Well-posedness in critical spaces for the system of Navier-Stokes compressible

TL;DR

This paper establishes well-posedness for large initial data in critical Besov spaces for the compressible Navier-Stokes equations, introducing new estimates that improve previous results and achieve uniqueness without gradient assumptions.

Contribution

It extends the well-posedness theory to more general initial densities in critical Besov spaces and introduces a novel a priori estimate that decouples density and velocity.

Findings

Proves well-posedness for large data in critical Besov spaces.

Develops a new a priori estimate for velocity that decouples density and velocity.

Achieves uniqueness without assumptions on the density gradient.

Abstract

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension $N\geq2$. We address the question of well-posedness for {\it large} data having critical Besov regularity. Our result improve the analysis of R. Danchin and of B. Haspot, by the fact that we choose initial density more general in $B^{\NN}_{p,1}$ with $1\leq p<+\infty$. Our result relies on a new a priori estimate for the velocity, where we introduce a new structure to \textit{kill} the coupling between the density and the velocity. In particular our result is the first where we obtain uniqueness without imposing hypothesis on the gradient of the density.