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

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 $1\le p<2d$ with $d\ge 2$. 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 $L_t^1{L}_x^{\infty }$. 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.