text:
\section{Geometric formulation}\label{sec:geomeqs}

The 3D incompressible Euler equations of fluid dynamics are
\begin{subequations}\label{eq:geomform}
\begin{align}
\mathcal{L}_S t & = 1 \label{transportt}\\
\mathcal{L}_S x^A & = v^A\label{transportx}\\
\mathcal{L}_S \Omega & = 0 \label{transportomega}\\
\mathrm{d} \Omega & = 0 \label{clom}\\
\delta_{AB}\; \mathrm{d} v^A \wedge \mathrm{d} x^B & = \Omega \label{elliptic2} \\
\varepsilon_{ABC}\; \mathrm{d} v^A \wedge \mathrm{d} x^B \wedge \mathrm{d} x^C & = 0 \label{elliptic3}
\end{align}
\end{subequations}
with the physical interpretation\,/\,terminology
\begin{alignat*}{4}
t &\;\;\;\;\;\; && \text{time}\\
x^A &&& \text{position, Cartesian components}\\
v^A &&& \text{velocity, Cartesian components} \displaybreak[0]\\
S &&& \text{material vector field}\\
\text{integral curve of $S$} &&& \text{fluid particle path}\\
\Omega &&& \text{vorticity}
\end{alignat*}

{\bf Preliminaries.}
Equations \eqref{eq:geomform}
are on a 4-dimensional manifold $M^4$, for seven
real valued functions $t$ and $x^A$ and $v^A$,
a vector field $S$, and a 2-form $\Omega$.
For the purpose of this section, they are smooth.
The operator $\mathcal{L}_S$ is the Lie derivative along $S$.
The indices $A,B,C$ each run over $1,2,3$.
By definition $\delta_{AB}$ is equal to $1$
if $A=B$ and $0$ otherwise, and $\varepsilon_{ABC}$ is totally
antisymmetric with $\varepsilon_{123}=1$.
The symmetries of \eqref{eq:geomform} are
discussed in Appendix \ref{app:sym}.

We require that $(t,x^1,x^2,x^3)$ is a diffeomorphism
$M^4 \to \text{(interval)} \times \mathbbm{R}^3$,
corresponding to a fluid that fills $\mathbbm{R}^3$.
Fluid particles are not allowed to disappear, that is,
each integral curve of $S$ must intersect all level sets of $t$.
Then the flow generated by $S$ maps level sets of $t$ to level sets of $t$
by \eqref{transportt}.

The exterior differential calculus in \eqref{eq:geomform}
is not the 4-dimensional one on $M^4$,
but the 3-dimensional one internal to each level set of $t$.
Therefore $\Omega$ is a 2-form on each such level set,
$\mathrm{d} x^A$ is a 1-form on each such level set, etc.

\mostimportant{Conservation of vorticity \eqref{transportomega}
is here a primitive;
it can be derived
from the more physical velocity-pressure formulation
of incompressible Euler, and is due to Helmholtz and Kelvin.
Conservation of volume
$\mathcal{L}_S (\mathrm{d} x^1 \wedge \mathrm{d} x^2 \wedge \mathrm{d} x^3) = 0$
follows from \eqref{transportx}, \eqref{elliptic3}.
The pressure at unit density is obtained from
$\delta_{AB}\,(\mathcal{L}_S v^A)\, \mathrm{d} x^B = -\mathrm{d} \text{(pressure)}$
by integration; the left hand side is closed by
\eqref{transportx}, \eqref{transportomega}, \eqref{elliptic2}.}
\vskip 3mm \noindent

{\bf Gauges.} A gauge for \eqref{eq:geomform}
is a choice of independent variables (coordinates).
Different parts of \eqref{eq:geomform} prefer different gauges:
\begin{center}
\begin{tabular}{l|l|l}
\emph{equations} & \emph{kind of problem} & \emph{preferred gauge}
\\
\hline
\eqref{transportt},\eqref{transportx},\eqref{transportomega}
& transport
& Lagrangian
\\
\eqref{clom},\eqref{elliptic2},\eqref{elliptic3}
& elliptic & Eulerian
\end{tabular}
\end{center}

The {\bf Eulerian gauge} promotes $t,x^A$ to independent variables.
In this gauge, the elliptic problem
\eqref{elliptic2}, \eqref{elliptic3} for $v$
is translation invariant.
Equations \eqref{transportt}, \eqref{transportx} imply $S = \partial_t + v^A \partial_{x^A}$,
which depends on the unknown $v$.

The {\bf Lagrangian gauge}
uses independent variables $t,\lambda^A$, where
the $\lambda^A$ are particle labels, that is,
they are constant along fluid particle paths:
$\mathcal{L}_S \lambda^A = 0$.
Then \eqref{transportt} fixes $S = \partial_t$,
which is convenient for the transport problem.
The $\lambda^A$ map each level set of
$t$ diffeomorphically to, by convention, $\mathbbm{R}^3$.
One can re-label particles
by acting on $\lambda^A$ with a
time independent diffeomorphism of $\mathbbm{R}^3$.

\mostimportant{A common approach uses both gauges:
the Lagrangian gauge for transport,
the Eulerian gauge for the elliptic problem,
with explicit transformations between the two.
Section \ref{sec:locex} pursues another,
consistently Lagrangian approach
to all parts of \eqref{eq:geomform},
including the elliptic problem.}

\section{Incompressible Euler is a real analytic ODE}\label{sec:locex}
We give a new proof of:
\emph{In the Lagrangian gauge, 3D incompressible Euler is an
autonomous real analytic ODE in Banach space.}

This ODE was already used in \cite{lichtenstein}, \cite{gunther}.
The qualification `real analytic' seems to be due to Serfati \cite{serfati},
with other accounts in \cite{shnirelman}, \cite{inci}.
This section contains a new proof
that is consistently Lagrangian.

\mostimportant{%
In the Lagrangian gauge,
the elliptic problem is not translation invariant, but close to
translation invariant, and the rest
can be controlled by a Neumann series.
Incompressible Euler becomes the ODE \eqref{eq:neumann} below.}

\mostimportant{%
In this paper we always refer to the Lagrangian gauge for local existence.
A solution of \eqref{eq:geomform} is then thought of
as an equivalence class of solutions in the Lagrangian gauge
(solution curves of a real analytic ODE),
two being equivalent iff related by a particle re-labeling.
}

\mostimportant{%
The Lagrangian gauge is naturally interpreted in terms
of particles. In Section \ref{sec:basicargXXXX} there is an equation for $\partial_t x$,
but none for $\partial_t v$, because we exploit
conservation of vorticity (2 conserved quantities per particle)
and conservation of volume (1 conserved quantity per particle).}

\subsection{Setup in the Lagrangian gauge}\label{sec:basicargXXXX}
\mostimportant{%
We abbreviate partial derivatives as usual:
if $a,b,c,d$ are used as independent variables,
then $\partial_a$ is an abbreviation for
$(\partial_a)_{b,c,d}$, the derivative w.r.t.~$a$ at constant $b,c,d$.}

We use Lagrangian coordinates $t,\lambda^A$ as independent variables,
$S = \partial_t$.
The vorticity
$\Omega = \Omega_{AB}(\lambda)\;\mathrm{d} \lambda^A\wedge \mathrm{d} \lambda^B$
has coefficients independent of $t$ by  \eqref{transportomega},
and is closed by \eqref{clom}. Fix such an $\Omega$.
System \eqref{eq:geomform} reduces to
\begin{subequations}\label{eq:lagformxxx}
\begin{align}
\partial_t x^A & = v^A\label{lagode}\\
\delta_{AB}\; \mathrm{d} v^A \wedge \mathrm{d} x^B & = \Omega \label{lagell2}\\
\varepsilon_{ABC}\; \mathrm{d} v^A \wedge \mathrm{d} x^B \wedge \mathrm{d} x^C & = 0\label{lagell3}
\end{align}
\end{subequations}

\mostimportant{The unknown is $x = x(t,\lambda)$,
while $v$
is merely a placeholder for the solution to the elliptic problem
\eqref{lagell2}, \eqref{lagell3},
which under suitable assumptions is a map:
given $x$ at time $t$ it determines $v$ at time $t$.
Now \eqref{lagode} is naturally interpreted as an ODE in function space.
System
\eqref{eq:lagformxxx} is invariant under re-labeling,
that is, applying a diffeomorphism of $\mathbbm{R}^3$ to $\lambda$.
Below we rewrite the equations, and then this symmetry will
be hidden.
}

Set $x^A = \lambda^A + y^A$, with $y$ the new unknown.
System \eqref{eq:lagformxxx} becomes
\begin{samepage}
\begin{subequations}\label{eq:lagformyyy}
\begin{alignat}{6}
\partial_t y^A & = v^A\label{lagodeyyy}\\
\delta_{AB}\; \mathrm{d} v^A \wedge \mathrm{d} \lambda^B & =
[\mathrm{d} v \wedge \mathrm{d} y]  && +  \Omega \label{lagell2yyy}\\
\varepsilon_{ABC}\; \mathrm{d} v^A \wedge \mathrm{d} \lambda^B \wedge \mathrm{d} \lambda^C & =
[\textstyle\sum_{Q = \lambda,y} \mathrm{d} v \wedge \mathrm{d} y \wedge \mathrm{d} Q] \label{lagell3yyy} && + 0
\end{alignat}
\end{subequations}
where within square brackets $[\;\;]$ we do not keep track of constant coefficients and indices.
\end{samepage}%
Note that \eqref{lagell2yyy}, \eqref{lagell3yyy}
is a linear system for $v$, arranged in the form
$$L_0v = L_y v + (\Omega,0)$$
where $L_0$ and $L_y$ are linear first-order differential operators.
The operator $L_0$ is translation invariant in $\lambda$
and can be inverted
with the Fourier transform.
The
operator $L_y$ has only terms linear and terms quadratic in $y$,
and therefore is small if $y$ is small;
$y$ can be made small initially without
loss of generality by re-labeling.
One can then hope to use a Neumann series,
\begin{equation}\label{eq:neumann}
(\partial_ty=)\qquad v = \frac{1}{1-L_0^{-1}L_y} L_0^{-1} (\Omega,0)
\end{equation}
\mostimportant{%
Mere `boundedness' of
$L_0^{-1}L_y$ is due to elliptic regularity:
The loss of a derivative through $L_y$ is regained by $L_0^{-1}$.
It allows one to move some terms that contain $v$ to the right,
without spoiling the analysis.
Elliptic regularity is encoded in assumptions (a.2) and (a.3) below.
}

\mostimportant{%
We are sloppy, especially in (a.3), about the fact that
$L_0$ only returns,
and $L_0^{-1}$ only takes,
pairs $(\omega^2,\omega^3)$ for which $\mathrm{d} \omega^2 = 0$.
In \eqref{eq:neumann} this is fine,
because $L_0^{-1}$ is only ever applied to $(\Omega,0)$
and to the image of $L_y$.}

\subsection{Argument}\label{sec:kksiuhikas}
Assumptions:
\begin{itemize}
\itemsep 1pt
\item[(a.1)] Two Banach spaces $\text{N}$ and $\text{DN}$
are fixed.
The norms $\NN{\,\cdot\,}$ and $\NDN{\,\cdot\,}$
take functions or tuples of functions
of $\lambda \in \mathbbm{R}^3$, depending on context.
\item[(a.2)]
$\NDN{\partial_{\lambda} f} \lesssim \NN{f}$.
\item[(a.3)]
$\NN{L_0^{-1}f} \lesssim \NDN{f}$.
\item[(a.4)]
$\NDN{fg} \lesssim \NDN{f} \NDN{g}$.
\item[(b)] $\text{BN} \subset \text{N}$ is the open subset
of all $f\in\text{N}$ with $\NDN{\partial_{\lambda} f} < \text{(a small constant)}$.
\item[(c)] The fixed vorticity satisfies $\NDN{\Omega} < \infty$.
\end{itemize}
\mostimportant{%
The norm of $\Omega$ is the norm of its coefficients with respect
to the $\mathrm{d} \lambda^A$. The small constant in (b) is chosen
below, and this choice depends only on
the structure of the equations and the norms,
but not on $\Omega$.
For each $\Omega$ there is an ODE on $\text{BN}$.
}

We show that \eqref{eq:neumann}
defines a map
$\text{BN} \to \text{N}, y \mapsto v$, with
the uniform bound
\begin{equation} \label{eq:boundvv31}
\NN{v}\,\lesssim\,\NDN{\Omega}
\end{equation}
This map is real analytic, in fact a Taylor series,
and \eqref{lagodeyyy} is an autonomous ODE on $\text{BN}$
with unique local real analytic solutions $t \mapsto y(t) \in \text{BN}$.

First note the operator norm
$\NNOP{L_0^{-1}L_y} \lesssim \NDN{\partial_{\lambda} y}
+ \NDN{\partial_{\lambda} y}^2$
by (a.2), (a.3), (a.4).
Choose the constant in (b) small enough to make
$\NNOP{L_0^{-1}L_y} \leq \tfrac{1}{2}$ for all $y \in \text{BN}$.
Then the Neumann series converges and we get \eqref{eq:boundvv31}.

To get a Taylor series,
decompose $L_y$ into parts linear
and quadratic in $y$, and then
reorganize the terms produced by the Neumann series
by their homogeneity:
\begin{equation}\label{eq:tyl}
1/(1-L_0^{-1}L_y) \;=\; \textstyle\sum_{n \geq 0} O_n(y,\ldots,y)
\end{equation}
with multilinear operators that for some $R > 0$ satisfy
$$\NNOP{O_n(y_1,\ldots,y_n)} \;\lesssim\; R^{-n}
\NDN{\partial_{\lambda}y_1}\cdots \NDN{\partial_{\lambda}y_n}$$
with an implicit constant independent of $n$.
Choose the constant in (b) smaller than $\tfrac{1}{2}R$,
to make the Taylor series \eqref{eq:tyl} converge for $y \in \text{BN}$.

\mostimportant{%
The argument above is abstract.
To ground the discussion,
let $\text{N}$ and $\text{DN}$ be subsets of the real-valued
tempered distributions $\mathcal{S}'$ on $\mathbbm{R}^3$,
with continuous embedding in
$C^0\subset \mathcal{S}'$, the bounded continuous functions.
Then partial differentiation $\partial_{\lambda}: \mathcal{S}' \to \mathcal{S}'$
and pointwise multiplication $C^0 \times C^0 \to \mathcal{S}'$
are defined as usual.
For all
$y \in \text{BN}$ the map $\lambda \mapsto x = \lambda + y(\lambda)$ is a $C^1$-diffeomorphism of $\mathbbm{R}^3$, if the constant in (b) is chosen small enough.

Explicit spaces $\text{N},\text{DN}$ that satisfy
the assumptions are in Appendix \ref{app:norms}.
}

\subsection{Re-labeling equivalence classes}\label{sec:RELAB}
Re-labeling is a right-action of the group of diffeomorphisms of $\mathbbm{R}^3$
on pairs $(x,\Omega)$ in Lagrangian coordinates $t,\lambda$
that maps solutions to solutions, see \eqref{eq:lagformxxx}.
The orbits are the re-labeling equivalence classes.

A diffeomorphism $\lambda = r(\lambda')$
acts by $(x,\Omega) \mapsto (x',\Omega')$
with $x'(t) = r^{\ast}x(t)$ and $\Omega' = r^{\ast}\Omega$.
The first is equivalent to $x'(t,\lambda') = x(t,\lambda)$, or
$$
\lambda' + y'(t,\lambda') = \lambda + y(t,\lambda)$$

\mostimportant{One can narrow the
action to the subgroup of those $r$
that act as continuous and continuously invertible maps
$\text{N} \to \text{N}$ and $\text{DN} \to \text{DN}$,
on $y$ and $\Omega$ respectively.}

\subsection{Re-labeling example}\label{relabalgXX}

\mostimportant{%
The construction in Section \ref{sec:kksiuhikas}
stops once $y$ reaches the boundary of $\text{BN}$,
but before that happens,
one can re-label to push $y$ far back into $\text{BN}$.
Here we discuss just one such re-labeling, without estimates.}

Fix a re-labeling time $t$.
To make $y' \approx 0$, stipulate
$\lambda' = \lambda + (y \ast m)(\lambda)$
with $m$ a smooth approximation of the identity.
Equivalently $r(\lambda')=\lambda'+c(\lambda')$ with
$c(\lambda') = - (y\ast m)(\lambda' + c(\lambda'))$,
which can be solved by the contraction mapping principle.
 Now
$y' = (y - y\ast m) \circ r$
and
$\Omega' = r^{\ast}\Omega$.

\mostimportant{
We have introduced $m$ to make $r$
smooth, to avoid pull-backs by rough functions.
One can let $m$ approach the identity.
Note that the re-labeling $r$
depends on the unknown $y$, and on $m$.
We will not use such re-labelings.}

\section{Approximate solution and nearby true solution}\label{sec:appref}

\mostimportant{%
We show how an approximate solution of \eqref{eq:geomform}
can be used as a reference,
to construct a nearby true solution,
with more control (on say the time of existence)
than a direct application of Section \ref{sec:locex}.
There is gauge freedom in how one identifies
the manifold on which the reference lives,
with the manifold on which the new solution
lives. The gauge used in this section
identifies the manifolds by \emph{identifying their fluid paths}.
In Section \ref{secxx4}
we allow for a class of singular references;
Section \ref{sec:hypoloc} will refer back to Section \ref{secxx4}.}

\subsection{System \eqref{eq:geomform} relative to a reference}\label{secxx1}
Fix a reference
$(M^4,t,X,V,S,\Omega)$
that satisfies
\eqref{eq:geomform}
up to errors $\mathbf{E}_2$, $\mathbf{E}_3$:
\begin{subequations}\label{eqsecxx1}
\begin{align}
\mathcal{L}_S t & = 1 \label{eq:owhkkjdkkkka}\\
\mathcal{L}_S X^A & = V^A \label{eq:XVref}\\
\mathcal{L}_S \Omega & = 0 \label{eq:somxx1}\\
\mathrm{d} \Omega & = 0\\
\delta_{AB}\; \mathrm{d} V^A \wedge \mathrm{d} X^B & = \Omega + \mathbf{E}_2\\
\varepsilon_{ABC}\; \mathrm{d} V^A \wedge \mathrm{d} X^B \wedge \mathrm{d} X^C & = \mathbf{E}_3
\end{align}
\end{subequations}

\mostimportant{By assumption, four equations
of \eqref{eq:geomform} hold identically.
They comprise
an underdetermined subproblem that can be solved in various ways.
The algebraic manipulations in this section are
intended for errors $\mathbf{E}_2$, $\mathbf{E}_3$ much smaller than $\Omega$.
}

Consider a true solution $(M^4,t,x,v,S,\Omega)$.
As suggested by the notation,
\emph{this solution uses the reference's $M^4$, $t$, $S$
(gauge that identifies fluid paths) as well as $\Omega$.}
Adopt $(t,X)$ as independent variables. Then
$$
S = \partial_t + V^A \partial_{X^A}
$$
by \eqref{eq:owhkkjdkkkka}, \eqref{eq:XVref}.
The vector field $S$ is fixed by the reference.
Set
\begin{alignat*}{6}
x(t,X) & = & X      &\,+ &\;y(t,X)&\\
v(t,X) & =\;& V(t,X) &\,+ &\;w(t,X)&
\end{alignat*}
with $y$ and $w$ the new unknown.
Equations
\eqref{transportt}, \eqref{transportomega}, \eqref{clom}
hold by construction, and
system \eqref{eq:geomform} is now equivalent to
\begin{subequations}\label{dkhkfhdlffffaa}
\begin{alignat}{6}
\mathcal{L}_S y^A & = w^A \label{ooop1}\\
 \delta_{AB}\; \mathrm{d} w^A \wedge \mathrm{d} X^B
& =
\big[\textstyle\sum_{P=V,w} \mathrm{d} P \wedge \mathrm{d} y\big] && -\mathbf{E}_2 \label{ooop2}\\
\varepsilon_{ABC}\; \mathrm{d} w^A \wedge \mathrm{d} X^B \wedge \mathrm{d} X^C
& =
\big[\textstyle\sum_{P=V,w}\sum_{Q=X,y} \mathrm{d} P \wedge \mathrm{d} y \wedge \mathrm{d} Q\big]
&& -\mathbf{E}_3 \label{ooop3}
\end{alignat}
\end{subequations}
Superficially similar to \eqref{eq:lagformyyy},
this system is not driven by a full-sized $\Omega$,
but by small $\mathbf{E}_2$ and $\mathbf{E}_3$.
Equation \eqref{ooop1}
can be recast as
\begin{equation}\label{eq:odeagain}
\tfrac{\mathrm{d}}{\mathrm{d} t}(y(t) \circ \phi(t,s)) = w(t)\circ \phi(t,s)
\end{equation}
for any fixed $s$.
Here $\phi(s,t): \mathbbm{R}^3 \to \mathbbm{R}^3$ is the flow map of the reference,
$\phi(t,t) = \text{(identity)}$ and
$\tfrac{\mathrm{d}}{\mathrm{d} s} \phi(s,t) = V(s) \circ \phi(s,t)$,
see Appendix \ref{app:flowmap} for details.
\vskip 3mm \noindent

\newcommand{\ref{sec:locex}}{\ref{sec:locex}}
\mostimportant{%
For local existence we appeal to Section {\ref{sec:locex}}, as follows.
Let $x(t),\Omega(t)$ be as in the current section.
For every $s$
let $x_{\ref{sec:locex},s}(t),\Omega_{\ref{sec:locex},s}$
be the solution produced by Section {\ref{sec:locex}}
with initial data $x_{\ref{sec:locex},s}(s) = x(s)$
and vorticity $\Omega_{\ref{sec:locex},s} = \Omega(s)$.
Then the correspondence is $x_{\ref{sec:locex},s}(t) = x(t) \circ \phi(t,s)$
and $\Omega_{\ref{sec:locex},s} = \phi(t,s)^{\ast}\Omega(t)$.
To rewrite this in terms of $y$'s,
use $x(t) = \mathbbm{1} + y(t)$
and $x_{\ref{sec:locex},s}(t) = \mathbbm{1} + y_{\ref{sec:locex},s}(t)$ with $\mathbbm{1}: \mathbbm{R}^3 \to \mathbbm{R}^3$ the identity.
 Then the correspondence is
$y(t) = (\phi(s,t)-\mathbbm{1}) + y_{\ref{sec:locex},s}(t)\circ \phi(s,t)$,
with $\phi$ fixed by the reference.
In particular $y_{\ref{sec:locex},s}(s) = y(s)$.

Basic assumptions about the reference needed for Section {\ref{sec:locex}}
are implicit below, e.g.~that $\NDN{\Omega(t)} < \infty$
and $\NN{\phi(s,t)-\mathbbm{1}} < \infty$, and
composition by $\phi(s,t)$ is a bounded linear map on $\text{N}$.
We assume that the reference is smooth.
To keep Section {\ref{sec:locex}} going
we need $y(t) \in \text{BN}$,
for which it suffices that $\NN{y(t)}$
is small.
Estimating $\NN{y(t)}$ is the purpose of Section \ref{secxx2}.
}

\subsection{Estimate}\label{secxx2}

Equation \eqref{eq:odeagain} implies
\begin{equation}\label{eq:kkkiq}
\big|\tfrac{\mathrm{d}}{\mathrm{d} t} \NN{y(t)}\big| \;\leq\; \NN{w(t)} +
\big|\tfrac{\mathrm{d}}{\mathrm{d} s}|_{s=t}
\NN{y(t) \circ \phi(t,s)}\big|
\end{equation}
\mostimportant{%
We use the notation
$|\tfrac{\mathrm{d}}{\mathrm{d} t} f(t)| = \limsup_{h \to 0} |(f(t+h)-f(t))/h|$.
Inequality \eqref{eq:kkkiq}
is obtained somewhat formally by
setting $a(t,s) = \NN{y(t)\circ \phi(t,s)}$ and using the chain rule
for $\tfrac{\mathrm{d}}{\mathrm{d} t} a(t,t)$.
}

In addition to (a) and (b) from Section \ref{sec:basicargXXXX},
 assume:
\newcommand{\text{DK}}{\text{DK}}
\newcommand{\NDK}[1]{\|#1\|_{\text{DK}}}
\begin{itemize}
\itemsep 0pt
\topsep 0pt
\parsep 0pt
\item[(e.1)] $\text{DK}$ is a Banach space.
\item[(e.2)] $\NDN{fg} \lesssim \NDK{f}\NDN{g}$.
\item[(e.3)]
$\big|\tfrac{\mathrm{d}}{\mathrm{d} s}|_{s=t} \NN{f \circ\textrm{(flow of $U$)}(t,s)}\big|
\;\lesssim\;
\big(|U(0)|
+ \NDK{\partial_X U}\big)\, \NN{f}$.
\item[(f)]
 $\NDN{(\mathbf{E}_2,\mathbf{E}_3)(t)}<\infty$ for all $t$.
\end{itemize}
\mostimportant{%
The space $\text{DK}$ plays a different role than $\text{N},\text{DN}$.
We anticipate that $\text{DK}$ appears in the
critical assumption (r.1) in Section \ref{secxx4}, and
that one will choose $\text{DK}$ to suit (r.1).
Estimates of the form (e.3)
for composition by the flow of a vector field $U$
can be obtained from Appendix \ref{app:compest}
if $\text{N}$ is a weighted Sobolev space;
one does not lose derivatives in $f$.
In (f) the norms of $\mathbf{E}_2$, $\mathbf{E}_3$
are the norms of their coefficients with respect the $\mathrm{d} X^A$.}

The linear system \eqref{ooop2}, \eqref{ooop3} for $w$ has the form
$$L_0w = L_yw + K_y - (\mathbf{E}_2,\mathbf{E}_3)$$
The operators $L_0$ and $L_y$ are the same as in Section \ref{sec:locex},
except that the independent variables now have a different name,
$X$ instead of $\lambda$. Terms without $w$ go into $K_y$,
they all contain one factor of $\mathrm{d} y$ at least.
We estimate the map
$\text{BN} \to \text{N}, y \mapsto w$ by using a Neumann series
as before,
\begin{equation}\label{eq:uuz}
\NN{w} \;\lesssim\; \NDK{\partial_X V} \NN{y} + \NDN{(\mathbf{E}_2,\mathbf{E}_3)}
\end{equation}
where the first term on the right bounds $\NDN{K_y}$.
From \eqref{eq:kkkiq} and \eqref{eq:uuz} we obtain,
uniformly on $\text{BN}$ and with implicit constant independent of $t$:
\begin{equation}\label{eq:xxxxx}
\big|\tfrac{\mathrm{d}}{\mathrm{d} t} \NN{y(t)}\big| \;\lesssim\;
\big(
|V(t,0)|
+ \NDK{(\partial_X V)(t)}\big)\, \NN{y(t)} + \NDN{(\mathbf{E}_2,\mathbf{E}_3)(t)}
\end{equation}

\mostimportant{The map $y \mapsto w$ is time dependent,
through $V$ and $\mathbf{E}_2$ and $\mathbf{E}_3$.
However $L_0$ and $L_y$ do not depend on time,
and neither do the estimates for the Neumann series, they
rely only on $y \in \text{BN}$.
Therefore the implicit constants in \eqref{eq:uuz}
and \eqref{eq:xxxxx}
are also independent of time, the time dependence is
explicit in the terms on the right hand side.}

\subsection{Rescaled estimate}
\newcommand{b}{b}
Set $\sigma(b): \mathbbm{R}^3 \to \mathbbm{R}^3, X \mapsto b X$.
For every constant $b  > 0$,
\begin{multline}\label{eq:dkhkdhdk}
\big|\tfrac{\mathrm{d}}{\mathrm{d} t} \NN{y(t) \circ \sigma(b )}\big|
\;\lesssim
\big(
b^{-1} |V(t,0)|
+ \NDK{(\partial_X V)(t) \circ \sigma(b )}\big)\, \NN{y(t) \circ \sigma(b )}\\ + b  \, \NDN{(\mathbf{E}_2,\mathbf{E}_3)(t) \circ \sigma(b)}
\end{multline}
provided $y(t) \circ \sigma(b ) \in \text{BN}$.
The implicit constant is independent of $b $.

\mostimportant{
Inequality \eqref{eq:dkhkdhdk}
follows from \eqref{eq:xxxxx} by
the scaling symmetry in $X$ of \eqref{dkhkfhdlffffaa}.
The notation
$\mathbf{E}_2(t) \circ \sigma(b )$ is short for
$\text{(coefficients of $\mathbf{E}_2$ w.r.t.~the $\mathrm{d} X$)(t)}
\circ \sigma(b )$, and similar for $\mathbf{E}_3$.
We assume that for every $b >0$, composition by $\sigma(b )$ is
a bounded linear map on $\text{N}$, $\text{DN}$, $\text{DK}$. We only state
this explicitly when we need a quantitative bound.
}

\subsection{Possibly singular reference}\label{secxx4}

\mostimportant{%
Suppose the reference is singular as $t \uparrow 0$.
Under some assumptions we show that
there is a nearby true solution that
is asymptotic to the reference as $t \uparrow 0$.
We do so by solving the equations in the negative time direction,
away from the singularity.
}

Assume that for some constant $A > 0$ the norm satisfies
\begin{itemize}
\item[(q)] $\NN{f} \leq b ^{-A} \NN{f \circ \sigma(b )}$ for all $b  \leq 1$.
\end{itemize}
Assume that the reference is defined for $t$ in some interval
$(-\text{const},0)$,
and that for some constants $B,E,R_1,R_2 > 0$ it satisfies
\begin{itemize}
\item[(r.1)]
 $
|t|^{-B} |V(t,0)|
+ \NDK{(\partial_X V)(t) \circ \sigma(|t|^B)} \,\leq\, R_1 |t|^{-1}$.
\item[(r.2)] $\NDN{(\mathbf{E}_2,\mathbf{E}_3)(t) \circ \sigma(|t|^B)} \,\leq\, R_2 |t|^E$.
\end{itemize}

Set
$\mathbf{J}(t) = |t|^{-AB} \NN{y(t) \circ \sigma(|t|^B)}$.
Then
\begin{equation}\label{eq:qoiwele}
\textstyle\limsup_{h \uparrow 0}\;(\mathbf{J}(t+h)-\mathbf{J}(t)) \big/ |h|
\;\lesssim\; R_1 |t|^{-1} \mathbf{J}(t) + R_2 |t|^{B(1-A) + E}
\end{equation}
provided $y(t) \circ \sigma(|t|^B) \in \text{BN}$.
This follows from
(q), \eqref{eq:dkhkdhdk}, (r.1), (r.2).

\mostimportant{This limits the growth of $\mathbf{J}(t)$
only in the negative time direction:
Given $\mathbf{J}(t)$ with $t < 0$, it yields a bound for
$\mathbf{J}(s)$
with $s < t$.
The implicit constants in \eqref{eq:qoiwele}
in front of the first and second terms,
respectively,
are the same as the implicit constants in \eqref{eq:xxxxx}:
they do not depend on $t$,
and they do not depend on the reference.
In \eqref{eq:qoiwele}
the dependence on the reference
is explicit in $B$, $E$, $R_1$, $R_2$.}

Let $C_1>0$ be the implicit constant in front of the first
term in \eqref{eq:qoiwele}.
Set $\mathbf{K}(t) = |t|^{-C_1R_1} \mathbf{J}(t)$. Then
\begin{equation}\label{eq:oeiuzeizwe}
\textstyle\limsup_{h \uparrow 0}\;(\mathbf{K}(t+h)-\mathbf{K}(t)) \big/ |h|
\;\lesssim\; R_2 |t|^{B(1-A) + E - C_1R_1}
\end{equation}
which is integrable over intervals
$(-\text{const},0)$ iff the exponent is $>-1$, that is
\begin{equation}\label{eq:sjkkkqn}
E \; > \; B(A-1) + C_1R_1 - 1
\end{equation}
If \eqref{eq:sjkkkqn} holds,
then by a limiting procedure there is a true
solution on some time interval $(-\text{const},0)$
that is asymptotic to the reference as $t \uparrow 0$,
$$\NN{y(t) \circ \sigma(|t|^B)} = \mathcal{O}(|t|^{B+E+1})$$
which includes
$y(t) \circ \sigma(|t|^B) \in \text{BN}$ for $t$ close to $0$.

\mostimportant{%
The limiting procedure works as follows.
Introduce a family $\{y_s\}$ of solutions parametrized by small $s<0$,
namely set $y_s(s) = 0$ and solve the equations
in the negative time direction. Let $(T(s),s]$
be the domain of $y_s$.
Then there is a common nontrivial time of existence,
$\limsup_{s\uparrow 0} T(s) < 0$, and furthermore $\lim_{s \uparrow 0}y_s$ exists.
To show convergence,
one estimates the distance between pairs
of solutions in the family.
}

\section{A hypothetical self-similar flow $\text{\bf S}$}\label{sec:sss}

We propose and motivate properties of a hypothetical self-similar
solution $\text{\bf S}$ of 3D incompressible Euler.
It would be interesting to know if this solution
exists,
or whether the properties are inconsistent.

\mostimportant{%
If one contemplates consistency,
then this section can be the starting point for
a numerical search (if a more conceptual approach does not come to mind).
The properties can be used to narrow the numerical search space,
and the section was written expressly with that purpose in mind.
Some properties seem forced,
while others are only motivated as being
the seemingly `simplest' among a set of alternatives,
the idea being that one may want to consider those first.

The perspective taken in this section is that one is looking
for an essentially unique object;
this is spelled out in property \sref{spec:localunique}.
This perspective motivates many of the proposals.

In this section,
we pretend that the hypothetical $\text{\bf S}$ exists.
For convenience, we will not endlessly repeat the word hypothetical.
The entire section is understood to be hypothetical.
}

\mostimportant{%
The solution $\text{\bf S}$ is self-similar, it develops a singularity,
and it has infinite energy, all of which follows from self-similarity.
Is the singularity peculiar to
this self-similar solution,
or does it encode a mechanism for singularity formation
that extends to finite energy solutions?
A tentative discussion is in Section \ref{sec:hypoloc}.
}

\vskip 3mm \noindent

This section
has features in common with
an interesting paper by Lathrop \cite{lathrop},
especially with the more mathematical Sections 2, 5, 6, 7 of \cite{lathrop}.

\mostimportant{
Lathrop emphasizes a vector field that he calls $H$,
which is the infinitesimal generator
of the dynamical system \eqref{eq:iiuuw2} below.
Lathrop considers a velocity that goes like
$r^{1-1/\alpha}$ as $r \to \infty$,
in our notation $|x|^{1-1/\text{KA}}$ ($\text{KA}$ is our symbol for $\alpha$)
which is consistent with this section.
The main differences with \cite{lathrop} are as follows.
We allow for
discrete self-similarity, see \sref{spec:selfsim}.
The expansions in and after \sref{spec:iiuu} are new.
The local uniqueness hypothesis \sref{spec:localunique} is new.
It motivates us to narrow the discussion in several places,
say in \sref{spec:conjclass}, and in \sref{spec:XSAY}.
}
\vskip 3mm \noindent

An interesting instance of self-similarity
is the general relativity solution found by
Choptuik \cite{choptuik}.

\mostimportant{%
 Choptuik's solution is a pair $(g,\phi)$
with $g$ the metric, $\phi$ a massless scalar field.
The self-similarity is discrete, with a generator that
not only rescales $g$, but also inverts the sign of $\phi$.
That is, the self-similarity is non-trivially coupled to other symmetries.
}

\mostimportant{%
A computer assisted proof of the existence and of the real analyticity
of Choptuik's solution is in \cite{trub}.
This is a case where numerics can be made into a rigorous proof.
It is the solution to an effectively 2-dimensional nonlinear PDE.
A purely conceptual construction is not known at this time.
}

\vskip 3mm \noindent

\begin{sproperty}[solution]\label{spec:u1}
The flow $\text{\bf S}$ solves incompressible Euler.
Its velocity
$$
(-\infty,0) \times \mathbbm{R}^3 \to \mathbbm{R}^3,\;\;
(t,x) \mapsto v_{\text{\bf S}}(t,x)
$$
with respect to Eulerian coordinates
is smooth, and likely real analytic.
\end{sproperty}

\mostimportant{Eulerian coordinates
are used to simplify the discussion;
the equations for these coordinates are in Appendix \ref{app:euleriangauge}.
It may be that this gauge is not the best
to search for $\text{\bf S}$, and that one should for example
use a gauge that `fixes the fluid paths', see below.}

\begin{sproperty}[nontrivial] \label{spec:u2}
The vorticity $\omega_{\text{\bf S}} = \nabla \times v_{\text{\bf S}}$ does not
vanish identically.
\end{sproperty}

\begin{sproperty}[sublinear growth]\label{sublineargrowth}
$v_{\text{\bf S}} = \mathcal{O}(|x|^{1-\delta})$
and $\nabla v_{\text{\bf S}} = \mathcal{O}(|x|^{-\delta})$
as $|x|\to \infty$, for fixed $t < 0$.
The constant $\delta > 0$ is independent of $t$.
\end{sproperty}

\mostimportant{
Here and in similar instances below,
`for fixed $t < 0$' is short for
`uniformly on every compact time subinterval of $(-\infty,0)$'.

\sref{sublineargrowth} excludes special
solutions of the form $v = |t|^{-1}
\text{(constant matrix)}\,x$.
They are doubly self-similar,
$v(at,bx) = (b/a) v(t,x)$
for all $a,b \in \mathbbm{R}^+$,
and they are the only such solutions.

\sref{sublineargrowth}
guarantees that the flow map (see Appendix \ref{app:flowmap}) exists,
and therefore that Lagrangian coordinates exist.
In fact, \sref{sublineargrowth} implies that the flow map is
asymptotic to the identity:
$\phi^{\text{\bf S}}(s,t)x = x + \mathcal{O}(|x|^{1-\delta})$
and its Jacobian is $\mathbbm{1} + \mathcal{O}(|x|^{-\delta})$,
for all fixed $s,t < 0$.
}

\begin{sproperty}[self-similar]\label{spec:selfsim}
There is
a constant $\text{KA} > 0$
and a nontrivial subgroup $Q \subset \mathbbm{R}^+$
such that
$v_{\text{\bf S}}(qt,\,q^{\text{KA}} x) \;=\; q^{\text{KA}-1}\,v_{\text{\bf S}}(t,x)$
for all $q \in Q$.
\end{sproperty}

\mostimportant{
Further symmetries are discussed later.
Most of the discussion uses only \sref{spec:selfsim}.

In a numerical search for $\text{\bf S}$, the number $\text{KA}$ is one of the unknowns.

We assume that either $Q = \mathbbm{R}^+$, or that $Q$ is isomorphic to $\mathbbm{Z}$.
For $Q = \mathbbm{R}^+$ the reduced equations are in Appendix \ref{app:css},
but we will not use them.
From now on, all occurrences of $q$ are
understood to be restricted by $q \in Q$.

\sref{spec:selfsim}
implies
$\phi^{\text{\bf S}}(qs,qt) \circ q^{\text{KA}} = q^{\text{KA}} \circ \phi^{\text{\bf S}}(s,t)$
where, abusing notation, $q^{\text{KA}}$ denotes the map $x \mapsto q^{\text{KA}}x$.
The vorticity satisfies
$\omega_{\text{\bf S}}(qt,q^{\text{KA}} x) = q^{-1} \omega_{\text{\bf S}}(t,x)$,
the vorticity 2-form
$(q^{\text{KA}})^{\ast} \Omega^{\text{\bf S}}(qt) = q^{2\,\text{KA}-1} \Omega^{\text{\bf S}}(t)$.
Also
$(\nabla^k v_{\text{\bf S}})(qt,q^{\text{KA}}x) = q^{(1-k)\text{KA}-1} (\nabla^k v_{\text{\bf S}})(t,x)$.
}

\begin{sproperty}[flow estimate]\label{hhzzeYY}
If $t < s < 0$ and $|t|^{-\text{KA}}|x| > \text{(big constant)}$ then
\begin{subequations}\label{eq:iiuBOTH}
\begin{align}
\label{eq:iiu11}
|\phi^{\text{\bf S}}(s,t)x - x| &
\;\lesssim\; (|t|^{-\text{KA}}|x|)^{-\delta} |x|\\
\label{eq:iiu22}
|\text{(Jacobian of $\phi^{\text{\bf S}}(s,t)$ at $x$)} - \mathbbm{1}| &
\;\lesssim\; (|t|^{-\text{KA}}|x|)^{-\delta}
\end{align}
\end{subequations}
\end{sproperty}

\mostimportant{%
Here $\delta$ is the constant in \sref{sublineargrowth}.
Here $A \lesssim B$ is short for $A \leq \text{(big constant)}\,B$.
It is understood that
the big constants do not depend on $s,t$ or $x$.

The assumptions and conclusions in \sref{hhzzeYY}
scale as in \sref{spec:selfsim}, and it therefore suffices to check
\sref{hhzzeYY} for fixed $t < 0$, say $t \approx -1$,
uniformly for all $s \in (t,0)$.
One can use the bound
$|\nabla^k v_{\text{\bf S}}|
\lesssim |t|^{\text{KA}\,\delta - 1} \max(|t|^{\text{KA}},|x|)^{1-k-\delta}$
for $k=0$ and $k=1$,
which follows from
\sref{sublineargrowth} and \sref{spec:selfsim}.
}

\begin{sproperty}[almost all particles go to self-similar infinity]\label{uuiew2113}
For almost all fluid paths $t \mapsto x(t)$
one has
$\textstyle\lim_{t \uparrow 0} |t|^{-\text{KA}}|x(t)| = \infty$.
\end{sproperty}

\mostimportant{%
By definition, a fluid path has property $H$ iff
$\textstyle\lim_{t \uparrow 0} |t|^{-\text{KA}}|x(t)| = \infty$.
Property $H$ is equivalent to $\liminf_{t \uparrow 0} |x(t)| > 0$
by \eqref{eq:iiu11}.
Property $H$ is also equivalent to
$\exists t < 0: |t|^{-\text{KA}}|x(t)|
\geq \text{(some threshold)}$, again by \eqref{eq:iiu11}.

Let $a(t)$ be the volume occupied at time $t$ by all particles
that \emph{do not}
have property $H$. Then $a(t)$ is at most the volume of a ball
with radius $\text{(threshold)} |t|^{\text{KA}}$, which goes to zero as
$t \uparrow 0$. Since $a(t)$ is independent of time by incompressibility,
it follows that $a(t)=0$.
}

\newcommand{X}{X}
\begin{sproperty}[limiting flow]\label{h9632Y}
Let $X(r) \subset \mathbbm{R}^3$ be the set of points with $|x| > r$.
Set $r(t) = \text{(big constant)} |t|^{\text{KA}}$.
Then
$\lim_{s \uparrow 0} \phi^{\text{\bf S}}(s,t)|_{X(r(t))}$
exists,
with $C^1$ convergence on compact subsets,
and is a $C^1$ diffeomorphism $X(r(t)) \to J$ whose image
satisfies $X(2r(t)) \subset J \subset X(r(t)/2)$.
Its Jacobian is uniformly close to $\mathbbm{1}$
and has determinant equal to $1$.
\end{sproperty}

\mostimportant{%
By \sref{spec:selfsim} is suffices to check
this for fixed $t < 0$.
Let $t < s' < s < 0$.
For $C^0$ convergence
use $|\phi^{\text{\bf S}}(s,t)x-\phi^{\text{\bf S}}(s',t)x|
= |\phi^{\text{\bf S}}(s,s')y - y|$ with $y = \phi^{\text{\bf S}}(s',t)x$
to apply \eqref{eq:iiu11} and let $s',s \uparrow 0$.
For $C^1$ convergence use \eqref{eq:iiu22} in a similar way,
with the chain rule for $\phi^{\text{\bf S}}(s,s')\circ \phi^{\text{\bf S}}(s',t)$.

One can derive more properties of
$\lim_{s \uparrow 0}\, \phi^{\text{\bf S}}(s,t)$
based on other properties proposed in this section.
Informally speaking, this limit is a degenerate endpoint in
V.I.~Arnold's variational interpretation
of incompressible Euler.
}

\newcommand{\Gamma^{\SSS}}{\Gamma^{\text{\bf S}}}
\newcommand{$\Gamma^{\SSS}$ }{$\Gamma^{\text{\bf S}}$ }

\mostimportant{%
To go on, introduce for all $q \in Q$ and $t < 0$ the diffeomorphism
\begin{equation}
\Gamma^{\SSS}_q(t) = q^{-\text{KA}} \circ \phi^{\text{\bf S}}(qt,t)\;:\; \mathbbm{R}^3 \to \mathbbm{R}^3
\end{equation}
By \sref{spec:selfsim} it is a homomorphism in $q$,
and different $t$ yield conjugate maps, explicitly:
\begin{subequations}\label{uausbbaksdq}
\begin{align}
\label{eq:iiuuw2}
\Gamma^{\SSS}_{pq}(t) & = \Gamma^{\SSS}_p(t) \circ \Gamma^{\SSS}_q(t) \qquad p,q \in Q\\
\Gamma^{\SSS}_q(t) & = \phi^{\text{\bf S}}(s,t)^{-1} \circ \Gamma^{\SSS}_q(s) \circ \phi^{\text{\bf S}}(s,t)
\end{align}
\end{subequations}
Vorticity and volume conservation imply identities internal to level sets of $t$:
\begin{subequations}
\begin{align}
\label{eq:zertzweefe}
\Gamma^{\SSS}_q(t)^{\ast}\Omega^{\text{\bf S}}(t) & = q^{1-2\,\text{KA}}\Omega^{\text{\bf S}}(t)\\
\label{eq:ooowihebf}
\Gamma^{\SSS}_q(t)^{\ast}(\mathrm{d} x^1 \wedge \mathrm{d} x^2 \wedge \mathrm{d} x^3) & = q^{-3\,\text{KA}}
(\mathrm{d} x^1 \wedge \mathrm{d} x^2 \wedge \mathrm{d} x^3)
\end{align}
\end{subequations}}

\mostimportant{%
To picture $\Gamma^{\SSS}_q$ geometrically,
quotient $(-\infty,0)\times \mathbbm{R}^3$
by identifying $(qt,q^{\text{KA}}x) \sim (t,x)$.
The quotient is topologically $S^1 \times \mathbbm{R}^3$ if $q \neq 1$,
and it is ruled by fluid paths
that monotonically wind around $S^1$.
Going once around $S^1$ along fluid paths permutes the fluid paths
(pick a transversal hypersurface and look at the return map)
and this permutation is  $\Gamma^{\SSS}_q$, up to conjugacy.
}

\begin{sproperty}[global vorticity bound]\label{SPEC:globalvorticitybound}
$|\omega_{\text{\bf S}}| \lesssim \max(|t|^{\text{KA}},|x|)^{-1/\text{KA}}$.
\end{sproperty}

\mostimportant{%
By \sref{spec:selfsim} it suffices to show
$|\omega_{\text{\bf S}}| \lesssim \max(1,|x|)^{-1/\text{KA}}$
for fixed $t<0$.
Let
$q \downarrow 0$ in $Q$ in \eqref{eq:zertzweefe}.
For $|x| > \text{(big constant)}$ one has
$|\Gamma^{\SSS}_q(t)x - q^{-\text{KA}}x| \lesssim q^{-\text{KA}} |x|^{1-\delta}$
uniformly for all $q \leq 1$ in $Q$ by \eqref{eq:iiu11},
and a similar bound for the Jacobian using \eqref{eq:iiu22}.
One obtains \sref{SPEC:globalvorticitybound}.
}

\begin{sproperty}[eigenvalues at fixed points]\label{spec:j2382hi}
If $\omega^{\text{\bf S}}(t) \neq 0$ at a fixed point of
$\Gamma^{\SSS}_q(t)$,
then the eigenvalues of the linearization satisfy
$\lambda_1 = q^{-1-\text{KA}}$ and $\lambda_2\lambda_3 = q^{1-2\,\text{KA}}$.
\end{sproperty}

\mostimportant{%
For a 3-dim linear map $A$
the identity
$p_{A \wedge A}(\mu) D = -\mu^3 p_A(D/\mu)$
for characteristic polynomials holds, with $D = \det A$.
Here $D = q^{-3\,\text{KA}}$ by \eqref{eq:ooowihebf},
and $p_{A \wedge A}(q^{1-2\,\text{KA}})=0$ by \eqref{eq:zertzweefe}
since $\Omega^{\text{\bf S}}(t) \neq 0$. Therefore $p_A(q^{-1-\text{KA}})=0$.
Then use $\lambda_2\lambda_3 = D/\lambda_1$.
}

\begin{sproperty}[classification of $\Gamma^{\SSS}$  up to conjugacy]\label{spec:conjclass}\rule{0pt}{0pt} These properties hold:
\begin{itemize}
\item[(a)] $\Gamma^{\SSS}_q(t)$ has exactly one fixed point $x_{\ast}(t)$
if $q \neq 1$.
\item[(b)] $\Gamma^{\SSS}_q(t)$
is conjugate in $\text{Diff}^{\infty}(\mathbbm{R}^3)$ to a linear map $A_q$.
\item[(c)] $A_q$ is repelling if $q<1$.
\item[(d)] $A_q$ is diagonalizable over $\mathbbm{C}$.
\item[(e)] $\omega_{\text{\bf S}}(t) \neq 0$ at $x_{\ast}(t)$, for some $t<0$.
\end{itemize}
\end{sproperty}

\mostimportant{%
\sref{spec:conjclass}
collects assumptions about the classification up to conjugacy,
it describes the seemingly simplest scenario.

As suggested by the notation, the fixed point in (a) does not depend on $q$,
and the linear map in (b) does not depend on $t$, by \eqref{uausbbaksdq}.
In (a) there is at least one fixed point since
$\Gamma^{\SSS}_q(t)x \sim q^{-\text{KA}} x$ as $|x|\to \infty$,
and to have exactly one fixed point seems to be  the simplest alternative.
Global linearizability (b) subsumes (a);
note that $q \mapsto A_q$ is a homomorphism.
 Repelling in (c) means
that $|A_q^n\text{(any nonzero vector)}| \to \infty$ as $n \to \infty$.
The non-degeneracy condition (e) seems reasonable,
assuming that there are no symmetries to prevent it.
}

\mostimportant{The classification of $\Gamma^{\SSS}$ up to conjugacy
is a practical matter if
one uses a gauge that `fixes the fluid paths' on the $S^1 \times \mathbbm{R}^3$
obtained by identifying $(qt,q^{\text{KA}}x) \sim (t,x)$.
Such a gauge is probably a fine choice,
because then \eqref{eq:zertzweefe}
and vorticity conservation essentially fix $\Omega^{\text{\bf S}}$,
but there is an obstruction:
by fixing the fluid paths,
one fixes the conjugacy class for $\Gamma^{\SSS}$.
If one goes along with \sref{spec:conjclass},
then the fluid paths can be fixed up to
a small number of parameters, namely the eigenvalues in (d),
and there are no other obstructions.
Note that \sref{spec:j2382hi} applies.

With a gauge that does not fix the fluid paths,
such as the Eulerian gauge,
one can in principle ignore the classification up to conjugacy.
}

\begin{sproperty}[all particles but one go to self-similar infinity]\label{spec:9u39hh93}
The map $t \mapsto x_{\ast}(t)$ from
 \sref{spec:conjclass}
is a fluid path and satisfies $|t|^{-\text{KA}}|x_{\ast}(t)| \lesssim 1$ uniformly
in time.
All other fluid paths $t \mapsto x(t)$ satisfy
$\lim_{t \uparrow 0} |t|^{-\text{KA}}|x(t)| = \infty$.
\end{sproperty}

\mostimportant{%
This is a consequence of \sref{spec:conjclass}.
The properties of $t \mapsto x_{\ast}(t)$ follow
from \eqref{uausbbaksdq}
and from the fixed point condition
$x_{\ast}(qt) = q^{\text{KA}} x_{\ast}(t)$.
For all other fluid particle paths,
(c)
implies that
$\Gamma^{\SSS}_q(t)x(t) = q^{-\text{KA}} x(qt)$
eventually leaves, as $q \downarrow 0$ in $Q$ at fixed $t$,
every compact subset of $\mathbbm{R}^3$, which gives
$\lim_{t \uparrow 0} |t|^{-\text{KA}}|x(t)| = \infty$.
This strengthens
\sref{uuiew2113}.}

\begin{sproperty}[lower bound for $\text{KA}$]\label{xxxuu12}
 $\text{KA} \,>\, \tfrac{1}{2}$.
\end{sproperty}

\mostimportant{%
By \sref{spec:j2382hi} and \sref{spec:conjclass}.
If $q<1$ then the eigenvalues of $A_q$ must satisfy
$|\lambda_1| > 1$, $|\lambda_2| > 1$, $|\lambda_3| > 1$
by (c), (d). Note that (e) holds for all $t$ if it holds for one $t$ by
vorticity conservation, since $t \mapsto x_{\ast}(t)$ is a fluid path.
Then $|\lambda_2\lambda_3| > 1$ and \sref{spec:j2382hi}
imply $1-2\,\text{KA} < 0$.

Incidentally, $\tfrac{1}{2}$ is the only
value compatible with Navier-Stokes diffusion.
A self-similar
solution of incompressible Euler with $\text{KA} > \tfrac{1}{2}$
 is in some sense
an approximate solution of Navier-Stokes at early times
$t \to -\infty$, but the relative strength of diffusion
grows as $t \uparrow 0$.
}

\mostimportant{%
To go on, we assume that $\text{\bf S}$ is in some sense
`regular at self-similar infinity'.
The ansatz \sref{spec:iiuu} accommodates
elliptic modes \sref{spec:elliptic}
and transport modes \sref{spec:transport},
and is `closed under multiplication'
in view of the nonlinear nature of incompressible Euler.
Other than that, the ansatz is intended to be minimal.
}

\begin{sproperty}[asymptotic expansion]\label{spec:iiuu}
Set $a_{mn}^{\alpha}(z) = z^{\alpha m+n}$.
Then asymptotically as $|x|^{-1/\text{KA}} |t| \to 0$, with a disclaimer below,
\begin{subequations}\label{eq:llzzuuBOTH}
\begin{alignat}{5}
\label{eq:llzzuu}
v_{\text{\bf S}}(t,x) \; & \sim\;\;& |x|/|t|
&\textstyle\sum_{m,n \geq 0}
c[v_{\text{\bf S}}]_{mn} (t,x)\, a_{mn}^{\text{KA}}(|x|^{-1/\text{KA}}|t|)\\
\label{eq:llzzuuvo}
\omega_{\text{\bf S}}(t,x) \; & \sim\;\;& 1/|t|
&\textstyle\sum_{m,n \geq 0}
c[\omega_{\text{\bf S}}]_{mn} (t,x)\, a_{mn}^{\text{KA}}(|x|^{-1/\text{KA}}|t|)
\end{alignat}
\end{subequations}
The coefficient functions
$c[\,\cdot\,]_{mn}: (-\infty,0) \times
(\mathbbm{R}^3 \setminus \{0\}) \to \mathbbm{R}^3$
are doubly periodic:
$c(qt,x) = c(t,q^{\text{KA}} x) = c(t,x)$ for all $q \in Q$.
\end{sproperty}

\mostimportant{%
Doubly periodic functions
live on a compact space, $S^2$ or $S^1\times S^1\times S^2$
depending on $Q$. In particular, doubly periodic implies bounded.
The dimensionless
$|t|\partial_t$ and $|x| \nabla$ map doubly periodic to doubly periodic.
Note that \eqref{eq:llzzuuBOTH} is
consistent with \sref{spec:selfsim}, and is
 both
an asymptotic expansion for fixed $t<0$ as $|x| \to \infty$,
and
for fixed $x \neq 0$ as $t \uparrow 0$.
}

\mostimportant{%
Disclaimer: The $a_{mn}^{\text{KA}}$ are linearly dependent
at rational $\text{KA}$,
which is unacceptable in \eqref{eq:llzzuuBOTH}.
One can replace the $a_{mn}^{\alpha}$ by a linearly independent
but otherwise equivalent family,
see for example Appendix \ref{app:specialfunc}.
To simplify the discussion, we keep using the $a_{mn}^{\alpha}$.
At rational $\text{KA}$, one can make some sense of this
by letting the $c[\,\cdot\,]_{mn}$
be meromorphic functions of $\alpha$ with rational poles
that cancel upon summation,
as in
$((\alpha-1)^{-1} a^{\alpha}_{10}(z)
  - (\alpha-1)^{-1} a^{\alpha}_{01}(z))|_{\alpha = \text{KA}}$.
Similarly for other coefficients and expansions below.

\sref{spec:iiuu} yields formally
a compactification of the problem:
there is only a discrete set of coefficients $c[\,\cdot\,]_{mn}$
even though the domain of $\text{\bf S}$ is unbounded.
It is conceivable that
the asymptotic expansions converge and are equal to $v_{\text{\bf S}}$, $\omega_{\text{\bf S}}$
 for small
$|x|^{-1/\text{KA}}|t|$.

In \sref{spec:iiuu} there are two indices $m,n$ for just
one geometric dimension, namely $|x|^{-1/\text{KA}}|t|$.
If correct, then this may rule out
certain numerical approaches to finding $\text{\bf S}$.
A numerical search that uses \eqref{eq:llzzuuBOTH},
or is at least aware of it, may have an advantage.

Complications for a numerical search that uses \eqref{eq:llzzuuBOTH}:
the $a_{mn}^{\text{KA}}$ depend on the unknown $\text{KA}$;
one may have to use something like
Appendix \ref{app:specialfunc};
two indices $m,n$ for just
one geometric dimension.
If $Q$ is discrete, then there are two more Fourier indices.
Separate expansions are needed in the bulk, away from self-similar infinity.
}

\begin{sproperty}[sublinear growth for coefficients]\label{spec:triang}
$c[v_{\text{\bf S}}]_{00}=0$ and $c[\omega_{\text{\bf S}}]_{00}=0$.
\end{sproperty}

\mostimportant{%
By \sref{sublineargrowth}.

Plugging \eqref{eq:llzzuuBOTH}
into incompressible Euler \eqref{rtriziugdu}
yields a problem for the coefficients $c_{mn}$.
We now discuss just this problem.
It is not by itself enough to determine all the $c_{mn}$:
there are `constants of integration'
that can presumably only be gotten by calculating $\text{\bf S}$ in the bulk.

This problem has a useful symmetry:
for all $\mu,\nu > 0$,
replacing all $c_{mn}$ by $\mu^m \nu^n c_{mn}$
both for $v_{\text{\bf S}}$ and $\omega_{\text{\bf S}}$ maps solutions to solutions.
That is, linear terms $c_{mn}$ and quadratic terms
$c_{m'n'}c_{m''n''}$ only appear together in an equation
when $m = m'+m''$ and $n=n'+n''$;
we have suppressed derivatives etc,
and each coefficient belongs to either $v_{\text{\bf S}}$ or $\omega_{\text{\bf S}}$.

Clearly $m',m'' \leq m$ and $n',n'' \leq n$,
however after using \sref{spec:triang} to drop terms
we also get $(m',n') \neq (m,n)$ and $(m'',n'') \neq (m,n)$.
This yields a recursion
where step $(m,n)$
has the form $A c_{mn} = B$,
with an expression
$B$ that is quadratic in coefficients constructed `earlier',
and with a linear operator $A$ that depends
on $m, n, \text{KA}$, but not on say any of the coefficients.

The linear operator $A$ has nontrivial kernel when
 $(m,n)$ has the form $(m,0)$ or $(0,1)$.
In these cases $B = 0$,
that is, these steps of the recursion are governed by the linear equations
$\partial_t \omega = 0$, $\nabla \cdot\omega = 0$, $\nabla \times v = \omega$,
$\nabla \cdot v = 0$ with respectively
\begin{alignat*}{9}
v & =\, &c[v_{\text{\bf S}}]_{m0}(t,x)&|x|^{1-m}&& |t|^{\text{KA}\,m - 1}
&\qquad\qquad
v & =\, &c[v_{\text{\bf S}}]_{01}(t,x)&|x|^{1-1/\text{KA}}\\
\omega & = &c[\omega_{\text{\bf S}}]_{m0}(t,x) &|x|^{-m} &&|t|^{\text{KA}\,m-1}
&
\omega & = &c[\omega_{\text{\bf S}}]_{01}(t,x)&|x|^{-1/\text{KA}}
\end{alignat*}
and this notation is not used elsewhere.
The kernels are in \sref{spec:elliptic} and \sref{spec:transport},
they are the `constants of integration' alluded to above.
The operator $A$ is invertible for all other $(m,n)$
if $\text{KA}$ is irrational, for rational $\text{KA}$
see the disclaimer after \sref{spec:iiuu}.}

\FloatBarrier
\begin{sproperty}[decaying elliptic modes]\label{spec:elliptic}
$c[\omega_{\text{\bf S}}]_{m0} = 0$,
and the $n=0$ subexpansion of \eqref{eq:llzzuu} is equal to
\begin{equation}\label{eq:kksw12}
(1/|t|)\,\textstyle\sum_{\ell \geq 1}\sum_{-\ell \leq k \leq \ell}
E_{\ell k}^{\text{\bf S}}(t)\,|t|^{\text{KA}(\ell+3)}\,
\nabla(|x|^{-\ell-1}Y_{\ell k})
\end{equation}
for some scalar-valued coefficients with $E(qt)=E(t)$.
\end{sproperty}

\mostimportant{%
The spherical harmonics $Y_{\ell k}$ are functions of $x/|x|$
with $\Delta_{S^2} Y_{\ell k} = -\ell(\ell+1)Y_{\ell k}$.
Expression \eqref{eq:kksw12} is for every $t$
the gradient of the multipole expansion of
a general decaying harmonic function in $x$.
There is no $\ell = 0$ term by incompressibility:
the flux of $v_{\text{\bf S}}$ through spheres of constant $|x|$
must vanish identically.
For each $\ell \geq 1$,
the $2\ell + 1$ functions $E_{\ell k}^{\text{\bf S}}$
determine $c[v_{\text{\bf S}}]_{\ell+3,0}$,
while $c[v_{\text{\bf S}}]_{m0} = 0$ for $m \leq 3$.
}

\mostimportant{%
Growing modes $\nabla(|x|^{\ell} Y_{\ell k})$
with $\ell \geq 2$
are suppressed by \sref{sublineargrowth},
but not the more subtle $\ell = 1$
which goes into $c[v_{\text{\bf S}}]_{10}$.
However, the symmetry `shake' in Appendix \ref{app:sym}, that takes
$v(t,x)$ to $v(t,x-\beta(t)) + \frac{\mathrm{d}}{\mathrm{d} t}\beta(t)$,
maps self-similar solutions to self-similar solutions if
$\beta(qt) = q^{\text{KA}} \beta(t)$.
Among the solutions related by this symmetry,
only one has no $\ell = 1$ growing mode,
and \sref{spec:elliptic} selects that one.
Thus `shake' is fixed, and the absence of growing elliptic modes
is a full set of `elliptic boundary conditions at infinity'
for $v_{\text{\bf S}}$.

These vanish: $c_{00}[v_{\text{\bf S}}]$ by \sref{sublineargrowth},
$c_{10}[v_{\text{\bf S}}]$ by `shake',
$c_{20}[v_{\text{\bf S}}]$ automatically,
$c_{30}[v_{\text{\bf S}}]$ by incompressibility.
}

\begin{sproperty}[transport modes]\label{spec:transport}
$c[v_{\text{\bf S}}]_{01}(t,x) = T^{\text{\bf S}}(x)$
for a vector-valued $T^{\text{\bf S}}$ that does not vanish identically,
$T^{\text{\bf S}}(q^{\text{KA}} x) = T^{\text{\bf S}}(x)$
and $\nabla \cdot(|x|^{1-1/\text{KA}}T^{\text{\bf S}})=0$.
\end{sproperty}

\mostimportant{
Then $c[v_{\text{\bf S}}]_{01}$ is
 the leading term of the expansion
assuming only $\text{KA} > \tfrac{1}{4}$, see \sref{xxxuu12}.
}

\begin{figure}[h]
\centering
\captionsetup{justification=centering}
\input{modes.pstex_t}
\caption*{%
\footnotesize
Coefficients $c[v_{\text{\bf S}}]_{mn}$
and $c[\omega_{\text{\bf S}}]_{mn}$
with $m\geq 0$ vertical, $n\geq 0$ horizontal.\\
White boxes vanish.
Black boxes are elliptic and transport modes.
}
\end{figure}

\FloatBarrier

\begin{sproperty}[global velocity bound]\label{spec:globalvelocitybound}
$|\nabla^k v_{\text{\bf S}}| \lesssim \max(|t|^{\text{KA}},|x|)^{1-1/\text{KA}-k}$
where the implicit constant depends on $k=0,1,2,\ldots$
\end{sproperty}

\mostimportant{
By \sref{spec:selfsim} it suffices to
show \sref{spec:globalvelocitybound} for
fixed $t < 0$, which holds by \sref{spec:iiuu}-\sref{spec:transport}
and \sref{xxxuu12}.

Note: If $\text{KA} > 1$ then $v_{\text{\bf S}}$ is bounded near
$(t,x)=(0,0)$, by \sref{spec:globalvelocitybound} with $k=0$.
If $\text{KA} < 1$ then $v_{\text{\bf S}}$ is unbounded near $(0,0)$,
 by  \sref{spec:selfsim}.
We take no stance on whether $\text{KA}>1$ or not.
}

\begin{sproperty}[final velocity]\label{spec:finalvelocity}
$\lim_{t \uparrow 0} v_{\text{\bf S}}(t,x) = |x|^{1-1/\text{KA}} T^{\text{\bf S}}(x)$ for
all $x \neq 0$.
\end{sproperty}

\begin{sproperty}[finite energy in bounded domains]
The energy in the ball with radius $r > 0$
stays bounded,
$\limsup_{t \uparrow 0} \int_{|x| < r} \mathrm{d}^3 x\,  |v_{\text{\bf S}}(t,x)|^2 < \infty$.
\end{sproperty}

\mostimportant{%
The integral is $\lesssim \int_{|x|<r}\mathrm{d}^3x\, \max(|t|^{\text{KA}},|x|)^{2-2/\text{KA}}$.
Split it into respectively $|x| < |t|^{\text{KA}}$ and $|t|^{\text{KA}} < |x| < r$.
One obtains $\lesssim |t|^{B\,\text{KA}} + r^B/B$
provided $B = 5-2/\text{KA}$ is positive.
}

\begin{sproperty}[local uniqueness]\label{spec:localunique}
Informally,
$\text{\bf S}$ is uniquely characterized,
locally in function space, by the properties
\sref{spec:u1}, \sref{spec:u2},
\sref{sublineargrowth}, \sref{spec:selfsim}.
Local uniqueness is understood modulo
the symmetries in Appendix \ref{app:sym}.
\end{sproperty}

\mostimportant{%
This makes searching for $\text{\bf S}$ a more attractive proposition.
\sref{spec:localunique} proposes that $\text{\bf S}$
is not part of a continuous family of solutions.
If there is a discrete family,
then there may still be one member that stands out as the most basic,
say the `most stable'.

\sref{spec:localunique} can be informally
motivated by counting degrees of freedom.
We discuss the case when $Q\subset \mathbbm{R}^+$ is discrete,
the case $Q=\mathbbm{R}^+$ is similar.
Let $q<1$ be the generator of $Q$.
Consider the fundamental region $[-1,-q] \times \mathbbm{R}^3$.
Sublinear growth \sref{sublineargrowth} only suppresses the
growing elliptic modes,
and yields a full set of `elliptic boundary conditions at infinity'
for $v_{\text{\bf S}}$;
see also remarks after \sref{spec:elliptic}.
Initial data at $t = -1$ maps,
assuming suitable existence and uniqueness,
to data at $t = -q$, but
only a discrete family of initial data
satisfies the self-similarity condition \sref{spec:selfsim}.
This counting has a flaw: \sref{spec:selfsim}
refers to $\text{KA}$ and the generator $q$,
two more unknowns that seem to spoil the counting.
The remedy are two residual symmetries,
scaling $x$
and scaling $t$,
which is an $\mathbbm{R}^+ \times \mathbbm{R}^+$ symmetry modulo the discrete $Q$.
Gauge fixing these two symmetries restores the counting.

With \sref{spec:localunique} it would seem that
$\text{\bf S}$ should have as much symmetry as possible.
However symmetry is not linearly ordered,
and it is not clear what `possible' means.
One should study symmetry systematically.
A very limited discussion is in
Appendix \ref{app:somestabcand}.
The symmetries proposed in \sref{spec:selfsim}
and \sref{spec:XSAY} are
among those in Appendix \ref{app:somestabcand}.
Prejudicing a numerical search to a more symmetric class makes
things easier resource-wise, but is more dangerous.
}

\begin{sproperty}[symmetry]\label{spec:XSAY}
Let $R_{\varphi} \in \SO(3)$ be rotation around
a fixed axis by angle $\varphi$.
Then the symmetries of $\text{\bf S}$ are a subgroup
of $\mathbbm{R}^+ \times \{\pm 1\} \times (\mathbbm{R}/2\pi \mathbbm{Z})$,
where each element $(p,\sigma,\varphi)$ is the symmetry\; $v_{\text{\bf S}}(pt,p^{\text{KA}} \sigma R_{\varphi}x) = p^{\text{KA}-1} \sigma R_{\varphi} v_{\text{\bf S}}(t,x)$.
\end{sproperty}

\mostimportant{%
\sref{spec:XSAY} proposes that
a numerical search should not assume symmetries beyond those given.
Such a search does not rule out other symmetries,
but does not use them either.

\sref{spec:XSAY} takes no stance on which subgroup the symmetries of $\text{\bf S}$
are. Some possibilities:
\begin{itemize}
\item[(i0)] all elements
\item[(i1)] all elements with $\mu \log p = \varphi$, for a constant $\mu \neq 0$
\item[(i2)] all elements with $p = (p_1)^n$ and $n \in \mathbbm{Z}$, for a constant $p_1 > 1$
\item[(i3)] like (i2), but with the additional condition $(-1)^n = \sigma$
\end{itemize}

They
 are all consistent with \sref{spec:selfsim}, explicitly $Q$ is
given by: (i0) $Q = \mathbbm{R}^+$, (i1) $ Q= (e^{2\pi k/\mu})_{k \in \mathbbm{Z}}$,
(i2) $Q=((p_1)^k)_{k \in \mathbbm{Z}}$, (i3) $Q=((p_1)^{2k})_{k \in \mathbbm{Z}}$.

The subgroup (i0) has continuous dimension 2,
while (i1), (i2), (i3) have dimension 1.

In (i0), (i1), (i2) there is no
restriction on $\sigma$. One obtains other subgroups by putting $\sigma = 1$.
Assuming the $\{\pm 1\}$ symmetry in a numerical search
yields a small gain at best.}

\section{A tentative
strategy to localize $\text{\bf S}$}\label{sec:hypoloc}

This entire section is under the
assumption that $\text{\bf S}$ in Section \ref{sec:sss} exists,
it is self-similar, it develops a singularity, and it has infinite energy.
Does it encode a
mechanism for singularity
formation that extends to finite energy solutions?
This section has no answer, but only a few remarks.

We sketch a strategy that modifies $\text{\bf S}$ to a
finite energy flow $\text{\bf S}'$
that does not itself solve incompressible Euler,
but is a good enough approximate solution to
invoke Section \ref{secxx4},
which (by solving incompressible Euler backwards)
produces a solution $\text{\bf S}''$ that has finite energy and
develops a singularity.

We do not know if this strategy can be implemented.
In step \stepref{underdetproblem} we
encounter a nonlinear system of equations that looks underdetermined, but
we do not know if there are obstructions to solving it.
The discussion is very informal.

\newcommand{\SSS|_{\text{ball}}}{\text{\bf S}|_{\text{ball}}}

\newcommand{\text{\bf EXT}}{\text{\bf EXT}}
\newcommand{\text{\bf L}}{\text{\bf L}}

\begin{sstep}\label{xosnfjlf} Let $\SSS|_{\text{ball}}$
be the restriction of $\text{\bf S}$ to a ball in $\mathbbm{R}^3$.
\end{sstep}

\mostimportant{%
Along the spherical boundary
and as $t \uparrow 0$,
the asymptotic expansion \eqref{eq:llzzuu} for $\text{\bf S}$ applies.}

\begin{sstep}
In the complement of the ball, introduce a formal series flow
\begin{equation}\label{eq:llzzuu333}
v_{\text{\bf L}}(t,x) \; = \; (|x|/|t|)
\textstyle\sum_{m,n \geq 0}
c[v_{\text{\bf L}}]_{mn} (t,x)\, a_{mn}^{\text{KA}}(|x|^{-1/\text{KA}}|t|)
\end{equation}
Require $c(qt,x) = c(t,x)$ for all $q \in Q$,
\emph{but do not require periodicity in $x$.}
\end{sstep}

\mostimportant{%
Without periodicity in $x$, one could absorb
all powers of $|x|$ in the coefficients, but keeping them
makes it easier to compare with
\eqref{eq:llzzuu}. The formal series \eqref{eq:llzzuu333} is an expansion at $t = 0$.
}

\begin{figure}[h]
\centering
\input{lok.pstex_t}
\end{figure}

\begin{sstep}\label{sstep:elliptic}
Set $c[v_{\text{\bf L}}]_{m0} = c[v_{\text{\bf S}}]_{m0}$.
\end{sstep}

\mostimportant{%
The corresponding terms in \eqref{eq:llzzuu333} decay like
$\mathcal{O}(|x|^{-3})$,
consistent with finite energy,
because $c[v_{\text{\bf L}}]_{m0}=0$ when $m \leq 3$,
see the discussion after \sref{spec:elliptic}.
In particular, $c[v_{\text{\bf L}}]_{00}=0$.}

\begin{sstep}
Data for $\text{\bf L}$ along $t = 0$ is a function
$c[v_{\text{\bf L}}]_{01}(t,x) = T^{\text{\bf L}}(x)$. It must
\begin{itemize}
\itemsep 0pt
\item be compatible with incompressibility $\nabla \cdot v_{\text{\bf L}} = 0$,
\item match $c[v_{\text{\bf S}}]_{01}(t,x) = T^{\text{\bf S}}(x)$ smoothly
at the spherical boundary,
\item decay, with its derivatives, sufficiently fast as $|x| \to \infty$.
\end{itemize}
Other than that, it can be chosen freely.
\end{sstep}

\begin{sstep}
Require that the formal series
$\text{\bf L}$ solves incompressible Euler,
and that the orthogonal components of
$v_{\text{\bf L}}$ and of the asymptotic series of $v_{\text{\bf S}}$
match continuously at the spherical boundary.
Now $T^{\text{\bf L}}$ determines $v_{\text{\bf L}}$
by a recursion.
\end{sstep}

\mostimportant{%
One can only prescribe the orthogonal velocity component
at the boundary.

The choice in \stepref{sstep:elliptic}
is compatible with both incompressible Euler and the boundary condition.
It seems that \stepref{sstep:elliptic} is forced,
because $T^{\text{\bf L}}$ does not affect the $c[v_{\text{\bf L}}]_{m0}$ components.
}

\begin{sstep}\label{underdetproblem}
Let $K > 0$ be a big integer.
\emph{Can one tweak $T^{\text{\bf L}}$ so that for all $|m| \leq K$ and $|n| \leq K$,
the coefficients $c[v_{\text{\bf L}}]_{mn}$ and
$c[v_{\text{\bf S}}]_{mn}$ match at the spherical boundary,
with $K$ continuous derivatives?}
\end{sstep}

\mostimportant{
This problem seems to be underdetermined.
For example if $Q = \mathbbm{R}^+$,
then $T^{\text{\bf L}}$ is a function of three independent variables
(exterior of a ball in $\mathbbm{R}^3$),
but the matching conditions concern only a finite number
of functions of two independent variables (sphere in $\mathbbm{R}^3$).}

\begin{sstep}
Assuming \stepref{underdetproblem} has a solution,
set
$\text{\bf S}' = \text{smoothing}(\SSS|_{\text{ball}},\text{truncation}(\text{\bf L}))$.
The truncation of $\text{\bf L}$ is the finite sum
over $|m|,|n| \leq K$.
It does not match $\SSS|_{\text{ball}}$ at the spherical boundary,
not even continuously,
but the mismatch decays quickly as $t \uparrow 0$
given \stepref{underdetproblem}.
To correct the mismatch, apply a smoothing near the boundary,
chosen so as to not destroy the $t \uparrow 0$ asymptotics.
\end{sstep}

\begin{sstep}
The flow $\text{\bf S}'$ is only an approximate solution of incompressible Euler,
with an error that decays quickly as $t \uparrow 0$.
One can now invoke Section \ref{secxx4}
(which solves the equations in the negative time direction)
to get a true solution $\text{\bf S}''$ on a small
time interval, with a singularity as $t \uparrow 0$.
\end{sstep}

\mostimportant{%
Disclaimer:
It is clear how
Section \ref{secxx4},
is intended to be applied,
but we have not checked all the assumptions there.
Certainly, $K$ must be big enough
in view of condition  \eqref{eq:sjkkkqn}.

Disclaimer: Section \ref{sec:appref} requires that references satisfy four
equations of \eqref{eq:geomform} identically,
see \eqref{eqsecxx1}, and one must adjust
$\text{\bf S}'$ accordingly. This should only be a technical matter.

Note that $\text{\bf S}'$ coincides with $\text{\bf S}$ for small $|x|$,
but $\text{\bf S}''$ does not.

We have used the Eulerian gauge to keep the discussion short.
For actual calculations
it may be easier to first put Lagrangian coordinates on
$\text{\bf S}$.
Instead of restricting $\text{\bf S}$ to a ball in Eulerian coordinates,
one uses a ball in Lagrangian coordinates.
The new boundary
is `characteristic' (ruled by fluid paths)
which may be helpful.

The strategy above, if it works at all,
only produces rather unnatural examples.
It would be interesting to study
more generic classes of finite energy solutions that are
singular like $\text{\bf S}$.
}

\begin{comment}

\begin{align*}
\NN{f} & =
  \|\langle x \rangle \partial f\|_{L^2}
+ \|\langle x \rangle \partial^2 f\|_{L^2}
+ \|\langle x \rangle \partial^3 f\|_{L^2}\\
\NDN{f} & =
  \|\langle x \rangle f\|_{L^2}
+ \|\langle x \rangle \partial f\|_{L^2}
+ \|\langle x \rangle \partial^2 f\|_{L^2}\\
\NDK{f} & =
  \| f\|_{\infty} + \|\partial f \|_{\infty} + \|\partial^2 f\|_{\infty}
\end{align*}
\end{comment}

\section*{Acknowledgments}
This paper is based on work at ETH Zurich and IAS Princeton.
I thank Eugene Trubowitz for many
useful discussions concerning this paper.
Our joint \cite{trub} was one of the motivations for this paper.
I enjoyed support from
The Giorgio and Elena Petronio Fellowship Fund,
and the (US) National Science Foundation.

\mostimportant{This material is based upon work supported by the National Science Foundation under agreement No.~DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.}

\mostimportant{%

redpajama_set_name: RedPajamaArXiv


Nuclear energy should be considered a clean energy source, Mr Montek Singh Ahluwalia, Deputy Chairman Planning Commission, said that at the end of the 23-Government Clean Energy Ministerial meeting that took place in London this week.
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India is set to host the fourth CEM meeting in next April. The London meeting brought together representative of governments from across the world, including China, Brazil, Australia and the US, and is meant to be an annual forum for some of the world's biggest emitters of greenhouse gases to work together on policies to increase their use of renewable energy.
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Maithreem Bhajata is a composition on world peace in Sanskrit by Jagadguru Chandrasekharendra Saraswati of Kanchi Kamakoti Peetam. The song is set to a Ragamalika in Yamuna Kalyani and Kapi by Vasant Desai.


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I am very prompt at email and can be reached at karan<at>karanbajaj.com but the best way to communicate with me is by signing up here.
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I love faux fur looks amazing with everything!
I'm all over fur this winter! I need to get myself a tartan scarf too.
I really do love your videos Lily, so keep them up and more fashiony ones like this would be good! Loving faux fur at the minute (more so this year than any other!?) Love how you added it to a leather jacket!Just wondering where you got your faux fur jacket?
Following your account, I think its a good idea.
love the sequin bib necklace and the fur stoles are amazing!
Don't worry so much about dictating and that... you're just saying what you like to wear and giving advice on how things can be worn.
great post, i love this.
too bad it is summer in australia.
Quite awesome. I also prefer to buy vintage jewelry Wholesale Jewelry Store.


text:
Craig Stephen's This Week in China
China's panic-buying of salt
By Craig Stephen
Published: Mar 20, 2011 8:28 p.m. ET
Commentary: A sign economy is at breaking point?
CraigStephen
HONG KONG (MarketWatch) — This week attention will remain on the spillover effects of Japan's nuclear crisis as a large degree of edginess remains.
Somewhat surprisingly it has been cities across China, including Hong Kong, that have witnessed panic in recent days, in contrast to the relative calm in Japan.
Last week, a salt-buying frenzy was unleashed across China that spread to Hong Kong, which quickly became a rather unsavory spectacle. Rumors had spread via the Internet from across the border that some salt contains iodine, which would provide protection from a looming radiation threat from Japan. Throngs of people began scrambling for salt, mobbing shops and wholesalers.
On Friday, the Hong Kong government had to resort to a mass text-message campaign imploring people not to believe rumors and to stop panic-buying salt. The fact you might have to eat kilos of the stuff to get sufficient iodine was of no consequence. Indeed, so chaotic was the behavior that some people were even seen bizarrely carting off bottles of soy sauce. Amid the buying, retailers reportedly raised the price of salt more than tenfold, seeing an opportunity to profit from this panic.
The whole episode has caused a certain degree of embarrassment and shame, particularly in contrast to the restrained behavior in Japan, where there is very real suffering. There the orderly and stoic lining up for water, transport and food in the middle of biblical-like devastation has shown the remarkable endurance and self-control of the Japanese people.
This makes the contrast with images on mainland China all the more stark. Some academics have suggested the willingness to believe bogus warnings reflects a lack of trust in the government's truthfulness in China. That could have some credence: Back in 2008, the government tried to keep the scandal of powdered milk mixed with melamine that was poisoning babies quiet until after the Olympic Games in Beijing. Perhaps one unintended consequence of a one-party government which controls the media with an army of censors is that it creates a fertile breeding ground for paranoia.
What is perhaps more surprising is that Hong Kong, which has a free flow of information and independent media, also joined the frenzy.
Another explanation for this out-of-control behavior is that those on the treadmill of the world's workshop are at breaking point as they struggle with a surge in living costs. That would be one way to interpret reports that many Guangdong factory workers did not bother to return to work from their Chinese New Year trip home.
The whole episode sits awkwardly next to the central government's newly introduced mantra of pursuing happiness, not just economic growth. The Economist featured a story on China this week "Don't worry be happy," reporting how various mainland provinces are now competing not on gross domestic product numbers, but to have the happiest citizens.
Incidentally, another case of failing the good-character test in the past week involved Singapore's media giant Media Corp. Shortly after the Japanese earthquake struck, it pitched clients with a new ad package saying it expected a surge in viewers on its news channel with coverage of the disaster. Not surprisingly, this clumsy attempt to make some extra money from the earthquake caused outrage, and Media Corp. had to later apologize.
Getting back to Japan's economy, China, as well as much of the region, will be watching carefully how quickly it recovers. If an economy the size of Japan's — which until last year was the second-largest in the world — goes into reverse, it potentially sets up a negative growth shock for the region.
But many are optimistic that can be avoided. Even before the aftershocks had subsided, various economists were already penciling in a strong rebound as rebuilding efforts got underway. But that optimism might be misplaced, argue analysts at Nomura. They say in a new report that despite initial confidence that reconstruction expenditure would feed into the economy by the year's end, this may be restrained by a lack of sustainable power.
Even harder to quantify is how the crisis impacts investor risk appetite by raising the level of uncertainty. For China, any shock from Japan's crises comes at a time when it has already been applying the brakes to its economy and is also dealing with rising oil and commodity prices. Now we are starting to see some impact, as bank lending slow markedly last month to 536.6 billion yuan ($81.7 billion), significantly lower than 1.04 trillion yuan in January.
Mainland Chinese policy makers have a delicate balancing act to steer the economy to slower growth and wean it off bank credit without tipping an overheated property market into bursting or killing jobs growth.
What we saw last week was not anything truly worrying, like a run on banks, but just a run on salt. Still, the manic scramble suggests a deeper level of anxiety on the ground over the direction of the economy — something that should not be taken with a pinch of salt.
Craig Stephen
Craig Stephen is a MarketWatch columnist and author of the column, "This Week in China".
Asia Markets

redpajama_set_name: RedPajamaCommonCrawl


At SKIN Cosmetic Dermatology we provide insurance billing and encourage you to confirm your coverage prior to making an appointment. Any co-payments are due at the time of your appointment. If the insurance company does not pay the full balance, or denies the claim, then any unpaid balance will be billed to you.
We also accept Visa, MasterCard, American Express and cash.


The Morning News calls it the best website ever… I would have to agree. Amazing amount of information, especially if you need building elevations – Skyscraper Page.
If haven't dropped by already, there is a video installation entitled "waiting" showing at Gallery 825 (825 La Cienega blvd) by Rise members Michele Jaquis and Jeremy Quinn.
Showing at Gallery 825 this thursday is "Take Me to the River", a 30 minute sneak preview of the upcoming feature length documentary of the largest human gathering in history – the Maha Kumbh Mela in India. Produced by Projectile Arts (New York), the showing will be accompanied by Q and A with Michele Jaquis. Also showing is another documentary on India, Sight Unseen – a travelogue.
Showing starts at 8pm, see Los Angeles Art Association website for more info.
Rise Industries Audio launches its first project, Crocket the Ironmonger. Two tracks available online. More to come.
For more info on Ninjas, check the official Ninja Web Page.
Ninjas again…. This time in an attempt to take over Detroit.


text:
PARANORMAL PARKING
Spooky CCTV footage purports to show a teen 'ghost' hovering in the shadows of a car park at night
The petrifying woman is clad in a white nightgown and her face is covered by her long black hair
Sam Webb
0:11, 4 May 2017
Updated: 13:13, 11 Oct 2017
A TERRIFYING ghost who haunts at a car park has been captured in this spine-tingling CCTV footage, it is claimed.
The supposedly supernatural incident was caught by cameras in the early morning on September 27, 2016 at an unknown location.
The 'ghost' is seen at the far right of the image hiding behind a cornerCredit: Scary Videos/Youtube
The video - which could easily be a hoax - shows the woman in a white nightgown with her face is covered by her long black hair - suspiciously similar to the popular Ring horror film franchise.
She mills about aimlessly at the right edge of the screen and then disappears into the wall behind her.
MOST READ IN WORLD NEWS
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BUM DEAL
China brings back anal swab testing for Covid two weeks before Winter Olympics
BEAST BREAKS LOOSE
French cannibal ESCAPES and attacks woman in random targeting
The female phantasm just sort of sways about in a cornerCredit: Scary Videos/Youtube
The video ends with the white-clad woman melting into the wallCredit: Scary Videos/Youtube
The spine-tingling footage, posted on Youtube, appears to have been taken using a night-vision camera.
It was posted on the Scary Videos Youtube channel, which claims to capture unearthly creatures in videos from across the world.
Last month a family in Melbourne, Australia, was left spooked after a ghost-like face "resembling their late grandmother" was apparently captured peering into their home.
Dayna Lynch, 18, said she took a pic of her dad Paul sitting with her step-mum Safiye Alev as they watched TV in Frankston – just days after the death of her grandmother.
Ghosts and the supernatural


Welcome to your personal 1:1 coaching sessions with Prestige!
Set milestones so we know when to celebrate!
The goal is to move the needle from having desires to accomplishing goals!
To become financially viable by making 7 k with both companies per month.
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Staging and taping of the videos and post 2 per week starting Friday.


Hopefully Charlie Montoyo takes note and finds a way to fire up the hitters as Martin Perez was quoted saying to MLB.com: "I think (Monday's) going to be a big day for me. I'm going to get the win ... I'm going long." Yeah, you want your SP to be confident, but that's pretty big talk to make public, especially after he's been hit pretty good so far this season. Sic 'im, boys.
Lourdes Gurriel Jr. has been sent down to Buffalo after a couple of brutal throws on Saturday and Sunday -- as Montoyo said, it's all up there in the head. It'd be good for Gurriel to clear the cobwebs and just focus on playing baseball for a while out of the limelight (that's what Vlad is for, to soak up the attention). In his place, Eric Sogard has been called up. To make room for Sogard on the 40-man, Ryan Borucki has been moved to the 60-day IL (see you in June, Ryan).
Take a look at this ERA leaderboard ... some of it helped, of course, by yesterday's throwing errors, but the pitchers might start thinking of suing for non-support.
This is your "just win, you assholes" game threat. No Gurriel or Stroman to bean spectators behind 1st base with balls gone awry, so you guys got no excuses. Just win.


text:
COLLECTION OVERVIEW - ΕΠΙΣΚΟΠΗΣΗ ΣΥΛΛΟΓΗΣ
Collection Number: GR ASCSA GENN REC
Name(s) of Creator(s): American School of Classical Studies at Athens
Title: ASCSA Gennadius Library Administrative Records
Date [inclusive]: 1923 – present
Language(s): English, Greek, German, French
Summary: This body of records documents the history, organization, and function of the Gennadius Library from 1922 until now.
Quantity: 41 linear meters (last measurement in March 2020)
Information about Access: Records created before 1980 are available for research, although researchers should be aware of the General Data Protection Regulation 2016/679 (GDPR), when accessing documents that may include personal data.
Cite as: American School of Classical Studies at Athens, Gennadius Library Administrative Records (Αμερικανική Σχολή Κλασικών Σπουδών στην Αθήνα, Διοικητικό Αρχείο Γενναδείου Βιβλιοθήκης)
Notes: The collection has been processed and catalogued by Eleftheria Daleziou, 2010 - present
For more information, please contact the Archives at:
The American School of Classical Studies at Athens
54 Souidias Street, Athens 106 76, Greece
phone: +30 213 000 2400 (ext. 425)
The records document the history, organization, and function of the Gennadius Library of the American School of Classical Studies at Athens: from the announcement of John Gennadius's gift of his library to the ASCSA in 1923 and the official opening of the building in 1926 until the present day. The organization of the material has followed Helen Samuel's Varsity Letters: Documenting Modern Colleges and Universities, while respecting any pre-existing order. The Gennadius Library records have been organized into eight thematic series.
Records created before 1980 are available for research, although researchers should be aware of the General Data Protection Regulation 2016/679 (GDPR), when accessing documents that may include personal data.
A pdf version of the catalogue is available HERE.


text:
John C. Servis (born October 25, 1958 in Charles Town, West Virginia) is an American Thoroughbred racehorse trainer who was relatively unknown until May 2004 when his horse Smarty Jones won the Kentucky Derby. The colt then went on to win the Preakness Stakes further increasing Servis' reputation. Servis' Cathryn Sophia won 2016 Kentucky Oaks, winning by 2-3/4 lengths over Land Over Sea.

In 2018, Servis won his first Breeders' Cup race with Jaywalk in the Breeders' Cup Juvenile Fillies.

Servis trains horses primarily out of Parx Casino and Racing in Bensalem, Pennsylvania.

He was born into a family involved in the Thoroughbred racing industry. As a boy, he moved to Philadelphia, Pennsylvania where his father worked as a jockey.

Servis trained his 1,000th winner on May 1, 2007 at Philadelphia Park.

John is also the brother of trainer Jason Servis and brother-in-law to Florida Thoroughbred trainer, Edward Plesa.

References

1958 births
Living people
American horse trainers
People from Charles Town, West Virginia

redpajama_set_name: RedPajamaWikipedia


text:
Q: JavaScript/jQuery add innerHtml in other html page I have two html pages, in the index.html file, I have an button and an input textbox. I want the value in the textbox to show on the next page. how should i go about doing this, should i have two javascript file? or what?
Javascript:
$('#searchBtn').click(function(){

    var search = document.getElementById('searchField');
    document.getElementById("demo").innerHTML = search.value;
  });

Index.html:
<!DOCTYPE html>

        <meta charset="utf-8">
        <link rel="stylesheet" href="css/index.css" type="text/css" />
        <title>The Lantzyfi Bay</title>

            <form>

                    <input id="searchField" type="text" placeholder="Lantzify Search" name="search" size="32"/>
                    Search

                    <input type="checkbox" name="music">Music</input>
                    <input type="checkbox" name="pics">Pics</input>
                    <input type="checkbox" name="videos">Videos</input>
                    <input type="checkbox" name="other">Other</input>

            </form>

                Userpolicy
                About
                Lorem Ipsum

results.html:
<!DOCHTYPE html>

        <meta charset="utf-8">
        <link rel="stylesheet" href="css/index.css" type="text/css" />
        <link rel="stylesheet" href="css/results.css" type="text/css" />
        <title>The Lantzyfi Bay</title>

            <form id="q">

                <input id="searchField" type="text" placeholder="User serch" name="search" size="30"/>
                Search
            </form>

Serach:

                            Type
                            Name
                            DB
                            RP

                        Test
                        Test
                        Test
                        Test

                        Test
                        Test
                        Test
                        Test

                        Test
                        Test
                        Test
                        Test

                        Test
                        Test
                        Test
                        Test

                        Test
                        Test
                        Test
                        Test

A: You could add it to the URL and than read it on the other side.
Index.html would something like this:
$("#searchBtn").on('click', function () {
   var url = "results.html?q=" + $("#searchField").val();
   window.location = url
}

Result.html would have something like:
$(document).read(function () {
    $("...").html(getParameterName('q'));
}

function getParameterByName(name) {
    name = name.replace(/[\[]/, "\\[").replace(/[\]]/, "\\]");
    var regex = new RegExp("[\\?&]" + name + "=([^&#]*)"),
        results = regex.exec(location.search);
   return results === null ? "" : decodeURIComponent(results[1].replace(/\+/g, " "));
}

Credit for getParameterName function
NOTE:
You do have to note that this is not the proper way of doing things as you will have a lot of problems like url encoding, strings that are too long. This is however a good way to start. All the best.
Also I was using some jquery in the code. If this is a problem let me know and I can change it to do without

A: Only with a simple JS and HTML can not. By means of storing value and Timer (1) or SignalR (2) it could do this.
1, store value somewear ( database or cookies if just for actual user) and at some time refrest Result Page, by JS or outher, and read values.
2, SignalR Async refresh content of page, but you nead ASP.NET (MCV).
http://www.asp.net/signalr/overview/getting-started/tutorial-getting-started-with-signalr#next

A: It appears two pages are served from the same origin.

*

*localStorage is an option:
The Mozilla demo containing an example

*If the first page has a reference to the window of the 2nd page, e.g. 2nd page is opened by calling window.open. window.postMessage is also an option, it also works for cross origin.

*If the web server is also under your control, you can of course use the server as a transmitter by sending data back to server through ajax then let the server notify the 2nd page through Server Sent Event or Web Socket and etc.

redpajama_set_name: RedPajamaStackExchange


Flame Series acrylic with black and silver metal base. Affordable and elegant. 3/8" thick upright that makes an excellent employee recognition award, achievement award, sales award or customer appreciation gift. Engraving is laser etched and a velvet presentation gift drawstring bag is included.


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text:
Egypt tycoon, ex-cop ordered hanged for Lebanese diva murder
CAIRO, May 21, 2009 (AFP) - A Cairo court on Thursday sentenced an Egyptian tycoon and a former policeman to death for the murder in Dubai last year of the businessman's former lover, Lebanese pop diva Suzanne Tamim.
The judge ordered Hisham Talaat Mustafa and retired policeman Mohsen al-Sukkari hanged for respectively ordering and carrying out the brutal slaying of the singer in a luxury Dubai apartment in July.
Hisham Talaat Mustafa reacts after hearing his sentence in a Cairo court on May 21, 2009 (AFP photo)
Mustafa, a stalwart of Egypt's ruling National Democratic Party, was found guilty of paying Sukkari two million dollars to cut the throat of his former lover Tamim, 30, at the flat they had bought together shortly before.
Chaos erupted in the packed courtroom after the judge read out a short statement and ordered the men's sentences referred to the mufti for confirmation. The men also have the right to appeal.
Sukkari's face went pale and family members burst into tears as the sentence was read out, ending a case whose mix of wealth, show business and politics has gripped Egypt, where powerful businessmen are rarely seen to face justice.
Scuffles erupted inside and outside the tense courtroom as the guilty men, who sat through the case in a divided cage, were taken away.
Mustafa ordered the killing after Tamim, who had previously married two men in the music business, tied the knot with Iraqi kick-boxing champion Riyad al-Azzawi whom she met at London's renowned Harrods department store.
Sukkari followed her to Dubai to stake out Tamim's flat. He then bought a knife, went to apartment saying he worked for the building owner and killed her when she opened the door.
Mustafa, 49, was arrested in September and had his immunity lifted as member of the Shura Council, Egypt's upper house of parliament. Media reports have said he offered Tamim 50 million dollars to marry him.
He ran the Talaat Mustafa Group real estate conglomerate that is worth several billion dollars, and is said to be close to President Hosni Mubarak's son and heir apparent, Gamal.
Talaat Mustafa Group appointed Tarek Talaat Mustafa, Hisham's brother, chairman of the board and chief executive officer following his arrest.
Tamim's life had been marred by domestic disputes, including a rocky marriage with her second husband and agent who had accused her in 2004 of being behind an attempt on his life.
Egyptian media said Tamim had a relationship with Mustafa over a three-year period that ended several months before her death.
Sukkari, who worked as head of security at one of Mustafa's hotels, was arrested after Dubai police found his footprint, tracked down the shop where he bought his shoes and identified him through his credit card.
Dubai authorities informed Egyptian police who arrested Sukkari aboard a ship off the Egyptian coast.
The judge imposed a media blackout on the case after Mustafa's defence team published a book, reportedly by the tycoon himself, proclaiming his innocence.
Five journalists and editors were fined 10,000 pounds each for defying the gag order.
Ahead of the blackout, state media reported that Sukkari said Mustafa asked for Tamim's severed head to be delivered before he would pay for the killing.