On the well-posedness of the compressible Navier-Stokes equations

TL;DR

This paper establishes optimal local well-posedness for the compressible Navier-Stokes equations in critical Besov spaces, proving the continuity of solution maps and bridging Eulerian and Lagrangian frameworks.

Contribution

It introduces a new difference estimate and demonstrates the continuity of solution maps, advancing the understanding of well-posedness in critical Besov spaces for these equations.

Findings

Proved optimal local well-posedness in critical Besov spaces.

Established continuity of solution maps from initial data to solutions.

Bridged Eulerian and Lagrangian methods via a continuous bijection.

Abstract

We consider the Cauchy problem to the barotropic compressible Navier-Stokes equations. We obtain optimal local well-posedness in the sense of Hadamard in the critical Besov space $\mathbb{X}_p=\dot{B}_{p,1}^{\frac{d}{p}}\times \dot{B}_{p,1}^{-1+\frac{d}{p}}$ for $1\leq p<2d$ with $d\geq2$. The main new result is the continuity of the solution maps from $\mathbb{X}_p$ to $C([0,T]: \mathbb{X}_p)$, which was not proved in previous works \cite{D2001, D2005, D2014}. To prove our results, we derive a new difference estimate in $L_t^1L_x^\infty$. Then we combine the method of frequency envelope (see \cite{Tao04}) but in the transport-parabolic setting and the Lagrangian approach for the compressible Navier-Stokes equations (see \cite{D2014}). As a by-product, the Lagrangian transform $(a,u)\to (\bar a, \bar u)=(a\circ X, u\circ X)$ used in \cite{D2014} is a continuous bijection and hence bridges the Eulerian and Lagrangian methods.