Existence and uniqueness of weak solutions of the compressible spherically symmetric Navier-Stokes equations

TL;DR

This paper proves the existence and uniqueness of weak solutions for the compressible spherically symmetric Navier-Stokes equations, overcoming previous obstacles related to the boundedness of the gradient of velocity.

Contribution

It introduces a new estimate for the gradient of radially symmetric vector functions, enabling the proof of uniqueness for weak solutions in this setting.

Findings

Established a new bound for bla u in terms of divergence for radially symmetric functions.

Proved the uniqueness of weak solutions for compressible spherically symmetric flows.

Resolved an open problem in the mathematical analysis of compressible fluid dynamics.

Abstract

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.