Well-posedness in critical spaces for barotropic viscous fluids

TL;DR

This paper proves well-posedness for large initial data of viscous compressible barotropic fluids in critical Besov spaces, introducing new estimates that allow for more general initial densities and establishing uniqueness without gradient assumptions.

Contribution

It extends the well-posedness theory to larger data with critical regularity, removing previous smallness and gradient constraints on the initial density.

Findings

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

Developed a new a priori estimate for the velocity to decouple density and velocity.

Achieved 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 sole additional assumption is that %the initial density be bounded away from zero. This improves the analysis of \cite{DL} where the smallness of $\rho_{0} %-\bar{\rho}$ for some positive constant $\bar{\rho}$ was needed. Our result improve the analysis of R. Danchin 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 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.