Existence of global strong solutions in critical spaces for barotropic viscous fluids

TL;DR

This paper proves the existence of global strong solutions for viscous compressible barotropic fluids with initial data near a constant state, introducing new estimates and extending previous results to broader initial conditions and viscosity coefficients.

Contribution

It provides a new a priori estimate for velocity, improves existing results by adding initial regularity, and generalizes to variable viscosity coefficients.

Findings

Established global strong solutions for initial data close to constant states.

Introduced a new variable called effective velocity to decouple density and velocity.

Extended results to include initial density in H^1 and variable viscosity coefficients.

Abstract

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension $N\geq2$. We address the question of the global existence of strong solutions for initial data close from a constant state having critical Besov regularity. In a first time, this article show the recent results of \cite{CD} and \cite{CMZ} with a new proof. 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 as in \cite{H2}. We study so a new variable that we call effective velocity. In a second time we improve the results of \cite{CD} and \cite{CMZ} by adding some regularity on the initial data in particular $\rho_{0}$ is in $H^{1}$. In this case we obtain global strong solutions for a class of large initial data on the density and the velocity which in particular improve the results of D. Hoff in \cite{5H4}. We conclude by generalizing these results for general viscosity coefficients.