Existence and uniqueness of weak solutions of the compressible spherically symmetric Navier-Stokes equations
One of the most influential fundamental tools in harmonic analysis is Riesz transform. It maps $L^p$ functions to $L^p$ functions for any $p\in (1,\infty)$ which plays an important role in singular operators. As an application in fluid dynamics, the norm equivalence between $\|\nabla u\|_{L^p}$ and $\|\mbox{div} u\|_{L^p}+\|\mbox{curl} u\|_{L^p}$ is well established for $p\in (1,\infty)$. However, since Riesz operators sent bounded functions only to BMO functions, there is no hope to bound $\|\nabla u\|_{L^\infty}$ in terms of $\|\mbox{div} u\|_{L^\infty}+\|\mbox{curl} u\|_{L^\infty}$. As pointed out by Hoff[{\it SIAM J. Math. Anal.} {\bf 37}(2006), No. 6, 1742-1760], this is the main obstacle to obtain uniqueness of weak solutions for isentropic compressible flows. Fortunately, based on new observations, see Lemma \ref{Riesz}, we derive an exact estimate for $\|\nabla u\|_{L^\infty}\le (2+1/N)\|\mbox{div} u\|_{L^\infty}$ for any N-dimensional radially symmetric vector functions $u$. As a direct application, we give an affirmative answer to the open problem of uniqueness of some weak solutions to the compressible spherically symmetric flows in a bounded ball.