Local well-posedness results for density-dependent incompressible fluids

Abstract

This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in ${\mathbb{R}}^N$ with $N\ge 2$. We address the question of well-posedness for {\it large} data having critical Besov regularity and we aim at stating well-posedness in functional spaces as close as possible to the ones imposed in the incompressible Navier Stokes system by Cannone, Meyer and Planchon where $u_{0}\in B^{\NN-1}_{p,\infty}$ with $1\le p<+\infty$. This improves the analysis of H. Abidi, R. Danchin and M. Paicu where $u_0$ is considered belonging to $B^{\NN-1}_{p,1}$ with $1\le p<2N$. Our result relies on a new a priori estimate for transport equation introduce by Bahouri, Chemin and Danchin when the velocity $u$ is not considered Lipschitz.

Cited by in corpus (4)

• Regularity of weak solutions of the compressible isentropic Navier-Stokes equation
• Well-posedness in critical spaces for barotropic viscous fluids
• Regularity of weak solutions of the compressible barotropic Navier-Stokes equations
• Well-posedness in critical spaces for the system of Navier-Stokes compressible