes equations. The construction of our semi-implicit scheme consists of two steps. First, we obtain a semiimplicit discretization which is proved to be unconditionally stable. This unconditionally stable semi-implicit scheme is still quite expensive to implement in practice. Next, we apply the small scale decomposition to the unconditionally stable semi-implicit scheme to construct our efficient semi-implicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit scheme can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme. Our extensive numerical experiments show that our semiimplicit scheme has much better stability property than an explicit scheme. This offers a substantial computational saving in using the immersed boundary method.
[48] Homogenization of interfaces moving in spatially random temporally periodic environment
Wenjia Jing Tsinghua University Hung V. Tran University of Wisconsin at Madison Panagiotis E. Souganidis The University of Chicago
We study the averaged behavior of interfaces moving with oscillatory normal velocity that is periodic in time and stationary ergodic in space. This problem can be interpreted as a homogenization problem of a Hamilton-Jacobi equation with a positively homogeneous and time dependent Hamiltonian. In an earlier work, we studied the setting when the environment is periodic in space and random in time, and established homogenization results through the averaging of reachable sets. In the present setting, we show that the minimal travel time between two spatial points, which now depends also on a starting time, has a deterministic averaging limit that is independent of the starting time. This averaged travel time function is then shown to determine the effective behavior of the moving interfaces.
[49] Random conformal weldings
Kari Astala Department of Mathematics and Statistics, University of Helsinki Antti Kupiainen Department of Mathematics and Statistics, University of Helsinki Eero Saksman Department of Mathematics and Statistics, University of Helsinki Peter Jones Department of Mathematics, Yale University
Analysis of PDEs Functional Analysis Probability mathscidoc:1701.03004
We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle. The homeomorphism is constructed using the exponential of$βX$, where X is the restriction of the 2-dimensional free field on the circle and the parameter$β$is in the "high temperature" regime $$ \beta < \sqrt {2} $$ . The welding problem is solved by studying a non-uniformly elliptic Beltrami equation with a random complex dilatation. For the existence a method of Lehto is used. This requires sharp probabilistic estimates to control conformal moduli of annuli and they are proven by decomposing the free field as a sum of independent fixed scale fields and controlling the correlations of the complex dilatation restricted to dyadic cells of various scales. For the uniqueness we invoke a result by Jones and Smirnov on conformal removability of Hölder curves. Our curves are closely related to SLE($ϰ$) for$ϰ$<4.
[50] On a doubly critical Schr odinger system in R4 with steep potential wells
Yuanze Wu China University of Mining and Technology Wenming Zou Tsinghua University
[ Download ] [ 2016-11-29 17:54:10 uploaded by wzou ] [ 647 downloads ] [ 0 comments ]
[51] Well-posedness for the motion of physical vacuum of the three-dimensional compressible Euler equations with or without self-gravitation
Tao Luo Georgetown University Zhouping Xin Chinese University of Hong Kong Huihui Zeng Tsinghua University
Arch. Ration. Mech. Anal., 213
This paper concerns the well-posedness theory of the motion of a physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in (Coutand et al., Commun Math Phys 296:559–587, 2010; Coutand and Shkoller, Arch Ration Mech Anal 206:515–616, 2012; Jang andMasmoudi,Well-posedness of compressible Euler equations in a physical vacuum, 2008) by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary.
[52] The Fu–Yau equation with negative slope parameter
Duong Phong Columbia University Sebastien Picard Columbia University Xiangwen Zhang University of California, Irvine
Analysis of PDEs Complex Variables and Complex Analysis Differential Geometry mathscidoc:1810.03001
[ Download ] [ 2018-10-30 14:45:57 uploaded by xiangwen ] [ 645 downloads ] [ 0 comments ]
[53] A volume preserving flow and the isoperimetric problem in warped product spaces
Pengfei Guan McGill University Junfang Li University of Alabama Mu-Tao Wang Columbia University
Analysis of PDEs Differential Geometry Geometric Analysis and Geometric Topology Metric Geometry mathscidoc:1804.03009
[ Download ] [ 2018-04-30 14:24:26 uploaded by mutaowang ] [ 642 downloads ] [ 0 comments ]
In this article, we continue the work in Guan-Li and study a normalized hypersurface flow in the more general ambient setting of warped product spaces. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. As an application, the isoperimetric problem in warped product spaces is solved for such domains.
[54] Nonlinear asymptotic stability of the Lane-Emden solutions for the viscous gaseous star problem with degenerate density dependent viscosities
Tao Luo City University of Hong Kong Zhouping Xin Chinese University of Hong Kong Huihui Zeng Tsinghua University, Harvard University CMSA
Comm. Math. Phy, 347
The nonlinear asymptotic stability of Lane-Emden solutions is proved in this paper for spherically symmetric motions of viscous gaseous stars with the density dependent shear and bulk viscosities which vanish at the vacuum, when the adiabatic exponent γ lies in the stability regime (4/3, 2), by establishing the global-in-time regularity uniformly up to the vacuum boundary for the vacuum free boundary problem of the compressible Navier-Stokes-Poisson systems with spherical symmetry, which ensures the global existence of strong solutions capturing the precise physical behavior that the sound speed is C1/2-Hölder continuous across the vacuum boundary, the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of Lane-Emden solutions with detailed convergence rates, and the detailed large time behavior of solutions near the vacuum boundary. Those uniform convergence are of fundamental importance in the study of vacuum free boundary problems which are missing in the previous results for global weak solutions. Moreover, the results obtained in this paper apply to much broader cases of viscosities than those in Fang and Zhang (Arch Ration Mech Anal 191:195–243, 2009) for the theory of weak solutions when the adiabatic exponent γ lies in the most physically relevant range. Finally, this paper extends the previous local-in-time theory for strong solutions to a global-in-time one.
[55] Fluctuations in the Homogenization of Semilinear Equations with Random Potentials
Guillaume Bal Columbia University Wenjia Jing The University of Chicago
Communications in Partial Differential Equations, 42, (12), 1839--1859, 2016
[56] Stability of Traveling Waves of Nonlinear Schrödinger Equation with Nonzero Condition at Infinity
Zhiwu Lin Georgia Institute of Technology 王征平 Wuhan University of Technology Chongchun Zeng Georgia Institute of Technology
Arch. Rational Mech. Anal., 222, (1), 143-212, 2016.10
[ Download ] [ 2017-02-07 18:04:25 uploaded by ZhengpingWang ] [ 629 downloads ] [ 0 comments ]
We study the stability of traveling waves of the nonlinear Schrödinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models for this are the Gross–Pitaevskii (GP) equation and the cubic-quintic equation. First, under a non-degeneracy condition we prove a sharp instability criterion for 3D traveling waves of (GP), which had been conjectured in the physical literature. This result is also extended for general nonlinearity and higher dimensions, including 4D (GP) and 3D cubic-quintic equations. Second, for cubic-quintic type nonlinearity, we construct slow traveling waves and prove their nonlinear instability in any dimension. For dimension two, the non-degeneracy condition is also proved for these slow traveling waves. For general traveling waves without vortices (that is nonvanishing) and with general nonlinearity in any dimension, we find a sharp condition for linear instability. Third, we prove that any 2D traveling wave of (GP) is transversally unstable, and we find the sharp interval of unstable transversal wave numbers. Near unstable traveling waves of all of the above cases, we construct unstable and stable invariant manifolds.
[57] Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles
Jann-Long Chern National central University Zhi-You Chen National Changhua University of Education Chang-Shou Lin National Taiwan University
Comm. Math. Phys. , 296, (2), 323–351. , 2010
The existence of topological solutions for the Chern-Simons equation with two Higgs particles has been proved by Lin, Ponce and Yang [16]. However, both the uniqueness problem and the existence of non-topological solutions have been left open. In this paper, we consider the case of one vortex at origin. Among others, we prove the uniqueness of topological solutions and give a complete study of the radial solutions, in particular, the existence of some non-topological solutions
[58] Convergence for elliptic equations in periodic perforated domains
[59] Minimizers of Caffarelli-Kohn-Nirenberg inequalities with the singularity on the boundary.
Jann-Long Chern National Central University Chang-Shou Lin National Taiwn University
Arch. Ration. Mech. Anal., 197, (2), 401–432, 2010
Li Y, Lin C. A Nonlinear Elliptic PDE with Two Sobolev–Hardy Critical Exponents[J]. Archive for Rational Mechanics and Analysis, 2011, 203(3): 943-968.
Fall M M. On the Hardy-Poincar\\\u0027e inequality with boundary singularities[J]. Communications in Contemporary Mathematics, 2010, 14(03).
Ishiwata M, Nakamura M, Wadade H, et al. On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form[J]. Annales De L Institut Henri Poincare-analyse Non Lineaire, 2014, 31(2): 297-314.
Shusen Yan · Jianfu Yang. Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy–Sobolev exponents. 2012.
Cerami G, Zhong X Y, Zou W, et al. On some nonlinear elliptic PDEs with Sobolev–Hardy critical exponents and a Li–Lin open problem[J]. Calculus of Variations and Partial Differential Equations, 2015, 54(2): 1793-1829.
Fall M M, Minlend I A, Thiam E H, et al. The role of the mean curvature in a Hardy-Sobolev trace inequality[J]. Nodea-nonlinear Differential Equations and Applications, 2014, 22(5): 1047-1066.
Yang J. Positive solutions for the Hardy–Sobolev–Maz\u0027ya equation with Neumann boundary condition[J]. Journal of Mathematical Analysis and Applications, 2015, 421(2): 1889-1916.
Jin T. Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy terms[J]. Annales De L Institut Henri Poincare-analyse Non Lineaire, 2011, 28(6): 965-981.
Chunhua Wang · Changlin Xiang. INFINITELY MANY SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING DOUBLE CRITICAL TERMS AND BOUNDARY GEOMETRY. 2016.
Mousomi Bhakta. Infinitely many sign changing solutions of an elliptic problem involving critical Sobolev and Hardy-Sobolev exponent. 2014.
[60] Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions
Hao Jia UNIVERSITY OF MINNESOTA Vladimir Sverak UNIVERSITY OF MINNESOTA
Invent math, 196, (1), 233-265, 2014.4
[ Download ] [ 2018-04-27 11:35:47 uploaded by HJia ] [ 624 downloads ] [ 0 comments ]
We show that the classical Cauchy problem for the incompressible 3d Navier-Stokes equations with (−1)-homogeneous initial data has a global scale-invariant solution which is smooth for positive times. Our main tech- nical tools are local-in-space regularity estimates near the initial time, which are of independent interest.
[61] Least action principles for incompressible flows and optimal transport between shapes
Jian-Guo Liu Duke University Robert L. Pego Department of Mathematical Sciences and Center for Nonlinear Analysis Dejan Slepčev Department of Mathematical Sciences and Center for Nonlinear Analysis
As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an Eulerian framework, we study incompressible flows of shapes as critical paths for action (kinetic energy) along transport paths constrained to be shape densities (characteristic functions). The formal geodesic equations for this problem are Euler equations for incompressible, inviscid potential flow of fluid with zero pressure and surface tension on the free boundary. The problem of minimizing this action exhibits an instability associated with microdroplet formation, with the following outcomes: Any two shapes of equal volume can be approximately connected by an Euler spray---a countable superposition of ellipsoidal geodesics. The infimum of the action is the Wasserstein distance squared, and is almost never attained except in dimension 1. Every Wasserstein geodesic between bounded densities of compact support provides a solution of the (compressible) pressureless Euler system that is a weak limit of (incompressible) Euler sprays. Each such Wasserstein geodesic is also the unique minimizer of a relaxed least-action principle for a two-fluid mixture theory corresponding to incompressible fluid mixed with vacuum.
[62] On the Finite-time Blowup of a 1D Model for the 3D Axisymmetric Euler Equations
KYUDONG CHOI Ulsan National Institute of Science and Technology THOMAS Y. HOU California Institute of Technology ALEXANDER KISELEV Rice University GUO LUO City University of Hong Kong VLADIMIR SVERAK University of Minnesota YAO YAO Georgia Institute of Technology
CPAM, 2017
In connection with the recent proposal for possible singularity formation at the boundary for solutions of three-dimensional axisymmetric incompressible Euler's equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)), we study models for the dynamics at the boundary and show that they exhibit a finite-time blowup from smooth data.
[63] Nonhexagonal lattices from a two species interacting system
Senping Luo Jiangxi Normal University Xiaofeng Ren The George Washington University Juncheng Wei University of British Columbia
SIAM Journal on Mathematical Analysis, 52, (2), 1903-1942, 2020
[ Download ] [ 2020-06-06 06:12:47 uploaded by SenpingLuo ] [ 622 downloads ] [ 0 comments ]
A two species interacting system motivated by the density functional theory for triblock copolymers contains long range interaction that affects the two species differently. In a two species periodic assembly of discs, the two species appear alternately on a lattice. The minimal two species periodic assembly is the one with the least energy per lattice cell area. There is a parameter $b$ in $[0,1]$ and the type of the lattice associated with the minimal assembly varies depending on $b$. There are several threshold defined by a number $B=0.1867...$ If $b \in [0, B)$, the minimal assembly is associated with a rectangular lattice whose ratio of the longer side and the shorter side is in $[\sqrt{3}, 1)$; if $b \in [B, 1-B]$, the minimal assembly is associated with a square lattice; if $b \in (1-B, 1]$, the minimal assembly is associated with a rhombic lattice with an acute angle in $[\frac{\pi}{3}, \frac{\pi}{2})$. Only when $b=1$, this rhombic lattice is the hexagonal lattice. None of the other values of $b$ yields the hexagonal lattice, a sharp contrast to the situation for one species interacting systems, where the hexagonal lattice is ubiquitously observed.
[64] On the role of Riesz potentials in Poisson's equation and Sobolev embeddings
Daniel Spector National Chiao Tung University Rahul Garg Technion
Indiana Univ. Math. J., 64, (6), 1697–1719., 2015
[ Download ] [ 2017-03-09 06:38:54 uploaded by spectda ] [ 620 downloads ] [ 0 comments ]
In this paper, we study the mapping properties of the classical Riesz potentials acting on $L^p$-spaces. In the supercritical exponent, we obtain new "almost" Lipschitz continuity estimates for these and related potentials (including, for instance, the logarithmic potential). Applications of these continuity estimates include the deduction of new regularity estimates for distributional solutions to Poisson's equation, as well as a proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.
[65] On a new class of partial differential equations
Daniel Spector National Chiao Tung University Tien-Tsan Shieh National Central University
Adv. Calc. Var. 8 , 8, (4), 321–336. , 2015
In this paper, we study a new class of fractional partial differential equations which are obtained by minimizing variational problems in fractional Sobolev spaces. We introduce a notion of fractional gradient which has the potential to extend to many classical results in the Sobolev spaces to the nonlocal and fractional setting in a natural way.
[66] Global uniqueness of the minimal sphere in the Atiyah--Hitchin manifold
Chung-Jun Tsai National Taiwan University Mu-Tao Wang Columbia University
Analysis of PDEs Differential Geometry Geometric Analysis and Geometric Topology Mathematical Physics mathscidoc:1804.03011
In this note, we study submanifold geometry of the Atiyah--Hitchin manifold, a double cover of the 2-monopole moduli space, which plays an important role in various settings such as the supersymmetric background of string theory. When the manifold is naturally identified as the total space of a line bundle over S^2, the zero section is a distinguished minimal 2-sphere of considerable interest. In particular, there has been a conjecture about the uniqueness of this minimal 2-sphere among all closed minimal 2-surfaces. We show that this minimal 2-sphere satisfies the ``strong stability condition" proposed in our earlier work, and confirm the global uniqueness as a corollary.
[67] Holder estimate for non-uniform parabolic equations in highly heterogeneous media
Yeh L. Elliptic and parabolic equations in fractured media[J]. Journal of Differential Equations, 2015, 259(12): 6887-6922.
Liming Yeh. Uniform bound and convergence for elliptic homogenization problems. 2015.
Liming Yeh. Convergence for elliptic equations in periodic perforated domains. 2013.
[68] Curvature estimates for immersed hypersurfaces in Riemannian manifolds
Pengfei Guan McGill University Siyuan Lu McGill University
[ Download ] [ 2018-11-23 01:12:18 uploaded by Siyuan ] [ 609 downloads ] [ 0 comments ]
We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general Riemannian manifold. The estimate has a direct consequence for the Weyl isometric embedding problem of $(\mathbb S^2, g)$ in $3$-dimensional warped product space $(N^3, \bar g)$. We also discuss isometric embedding problem in spaces with horizon in general relativity, like the Anti-de Sitter-Schwarzschild manifolds and the Reissner-Nordstr\"om manifolds.
[69] Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces
Jean Bourgain Dong Li
Inventiones Mathematicae, 201, (1), 97-157, 2015
[ Download ] [ 2017-02-26 22:01:41 uploaded by madli ] [ 608 downloads ] [ 0 comments ]
For the d-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space with regularity above some critical threshold. The borderline case was a folklore open problem. In this paper we consider the physical dimension d=2 and show that if we perturb any given smooth initial data in critical norm, then the corresponding solution can have infinite critical norm instantaneously at t>0. In a companion paper we settle the 3D and more general cases. The constructed solutions are unique and even infinitely-smooth (locally) in some cases. To prove these results we introduce a new strategy: large Lagrangian deformation induces critical norm inflation. As an application we also settle several closely related open problems.
[70] Spectral analysis of the subelliptic oblique derivative problem
Kazuaki Taira University of Tsukuba
Arkiv for Matematik, 55, 2017
This paper is devoted to a functional analytic approach to the subelliptic oblique derivative problem for the usual Laplacian with a complex parameter λ. We solve the long-standing open problem of the asymptotic eigenvalue distribution for the homogeneous oblique derivative problem when |λ| tends to ∞. We prove the spectral properties of the closed realization of the Laplacian similar to the elliptic (non-degenerate) case. In the proof we make use of Boutet de Monvel calculus in order to study the resolvents and their adjoints in the framework of L2 Sobolev spaces.
[71] Accurate Solution and Gradient Computation for Elliptic Interface Problems with Variable Coefficients
Zhilin Li North Carolina State University Haifeng Ji Nanjing University of Posts and Telecommunication Xiaohong Chen North Carolina State University
SIAM Journal on Numerical Analysis, 55, (2), 570–597, 2017.3
[ Download ] [ 2018-04-05 02:43:00 uploaded by xiaohong_chen1991 ] [ 601 downloads ] [ 0 comments ]
A new augmented method is proposed for elliptic interface problems with a piecewise variable coefficient that has a finite jump across a smooth interface. The main motivation is to get not only a second order accurate solution but also a second order accurate gradient from each side of the interface. Key to the new method is introducing the jump in the normal derivative of the solution as an augmented variable and rewriting the interface problem as a new PDE that consists of a leading Laplacian operator plus lower order derivative terms near the interface. In this way, the leading second order derivative jump relations are independent of the jump in the coefficient that appears only in the lower order terms after the scaling. An upwind type discretization is used for the finite difference discretization at the irregular grid points on or near the interface so that the resulting coefficient matrix is an M-matrix. A multigrid solver is used to solve the linear system of equations, and the GMRES iterative method is used to solve the augmented variable. Second order convergence for the solution and the gradient from each side of the interface is proved in this paper. Numerical examples for general elliptic interface problems confirm the theoretical analysis and efficiency of the new method.
[72] Continuum dynamics of the intention field under weakly cohesive social interaction
Pierre Degond South Kensington Campus, Jian-Guo Liu Department of Physics, Duke University Sara Merino-Aceituno South Kensington Campus Thomas Tardiveau Ecole Polytechnique, Route de Saclay
Mathematical Models and Methods in Applied Sciences, 27, (1), 159-182, 2017.1
We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghesi, Bouchaud and Jensen. First, we derive a Fokker–Planck-type equation under the assumption that interactions between individuals produce little consensus of opinion (grazing collision approximation). Second, we study conditions under which the Fokker–Planck equation has non-trivial equilibria and derive the macroscopic limit (corresponding to the long-time dynamics and spatially localized interactions) for the evolution of the mean opinion. Finally, we compare two different types of interaction rates: the original one given in the work of Borghesi, Bouchaud and Jensen (symmetric binary interactions) and one inspired from works by Motsch and Tadmor (non-symmetric binary interactions). We show that the first case leads to a conservative model for the density of the mean opinion whereas the second case leads to a non-conservative equation. We also show that the speed at which consensus is reached asymptotically for these two rates has fairly different density dependence.
[73] Some compactness criteria for weak solutions of time fractional PDEs
Lei Li Jian-Guo Liu
SIAM J. Math. Anal., 50, 3963-3995, 2018
[74] Asymptotic behavior of positive singular solutions to fractional Hardy-Henon equations
Hui Yang Tsinghua University Wenming Zou Tsinghua University
[ Download ] [ 2019-01-09 08:51:56 uploaded by huiy ] [ 590 downloads ] [ 0 comments ]
In this paper, we study the asymptotic behavior of positive solutions of fractional Hardy-Henon equation with an isolated singularity at the origin. We give a classification of isolated singularities of positive solutions near x = 0. Further, we prove the asymptotic behavior of positive singular solutions. These results parallel those known for the Laplacian counterpart proved by Caffarelli, Gidas and Spruck (Caffarelli, Gidas and Spruck in Comm Pure Appl Math, 1981, 1989), but the technique is very different, since the ODEs analysis is a missing ingredient in the fractional case. Our proofs are based on a monotonicity formula, combined with a blow up (down) argument, the Kelvin transformation and uniqueness of solutions of related degenerate equations on semi spherical surface. We also investigate isolated singularities located at infinity of fractional Hardy-Henon equation.
[75] Uniqueness of topological multivortex solutions in the Maxwell-Chern-Simons model
Zhi-You Chen National Changhua University of Education Jann-Long Chern National Central University
J. Funct. Anal. , 270, (6), 2073–2125, 2016.2
In this paper, we prove the uniqueness of topological multivortex solutions for the self-dual Maxwell–Chern–Simons U(1)U(1) model if the Chern–Simons coupling parameter is sufficiently large and the charge of electron is sufficiently small or large. On the other hand, we also establish the sharp region of the flux for non-topological solutions and provide the classification of radial solutions of all types in the case of one vortex point.
[76] A proof of Bondesson's conjecture on stable densities
Pierre Bosch Laboratoire Paul Painlevé, Université Lille 1 Thomas Simon Laboratoire Paul Painlevé, Université Lille 1
Arkiv for Matematik, 54, (1), 31-38, 2014.11
We show that positive$α$-stable densities are hyperbolically completely monotone if and only if$α$≤1/2. This gives a positive answer to a question raised by L. Bondesson in 1977.
[77] A Nonlocal Stokes system with volume constraints
Qiang Du Columbia University, Zuoqiang Shi Tsinghua University
In this paper, we introduce a nonlocal model for linear steady Stokes system with physical no-slip boundary condition. We use the idea of volume constraint to enforce the no-slip boundary condition. We prove that the nonlocal model is well-posed and the solution of the nonlocal system converges to the solution of the original Stokes system when the nonlocality vanishes.
[78] Addendum: 2D homogeneous solutions to the Euler equation
Comm.Partial DifferentialEquations, 42, (3), 491-493, 2017
In this addendum note we fill in the gap left in the description of 2D homogeneous solutions to the stationary Euler system initiated in the previous publication. This completes the classification of all homogeneous stationary solutions. The note includes updated classification tables.
[79] The explicit formulae for scaling limits in the ergodic decomposition of infinite Pickrell measures
Alexander I. Bufetov Aix-Marseille Université, Centrale Marseille, CNRS, I2M, UMR7373, 39 Rue F. Joliot Curie, Marseille Cedex 13, France Yanqi Qiu Aix-Marseille Université, Centrale Marseille, CNRS, I2M, UMR7373, 39 Rue F. Joliot Curie, Marseille Cedex 13, France
The main result of this paper, Theorem 1.5, gives explicit formulae for the kernels of the ergodic decomposition measures for infinite Pickrell measures on the space of infinite complex matrices. The kernels are obtained as the scaling limits of Christoffel-Uvarov deformations of Jacobi orthogonal polynomial ensembles.
[80] Kazdan–Warner equation on graph
Alexander Grigor'yan University of Bielefeld Yong Lin Renmin University of China Yunyan Yang Renmin University of China
[81] Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations
Xiangdi Huang Institute of Mathematics, Academy of Mathematics and Systems Science
Arch. Ration. Mech. Anal., 227, (3), 995-1059, 2018.3
[ Download ] [ 2018-03-02 11:28:34 uploaded by Evan ] [ 577 downloads ] [ 0 comments ]
For the three-dimensional full compressible Navier–Stokes system describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid, we establish the global existence and uniqueness of classical solutions with smooth initial data which are of small energy but possibly large oscillations where the initial density is allowed to vanish. Moreover, for the initial data, which may be discontinuous and contain vacuum states, we also obtain the global existence of weak solutions. These results generalize previous ones on classical and weak solutions for initial density being strictly away from a vacuum, and are the first for global classical and weak solutions which may have large oscillations and can contain vacuum states.
[82] Yamabe type equations on graphs
Journal of Differential Equations, 261, 4924-4943, 2016.2
[83] Nonlinear stability of discrete shocks for systems of conservation laws
Jian-Guo Liu Courant Institute of Mathematical Sciences Zhouping Xin Courant Institute of Mathematical Sciences
Archive for Rational Mechanics and Analysis, 125, (3), 217–256, 1993.9
In this paper we study the asymptotic nonlinear stability of discrete shocks for the Lax-Friedrichs scheme for approximating general m×m systems of nonlinear hyperbolic conservation laws. It is shown that weak single discrete shocks for such a scheme are nonlinearly stable in the Lp-norm for all p ≧ 1, provided that the sums of the initial perturbations equal zero. These results should shed light on the convergence of the numerical solution constructed by the Lax-Friedrichs scheme for the single-shock solution of system of hyperbolic conservation laws. If the Riemann solution corresponding to the given far-field states is a superposition of m single shocks from each characteristic family, we show that the corresponding multiple discrete shocks are nonlinearly stable in Lp (P ≧ 2). These results are proved by using both a weighted estimate and a characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the Lax-Friedrichs scheme.
[84] Global well-posedness of the three-dimensional primitive equations with only horizontal viscosity and diffusion
Chongsheng Cao Florida International Uni Jinkai Li South China Normal University Edriss S. Titi Texas A&M University
Gold Award Paper in 2020
Communications on Pure and Applied Mathematics, 69, (8), 2016
[ Download ] [ 2019-05-20 11:41:04 uploaded by jklimath ] [ 575 downloads ] [ 55 comments ]
In this paper, we consider the initial-boundary value problem of the 3D primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well-posedness of strong solution is established for any $H^2$ initial data. An $N$-dimensional logarithmic Sobolev embedding inequality, which bounds the $L^\infty$ norm in terms of the $L^q$ norms up to a logarithm of the $L^p$-norm, for $p>N$, of the first order derivatives, and a system version of the classic Gronwall inequality are exploited to establish the required a priori $H^2$ estimates for the global regularity.
[85] Global existence of solutions to a tear film model with locally elevated evaporation rates
Yuan Gao Fudan University Hangjie Ji Duke University Jian-Guo Liu Duke University Thomas P. Witelski Duke University
Motivated by a model proposed by Peng et al. [Advances in Coll. and Interf. Sci. 206 (2014)] for break-up of tear films on human eyes, we study the dynamics of a generalized thin film model. The governing equations form a fourth-order coupled system of nonlinear parabolic PDE for the film thickness and salt concentration subject to nonconservative effects representing evaporation. We analytically prove the global existence of solutions to this model with mobility exponents in several different ranges and the results are then validated against PDE simulations. We also numerically capture other interesting dynamics of the model, including finite-time rupture-shock phenomenon due to the instabilities caused by locally elevated evaporation rates, convergence to equilibrium and infinite-time thinning.
[86] Convergence of vortex methods for weak solutions to the 2-D Euler equations with vortex sheets data
Jian-Guo Liu Temple University Zhouping Xin Courant Institute
Communications on Pure & Applied Mathematics, 48, (6), 611-628, 1995.1
We prove the convergence of vortex blob methods to classical weak solutions for the twodimensional incompressible Euler equations with initial data satisfying the conditions that the vorticity is a finite Radon measure of distinguished sign and the kinetic energy is locally bounded. This includes the important example of vortex sheets. The result is valid as long as the computational grid size h does not exceed the smoothing blob size E, i.e., h/~ 5 C. 0 1995 John Wiley & Sons, Inc.
[87] The p-capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems
Han Hong Memorial University of Newfoundland Deping Ye Memorial University of Newfoundland Ning Zhang University of Alberta
Analysis of PDEs Functional Analysis Convex and Discrete Geometry mathscidoc:1805.03001
Calculus of Variations and Partial Differential Equations, 57, 5, 2018.2
[ Download ] [ 2018-05-29 23:23:16 uploaded by ydpzjumn ] [ 570 downloads ] [ 0 comments ]
In this paper, combining the $p$-capacity for $p\in (1, n)$ with the Orlicz addition of convex domains, we develop the $p$-capacitary Orlicz-Brunn-Minkowski theory. In particular, the Orlicz $L_{\phi}$ mixed $p$-capacity of two convex domains is introduced and its geometric interpretation is obtained by the $p$-capacitary Orlicz-Hadamard variational formula. The $p$-capacitary Orlicz-Brunn-Minkowski and Orlicz-Minkowski inequalities are established, and the equivalence of these two inequalities are discussed as well. The $p$-capacitary Orlicz-Minkowski problem is proposed and solved under some mild conditions on the involving functions and measures. In particular, we provide the solutions for the normalized $p$-capacitary $L_q$ Minkowski problems with $q>1$ for both discrete and general measures.
[88] Heat equations defined by a class of fractal measures
Wei Tang Hunan First Normal University Sze-Man Ngai Georgia Southern University and Hunan Normal University
[ Download ] [ 2018-09-30 06:19:01 uploaded by smngai ] [ 567 downloads ] [ 0 comments ]
We set up a framework to study one-dimensional heat equations defined by fractal Laplacians associated with self-similar measures with overlaps. We show that for a class of such self-similar measures, a heat equation can be discretized and the finite element method can be applied to yield a system of linear differential equations. We show that the numerical solutions converge to the actual solution and obtain the rate of convergence. We also study some properties of the solutions of the heat equation.
[89] Removable singularity of positive solutions for a critical elliptic system with isolated singularity
Zhijie Chen Taiwan University Chang-Shou Lin Taiwan University
Mathematische Annalen, 363, 501-523, 2015.2
[ Download ] [ 2018-03-18 13:52:10 uploaded by zjchen ] [ 566 downloads ] [ 0 comments ]
[90] The solution structure of the O(3) sigma model in a Maxwell-Chern-Simons theory
JOURNAL OF MATHEMATICAL PHYSICS, 58, 071503, 2017
In this paper, a system of semilinear elliptic equations arising from a relativistic self-dual Maxwell-Chern-Simons O(3) sigma model is considered. We reveal the uniqueness aspect of the topological solutions for the model. The uniqueness result is associated with a clear solution structure of the equations of the radially symmetric case. We locate each solution set denoted by a planar diagram. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4994060]
[91] A strong stability condition on minimal submanifolds and its implications
Analysis of PDEs Differential Geometry Geometric Analysis and Geometric Topology mathscidoc:1804.03010
We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that the mean curvature flow of any other submanifold in a neighborhood of such a minimal submanifold exists for all time, and converges exponentially to the minimal one. This extends our previous uniqueness and stability theorem which applies only to calibrated submanifolds of special holonomy ambient manifolds.
[92] An asymptotic propertiy of the Camassa-Holm equation
Jia Jia Tarim University Shun-Guang Kang Tarim University Tai-Man Tang Xiangtan University
Nonlinear Analysis: Theory, Methods and Applications, 97, 55-64, 2014
[93] Partial differential equations in several complex variables
Shing-Tung Yau Mei-Chi Shaw
[ Download ] [ 2019-12-24 20:34:45 uploaded by yaust ] [ 558 downloads ] [ 0 comments ]
This work is intended as both an introductory text and a reference book for those interested in studying several complex variables in the context of partial differential equations. Significant progress has been made in the study of Cauchy-Riemann and tangential Cauchy-Riemann operators; this progress greatly influenced the development of PDEs and several complex variables.
[94] $R$ -Boundedness versus $\gamma$ -boundedness
Stanislaw Kwapień Institute of Mathematics, Warsaw University Mark Veraar Delft Institute of Applied Mathematics, Delft University of Technology Lutz Weis Institut für Analysis, Karlsruhe Institute of Technology
It is well-known that in Banach spaces with finite cotype, the $R$ -bounded and $\gamma $ -bounded families of operators coincide. If in addition $X$ is a Banach lattice, then these notions can be expressed as square function estimates. It is also clear that $R$ -boundedness implies $\gamma $ -boundedness. In this note we show that all other possible inclusions fail. Furthermore, we will prove that $R$ -boundedness is stable under taking adjoints if and only if the underlying space is $K$ -convex.
[95] Uniqueness of positive solution to some coupled cooperative variational elliptic systems
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 2018
The uniqueness of positive solutions to some semilinear elliptic systems with variational structure arising from mathematical physics is proved. The key ingredient of the proof is the oscillatory behavior of solutions to linearized equations for cooperative semilinear elliptic systems of two equations on one-dimensional domains, and it is shown that the stability of the positive solutions for such semilinear system is closely related to the oscillatory behavior.
[96] Spectral asymptotics of one-dimensional graph-directed self-similar measures with overlaps
Sze-Man Ngai Georgia Southern University and Georgia Southern University Yuanyuan Xie Renmin University of China
[97] An accurate front capturing scheme for tumor growth models with a free boundary limit
Jian-Guo Liu Min Tang
J. Comput. Phy,, 364, 73-94, 2018
[98] Flow on sweeping networks
Pierre Degond Institut de Math´ematiques de Toulouse Michael Herty RWTH Aachen Universit Jian-Guo Liu Duke University
MULTISCALE MODEL. SIMUL, 12, (2), 538–565, 2014.2
We introduce a cellular automaton model coupled with a transport equation for flows on graphs. The direction of the flow is described by a switching process where the switching probability dynamically changes according to the value of the transported quantity in the neighboring cells. A motivation is pedestrian dynamics in a small corridor where the propagation of people in a part of the corridor can be either left or rightgoing. Under the assumptions of propagation of chaos and mean-field limit, we derive a master equation and the corresponding meanfield kinetic and macroscopic models. Steady--states are computed and analyzed analytically and exhibit the possibility of multiple meta-stable states and hysteresis.
[99] Regularity for a fractional $p$-Laplace equation
Daniel Spector National Chiao Tung University Armin Schikorra University of Frieberg Tien-Tsan Shieh National Central University
Communications in Contemporary Mathematics
In this note we consider regularity theory for a fractional p-Laplace operator which arises in the complex interpolation of the Sobolev spaces, the $H^{s,p}$-Laplacian. We obtain the natural analogue to the classical p-Laplacian situation, namely $C^{s+α}$-regularity for the loc homogeneous equation.
[100] On Weyl's embedding problem in Riemannian manifolds
Siyuan Lu Rutgers University
[101] Operator-Lipschitz functions in Schatten–von Neumann classes
Denis Potapov School of Mathematics & Statistics, University of New South Wales Fedor Sukochev School of Mathematics & Statistics, University of New South Wales
Analysis of PDEs Spectral Theory and Operator Algebra mathscidoc:1701.03007
This paper resolves a number of problems in the perturbation theory of linear operators, linked with the 45-year-old conjecure of M. G. Kreĭn. In particular, we prove that every Lipschitz function is operator-Lipschitz in the Schatten–von Neumann ideals$S$^{$α$}, 1 <$α$< ∞. Alternatively, for every 1 <$α$< ∞, there is a constant$c$_{$α$}> 0 such that $$ {\left\| {f(a) - f(b)} \right\|_{\alpha }} \leqslant {c_{\alpha }}{\left\| f \right\|_{{{\text{Lip}}\,{1}}}}{\left\| {a - b} \right\|_{\alpha }}, $$ where$f$is a Lipschitz function with $$ {\left\| f \right\|_{{{\text{Lip}}\,{1}}}}: = \mathop{{\sup }}\limits_{{_{{\lambda \ne \mu }}^{{\lambda, \mu \in \mathbb{R}}}}} \left| {\frac{{f\left( \lambda \right) - f\left( \mu \right)}}{{\lambda - \mu }}} \right| < \infty, $$ $$ {\left\| \cdot \right\|_{\alpha }} $$ is the norm is$S$^{$α$}, and$a$and$b$are self-adjoint linear operators such that $$ a - b \in {S^{\alpha }} $$ .
[102] The convergence Newton polygon of a$p$-adic differential equation II: Continuity and finiteness on Berkovich curves
Jérôme Poineau Laboratoire de mathématiques Nicolas Oresme, Université de Caen Andrea Pulita Département de mathématiques, Université de Montpellier II, CC051
We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic zero. Relying on work of the second author who investigated its properties on affinoid domains of the affine line, we prove that its slopes give rise to continuous functions that factorise by the retraction through a locally finite subgraph of the curve.
[103] Existence and blowup behavior of global strong solutions to the two-dimensional barotropic compressible Navier–Stokes system with vacuum and large initial data
J. Math. Pures Appl., 106, (1), 123–154, 2016.2
For periodic initial data with density allowed to vanish initially, we establish the global existence of strong and weak solutions to the two-dimensional barotropic compressible Navier–Stokes equations with no restrictions on the size of initial data provided the shear viscosity is a positive constant and the bulk one is λ =ρ^β with β>4/3. These results generalize and improve the previous ones due to Vaigant–Kazhikhov [Sib. Math. J. 36 (1995) 1283–1316] who required β>3. Moreover, we also prove that the densities for both the strong and weak solutions remain bounded from above independently of time. As a consequence, it is shown that both the strong and weak solutions converge to the equilibrium state as time tends to infinity.
[104] Stable hypersurfaces with zero scalar curvature in Euclidean space
Hilário Alencar Instituto de Matemática, Universidade Federal de Alagoas Manfredo do Carmo Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, RJ, Brazil Gregório Silva Neto Instituto de Matemática, Universidade Federal de Alagoas
Arkiv for Matematik, 1-9, 2015.9
In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bounded away from zero. We conclude this paper giving a sufficient condition for a regular domain to be stable in terms of the mean and the Gauss-Kronecker curvatures of the hypersurface and the radius of the smallest extrinsic ball which contains the domain.
[105] Monge–Ampère equations in big cohomology classes
Sébastien Boucksom Institut de Mathématiques de Jussieu, CNRS–Université Pierre et Marie Curie Philippe Eyssidieux Institut Fourier, Université Grenoble 1 Vincent Guedj Institut de Mathématiques de Toulouse, Université Paul Sabatier Ahmed Zeriahi Institut de Mathématiques de Toulouse, Université Paul Sabatier
We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold$X$. Given a big (1, 1)-cohomology class$α$on$X$(i.e. a class that can be represented by a strictly positive current) and a positive measure$μ$on$X$of total mass equal to the volume of$α$and putting no mass on pluripolar sets, we show that$μ$can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in$α$. We then extend Kolodziedj's approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if$μ$has$L$^{1+$ε$}-density with respect to Lebesgue measure. If$μ$is smooth and positive everywhere, we prove that$T$is smooth on the ample locus of$α$provided$α$is nef. Using a fixed point theorem, we finally explain how to construct singular Kähler–Einstein volume forms with minimal singularities on varieties of general type.
[106] Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton
Yvan Martel Université de Versailles St-Quentin and Institut Universitaire de France, LMV CNRS UMR8100, Versailles Cedex, France Frank Merle Université de Cergy Pontoise and Institut des Hautes Études Scientifiques, AGM CNRS UMR8088, Paris, France Pierre Raphaël Université Paul Sabatier and Institut Universitaire de France, IMT CNRS UMR 5219, Toulouse, France
Acta Mathematica, 212, (1), 59-140, 2012.2
We consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in$H$^{1}close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18].
[107] Mean field equations, hyperelliptic curves and modular forms: I
Ching-Li Chai U. Pennsylvania Chang-Shou Lin National Taiwan U. Chin-Lung Wang National Taiwan U.
Cambridge J. Math., 3, 127-274, 2015
[ Download ] [ 2016-10-03 23:33:27 uploaded by dragon ] [ 534 downloads ] [ 0 comments ]
We develop a theory to connect the following three areas: (a) the mean field equations on flat tori, (b) the classical Lame equations and (c) modular forms. A major theme in part I is a classification of developing maps f attached to solutions of the mean field equation according to the types of transformation laws (or monodromy) with respect to the period lattice satisfied by f.
[108] Front Quenching in G-equation Model Induced by Straining of Cellular Flow
Jack Xin University of California at Irvine Yifeng Yu University of California at Irvine
Archive for Rational Mechanics and Analysis, 214, (1), 2014
We study homogenization of G-equation with a ow straining term (or the strain G-equation) in two dimensional periodic cellular ow. The strain G-equation is a highly non-coercive and non-convex level set Hamilton-Jacobi equation. The main objective is to investigate how the ow induced straining (the nonconvex term) in uences front propagation as the ow intensity A increases. Three distinct regimes are identified. When A is below the critical level, homogenization holds and the turbulent ame speed sT (effective Hamiltonian) is well-defined for any periodic ow with small divergence and is enhanced by the cellular ow as $s_T \ge O(A/logA)$. In the second regime where A is slightly above the critical value, homogenization breaks down, and $s_T$ is not well defined along any direction. Solutions become a mixture of fast moving part and a stagnant part. When $A$ is sufficiently large, the whole ame front ceases to propagate forward due to the flow induced straining. In particular, along directions $p = ( 1; 0)$ and $(0; 1)$, $s_T$ is well-defined again with a value of zero (trapping). A partial homogenization result is also proved. If we consider a similar but relatively simpler Hamiltonian, the trapping occurs along all directions. The analysis is based on the two-player di erential game representation of solutions, selection of game strategies and trapping regions, and construction of connecting trajectories.
[109] Curvature effect in shear flow: slowdown of turbulent flame speeds with Markstein number
Jiancheng Lyu University of California at Irvine Jack Xin University of California at Irvine Yifeng Yu University of California at Irvine
Comm. Math Physics, 2017
[ Download ] [ 2018-02-14 11:00:12 uploaded by jack ] [ 525 downloads ] [ 0 comments ]
It is well-known in the combustion community that curvature effect in general slows down ame propagation speeds because it smooths out wrinkled ames. However, such a folklore has never been justi ed rigorously. In this paper, as the rst theoretical result in this direction, we prove that the turbulent ame speed (an e ective burning velocity) is decreasing with respect to the curvature di usivity (Markstein number) for shear flows in the well-known G-equation model. Our proof involves several novel and rather sophisticated inequalities arising from the nonlinear structure of the equation. On a related fundamental issue, we solve the selection problem of weak solutions or nd the "physical fluctuations" when the Markstein number goes to zero and solutions approach those of the inviscid G-equation model. The limiting solution is given by a closed form analytical formula.
[110] Local well-posedness of isentropic compressible Navier-Stokes equations with vacuum
Huajun Gong Shenzhen University Jinkai Li South China Normal University Xian-Gao Liu Fudan University Xiaotao Zhang South China Normal University
In this paper, the local well-posedness of strong solutions to the Cauchy problem of the isentropic compressible Navier-Stokes equations is proved with the initial date being allowed to have vacuum. The main contribution of this paper is that the well-posedness is established without assuming any compatibility condition on the initial data, which was widely used before in many literatures concerning the well-posedness of compressible Navier-Stokes equations in the presence of vacuum.
[111] On the formation of trapped surfaces
Sergiu Klainerman Department of Mathematics, Princeton University Igor Rodnianski Department of Mathematics, Princeton University
In a recent important breakthrough D. Christodoulou has solved a long standing problem of General Relativity of evolutionary formation of trapped surfaces in the Einstein-vacuum space-times. He has identified an open set of regular initial conditions on an outgoing null hypersurface (both finite and at past null infinity) leading to a formation a trapped surface in the corresponding vacuum space-time to the future of the initial outgoing hypersurface and another incoming null hypersurface with the prescribed Minkowskian data. In this paper we give a simpler proof for a finite problem by enlarging the admissible set of initial conditions and, consistent with this, relaxing the corresponding propagation estimates just enough that a trapped surface still forms. We also reduce the number of derivatives needed in the argument from two derivatives of the curvature to just one. More importantly, the proof, which can be easily localized with respect to angular sectors, has the potential for further developments.
[112] Non-uniqueness of Weak Solutions to Hyperviscous Navier-Stokes Equations - On Sharpness of J.-L. Lions Exponent
Tianwen Luo Tsinghua University Edriss S. Titi Texas A\&M University; University of Cambridge; The Weizmann Institute of Science
[ Download ] [ 2019-06-28 07:16:42 uploaded by twluo ] [ 518 downloads ] [ 0 comments ]
Using the convex integration technique for the three-dimensional Navier-Stokes equations introduced by T. Buckmaster and V. Vicol, it is shown the existence of non-unique weak solutions for the 3D Navier-Stokes equations with fractional hyperviscosity $(-\Delta)^{\theta}$, whenever the exponent $\theta$ is less than J.-L. Lions' exponent $5/4$, i.e., when $\theta < 5/4$.
[113] Exact number and non-degeneracy of critical points of multiple Green functions on rectangular tori
Zhijie Chen Tsinghua University Chang-Shou Lin Taiwan University
Journal of Differential Geometry, 118, 457-485, 2021.7
[114] A random particle blob method for the Keller-Segel equation and convergence analysis
Jian-Guo Liu Duke University Rong Yang Beijing University of Technology
MATHEMATICS OF COMPUTATION, 86, (304), 725–745, 2017.2
In this paper, we introduce a random particle blob method for the Keller-Segel equation (with dimension ) and establish a rigorous convergence analysis.
[115] Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows
Commun. Math. Phys, 324, (1), 147–171, 2013.2
This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magneto-hydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies Serrin's condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanishing or vacuum appearing in the non-vacuum region or even milder singularities) to form before the density becomes unbounded. This criterion is analogous to the well known Serrin's blowup criterion for the three-dimensional incompressible Navier-Stokes equations, in particular, it is independent of the temperature and magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for the strong or smooth solutions to the three-dimensional full compressible Navier-Stokes system describing the motion of a viscous, compressible, and heat conducting fluid.
[116] Global integral gradient bounds for quasilinear equations below or near the natural exponent
Nguyen Cong Phuc Department of Mathematics, Louisiana State University
Nguyen Cong Phuc. Nonlinear Muckenhoupt-Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations. 2013.
Baroni P, Castro A D, Palatucci G, et al. Global estimates for nonlinear parabolic equations[J]. Journal of Evolution Equations, 2013, 13(1): 163-195.
Phuc N C. Morrey global bounds and quasilinear Riccati type equations below the natural exponent[J]. Journal de Mathématiques Pures et Appliquées, 2014, 102(1): 99-123.
Adimurthi K, Phuc N C. Global Lorentz and Lorentz–Morrey estimates below the natural exponent for quasilinear equations[J]. Calculus of Variations and Partial Differential Equations, 2014, 54(3): 3107-3139.
Mengesha T, Phuc N C. Quasilinear Riccati type equations with distributional data in Morrey space framework[J]. Journal of Differential Equations, 2016, 260(6): 5421-5449.
Paolo Baroni · Agnese Di Castro · Giampiero Palatucci. FOR NONLINEAR PARABOLIC EQUATIONS. 2013.
We obtain sharp integral potential bounds for gradients of solutions to a wide class of quasilinear elliptic equations with measure data. Our estimates are global over bounded domains that satisfy a mild exterior capacitary density condition. They are obtained in Lorentz spaces whose degrees of integrability lie below or near the natural exponent of the operator involved. As a consequence, nonlinear Calderón–Zygmund type estimates below the natural exponent are also obtained for $\mathcal{A}$ -superharmonic functions in the whole space ℝ^{$n$}. This answers a question raised in our earlier work (On Calderón–Zygmund theory for$p$- and $\mathcal{A}$ -superharmonic functions, to appear in$Calc. Var. Partial Differential Equations$, DOI10.1007/s00526-011-0478-8) and thus greatly improves the result there.
[117] Hölder regularity for degenerate parabolic obstacle problems
Verena Bögelein Universität Salzburg Christoph Scheven Universität Duisburg-Essen Teemu Lukkari Aalto University
We prove that weak solutions to the obstacle problem for the porous medium equation are locally Hölder continuous, provided that the obstacle is Hölder continuous.
[118] Starshaped compact hypersurfaces with prescribed Weingarden curvature in warped product manifolds
Daguang Chen Tsinghua University Haizhong Li Tsinghua University Zhizhang Wang Fudan University
Calc. Var. Partial Differential Equations, 57, (2), 2018.2
Given a compact Riemannian manifold $M$, we consider a warped product manifold $\bar M = I \times_h M$, where $I$ is an open interval in $\Bbb R$. For a positive function $\psi$ defined on $\bar M$, we generalize the arguments in \cite{GRW2015} and \cite{RW16}, to obtain the curvature estimates for Hessian equations $\sigma_k(\kappa)=\psi(V,\nu(V))$. We also obtain some existence results for the starshaped compact hypersurface $\Sigma$ satisfying the above equation with various assumptions.
[119] Embedded minimal tori in$S$^{3}and the Lawson conjecture
Simon Brendle Department of Mathematics, Stanford University
We show that any embedded minimal torus in$S$^{3}is congruent to the Clifford torus. This answers a question posed by H. B. Lawson, Jr., in 1970.
[120] Spikes of the two-component elliptic system in R4 with the critical Sobolev exponent
Calculus of Variations and Partial Differential Equations, 58, (1), 24, 2019.1
[ Download ] [ 2019-01-09 00:34:57 uploaded by wuyz850306 ] [ 508 downloads ] [ 0 comments ]
Consider the following elliptic system: \begin{equation*} \left\{\aligned&-\ve^2\Delta u_1+\lambda_1u_1=\mu_1u_1^3+\alpha_1u_1^{p-1}+\beta u_2^2u_1\quad&\text{in }\Omega,\\ &-\ve^2\Delta u_2+\lambda_2u_2=\mu_2u_2^3+\alpha_2u_2^{p-1}+\beta u_1^2u_2\quad&\text{in }\Omega,\\ &u_1,u_2>0\quad\text{in }\Omega,\quad u_1=u_2=0\quad\text{on }\partial\Omega,\endaligned\right. \end{equation*} where $\Omega\subset\bbr^4$ is a bounded domain, $\lambda_i,\mu_i,\alpha_i>0$ $(i=1,2)$ and $\beta\not=0$ are constants, $\ve>0$ is a small parameter and $2<p<2^*=4$. By using variational methods, we study the existence of ground state solutions to this system for sufficiently small $\ve>0$. The concentration behaviors of least-energy solutions as $\ve\to0^+$ are also studied. Furthermore, by combining elliptic estimates and local energy estimates, we obtain the locations of these spikes as $\ve\to0^+$.
[121] Error estimate of a random particle blob method for the Keller-Segel equation
Hui Huang Tsinghua Universit Jian-Guo Liu Duke University
MATHEMATICS OF COMPUTATION, 2017.1
We establish an optimal error estimate for a random particle blob method for the Keller-Segel equation in Rd (d 2). With a blob size " = h
[122] Entropy bounded solutions to the one-dimensional compressible Navier-Stokes equations with zero heat conduction and far field vacuum
Jinkai Li South China Normal University Zhouping Xin The Chinese University of Hong Kong
AdvancesinMathematics, 361, 106923, 2020
[ Download ] [ 2020-02-11 20:24:02 uploaded by jklimath ] [ 504 downloads ] [ 0 comments ]
The entropy is one of the fundamental states of a fluid and, in the viscous case, the equation that it satisfies is highly singular in the region close to the vacuum. In spite of its importance in the gas dynamics, the mathematical analyses on the behavior of the entropy near the vacuum region, were rarely carried out; in particular, in the presence of vacuum, either at the far field or at some isolated interior points, it was unknown whether the entropy remains its boundedness. The results obtained in this paper indicate that the ideal gases retain their uniform boundedness of the entropy, locally or globally in time, if the vacuum occurs at the far field only and the density decays slowly enough at the far field. Precisely, we consider the Cauchy problem to the one-dimensional full compressible Navier-Stokes equations without heat conduction, and establish the local and global existence and uniqueness of entropy-bounded solutions, in the presence of vacuum at the far field only. It is also shown that, different from the case that with compactly supported initial density, the compressible Navier-Stokes equations, with slowly decaying initial density, can propagate the regularities in inhomogeneous Sobolev spaces.
[123] Topological multivortex solutions for the Chern-Simons system with two Higgs particles
Comm. Partial Differential Equations, 41, (4), 705–731, 2016.4
In this paper, we prove the uniqueness of topological multivortex solutions to the self-dual abelian Chern–Simons model if either the Chern–Simons coupling parameter is sufficiently small or sufficiently large. In addition, we also establish the sharp region of the flux for nontopological solutions with a single vortex point.
[124] Basic themes and pretty problems of nonlinear solid mechanics
Stuart S. Antman University of Maryland Jian-Guo Liu University of Maryland
Milan Journal of Mathematics , 75, (1), 135-176, 2007.12
The first part of this paper describes some important underlying themes in the mathematical theory of continuum mechanics that are distinct from formulating and analyzing governing equations. The main part of this paper is devoted to a survey of some concrete, conceptually simple, pretty problems that help illuminate the underlying themes. The paper concludes with a discussion of the crucial role of invariant constitutive equations in computation.
[125] A simple proof of the Cucker-Smale flocking dynamics and mean-field limit
Seung Yeal Ha Seoul National University Jian-Guo Liu Duke University
Communications in Mathematical Sciences, 7, (2), 297-325, 2009.4
We present a simple proof on the formation of flocking in the F. Cucker and S. Smale system [Jpn. J. Math. (3) 2, No. 1, 197–227 (2007; Zbl 1166.92323)] based on the explicit construction of a Lyapunov functional. Our results also provide a unified condition on the initial states in which the exponential convergence to the flocking state will occur. For large particle systems, we give a rigorous justification for the mean-field limit from the many particle Cucker-Smale system to the Vlasov equation with flocking dissipation as the number of particles goes to infinity.
[126] On the quenching behavior of the MEMS with fringing field
Xue Luo Beihang University Stephen S.-T. Yau Tsinghua University
Quarterly of Applied Mathematics, 73, (4), 629-659, 2015.12
Qi Wang. Dynamical solutions of singular parabolic equations modeling electrostatic MEMS. 2014.
Hongjing Pan · Ruixiang Xing. Fringing field can prevent infinite time quenching. 2016.
The singular parabolic problem $u_t-\triangle u=\l{\frac{1+\d|\nabla u|^2}{(1-u)^2}}$ on a bounded domain $\O$ of $\Rn$ with Dirichlet boundary condition, models the Microelectromechanical systems (MEMS) device with fringing field. In this paper, we focus on the quenching behavior of the solution to this equation. We first show that there exists a critical value $\l_\d^*>0$ such that if $0<\l<\l_\d^*$, all solutions exist globally; while for $\l>\l_\d^*$, all the solution will quench in finite time. The estimate of the quenching time in terms of large voltage $\l$ is investigated. Furthermore, the quenching set is a compact subset of $\O$, provided $\O$ is a convex bounded domain in $\mathbb{R}^n$. In particular, if the domain $\O$ is radially symmetric, then the origin is the only quenching point. We not only derive the one-side estimate of the quenching rate, but also further study the refined asymptotic behavior of the finite quenching solution.
[127] Compression bounds for Lipschitz maps from the Heisenberg group to$L$_{1}
Jeff Cheeger Courant Institute, New York University Bruce Kleiner Courant Institute, New York University Assaf Naor Courant Institute, New York University
We prove a quantitative bi-Lipschitz non-embedding theorem for the Heisenberg group with its Carnot–Carathéodory metric and apply it to give a lower bound on the integrality gap of the Goemans–Linial semidefinite relaxation of the sparsest cut problem.
[128] On a K-component elliptic system with the Sobolev critical exponent in high dimensions: the repulsive case
Yuanze Wu China University of Mining and Technology
Calculus of Variations and Partial Differential Equations, 56, (1), 151, 2017.9
[129] Uniform $L^{\infty}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model
Wenting Cong Jilin University Jian-Guo Liu Duke University
Discrete and Continuous Dynamical Systems - Series B, 22, (2), 307-338, 2017.2
This paper investigates the existence of a uniform in time $L^{\infty}$ bounded weak entropy solution for the quasilinear parabolic-parabolic Keller-Segel model with the supercritical diffusion exponent $0 < m < 2-\frac{2}{d}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(2-m)}{2}}$ norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution $u(x,t)$ satisfies mass conservation when $m > 1-\frac{2}{d}$. We also prove the local existence of weak entropy solutions and a blow-up criterion for general $L^1\cap L^{\infty}$ initial data.
[130] Directional Poincaré inequalities along mixing flows
Stefan Steinerberger Department of Mathematics, Yale University
Arkiv for Matematik, 1-15, 2015.11
We provide a refinement of the Poincaré inequality on the torus $\mathbb{T}^{d}$ : there exists a set $\mathcal{B} \subset \mathbb{T} ^{d}$ of directions such that for every $\alpha \in \mathcal{B}$ there is a $c_{\alpha } > 0$ with $$\begin{aligned} \|\nabla f\|_{L^{2}(\mathbb{T}^{d})}^{d-1} \| \langle \nabla f, \alpha \rangle \|_{L^{2}(\mathbb{T}^{d})} \geq c_{\alpha }\|f\| _{L^{2}(\mathbb{T}^{d})}^{d} \quad \mbox{for all}~f\in H^{1}\bigl( \mathbb{T}^{d}\bigr)~ \mbox{with mean 0.} \end{aligned}$$ The derivative $\langle \nabla f, \alpha \rangle $ does not detect any oscillation in directions orthogonal to $\alpha $ , however, for certain $\alpha $ the geodesic flow in direction $\alpha $ is sufficiently mixing to compensate for that defect. On the two-dimensional torus $\mathbb{T}^{2}$ the inequality holds for $\alpha = (1, \sqrt{2})$ but is not true for $\alpha = (1,e)$ . Similar results should hold at a great level of generality on very general domains.
[131] Self-dual radial non-topological solutions to a competitive Chern-Simons model
Advances in Mathematics, 331, 484-541, 2018
[132] Asymptotic behavior of solutions to a class of diffusive predator-prey systems
Arnaud Ducrot University of Bordeaux Jong-Shenq Guo Tamkang University
[133] Limiting distribution of elliptic homogenization error with periodic diffusion and random potential
Wenjia Jing Tsinghua University
Analysis & PDE, 9, (1), 193--228., 2016
We study the limiting probability distribution of the homogenization error for second order elliptic equations in divergence form with highly oscillatory periodic conductivity coefficients and highly oscillatory stochastic potential. The effective conductivity coefficients are the same as those of the standard periodic homogenization, and the effective potential is given by the mean. We show that the limiting distribution of the random part of the homogenization error, as random elements in proper Hilbert spaces, is Gaussian and can be characterized by the homogenized Green's function, the homogenized solution and the statistics of the random potential. This generalizes previous results in the setting with slowly varying diffusion coefficients, and the current setting with fast oscillations in the differential operator requires new methods to prove compactness of the probability distributions of the random fluctuation.
[134] Global small solutions of heat conductive compressible Navier-Stokes equations with vacuum: smallness on scaling invariant quantity
Jinkai Li South China Normal University
[ Download ] [ 2019-06-21 22:56:08 uploaded by jklimath ] [ 490 downloads ] [ 177 comments ]
In this paper, we consider the Cauchy problem to the heat conductive compressible Navier-Stokes equations in the presence of vacuum and with vacuum far field. Global well-posedness of strong solutions is established under the assumption, among some other regularity and compatibility conditions, that the scaling invariant quantity $\|\rho_0\|_\infty(\|\rho_0\|_3+\|\rho_0\|_\infty^2\|\sqrt{\rho_0}u_0\|_2^2)(\|\nabla u_0\|_2^2+ \|\rho_0\|_\infty\|\sqrt{\rho_0}E_0\|_2^2)$ is sufficiently small, with the smallness depending only on the parameters $R, \gamma, \mu, \lambda,$ and $\kappa$ in the system. Notably, the smallness assumption is imposed on the above scaling invariant quantity exclusively, and it is independent of any norms of the initial data, which is different from the existing papers, see, e.g., \cite{HLX12,HUANGLI11,WENZHU17}. The total mass can be either finite or infinite. A new equation for the density, more precisely for the cubic of the density, derived from combing the continuity and momentum equations, is employed to get the $L^\infty_t(L^3)$ type estimate of the density.
[135] Homogenization and boundary layers
David Gérard-Varet Université Denis Diderot Paris 7, Institut de Mathématiques de Jussieu, 175 rue du Chevaleret, Paris, France Nader Masmoudi New York University, Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, U.S.A.
[136] Some remarks on boundary operators of Bessel extensions
Daniel Spector National Chiao Tung University Jesse Goodman
Discrete Contin. Dyn. Syst. Ser. S, 11, (3), 2018
In this paper we study some boundary operators of a class of Bessel-type Littlewood-Paley extensions whose prototype is $$ \alignat 2 \Delta_xu(x,y)+\frac{1-2s}{y}\frac{\partial u}{\partial y}(x,y)+\frac{\partial^2u}{\partial y^2}(x,y)&=0&&\qquad\text{for}\ x\in\Bbb{R}^dd,\ y>0,\\u(x,0)&=f(x)&&\qquad\text{for}\ x\in\Bbb{R}^d. \endalignat $$ In particular, we show that with a logarithmic scaling one can capture the failure of analyticity of these extensions in the limiting cases $s=k\in\Bbb{N}$.
[137] A generalized Sz. Nagy inequality in higher dimensions and the critical thin film equation
Jian-Guo Liu Duke University Jinhuan Wang Liaoning University
IOPscience, 30, (1), 35–60, 2017.1
Abstract In this paper, we provide an alternative proof for the classical Sz. Nagy inequality in one dimension by a variational method and generalize it to higher dimensions J(h):=(∫Rd|h| dx)a611∫Rd|63h|2 dx(∫Rd|h|m+1 dx)a+1m+1≥β0, where m > 0 for d = 1, 2, for , and . The Euler–Lagrange equation for critical points of in the non-negative radial decreasing function space is given by a free boundary problem for a generalized Lane–Emden equation, which has a unique solution (denoted by h c ) and the solution determines the best constant for the above generalized Sz. Nagy inequality. The connection between the critical mass for the thin-film equation and the best constant of the Sz. Nagy inequality in one dimension was first noted by Witelski et al (2004 Eur. J. Appl. Math. 15 223–56). For the following critical thin film equation in multi-dimension ht+6366(h 63Δh)+6366(h 63hm)=0,x∈Rd, where m = 1 + 2/d, the critical mass is also given by . A finite time blow-up occurs for solutions with the initial mass larger than M c . On the other hand, if the initial mass is less than M c and a global non-negative entropy weak solution exists, then the second moment goes to infinity as or in for some subsequence . This shows that a part of the mass spreads to infinity.
[138] Entropy-bounded solutions to the one-dimensional heat conductive compressible Navier--Stokes equations with far field vacuum
Comm. Pure Appl. Math., 75, (11), 2393–2445, 2022.11
In the presence of vacuum, the physical entropy for polytropic gases behaves singularly and it is thus a challenge to study its dynamics. It is shown in this paper that the boundedness of the entropy can be propagated up to any finite time provided that the initial vacuum presents only at far fields with sufficiently slow decay of the initial density. More precisely, for the Cauchy problem of the onedimensional heat conductive compressible Navier--Stokes equations, the global well-posedness of strong solutions and uniform boundedness of the corresponding entropy are established, as long as the initial density vanishes only at far fields with a rate no more than $O(\frac{1}{x^2})$. The main tools of proving the uniform boundedness of the entropy are some singularly weighted energy estimates carefully designed for the heat conductive compressible Navier--Stokes equations and an elaborate De Giorgi type iteration technique for some classes of degenerate parabolic equations. The De Giorgi type iterations are carried out to different equations in establishing the lower and upper bounds of the entropy.
[139] Stochastic homogenization of viscous superquadratic Hamilton–Jacobi equations in dynamic random environment
Wenjia Jing The University of Chicago Panagiotis E. Souganidis The University of Chicago Hung V. Tran University of Wisconsin at Madison
Research in the Mathematical Sciences, 4, 2017
[140] Quermassintegral preserving curvature flow in hyperbolic space
Ben Andrews Australian National University Yong Wei Australian National University
Geometric and Functional Analysis, 28, 1183–1208, 2018.10
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We consider the quermassintegral preserving flow of closed \emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function $f$ of the principal curvatures which is inverse concave and has dual $f_*$ approaching zero on the boundary of the positive cone. We prove that if the initial hypersurface is \emph{h-convex}, then the solution of the flow becomes strictly \emph{h-convex} for $t>0$, the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology.
[141] Non-uniqueness of Admissible Weak Solutions to Compressible Euler Systems with Source Terms
Tianwen Luo Tsinghua University Chunjing Xie Shanghai Jiao Tong University Zhouping Xin The Chinese University of Hong Kong
Advances in Mathematics, 291, 542-583, 2016.3
We consider admissible weak solutions to the compressible Euler system with source terms, which include rotating shallow water system and the Euler system with damping as special examples. In the case of anti-symmetric sources such as rotations, for general piecewise Lipschitz initial densities and some suitably constructed initial momentum, we obtain infinitely many global admissible weak solutions. Furthermore, we construct a class of finite-states admissible weak solutions to the Euler system with anti-symmetric sources. Under the additional smallness assumption on the initial densities, we also obtain multiple global-in-time admissible weak solutions for more general sources including damping. The basic framework are based on the convex integration method developed by De~Lellis and Sz\'{e}kelyhidi \cite{dLSz1,dLSz2} for the Euler system. One of the main ingredients of this paper is the construction of specified localized plane wave perturbations which are compatible with a given source term.
[142] A four-dimensional Neumann ovaloid
Lavi Karp ORT Braude College Erik Lundberg Florida Atlantic University
Analysis of PDEs Geometric Analysis and Geometric Topology Numerical Analysis and Scientific Computing mathscidoc:1803.43015
What is the shape of a uniformly massive object that generates a gravitational potential equivalent to that of two equal point-masses? If the weight of each point-mass is sufficiently small compared to the distance between the points then the answer is a pair of balls of equal radius, one centered at each of the two points, but otherwise it is a certain domain of revolution about the axis passing through the two points. The existence and uniqueness of such a domain is known, but an explicit parameterization is known only in the plane where the region is referred to as a Neumann oval. We construct a four-dimensional "Neumann ovaloid", solving explicitly this inverse potential problem.
[143] The quenching rate for a nonlocal problem arising in the micro-electro mechanical system
Jong-Shenq Guo Tamkang University Bei Hu University of Notre Dame
[144] Extrapolation from $A_{\infty}^{\rho,\infty}$ , vector-valued inequalities and applications in the Schrödinger settings
Lin Tang LMAM, School of Mathematical Science, Peking University
In this paper, we generalize the$A$_{∞}extrapolation theorem ($Cruz-Uribe–Martell–Pérez$, Extrapolation from$A$_{∞}weights and applications,$J. Funct. Anal.$$213$(2004), 412–439) and the$A$_{$p$}extrapolation theorem of Rubio de Francia to Schrödinger settings. In addition, we also establish weighted vector-valued inequalities for Schrödinger-type maximal operators by using weights belonging to $A_{p}^{\rho,\infty }$ which includes$A$_{$p$}. As applications, we establish weighted vector-valued inequalities for some Schrödinger-type operators.
[145] Multiplication in Sobolev spaces, revisited
A. Behzadan Department of Mathematics and Statistics. California State University, Sacramento, Calif., U.S.A. M. Holst Department of Mathematics, University of California at San Diego, La Jolla, Calif., U.S.A.
In this article, we re-examine some of the classical pointwise multiplication theorems in Sobolev–Slobodeckij spaces, in part motivated by a simple counter-example that illustrates how certain multiplication theorems fail in Sobolev–Slobodeckij spaces when a bounded domain is replaced by R^n. We identify the source of the failure, and examine why the same failure is not encountered in Bessel potential spaces. To analyze the situation, we begin with a survey of the classical multiplication results stated and proved in the 1977 article of Zolesio, and carefully distinguish between the case of spaces defined on the all of R^n and spaces defined on a bounded domain (with e.g. a Lipschitz boundary). However, the survey we give has a few new wrinkles; the proofs we include are based almost exclusively on interpolation theory rather than Littlewood–Paley theory and Besov spaces, and some of the results we give and their proofs, including the results for negative exponents, do not appear in the literature in this form. We also include a particularly important variation of one of the multiplication theorems that is relevant to the study of nonlinear PDE systems arising in general relativity and other areas. The conditions for multiplication to be continuous in the case of Sobolev–Slobodeckij spaces are somewhat subtle and intertwined, and as a result, the multiplication theorems of Zolesio in 1977 have been cited (more than once) in the standard literature in slightly more generality than what is actually proved by Zolesio, and in cases that allow for construction of counter-examples such as the one included here.
[146] On the Free Surface Motion of Highly Subsonic Heat-conducting Inviscid Flows
Tao Luo City University of Hong Kong Huihui Zeng Tsinghua University
For the free surface problem of the highly subsonic heat-conducting inviscid flow in 2-D and 3-D, a priori estimates for geometric quantities of free surfaces, such as the second fundamental form and the injectivity radius of the normal exponential map, and the Sobolev norms of fluid variables are proved by investigating the coupling of the boundary geometry and the interior solutions. An interesting feature for the free surface problem studied in this paper is the loss of one more derivative than the problem of incompressible Euler equations for which a geometric approach was introduced by Christodoulou and Lindblad in [11]. Due to the loss of one more derivative and loss of symmetry of equations, the geometric approach in [11] needs to be substantially developed by exploring the interaction of large variation of temperature, heat-conduction, non-zero divergence of the fluid velocity and the evolution of free surfaces.
[147] A sharp lower bound for the log canonical threshold
Jean-Pierre Demailly Université de Grenoble I, Département de Mathématiques, Institut Fourier, Saint-Martin d'Hères, France Hoàng Hiệp Phạm Department of Mathematics, Hanoi National University of Education
Acta Mathematica, 212, (1), 1-9, 2012.1
In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function $${\varphi}$$ with an isolated singularity at 0 in an open subset of $${\mathbb{C}^n}$$ . This threshold is defined as the supremum of constants$c$> 0 such that $${e^{-2c\varphi}}$$ is integrable on a neighborhood of 0. We relate $${c(\varphi)}$$ to the intermediate multiplicity numbers $${e_j(\varphi)}$$ , defined as the Lelong numbers of $${(dd^c\varphi)^j}$$ at 0 (so that in particular $${e_0(\varphi)=1}$$ ). Our main result is that $${c(\varphi)\geqslant\sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)}$$ . This inequality is shown to be sharp; it simultaneously improves the classical result $${c(\varphi)\geqslant 1/e_1(\varphi)}$$ due to Skoda, as well as the lower estimate $${c(\varphi)\geqslant n/e_n(\varphi)^{1/n}}$$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals.
[148] Energy method for Boltzmann equation
Tai-Ping Liu Tong Yang Shih-Hsien Yu
Physica D: Nonlinear Phenomena, 188, 178-192, 2004.2
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A basic, simple energy method for the Boltzmann equation is presented here. It is based on a new macromicro decomposition of the Boltzmann equation as well as the H-theorem. This allows us to make use of the ideas from hyperbolic conservation laws and viscous conservation laws to yield the direct energy method. As an illustration, we apply the method for the study of the time-asymptotic, nonlinear stability of the global Maxwellian states. Previous energy method, starting with Grad and finishing with Ukai, involves the spectral analysis and regularity of collision operator through sophisticated weighted norms.
[149] Quasiconformality, homeomorphisms between metric measure spaces preserving quasiminimizers, and uniform density property
Riikka Korte Department of Mathematics and Statistics, University of Helsinki Niko Marola Department of Mathematics and Statistics, University of Helsinki Nageswari Shanmugalingam Department of Mathematical Sciences, University of Cincinnati
We characterize quasiconformal mappings as those homeomorphisms between two metric measure spaces of locally bounded geometry that preserve a class of quasiminimizers. We also consider quasiconformal mappings and densities in metric spaces and give a characterization of quasiconformal mappings in terms of the uniform density property introduced by Gehring and Kelly.
[150] Invertibility of nonsmooth mappings
Marcelo Montenegro Universidade Estadual de Campinas Adilson Presoto Universidade Federal de São Carlos
Let F :RN→RN be a locally Lipschitz continuous function. We prove that F is a global homeomorphism or only injective, under suitable assumptions on the subdifferential ∂F(x). We use variational methods, nonsmooth inverse function theorem and extensions of the Hadamard-Levy Theorem. We also address questions on the Markus-Yamabe conjecture.
[151] MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well-Posedness Theory
Liu Cheng-jie Shanghai Jiao Tong University Xie Feng Shanghai Jiao Tong University Yang Tong City University of Hong Kong
Communications on Pure and Applied Mathematics, 72, (1), 0063–0121, 2019.1
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We study the well-posedness theory for the MHD boundary layer. The boundary layer equations are governed by the Prandtl-type equations that are derived from the incompressible MHD system with non-slip boundary condition on the velocity and perfectly conducting condition on the magnetic field. Under the assumption that the initial tangential magnetic field is not zero, we establish the local-in time existence, uniqueness of solutions for the nonlinear MHD boundary layer equations. Compared with the well-posedness theory of the classical Prandtl equations for which the monotonicity condition of the tangential velocity plays a crucial role, this monotonicity condition is not needed for the MHD boundary layer. This justifies the physical understanding that the magnetic field has a stabilizing effect on MHD boundary layer in rigorous mathematics.
[152] Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos
Hui Huang Tsinghua University Jian-Guo Liu Duke University
Kinetic and Related Models, 9, (4), 715 - 748, 2016.12
This paper investigates the generalized Keller-Segel (KS) system with a nonlocal diffusion term -ν(-Δ) α/2 ρ (1 < α < 2). Firstly, the global existence of weak solutions is proved for the initial density ρ0 ∈ L01∩L d/α (60d) (d ≥ 2) with [norm of matrix]ρ0[norm of matrix] d/α < K, where K is a universal constant only depending on d, α, ν. Moreover, the conservation of mass holds true and the weak solution satisfies some hyper-contractive and decay estimates in Lr for any 1 < r < ∞. Secondly, for the more general initial data ρ0 ∈ L01 ∩ L05(60d) (d = 2, 3), the local existence is obtained. Thirdly, for ρ0 ∈ L01 (60d; (1 + |x|)dx ∩ L∞(60d)( d ≥ 2) with [norm of matrix]ρ0[norm of matrix]d/α < K, we prove the uniqueness and stability of weak solutions under Wasserstein metric through the method of associating the KS equation with a self-consistent stochastic process driven by the rotationally invariant α-stable Lévy process Lα(t). Also, we prove the weak solution is L1 bounded uniformly in time. Lastly, we consider the N-particle interacting system with the Lévy process Lα(t) and the Newtonian potential aggregation and prove that the expectation of collision time between particles is below a universal constant if the moment ∫60d |x| γρ0dx for some 1 < γ < α is below a universal constant K γ and ν is also below a universal constant. Meanwhile, we prove the propagation of chaos as N → ∞ for the interacting particle system with a cut-off parameter ε ~ (ln N)-1/d, and show that the mean field limit equation is exactly the generalized KS equation.
[153] Uniform exponential stability of the Ekman spiral
Yoshikazu Giga Graduate School of Mathematical Sciences, University of Tokyo Jürgen Saal Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf
This paper studies stability of the Ekman boundary layer. We utilize a new approach, developed by the authors in a precedent paper, based on Fourier transformed finite vector Radon measures, which yields exponential stability of the Ekman spiral. By this method we can also derive very explicit bounds for solutions of the linearized and the nonlinear Ekman systems. For example, we can prove these bounds to be uniform with respect to the angular velocity of rotation, which has proved to be relevant for several aspects. Another advantage of this approach is that we obtain well-posedness in classes containing nondecaying vector fields such as almost periodic functions. These outcomes give respect to the nature of boundary layer problems and cannot be obtained by approaches in standard function spaces such as Lebesgue, Bessel-potential, Hölder or Besov spaces.
[154] Evaluating solutions on an elliptic problem in a gravitational gauge field theory
Jann-Long Chern National Central Universiy Sze-Guang Yang National Central University
J. Funct. Anal. , 265, (7), 1240–1263, 2010
An elliptic equation arising from the study o fstatic solutions with prescribed zeros and poles of the Einstein equations coupled with the classical sigma model and an Abelian gauge field, is considered. We classify the solutions and establish the uniqueness of radially symmetric solutions. We also complete a classification of symmetric solutions of an elliptic equation on the sphere.
[155] On a Class of Functional Inequalities and their Applications to Fourth-Order Nonlinear Parabolic Equations
Jian-Guo Liu Department of Physics and Department of Mathematics, Duke University Xiangsheng Xu Department of Mathematics and Statistics, Mississippi State University
Abstract We study a class of fourth order nonlinear parabolic equations which include the thin-film equation and the quantum drift-diffusion model as special cases. We investigate these equations by first developing functional inequalities of the type $$ \int_\Omega u^{2\gamma-\alpha-\beta}\Delta\ua\Delta\ub dx \geq c\int_\Omega|\Delta \ur |^2dx, $$ which seem to be of interest on their own right.
[156] Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent m > 0
Shen Bian Tsinghua University Jian-Guo Liu Duke University
Communications in Mathematical Physics, 323, (3), 1017–1070, 2013.11
This paper investigates infinite-time spreading and finite-time blow-up for the Keller-Segel system. For 0 < m ≤ 2 − 2 / d, the L p space for both dynamic and steady solutions are detected with ${p:=\frac{d(2-m)}{2} }$ . Firstly, the global existence of the weak solution is proved for small initial data in L p . Moreover, when m > 1 − 2 / d, the weak solution preserves mass and satisfies the hyper-contractive estimates in L q for any p < q < ∞. Furthermore, for slow diffusion 1 < m ≤ 2 − 2/d, this weak solution is also a weak entropy solution which blows up at finite time provided by the initial negative free energy. For m > 2 − 2/d, the hyper-contractive estimates are also obtained. Finally, we focus on the L p norm of the steady solutions, it is shown that the energy critical exponent m = 2d/(d + 2) is the critical exponent separating finite L p norm and infinite L p norm for the steady state solutions.
[157] Large-scale dynamics of Mean-Field Games driven by local Nash equilibria
Pierre Degond Institut de Mathématiques de Toulouse Jian-Guo Liu Duke University Christian Ringhofer School of Mathematics and Statistical Sciences
Journal of Nonlinear Science, 24, (1), 93-115, 2014.1
We introduce a new mean field kinetic model for systems of rational agents interacting in a game-theoretical framework. This model is inspired from non-cooperative anonymous games with a continuum of players and Mean-Field Games. The large time behavior of the system is given by a macroscopic closure with a Nash equilibrium serving as the local thermodynamic equilibrium. An application of the presented theory to a social model (herding behavior) is discussed.
[158] Boundary integral operators and boundary value problems for Laplace's equation
TongKeun Chang Department of Mathematics, Korea Institute of Advanced Studies John L. Lewis Mathematics Department, University of Kentucky
In this paper, we define boundary single and double layer potentials for Laplace's equation in certain bounded domains with$d$-Ahlfors regular boundary, considerably more general than Lipschitz domains. We show that these layer potentials are invertible as mappings between certain Besov spaces and thus obtain layer potential solutions to the regularity, Neumann, and Dirichlet problems with boundary data in these spaces.
[159] Perturbation theorems for Hele-Shaw flows and their applications
Yu-Lin Lin Institute of Mathematics, Academia Sinica
In this work, we give a perturbation theorem for strong polynomial solutions to the zero surface tension Hele-Shaw equation driven by injection or suction, the so called Polubarinova–Galin equation. This theorem enables us to explore properties of solutions with initial functions close to polynomials. Applications of this theorem are given in the suction and injection cases. In the former case, we show that if the initial domain is close to a disk, most of the fluid will be sucked before the strong solution blows up. In the latter case, we obtain precise large-time rescaling behaviors for large data to Hele-Shaw flows in terms of invariant Richardson complex moments. This rescaling behavior result generalizes a recent result regarding large-time rescaling behavior for small data in terms of moments. As a byproduct of a theorem in this paper, a short proof of existence and uniqueness of strong solutions to the Polubarinova–Galin equation is given.
[160] Well-posedness for Hall-magnetohydrodynamics
Dongho Chae Chung-Ang University Pierre Degond Institut de Mathématiques de Toulouse Jian-Guo Liu Duke University
Annales De Linstitut Henri Poincare Non Linear Analysis, 31, (3), 555–565, 2014.5
We prove local existence of smooth solutions for large data and global smooth solutions for small data to the incompressible, resitive, viscous or inviscid Hall-MHD model. We also show a Liouville theorem for the stationary solutions.
[161] On Singularity Formation of a Nonlinear Nonlocal System
Thomas Y. Hou Caltech Congming Li University of Colorado Zuoqiang Shi Tsinghua University Shu Wang Beijing University of Technology Xinwei Yu University of Alberta
Archive for Rational Mechanics and Analysis, 199, 117-144, 2011
[162] Asymptotic-Preserving schemes for kinetic-fluid modeling of disperse two-phase flows
Thierry Goudon CNRS & Université Nice Sophia Antipolis, France Shi Jin Ministry of Education Key Laboratory of Scientific and Engineering Computing Jian-Guo Liu Duke University Bokai Yan University of California
Journal of Computational Physics, 246, (1), 145–164, 2013.8
[163] Ground-state solutions of a two-component elliptic system in R4 with the Sobolev critical exponent
[164] On a two-component Bose-Einstein condensate with steep potential wells
Yuanze Wu China University of Mining and Technology Tsung-fang Wu National University of Kaohsiung Wenming Zou Tsinghua University
[165] Asymptotic analysis for time harmonic wave problems with small wave number
Houde Han Tsinghua University
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[166] Global solutions of the gravity-capillary water-wave system in three dimensions
Yu Deng New York University Alexandru D. Ionescu Princeton University Benoît Pausader Princeton University Fabio Pusateri Princeton University
In this paper we prove global regularity for the full water-wave system in three dimensions for small data, under the influence of both gravity and surface tension. This problem presents essential difficulties which were absent in all of the earlier global regularity results for other water-wave models. To construct global solutions, we use a combination of energy estimates and matching dispersive estimates. There is a significant new difficulty in proving energy estimates in our problem, namely the combination of slow pointwise decay of solutions (no better than |t|−5/6) and the presence of a large, codimension-1, set of quadratic time-resonances. To deal with such a situation, we propose here a new mechanism, which exploits a non-degeneracy property of the time-resonant hypersurfaces and some special structure of the quadratic part of the non-linearity, connected to the conserved energy of the system. The dispersive estimates rely on analysis of the Duhamel formula in the Fourier space. The main contributions come from the set of space-time resonances, which is a large set of dimension 1. To control the corresponding bilinear interactions, we use harmonic analysis techniques, such as orthogonality arguments in the Fourier space and atomic decompositions of functions. Most importantly, we construct and use a refined norm which is well adapted to the geometry of the problem.
[167] Existence of solutions to the even dual minkowski problem
Yiming Zhao New York University
Journal of Differential Geometry, 110, (3), 543-572, 2018
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Recently, Huang, Lutwak, Yang & Zhang discovered the duals of Federer's curvature measures within the dual Brunn-Minkowski theory and stated the "Minkowski problem" associated with these new measures. As they showed, this dual Minkowski problem has as special cases the Aleksandrov problem (when the index is 0) and the logarithmic Minkowski problem (when the index is the dimension of the ambient space)—two problems that were never imagined to be connected in any way. Huang, Lutwak, Yang & Zhang established sufficient conditions to guarantee existence of solution to the dual Minkowski problem in the even setting. In this work, existence of solution to the even dual Minkowski problem is established under new sufficiency conditions. It was recently shown by B ̈or ̈oczky, Henk & Pollehn that these new sufficiency conditions are also necessary.
[168] Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model
Jian-Guo Liu Jianfeng Lu Dionisios Margetis Jeremy L. Marzuola
Physica D, 393, 54-67, 2019
[169] Existence of local solutions for the Boltzmann equation without angular cutoff
Radjesvarane Alexandre Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang
Analysis of PDEs mathscidoc:1912.431012
Comptes Rendus Mathematique, 347, 1237-1242, 2009.11
We consider the spatially inhomogeneous Boltzmann equation without angular cutoff. We prove the existence and uniqueness of local classical solutions to the Cauchy problem, in the function space with Maxwellian type exponential decay with respect to the velocity variable. <b><i>To cite this article: R. Alexandre et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).</i></b>
[170] Qualitative analysis for an elliptic system in the punctured space
In this paper, we investigate the qualitative properties of positive solutions for a two-coupled elliptic system in the punctured space. We establish a monotonicity formula that completely characterizes the singularity of positive solutions. We prove a sharp global estimate for both components of positive solutions. We also prove the nonexistence of positive semi-singular solutions, which means that one component is bounded near the singularity and the other component is unbounded near the singularity.
[171] Self-similar spreading in a merging-splitting model of animal group size
Jian-Guo Liu B. Niethammer Robert L. Pego
Journal of Statistical Physics, 2019
[172] Numerical methods for oscillatory solutions to hyperbolic problems
Bjorn Engquist University of California, Los Angeles Jian-Guo Liu Courant Institute
Communications on Pure and Applied Mathematics, 46, (10), 1327-1361, 1993.10
Difference approximations of hyperbolic partial differential equations with highly oscillatory coefficients and initial values are studied. Analysis of strong and weak convergence is carried out in the practically interesting case when the discretization step sizes are essentially independent of the oscillatory wave lengths. © 1993 John Wiley & Sons, Inc.
[173] The dual Orlicz-Minkowski problem
Baocheng Zhu Hubei University for Nationalities Sudan Xing Memorial University of Newfoundland Deping Ye Memorial University of Newfoundland
Analysis of PDEs Functional Analysis Geometric Analysis and Geometric Topology Convex and Discrete Geometry mathscidoc:1904.03005
J. Geom. Anal., 28, (14), 3829–3855, 2018.12
In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the dual Orlicz-Minkowski problem regarding the dual Orlicz curvature measure is provided.
[174] Global solution for the 3D gravity water waves system above a flat bottom
Xuecheng Wang Tsinghua University
Advances in Mathematics , 346, 805–886, 2019
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Given any suitably small, localized, and smooth initial data, in this paper, we prove global regularity for the 3D finite depth gravity water wave system. As a byproduct, we rule out the small, localized traveling waves in 3D, which do exist for the same system in 2D.
[175] On the Carleson duality
Tuomas Hytönen Department of Mathematics and Statistics, P.O. Box 68 (Gustaf Hällströms gata 2b), Helsingfors universitet, Finland Andreas Rosén Department of Mathematics, Linköpings universitet
Auscher P, Rosen A. Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II[J]. Analysis \u0026 PDE, 2010, 5(5): 983-1061.
Hofmann S, Kenig C E, Mayboroda S, et al. The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients[J]. Mathematische Annalen, 2013: 863-907.
Rosen A. Cauchy non-integral formulas[C]., 2012: 163-178.
Pascal Auscher · Dorothee Frey. ON THE WELL-POSEDNESS OF PARABOLIC EQUATIONS OF NAVIER–STOKES TYPE WITH DATA. 2015.
As a tool for solving the Neumann problem for divergence-form equations, Kenig and Pipher introduced the space ${\mathcal{X}}$ of functions on the half-space, such that the non-tangential maximal function of their$L$_{2}Whitney averages belongs to$L$_{2}on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of ${\mathcal{X}}$ , and characterize the pointwise multipliers from ${\mathcal{X}}$ to$L$_{2}on the half-space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to$L$_{$p$}generalizations of the space ${\mathcal{X}}$ . Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.
[176] An improved Combes–Thomas estimate of magnetic Schrödinger operators
Zhongwei Shen Department of Mathematics and Statistics, Auburn University
In the present paper, we prove an improved Combes–Thomas estimate, viz. the Combes–Thomas estimate in trace-class norms, for magnetic Schrödinger operators under general assumptions. In particular, we allow for unbounded potentials. We also show that for any function in the Schwartz space on the reals the operator kernel decays, in trace-class norms, faster than any polynomial.
[177] Thin film epitaxy with or without slope selection
BO LI University of Maryland Jian-Guo Liu University of Maryland
European Journal of Applied Mathematics, 14, (6), 713–743, 2003.2
Two nonlinear diffusion equations for thin film epitaxy, with or without slope selection, are studied in this work. The nonlinearity models the Ehrlich–Schwoebel effect – the kinetic asymmetry in attachment and detachment of adatoms to and from terrace boundaries. Both perturbation analysis and numerical simulation are presented to show that such an atomistic effect is the origin of a nonlinear morphological instability, in a rough-smooth-rough pattern, that has been experimentally observed as transient in an early stage of epitaxial growth on rough surfaces. Initial-boundary-value problems for both equations are proven to be well-posed, and the solution regularity is also obtained. Galerkin spectral approximations are studied to provide both a priori bounds for proving the well-posedness and numerical schemes for simulation. Numerical results are presented to confirm part of the analysis and to explore the difference between the two models on coarsening dynamics.
[178] Global well-posedness of the one-dimensional compressible navier-stokes equations with constant heat conductivity and nonnegative density
Jinkai Li South Chine Normal University
SIAM J. MATH. ANAL., 51, (5), 366-{3693, 2019
In this paper we consider the initial-boundary value problem to the one-dimensional compressible Navier{Stokes equations for ideal gases. Both the viscous and heat conductive coeffcients are assumed to be positive constants, and the initial density is allowed to have vacuum. Global existence and uniqueness of strong solutions is established for any H2 initial data, which generalizes the well-known result of Kazhikhov and Shelukhin [J. Appl. Math. Mech., 41 (1977), pp. 273{282] to the case that with nonnegative initial density. An observation to overcome the di culty caused by the lack of the positive lower bound of the density is that the ratio of the density to its initial value is inversely proportional to the time integral of the upper bound of the temperature along the trajectory.
[179] Maximal Marcinkiewicz multipliers
Petr Honzík Faculty of Mathematics and Physics, Charles University
Woocheol Choi. Maximal functions for multipliers on stratified groups: Maximal functions for multipliers on stratified groups. 2015.
Let $\mathcal{M} =\{m_{j}\}_{j=1}^{\infty}$ be a family of Marcinkiewicz multipliers of sufficient uniform smoothness in $\mathbb{R}^{n}$ . We show that the$L$^{$p$}norm, 1<$p$<∞, of the related maximal operator $$M_Nf(x)= \sup_{1\leq j \leq N} |\mathcal{F}^{-1} ( m_j \mathcal{F} f)|(x) $$ is at most$C$(log($N$+2))^{$n$/2}. We show that this bound is sharp.
[180] Polynomial hulls and proper analytic disks
Armen Edigarian Institute of Mathematics, Jagiellonian University
We show how to construct the Perron–Bremermann function by using proper analytic disks. We apply this result to the polynomial hull of a compact set$K$defined on the boundary of the unit ball.
[181] On Singularity Formation of a 3D Model for Incompressible Navier-Stokes Equations
Thomas Y. Hou Caltech Zuoqiang Shi Tsinghua University Shu Wang Beijing University of Technology
We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei (2009) in [15] for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. This model shares many properties of the 3D incompressible Navier– Stokes equations. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3D inviscid model for a class of initial boundary value problems with smooth initial data of finite energy. We also prove the global regularity of the 3D inviscid model for a class of small smooth initial data.
[182] Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics
Pierre Degond Université de Toulouse Amic Frouvelle Université Paris-Dauphine Jian-Guo Liu Duke University
Archive for Rational Mechanics and Analysis, 216, (1), 63-115, 2015.1
We provide a complete and rigorous description of phase transitions for kinetic models of self-propelled particles interacting through alignment. These models exhibit a competition between alignment and noise. Both the alignment frequency and noise intensity depend on a measure of the local alignment. We show that, in the spatially homogeneous case, the phase transition features (number and nature of equilibria, stability, convergence rate, phase diagram, hysteresis) are totally encoded in how the ratio between the alignment and noise intensities depend on the local alignment. In the spatially inhomogeneous case, we derive the macroscopic models associated to the stable equilibria and classify their hyperbolicity according to the same function.
[183] Regularity for a fractional p-Laplace equation
Daniel Spector National Chiao Tung University Tien-Tsan Shieh National Center for Theoretical Science
Commun. Contemp. Math., 20, (1), 2017.10
In this note, we consider regularity theory for a fractional p-Laplace operator which arises in the complex interpolation of the Sobolev spaces, the Hs,p-Laplacian. We obtain the natural analogue to the classical p-Laplacian situation, namely Cs+αloc-regularity for the homogeneous equation.
[184] Linear elliptic equations in composite media with anisotropic fibres
Journal of Differential Equations, 266, (10), 6580--6620, 2019.5
Linear elliptic equations in composite media with anisotropic fibres are concerned. The media consist of a periodic set of anisotropic fibres with low conductivity, included in a connected matrix with high conductivity. Inside the anisotropic fibres, the conductivity in the longitudinal direction is relatively high compared with that in the transverse directions. The coefficients of the elliptic equations depend on the conductivity. This work is to derive the H\"older and the gradient $L^p$ estimates (uniformly in the period size of the set of anisotropic fibres as well as in the conductivity ratio of the fibres in the transverse directions to the connected matrix) for the solutions of the elliptic equations. Furthermore, it is shown that, inside the fibres, the solutions have higher regularity along the fibres than in the transverse directions.
[185] General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski Problem I
Richard J. Gardner Western Washington University Daniel Hug Karlsruhe Institute of Technology Wolfgang Weil Karlsruhe Institute of Technology Sudan Xing Memorial University of Newfoundland Deping Ye Memorial University of Newfoundland
Calc. Var. PDE., 58, 12, 2019
The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski problem posed by Lutwak, Yang, and Zhang for the $(p,q)$-dual curvature measures when $p>0$ and $q<0$. A dual Orlicz-Brunn-Minkowski inequality for general volumes is obtained, as well as dual Orlicz-Minkowski-type inequalities and uniqueness results for star bodies. Finally, a very general Minkowski-type inequality, involving two Orlicz functions, two convex bodies, and a star body, is proved, that includes as special cases several others in the literature, in particular one due to Lutwak, Yang, and Zhang for the $(p,q)$-mixed volume.
[186] On the solutions to a Liouville-type system involving singularity
Jann-Long Chern National Central University Zhi-You Chen National Changhua University of Education Yong-Li Tang National Central University
Calc. Var. Partial Differential Equations, 43, (2), 57–81, 2012
Chern J, Chen Z, Tang Y, et al. Structure of solutions to a singular Liouville system arising from modeling dissipative stationary plasmas[J]. Discrete and Continuous Dynamical Systems, 2012, 33(6): 2299-2318.
Zhiyou Chen · Jannlong Chern. Topological multivortex solutions for the Chern–Simons system with two Higgs particles. 2016.
[187] A degenerate p-Laplacian Keller-Segel model
Kinetic and Related Models , 9, (4), 687-714, 2016.12
where p > 1. 1 < p < 2 is called the fast p-Laplacian diffusion, while p > 2 is called the slow p-Laplacian diffusion. Especially, the p-Laplacian Keller-Segel model turns to the original model when p = 2.The Keller-Segel model was firstly presented in 1970 to describe the chemotaxis of cellular slime molds [13, 14]. The original model was considered in 2D,
[188] H\"{o}lder Continuous Solutions to the Three-dimensional Prandtl System
Tianwen Luo Tsinghua University Zhouping Xin The Chinese University of Hong Kong
Adapting the convex integration technique introduced in \cite{dLSz5} and subsequently developed in \cite{IsettVicol,IsettOh16,Buckmaster2013transporting}, we construct H\"{o}lder continuous weak solutions to the three dimensional Prandtl system and some other models with vertical viscosity.
[189] Spectral theory for the Weil–Petersson Laplacian on the Riemann moduli space
Lizhen Ji University of Michigan Rafe Mazzeo Stanford University Werner Müller Universität Bonn Andras Vasy Stanford University
Comment. Math. Helv, 89, 867–894, 2014
We study the spectral geometric properties of the scalar Laplace–Beltrami operator associated to the Weil–Petersson metric $g_{WP}$ on $\mathcal{M}_{\gamma}$ , the Riemann moduli space of surfaces of genus $\gamma>1$. This space has a singular compactification with respect to $g_{WP}$, and this metric has crossing cusp-edge singularities along a finite collection of simple normal crossing divisors. We prove first that the scalar Laplacian is essentially self-adjoint, which then implies that its spectrum is discrete.
[190] Stability of the Superposition of a Viscous Contact Wave with Two Rarefaction Waves to the Bipolar Vlasov--Poisson--Boltzmann System
Hai-liang Li Teng Wang Yi Wang
SIAM Journal on Mathematical Analysis, 50, (2), 1829-1876, 2018.4
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We investigate the nonlinear stability of the superposition of a viscous contact wave and two rarefaction waves for a one-dimensional (1D) bipolar Vlasov--Poisson--Boltzmann (VPB) system, which can be used to describe the transportation of charged particles under the additional electrostatic potential force. Based on a new micro-macro type of decomposition around the local Maxwellian related to the bipolar VPB system in the previous work [H.-L. Li, Y. Wang, T. Yang, and M.-Y. Zhong, <i>Arch. Ration. Mech. Anal.</i>, 228 (2018), pp. 39--127], we prove that the superposition of a viscous contact wave and two rarefaction waves is time-asymptotically stable to the 1D bipolar VPB system under some smallness conditions on the initial perturbations and wave strength, which implies that this typical superposition wave pattern is nonlinearly stable under the combined effects of the binary collisions, the electrostatic potential
[191] Compressible Euler equations with vacuum
Tai-Ping Liu Tong Yang
Journal of differential equations, 140, (2), 223-237, 1997.11
In this paper, we first show that the regular solutions of compressible Euler equations in<i>R</i><sup>3</sup>with damping will not be global if the initial density function has compact support. This implies that<i>c</i><sup>2</sup>cannot be smooth across the boundary<i></i>separating the gas and the vacuum after a finite time, where<i>c</i>is the speed of sound. Then we study the local existence of solutions for isentropic gas flow in<i>R</i>when<i>c<sup></sup></i>, 0<<i></i>1, is smooth across<i></i>, using the energy method and the characteristics method.
[192] A note on Monge–Amp`ere Keller–Segel equation
Applied Mathematics Letters, 11, 26–34, 2016.11
This note studies the Monge-Ampère Keller–Segel equation in a periodic domain , a fully nonlinear modification of the Keller–Segel equation where the Monge-Ampère equation substitutes for the usual Poisson equation . The existence of global weak solutions is obtained for this modified equation. Moreover, we prove the regularity in for some .
[193] Existence and nonexistence to exterior Dirichlet problem for Monge-Ampere equation
YanYan Li Rutgers University Siyuan Lu Rutgers University
[194] A new type of non-topological bubbling solutions to a competitive Chern-Simons model
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 19, 65-108, 2019
[195] A coupled chemotaxis-fluid model: Global existence
Jian-Guo Liu Duke University Alexander Lorz University of Cambridge
Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 28, (5), 643–652, 2011.9
Nous considérons un modèle provenant de la biologie, composé d03équations de chimiotactisme couplées aux équations de fluide visqueux incompressible par le transport et le for04age externe. L03existence globale des solutions du problème de Cauchy est étudiée sous certaines conditions. Précisément, pour le système chimiotactisme–Navier–Stokes en deux dimensions d03espace, nous obtenons l03existence globale pour des données grandes. En trois dimensions d03espace, nous démontrons l03existence globale des solutions faibles pour le système chimiotactisme–Stokes avec une diffusion non-linéaire de la densité des cellules.
[196] Instability of spherical naked singularities of a scalar field under gravitational perturbations
Junbin Li Sun Yat-sen University Jue Liu Sun Yat-sen University
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In this paper, we initiate the study of the instability of naked singularities without symmetries. In a series of papers, Christodoulou proved that naked singularities are not stable in the context of the spherically symmetric Einstein equations coupled with a massless scalar field. We study in this paper the next simplest case: a characteristic initial value problem of this coupled system with the initial data given on two intersecting null cones, the incoming one of which is assumed to be spherically symmetric and singular at its vertex, and the outgoing one of which has no symmetries. It is shown that, arbitrarily fixing the initial scalar field, the set of the initial conformal metrics on the outgoing null cone such that the maximal future development does not have any sequences of closed trapped surfaces approaching the singularity, is of first category in the whole space in which the shear tensors are continuous. Such a set can then be viewed as exceptional, although the exceptionality is weaker than the at least 1 co-dimensionality in spherical symmetry. Almost equivalently, it is also proved that, arbitrarily fixing an incoming null cone C––ε to the future of the initial incoming null cone, the set of the initial conformal metrics such that the maximal future development has at least one closed trapped surface before C––ε, contains an open and dense subset of the whole space. Since the initial scalar field can be chosen such that the singularity is naked if the initial shear is set to be zero, we may say that the spherical naked singularities of a self-gravitating scalar field are not stable under gravitational perturbations. This in particular gives new families of non-spherically symmetric gravitational perturbations different from the original spherically symmetric scalar perturbations given by Christodoulou.
[197] Existence and uniqueness of global weak solution to a kinetic model for the sedimentation of rod-like particles
Xiuqing Chen Beijing University of Posts and Telecommunications Xiaolong Li Beijing University of Posts and Telecommunications Jian-Guo Liu Duke University
Communications in Mathematical Sciences, 12, (8), 1579–1601, 2014.1
ABSTRACT We investigate a kinetic model for the sedimentation of dilute suspensions of rod-like particles under gravity, deduced by Helzel, Otto, and Tzavaras (2011), which couples the impressible (Navier-)Stokes equation with the Fokker-Planck equation. With a no-flux boundary condition for the distribution function, we establish the existence and uniqueness of a global weak solution to the two dimensional model involving the Stokes equation.
[198] Global stability for solutions to the exponential PDE describing epitaxial growth
Jian-Guo Liu ROBERT M. STRAIN
Interfaces and Free Boundaries, 21, 61–86, 2019
[199] A note on $L^{\infty}$-bound and uniqueness to a degenerate Keller-Segel model
Acta Applicandae Mathematicae, 142, (1), 173–188, 2016.4
Abstract In this note we establish the uniform (Formula presented.)-bound for the weak solutions to a degenerate Keller-Segel equation with the diffusion exponent (Formula presented.) under a sharp condition on the initial data for the global existence. As a consequence, the uniqueness of the weak solutions is also proved.
[200] Analysis of polymeric flow models and related compactness theorems in weighted spaces
XIUQING CHEN Beijing University of Posts and Telecommunications Jian-Guo Liu Duke University
Siam Journal on Mathematical Analysis, 45, (3), 1179–1215, 2013.5
We studied coupled systems of the Fokker--Planck equation and the Navier--Stokes equation modeling the Hookean and the finitely extensible nonlinear elastic (FENE)-type polymeric flows. We proved the continuous embedding and compact embedding theorems in weighted spaces that naturally arise from related entropy estimates. These embedding estimates are shown to be sharp. For the Hookean polymeric system with a center-of-mass diffusion and a superlinear spring potential, we proved the existence of a global weak solution. Moreover, we were able to tackle the FENE model with $L^2$ initial data for the polymer density instead of the $L^\infty$ counterpart in the literature.
[201] Relaxation and diffusion enhanced dispersive waves
Shi Jin Courant Institute of Mathematical Sciences Jian-Guo Liu Temple University
Mathematical and Physical Sciences, 446, (1928), 555-563, 1994.9
The development of shocks in nonlinear hyperbolic conservation laws may be regularized through either diffusion or relaxation. However, we have observed surprisingly that for some physical problems, when both of the smoothing factors - diffusion and relaxation - coexist, under appropriate asymptotic assumptions, the dispersive waves are enhanced. This phenomenon is studied asymptotically in the sense of the Chapman-Enskog expansion and demonstrated numerically.
[202] 1-minimization method for link flow correction
Penghang Yin Zhe Sun Wen-Long Jin Jack Xin
Transportation Research Part B: Methodological, 104, 398-408, 2017.10
A computational method, based on <sub>1</sub>-minimization, is proposed for the problem of link flow correction, when the available traffic flow data on many links in a road network are inconsistent with respect to the flow conservation law. Without extra information, the problem is generally ill-posed when a large portion of the link sensors are unhealthy. It is possible, however, to correct the corrupted link flows accurately with the proposed method under a recoverability condition if there are only a few bad sensors which are located at certain links. We analytically identify the links that are robust to miscounts and relate them to the geometric structure of the traffic network by introducing the recoverability concept and an algorithm for computing it. The recoverability condition for corrupted links is simply the associated recoverability being greater than 1. In a more realistic setting, besides the unhealthy link sensors, small
[203] Evolutions of the momentum density, deformation tensor and the nonlocal term of the Camassa-Holm equation
Wei-Wei Guo Ningxia Institute of Science and Technology Tai-Man Tang Xiangtan University
[204] Inverse curvature flow in anti-de Sitter-Schwarzschild manifold
Siyuan Lu McGill University
[205] Maximization of the second non-trivial Neumann eigenvalue
Dorin Bucur Université Grenoble Alpes Antoine Henrot Université de Lorraine
Acta Mathematica, 222, (2), 337-361, 2019
In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of RN with prescribed measure m attains its maximum on the union of two disjoint balls of measure m/2. As a consequence, the Pólya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.
[206] On a two-component Bose–Einstein condensate with steep potential wells
Annali di Matematica, 196, (1), 1695–1737, 2017.2
[207] Macroscopic fluid modes with localized kinetic upscaling effects
PIERRE DEGOND Universit´e Paul Sabatier Jian-Guo Liu University of Maryland LUC MIEUSSENS Universit´e Paul Sabatier
MULTISCALE MODEL. SIMUL, 5, (3), 940–979, 2006.6
This paper presents a general methodology to design macroscopic fluid models that take into account localized kinetic upscaling effects. The fluid models are solved in the whole domain together with a localized kinetic upscaling that corrects the fluid model wherever it is necessary. This upscaling is obtained by solving a kinetic equation on the nonequilibrium part of the distri-bution function. This equation is solved only locally and is related to the fluid equation through a downscaling effect. The method does not need to find an interface condition as do usual domain decomposition methods to match fluid and kinetic representations. We show our approach applies to problems that have a hydrodynamic time scale as well as to problems with diffusion time scale. Simple numerical schemes are proposed to discretize our models, and several numerical examples are used to validate the method.
[208] Normalized solutions for a coupled Schr\"odinger system
Thomas Bartsch Mathematisches Institut, Justus-Liebig-Universit\"at Giessen, Arndtstrasse 2, 35392 Giessen, Germany Xuexiu Zhong South China Research Center for Applied Mathematics and Interdisciplinary Studies,South China Normal University, Guangzhou 510631, China. Wenming Zou Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China.
Math. Ann. , 2020
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In the present paper, we prove the existence of solutions $(\lambda_1,\lambda_2,u,v)\in\R^2\times H^1(\R^3,\R^2)$ to systems of coupled Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1u=\mu_1 u^3+\beta uv^2\quad &\hbox{in}\;\R^3\\ -\Delta v+\lambda_2v=\mu_2 v^3+\beta u^2v\quad&\hbox{in}\;\R^3\\ u,v>0&\hbox{in}\;\R^3 \end{cases} $$ satisfying the normalization constraint $ \displaystyle\int_{\R^3}u^2=a^2\quad\hbox{and}\;\int_{\R^3}v^2=b^2, $ which appear in binary mixtures of Bose-Einstein condensates or in nonlinear optics. The parameters $\mu_1,\mu_2,\beta>0$ are prescribed as are the masses $a,b>0$. The system has been considered mostly in the case of fixed frequencies $\lambda_1,\lambda_2$. When the masses are prescribed, the standard approach to this problem is variational with $\lambda_1,\lambda_2$ appearing as Lagrange multipliers. Here we present a new approach based on the fixed point index in cones, bifurcation theory, and the continuation method. We obtain the existence of normalized solutions for any given $a,b>0$ for $\beta$ in a large range. We also have a result about the nonexistence of positive solutions which shows that our existence theorem is almost optimal. Especially, if $\mu_1=\mu_2$ we prove that normalized solutions exist for all $\beta>0$ and all $a,b>0$.
[209] On a Schrodinger-Landau-Lifshitz system: Variational structure and numerical methods
Jingrun Chen Soochow University Jian-Guo Liu Duke University Zhennan Zhou Duke University
Multiscale Modeling and Simulation, 14, (4), 1463-1487, 2016.2
From a variational perspective, we derive a series of magnetization and quantum spin current systems coupled via an "s-d" potential term, including the Schrödinger--Landau--Lifshitz--Maxwell system, the Pauli--Landau--Lifshitz system, and the Schrödinger--Landau--Lifshitz system with successive simplifications. For the latter two systems, we propose using the time splitting spectral method for the quantum spin current and the Gauss--Seidel projection method for the magnetization. Accuracy of the time splitting spectral method applied to the Pauli equation is analyzed and verified by numerous examples. Moreover, behaviors of the Schrödinger--Landau--Lifshitz system in different "s-d" coupling regimes are explored numerically.
[210] Compressible NavierStokes equations with degenerate viscosity coefficient and vacuum
Tong Yang Changjiang Zhu
Communications in Mathematical Physics, 230, (2), 329-363, 2002.10
In this paper, we consider the compressible NavierStokes equations for isentropic flow of finite total mass when the initial density is either of compact or infinite support. The viscosity coefficient is assumed to be a power function of the density so that the Cauchy problem is well-posed. New global existence results are established when the density function connects to the vacuum states continuously. For this, some new <i>a priori</i> estimates are obtained to take care of the degeneracy of the viscosity coefficient at vacuum. We will also give a non-global existence theorem of regular solutions when the initial data are of compact support in Eulerian coordinates which implies singularity forms at the interface separating the gas and vacuum.
[211] On incompressible Navier-Stokes dynamics: a new approach for analysis and computation
Jian-Guo Liu University of Maryland Jie Liu University of Maryland Robert L. Pego Carnegie Mellon University
We show that in bounded domains with no-slip boundary conditions, the Navier-Stokes pressure can be determined in a such way that it is strictly dominated by viscosity. As a consequence, in a general domain we can treat the Navier-Stokes equations as a perturbed vector diffusion equation, instead of as a perturbed Stokes system. To illustrate the advantages of this view, we provide a simple proof of the unconditional stability of a difference scheme that is implicit only in viscosity and explicit in both pressure and convection terms, requiring no solution of stationary Stokes systems or inf-sup conditions.
[212] Convergence of point vortex method for 2-D vortex sheet
Jian-Guo Liu University of Maryland Zhouping Xin New York University and IMS and Dept
595 - 606 , 70, (234), 595-606 , 2001.1
We give an elementary proof of the convergence of the point vortex method (PVM) to a classical weak solution for the two-dimensional incompressible Euler equations with initial vorticity being a finite Radon measure of distinguished sign and the initial velocity of locally bounded energy. This includes the important example of vortex sheets, which exhibits the classical Kelvin-Helmholtz instability. A surprise fact is that although the velocity fields generated by the point vortex method do not have bounded local kinetic energy, the limiting velocity field is shown to have a bounded local kinetic energy.
[213] Well-posedness in Gevrey function space for the three-dimensional Prandtl equations
Wei-Xi Li Tong Yang
arXiv preprint arXiv:1708.08217, 2017.8
In the paper, we study the three-dimensional Prandtl equations, and prove that if one component of the tangential velocity field satisfies the monotonicity assumption in the normal direction, then the system is locally well-posed in the Gevrey function space with Gevrey index in ] 1, 2]. The proof relies on some new cancellation mechanism in the system in addition to those observed in the two-dimensional setting.
[214] Hydrodynamic models of self-organized dynamics: derivation and existence theory
Pierre Degond Institut de Math´ematiques de Toulouse Jian-Guo Liu Duke University Sébastien Motsch University of Maryland Vladislav Panferov California State University, Northridge
Methods & Applications of Analysis, 20, (2), 089–114, 2013.6
This paper is concerned with the derivation and analysis of hydrodynamic models for systems of self-propelled particles subject to alignment interaction and attraction-repulsion. The starting point is the kinetic model considered in earlier work of Degond & Motsch with the addition of an attraction-repulsion interaction potential. Introducing different scalings than in Degond & Motsch, the non-local effects of the alignment and attraction-repulsion interactions can be kept in the hydrodynamic limit and result in extra pressure, viscosity terms and capillary force. The systems are shown to be symmetrizable hyperbolic systems with viscosity terms. A local-in-time existence result is proved in the 2D case for the viscous model and in the 3D case for the inviscid model. The proof relies on the energy method.
[215] Continuum Limit of a Mesoscopic Model with Elasticity of Step Motion on Vicinal Surfaces
Journal of Nonlinear Science, 1-54, 2017.1
This work considers the rigorous derivation of continuum models of step motion starting from a mesoscopic Burton-Cabrera-Frank (BCF) type model following the work [Xiang, SIAM J. Appl. Math. 2002]. We prove that as the lattice parameter goes to zero, for a finite time interval, a modified discrete model converges to the strong solution of the limiting PDE with first order convergence rate.
[216] Dynamics in a kinetic model of oriented particles with phase transition
Amic Frouvelle †Institut de Math´ematiques de Toulouse Jian-Guo Liu Duke University
SIAM Journal on Mathematical Analysis, 44, (2), 791-826, 2012.2
Motivated by a phenomenon of phase transition in a model of alignment of self-propelled particles, we obtain a kinetic mean-field equation which is nothing else than the Doi equation (also called Smoluchowski equation) with dipolar potential. In a self-contained article, using only basic tools, we analyze the dynamics of this equation in any dimension. We first prove global well-posedness of this equation, starting with an initial condition in any Sobolev space. We then compute all possible steady-states. There is a threshold for the noise parameter: over this threshold, the only equilibrium is the uniform distribution, and under this threshold, there is also a family of non-isotropic equilibria. We give a rigorous prove of convergence of the solution to a steady-state as time goes to infinity. In particular we show that in the supercritical case, the only initial conditions leading to the uniform distribution in large time are those with vanishing momentum. For any positive value of the noise parameter, and any initial condition, we give rates of convergence towards equilibrium, exponentially for both supercritical and subcritical cases and algebraically for the critical case.
[217] A reverse quasiconformal composition problem for Qα(Rn)
Jie Xiao Memorial University Yuan Zhou Beihang University
We give a partial converse to [8, Theorem 1.3] (as a resolution of [2, Problem 8.4] for the quasiconformal Q-composition) for Q0<α<2−1(Rn≥2), and yet demonstrate that if f:R2→R2 is a homeomorphism then the boundedness of u↦u∘f on Q2−1<α<1(R2)⊂BMO(R2) yields the quasiconformality of f.
[218] A vicinal surface model for epitaxial growth with logarithmic free energy
Yuan Gao Hangjie Ji Jian-Guo Liu Thomas P. Witelski
Discrete Contin. Dyn. Syst. Ser. B, 23, 4433-4453, 2017
[219] Partial regularity for weak solutions to a PDE system with cubic nonlinearities
Jian-Guo Liu Xiangsheng Xu
Journal of Differential Equations, 264, 5489-5526, 2018
[220] Eptaxial growth without slope selection: energetics, coarsening, and dynamic scaling
Bo Li University of California at San Diego Jian-Guo Liu University of Maryland
Journal of Nonlinear Science, 14, (5), 429–451, 2004.10
We study a continuum model for epitaxial growth of thin films in which the slope of mound structure of film surface increases. This model is a diffusion equation for the surface height profile h which is assumed to satisfy the periodic boundary condition. The equation happens to possess a Liapunov or �free-energy� functional. This functional consists of the term |? h|2, which represents the surface diffusion, and - log (1 + |? h|2), which describes the effect of kinetic asymmetry in the adatom attachment-detachment. We first prove for large time t that the interface width---the standard deviation of the height profile---is bounded above by O(t1/2), the averaged gradient is bounded above by O(t1/4), and the averaged energy is bounded below by O(- log t). We then consider a small coefficient e2 of |? h|2 with e = 1/L and L the linear size of the underlying system, and study the energy asymptotics in the large system limit e ? 0. We show that global minimizers of the free-energy functional exist for each e>0, the L2-norm of the gradient of any global minimizer scales as O(1/e), and the global minimum energy scales as O( log e). The existence of global energy minimizers and a scaling argument are used to construct a sequence of equilibrium solutions with different wavelengths. Finally, we apply our minimum energy estimates to derive bounds in terms of the linear system size L for the saturation interface width and the corresponding saturation time.
[221] Global existence for a thin film equation with subcritical mass
discrete and continuous dynamical systems, 22, (4), 1461-1492, 2017.1
In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin lm equation in one-dimensional case ht + @x(h n @xxxh) + @x(h n+2@xh) = 0; where n 1. There exists a critical mass Mc = 2 p 6 3 found by Witelski etal. (2004 Euro. J. of Appl. Math. 15, 223{256) for n = 1. We obtain global existence of a non-negative entropy weak solution if initial mass is less than Mc. For n 4, entropy weak solutions are positive and unique. For n = 1,a nite time blow-up occurs for solutions with initial mass larger than Mc.For the Cauchy problem with n = 1 and initial mass less than Mc, we show that at least one of the following long-time behavior holds: the second moment goes to in nity as the time goes to in nity or h( ; tk) * 0 in L1 (R) for some subsequence tk ! 1.
[222] Boundary layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation
Jian-Guo Liu Temple University Zhouping Xin Courant Institute of Mathematical Sciences
Archive for Rational Mechanics and Analysis, 135, (1), 61–105, 1996.10
In this paper, we study the fluid-dynamic limit for the one-dimensional Broadwell model of the nonlinear Boltzmann equation in the presence of boundaries. We consider an analogue of Maxwell's diffusive and reflective boundary conditions. The boundary layers can be classified as either compressive or expansive in terms of the associated characteristic fields. We show that both expansive and compressive boundary layers (before detachment) are nonlinearly stable and that the layer effects are localized so that the fluid dynamic approximation is valid away from the boundary. We also show that the same conclusion holds for short time without the structural conditions on the boundary layers. A rigorous estimate for the distance between the kinetic solution and the fluid-dynamic solution in terms of the mean-free path in theL∞-norm is obtained provided that the interior fluid flow is smooth. The rate of convergence is optimal.
[223] Convergence analysis of the vortex blob method for the b-equation
Yong Duan University of Electronic Science and Technology of China Jian-Guo Liu Duke University
Discrete & Continuous Dynamical Systems, 34, (5), 1995 - 2011, 2014.5
Abstract We prove the convergence of the vortex blob method for a family of nonlinear evolutionary partial differential equations (PDEs), the so-called b-equation. This kind of PDEs, including the Camassa-Holm equation and the Degasperis-Procesi equation, has many applications in diverse scientific fields. Our convergence analysis also provides a proof for the existence of the global weak solution to the b-equation when the initial data is a nonnegative Radon measure with compact support.
[224] Global weak entropy solution to Doi-Saintillan-Shelley model for active and passive rod-like and ellipsoidal particle suspensions
Journal of Differential Equations, 254, (7), 2764–2802, 2013.4
We prove the existence of the global weak entropy solution to the Doi–Saintillan–Shelley model for active and passive rod-like particle suspensions, which couples a Fokker–Planck equation with the incompressible Navier–Stokes or Stokes equation, under the no-flux boundary conditions, L2(Ω;L1(Sd611))L2(Ω;L1(Sd611)) initial data, and finite initial entropy for the particle distribution function in two and three dimensions. Furthermore, for the model with the Stokes equation, we obtain the global L2(Ω×Sd611)L2(Ω×Sd611) weak solution in two and three dimensions and the uniqueness in two dimension.
[225] Regularizing effect and local existence for the non-cutoff Boltzmann equation
Archive for rational mechanics and analysis, 198, (1), 39-123, 2010.10
The Boltzmann equation without Grads angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial
[226] Time regularity of the solutions to second order hyperbolic equations
Tamotu Kinoshita Institute of Mathematics, University of Tsukuba Giovanni Taglialatela Dipartimento di Scienze Economiche e Metodi Matematici, University of Bari
We consider the Cauchy problem for a second order weakly hyperbolic equation, with coefficients depending only on the time variable. We prove that if the coefficients of the equation belong to the Gevrey class $\gamma^{s_{0}}$ and the Cauchy data belong to $\gamma^{s_{1}}$ , then the Cauchy problem has a solution in $\gamma^{s_{0}}([0,T^{*}];\gamma^{s_{1}}(\mathbb{R}))$ for some$T$^{*}>0, provided 1≤$s$_{1}≤2−1/$s$_{0}. If the equation is strictly hyperbolic, we may replace the previous condition by 1≤$s$_{1}≤$s$_{0}.
[227] P-Euler equations and p-Navier-Stokes equations,
[228] On the Expanding Configurations of Viscous Radiation Gaseous Stars: Isentropic and Thermodynamic Models
Kefeng Liu
arXiv preprint arXiv:1710.06527, 2017.10
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In this work, we study the stability of the expanding configurations for radiation gaseous stars. Such expanding configurations exist for the isentropic model and the thermodynamic model. In particular, we focus on the effect of the monatomic gas viscosity tensor on the expanding configurations. With respect to small perturbation, it is shown the self-similarly expanding homogeneous solutions of the isentropic model is unstable, while the linearly expanding homogeneous solutions of the isentropic model and the thermodynamic model are stable. This is an extensive study of the result in arXiv: 1605.08083 by Hadzic and Jang.
[229] A normality criterion involving rotations and dilations in the argument
Jürgen Grahl Department of Mathematics, University of Würzburg
We show that a family $\mathcal{F}$ of analytic functions in the unit disk ${\mathbb{D}}$ all of whose zeros have multiplicity at least$k$and which satisfy a condition of the form $$f^n(z)f^{(k)}(xz)\ne1$$ for all $z\in{\mathbb{D}}$ and $f\in\mathcal{F}$ (where$n$≥3,$k$≥1 and 0<|$x$|≤1) is normal at the origin. The proof relies on a modification of Nevanlinna theory in combination with the Zalcman–Pang rescaling method. Furthermore we prove the corresponding Picard-type theorem for entire functions and some generalizations.
[230] Global infinite energy solutions for the 2D gravity water waves system
Communications on pure and applied mathematics, 71, (1), 90-162, 2018
We prove global existence and a modified scattering property for the solutions of the 2D gravity water waves system in the infinite depth setting for a class of initial data, which is only required to be small above the level \dot{H}^{1/5}\times \dot{H}^{1/2+1/5}. No assumption is made below this level. Therefore, the nonlinear solution can have infinite energy. As a direct consequence, the momentum condition assumed on the physical velocity in all previous small energy results by Ionescu‐Pusateri, Alazard‐Delort, and Ifrim‐Tataru is removed.
[231] Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate
Jian-Guo Liu University of Maryland, College Park JIE LIU University of Maryland, College Park ROBERT L. PEGO Carnegie Mellon University
Communications on Pure and Applied Mathematics, 60, (10), 1443-1487, 2007.3
For strong solutions of the incompressible Navier-Stokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure. These schemes require no solution of stationary Stokes systems, nor any compatibility between velocity and pressure spaces to ensure an inf-sup condition, and are representative of a class of highly efficient computational methods that have recently emerged. The proofs are simple, based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators. This allows us to treat an unconstrained formulation of the Navier-Stokes equations as a perturbed diffusion equation. © 2007 Wiley Periodicals, Inc.
[232] Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow
Michael Ghil D´epartement Terre-Atmosph´ere-oc´ean, Erole Normale Sup´erieure Jian-Guo Liu University of Maryland Cheng Wang The University of Tennessee Shouhong Wang Indiana University
Physica D Nonlinear Phenomena, 197, (1), 149–173, 2004.10
We study the detailed process of bifurcation in the flow's topological structure for a two-dimensional (2-D) incompressible flow subject to no-slip boundary conditions and its connection with boundary-layer separation. The boundary-layer separation theory of M. Ghil, T. Ma and S. Wang, based on the structural-bifurcation concept, is translated into vorticity form. The vorticity formulation of the theory shows that structural bifurcation occurs whenever a degenerate singular point for the vorticity appears on the boundary; this singular point is characterized by nonzero tangential second-order derivative and nonzero time derivative of the vorticity. Furthermore, we prove the presence of an adverse pressure gradient at the critical point, due to reversal in the direction of the pressure force with respect to the basic shear flow at this point. A numerical example of 2-D driven-cavity flow, governed by the Navier Stokes equations, is presented; boundary-layer separation occurs, the bifurcation criterion is satisfied, and an adverse pressure gradient is shown to be present.
[233] Blow-up, zero $\alpha$ limit and the Liouville type theorem for the Euler-Poincar\'{e} equations
Dongho Chae Chung-Ang University Jian-Guo Liu Duke University
Communications in Mathematical Physics, 314, (3), 671–687, 2012.3
In this paper we study the Euler-Poincar\'{e} equations in $\Bbb R^N$. We prove local existence of weak solutions in $W^{2,p}(\Bbb R^N),$ $p>N$, and local existence of unique classical solutions in $H^k (\Bbb R^N)$, $k>N/2+3$, as well as a blow-up criterion. For the zero dispersion equation($\alpha=0$) we prove a finite time blow-up of the classical solution. We also prove that as the dispersion parameter vanishes, the weak solution converges to a solution of the zero dispersion equation with sharp rate as $\alpha\to0$, provided that the limiting solution belongs to $C([0, T);H^k(\Bbb R^N))$ with $k>N/2 +3$. For the {\em stationary weak solutions} of the Euler-Poincar\'{e} equations we prove a Liouville type theorem. Namely, for $\alpha>0$ any weak solution $\mathbf{u}\in H^1(\Bbb R^N)$ is $\mathbf{u}=0$; for $\alpha=0$ any weak solution $\mathbf{u}\in L^2(\Bbb R^N)$ is $\mathbf{u}=0$.
[234] A model of radiational gaseous stars
SIAM Journal on Mathematical Analysis, 50, (6), 6100-6155, 2018.12
We introduce a model concerning radiational gaseous stars and establish the existence theory of stationary solutions to the free boundary problem of hydrostatic equations describing the radiative equilibrium. We also establish in the spherically symmetric case the local well-posedness of the moving boundary problem of the corresponding Navier--Stokes--Fourier--Poisson system and construct a priori estimates of strong and classical solutions. Our results explore the vacuum behavior of density and temperature near the free boundary for the equilibrium and capture such degeneracy in the evolutionary problem.
[235] Time periodic solution to the compressible Navier-Stokes equations in a periodic domain
JIN Chunhua Tong Yang
Acta Mathematica Scientia, 36, (4), 1015-1029, 2016.7
This article is concerned with the time periodic solution to the isentropic compressible Navier-Stokes equations in a periodic domain. Using an approach of parabolic regularization, we first obtain the existence of the time periodic solution to a regularized problem under some smallness and symmetry assumptions on the external force. The result for the original compressible Navier-Stokes equations is then obtained by a limiting process. The uniqueness of the periodic solution is also given.
[236] Multidimensional Degenerate Keller–Segel System with Critical Diffusion Exponent $2n/(n+2)$ Read More: http://epubs.siam.org/doi/abs/10.1137/110839102
Li Chen Tsinghua University Jian-Guo Liu Duke University Jinhuan Wang Liaoning University
This paper deals with a degenerate diffusion Patlak–Keller–Segel system in $n\geq 3$ dimension. The main difference between the current work and many other recent studies on the same model is that we study the diffusion exponent $m=2n/(n+2)$, which is smaller than the usual exponent $m^{*}=2-2/n$ used in other studies. With the exponent $m=2n/(n+2)$, the associated free energy is conformal invariant, and there is a family of stationary solutions $U_{\lambda,x_0}(x)=C(\frac{\lambda} {\lambda^2+|x-x_0|^2})^{\frac{n+2}{2}}$ $\forall \lambda>0$, $x_0\in {\mathbb R}^n$. For radially symmetric solutions, we prove that if the initial data are strictly below $U_{\lambda,0}(x)$ for some $\lambda$, then the solution vanishes in $L^1_{loc}$ as $t\to\infty$; if the initial data are strictly above $U_{\lambda,0}(x)$ for some $\lambda$, then the solution either blows up at a finite time or has a mass concentration at $r=0$ as time goes to infinity. For general initial data, we prove that there is a global weak solution provided that the $L^m$ norm of initial density is less than a universal constant, and the weak solution vanishes as time goes to infinity. We also prove a finite time blow-up of the solution if the $L^m$ norm for initial data is larger than the $L^m$ norm of $U_{\lambda,x_0}(x)$, which is constant independent of $\lambda$ and $x_0$, and the free energy of initial data is smaller than that of $U_{\lambda,x_0}(x)$.
[237] Cauchy problems for Keller–Segel type time–space fractional diffusion equation
Lei Li Jian-Guo Liu Lizhen Wang
[238] Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system
Tong Yang Huijiang Zhao
The time evolution of the distribution function for the charged particles in a dilute gas is governed by the VlasovPoissonBoltzmann system when the force is self-induced and its potential function satisfies the Poisson equation. In this paper, we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global Maxwellian. Moreover, the convergence rate in time to the global Maxwellian is also obtained through the energy method. The proof is based on the theory of compressible NavierStokes equations with forcing and the decomposition of the solutions to the Boltzmann equation with respect to the local Maxwellian introduced in [23] and elaborated in [31].
[239] Well-posedness of thermal layer equations for inviscid compressible flows
Cheng-Jie Liu Ya-Guang Wang Tong Yang
A semi-explicit formula of solution to the boundary layer system for thermal layer derived from the compressible Navier-Stokes equations with the non-slip boundary condition when the viscosity coefficients vanish is given, in particular in three space dimension. In contrast to the inviscid Prandtl system studied by [7] in two space dimension, the main difficulty comes from the coupling of the velocity field and the temperature field through a degenerate parabolic equation. The convergence of these boundary layer equations to the inviscid Prandtl system is justified when the initial temperature goes to a constant. Moreover, the time asymptotic stability of the linearized system around a shear flow is given, and in particular, it shows that in three space dimension, the asymptotic stability depends on whether the direction of tangential velocity field of the shear flow is invariant in the normal direction respective to the boundary.
[240] Refined hyper-contractivity and uniqueness for the Keller-Segel equations
Applied Mathematics Letters, 52, 212–219, 2016.2
This note considers the hyper-contractivity and the uniqueness of the weak solutions to the two dimensional Keller–Segel equations. We prove a refined hyper-contractive property and consequently obtain the uniqueness of global weak solutions provided that the initial data satisfy , and . We also extend the result to higher dimensions.
[241] Front propagation in heterogeneous media
SIAM review, 42, (2), 161-230, 2000
A review is presented of recent results on front propagation in reaction-diffusion-advection equations in homogeneous and heterogeneous media. Formal asymptotic expansions and heuristic ideas are used to motivate the results wherever possible. The fronts include constant-speed monotone traveling fronts in homogeneous media, periodically varying traveling fronts in periodic media, and fluctuating and fractal fronts in random media. These fronts arise in a wide range of applications such as chemical kinetics, combustion, biology, transport in porous media, and industrial deposition processes. Open problems are briefly discussed along the way.
[242] Global solutions to the Boltzmann equation with external forces
Seiji Ukai Tong Yang Huijiang Zhao
Analysis and Applications, 3, (2), 157-193, 2005.4
For the Boltzmann equation with an external potential force depending only on the space variables, there is a family of stationary solutions, which are local Maxwellians with space dependent density, zero velocity and constant temperature. In this paper, we will study the nonlinear stability of these stationary solutions by using the energy method. The analysis combines the analytic techniques used for the conservation laws using the fluid-type system derived from the Boltzmann equation (cf. [14]) and the dissipative effects on the fluid and non-fluid components of the Boltzmann equation through the celebrated H-theorem. To our knowledge, this is the first result on the global classical solutions to the Boltzmann equation with external force and non-trivial large time behavior in the whole space.
[243] Gradient estimates for parabolic and elliptic systems from linear laminates
Hongjie Dong
Archive for Rational Mechanics and Analysis, 205, (1), 119-149, 2012.7
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We establish several gradient estimates for second-order divergence type parabolic and elliptic systems. The coefficients and data are assumed to be Hlder or Dini continuous in the time variable and all but one spatial variable. This type of system arises from the problems of linearly elastic laminates and composite materials. For the proof, we use Campanatos approach in a novel way. Non-divergence type equations under similar conditions are also discussed.
[244] Estimates on the Stokes pressure by partitioning the energy of harmonic functions
Rims Kokyuroku Bessatsu, 1, 251-270, 2007.3
We show that in a tubular domain with sufficiently small width, the normal and tangential gradients of a harmonic function have almost the same L 2 norm. This estimate yields a sharp estimate of the pressure in terms of the viscosity term in the Navier-Stokes equation with no-slip boundary condition. By consequence, one can analyze the Navier-Stokes equations simply as a perturbed vector diffusion equation instead of as a perturbed Stokes system. As an application, we describe a rather easy approach to establish a new isomorphism theorem for the non-homogeneous Stokes system.
[245] Maximal monotone operator theory and its applications to thin film equation in epitaxial growth on vicinal surface
Yuan Gao Jian-Guo Liu Xin Yang Lu Xiangsheng Xu
Calc. Var, 57:55, 2018
[246] Propagation of uniform boundedness of entropy and inhomogeneous regularities for viscous and heat conductive gases with far field vacuum in three dimensions
Due to the highly degeneracy and singularities of the entropy equation, the physical entropy for viscous and heat conductive polytropic gases behave singularly in the presence of vacuum and it is thus a challenge to study its dynamics. It is shown in this paper that the uniform boundedness of the entropy and the inhomogeneous Sobolev regularities of the velocity and temperature can be propagated for viscous and heat conductive gases in $\mathbb R^3$, provided that the initial vacuum occurs only at far fields with suitably slow decay of the initial density. Precisely, it is proved that for any strong solution to the Cauchy problem of the heat conductive compressible Navier--Stokes equations, the corresponding entropy keeps uniformly bounded and the $L^2$ regularities of the velocity and temperature can be propagated, up to the existing time of the solution, as long as the initial density vanishes only at far fields with a rate no more than $O(\frac{1}{|x|^2})$. The main tools are some singularly weighted energy estimates and an elaborate De Giorgi type iteration technique. The De Giorgi type iterations are carried out to different equations in establishing the lower and upper bounds of the entropy.
[247] Large oscillations arising from a dispersive numerical scheme
C.David Levermore University of Arizona Jian-Guo Liu New York University
Physica D: Nonlinear Phenomena, 99, (2), 191-216, 1996.11
We study the oscillatory behavior that arises in solutions of a dispersive numerical scheme for the Hopf equation whenever the classical solution of the equation develops a singularity. Modulation equations are derived that describe period-two oscillations so long as the solution of those equations takes values for which the equations are hyperbolic. However, those equations have an elliptic region that may be entered by its solutions in a finite time, after which the corresponding period-two oscillations are seen to break down. This kind of phenomenon has not been observed for integrable schemes. The generation and propagation of period-two oscillations are asymptotically analyzed and a matching formula is found for the transition between oscillatory and nonoscillatory regions. Modulation equations are also presented for period-three oscillations. Numerical experiments are carried out that illustrate our analysis.
[248] Weak solutions of general systems of hyperbolic conservation laws
In this paper, we establish the existence theory for general system of hyperbolic conservation laws and obtain the uniform <i>L</i> <sub> <i>1</i> </sub> boundness for the solutions. The existence theory generalizes the classical Glimm theory for systems, for which each characteristic field is either genuinely nonlinear or linearly degenerate in the sense of Lax. We construct the solutions by the Glimm scheme through the wave tracing method. One of the key elements is a new way of measuring the potential interaction of the waves of the same characteristic family involving the angle between waves. A new analysis is introduced to verify the consistency of the wave tracing procedure. The entropy functional is used to study the <i>L</i> <sub> <i>1</i> </sub> boundedness.
[249] Zeros of the deformed exponential function
Liuquan Wang Wuhan University Cheng Zhang Johns Hopkins University
Analysis of PDEs Classical Analysis and ODEs Combinatorics Number Theory mathscidoc:2103.03003
Advances in Mathematics, 332, 311–348, 2018
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Let $f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}q^{n(n-1)/2}x^n$ ($0<q<1$) be the deformed exponential function. It is known that the zeros of $f(x)$ are real and form a negative decreasing sequence $(x_k)$ ($k\ge 1$). We investigate the complete asymptotic expansion for $x_{k}$ and prove that for any $n\ge1$, as $k\to \infty$, \begin{align*} x_k=-kq^{1-k}\Big(1+\sum_{i=1}^{n}C_i(q)k^{-1-i}+o(k^{-1-n})\Big), \end{align*} where $C_i(q)$ are some $q$ series which can be determined recursively. We show that each $C_{i}(q)\in \mathbb{Q}[A_0,A_1,A_2]$, where $A_{i}=\sum_{m=1}^{\infty}m^i\sigma(m)q^m$ and $\sigma(m)$ denotes the sum of positive divisors of $m$. When writing $C_{i}$ as a polynomial in $A_0, A_1$ and $A_2$, we find explicit formulas for the coefficients of the linear terms by using Bernoulli numbers. Moreover, we also prove that $C_{i}(q)\in \mathbb{Q}[E_2,E_4,E_6]$, where $E_2$, $E_4$ and $E_6$ are the classical Eisenstein series of weight 2, 4 and 6, respectively.
[250] Ultra-contractivity for Keller-Segel model with diffusion exponent m>1-2/d
Shen Bian Ocean University of China Jian-Guo Liu Duke University Chen Zou Duke University
Kinetic & Related Models, 7, (1), 9 - 28, 2014.3
This paper establishes the hyper-contractivity in L∞ (ℝd) (it's known as ultra-contractivity) for the multi-dimensional Keller-Segel systems with the dif;fusion exponent m > 1 -2/d. The results show that for the supercritical and critical case 1 - 2/d < m ≤ 2 - 2/d, if || U0||d(2-m)/2) < Cd, m where Cd, m is a universal constant, then for any t > 0, || u (⋅, t)|| ... is bounded and decays as t goes to infinity, For the subcritical case m > 2 - 2/d, the solution u (⋅, t) ∈ L∞ (ℝd) with any initial data U0 ∈ L1+ (ℝd) for any positive time.
[251] Kinetic and viscous boundary layers for the Broadwell equations
Transport Theory and Statistical Physics, 25, (3), 447-461, 1996.4
In this paper, we investigate the boundary layer behavior of solutions to the one dimensional Broadwell model of the nonlinear Boltzmann equation for small mean free path. We consider the analogue of Maxwell's diffusive and the reflexive boundary conditions. It is found that even for such a simple model, there are boundary layers due to purely kinetic effects which cannot be detected by the corresponding Navier-Stokes system. It is also found numerically that a compressive boundary layer is not always stable in the sense that it may detach from the boundary and move into the interior of the gas as a shock layer.
[252] On $l_p$-estimates for elliptic and parabolic equations with $a_p$ weights
Hongjie Dong Brown University Doyoon Kim Korea University
Trans. Amer. Math. Soc., 370, (7), 5081–5130, 2018.7
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We prove generalized Fefferman-Stein type theorems on sharp functions with Ap weights in spaces of homogeneous type with either finite or infinite underlying measure. We then apply these results to establish mixed-norm weighted Lp-estimates for elliptic and parabolic equations/systems with (partially) BMO coefficients in regular or irregular domains.
[253] Structural stability and bifurcation for 2D incompressible flows with symmetry
Chun-Hsiung Hsia National Taiwan University Jian-Guo Liu Duke University Cheng Wang University of Massachusetts Dartmouth
Methods and Applications of Analysis, 15, (4), 495–512, 2008.4
This article studies the structure and its evolution of incompressible flows with the anti-symmetry using a combination of rigorous analysis and numerical simulations, with an application to an example of oceanic flow. In particular, necessary and sufficient conditions for 2D divergence-free vector fields with anti-symmetry are obtained, and a detailed numerical simulation for a simplified model of Marsigli oceanic flow is provided to explore and verify the structure and its transitions of the flow. It is expected that the study will lead to useful insights to the understanding of the flow dynamics from both the mathematical and physical points of view.
[254] Almost Global Solutions to the Three-dimensional Isentropic Inviscid Flows with Damping in Physical Vacuum Around Barenlatt Solutions
Huihui Zeng Tsinghua University
Arch. Rational Mech. Anal., 2020
[255] Existence of global smooth solutions for Euler equations with symmetry (II)
Tong Yang Changjiang Zhu Yongshu Zheng
Nonlinear Analysis, Theory, Methods and Applications, 41, (1), 187-203, 2000.7
The compressible Euler equations, which govern the gas flow surrounding a solid ball with mass M and frictional damping in n dimensions, t+(u)= 0,(u) t+ (u u)+ P ()=-Mx/| x| n-2u, where , u, P and M are the density, velocity, pressure and mass of the gas, respectively, n 3 is the dimension of x, and > 0 is the frictional constant, are examined. The pressure is assumed to satisfy the law and 12 , K is a positive constant. The existence and non-existence of global smooth solutions are studied for the initial boundary problem of the Euler equations.
[256] Monge-Ampere equation with bounded periodic data
YanYan Li Rutgers University Siyuan Lu McMaster University
[257] Coagulation-fragmentation model for animal group-size statistics
Pierre Degond , Imperial College London Jian-Guo Liu Duke University Robert L. Pego Center for Nonlinear Analysis, Carnegie Mellon University
ABSTRACT We study coagulation-fragmentation equations inspired by a simple model proposed in fisheries science to explain data for the size distribution of schools of pelagic fish. Although the equations lack detailed balance and admit no $H$-theorem, we are able to develop a rather complete description of equilibrium profiles and large-time behavior, based on recent developments in complex function theory for Bernstein and Pick functions. In the large-population continuum limit, a scaling-invariant regime is reached in which all equilibria are determined by a single scaling profile. This universal profile exhibits power-law behavior crossing over from exponent $-\frac23$ for small size to $-\frac32$ for large size, with an exponential cut-off.
[258] Global stability of wave patterns for compressible Navier-Stokes system with free boundary
QIN Xiaohong WANG Teng Yi Wang
In this article, we investigate the global stability of the wave patterns with the superposition of viscous contact wave and rarefaction wave for the one-dimensional compressible Navier-Stokes equations with a free boundary. It is shown that for the ideal polytropic gas, the superposition of the viscous contact wave with rarefaction wave is nonlinearly stable for the free boundary problem under the large initial perturbations for any > 1 with being the adiabatic exponent provided that the wave strength is suitably small.
[259] Modeling light bullets with the two-dimensional sineGordon equation