Well-posedness in critical spaces for barotropic viscous fluids

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 $\rho_{0}$ 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

Weak-Strong uniqueness for compressible Navier-Stokes system with degenerate viscosity coefficient and vacuum in one dimension

We prove weak-strong uniqueness results for the compressible Navier-Stokes system with degenerate viscosity coefficient and with vacuum in one dimension. In other words, we give conditions on the weak solution constructed in \cite{Jiu} so that it is unique. The novelty consists in dealing with initial density $\rho_0$ which contains vacuum. To do this we use the notion of relative entropy developed recently by Germain, Feireisl et al and Mellet and Vasseur (see \cite{PG,Fei,15}) combined with a new formulation of the compressible system (\cite{cras,CPAM,CPAM1,para}) (more precisely we introduce a new effective velocity which makes the system parabolic on the density and hyperbolic on this velocity).Comment: arXiv admin note: text overlap with arXiv:1411.550

New formulation of the compressible Navier-Stokes equations and parabolicity of the density

In this paper we give a new formulation of the compressible Navier-Stokes by introducing an suitable effective velocity v=u+\n\va(\rho) provided that the viscosity coefficients verify the algebraic relation of \cite{BD}. We give in particular a very simple proof of the entropy discovered in \cite{BD}, in addition our argument show why the algebraic relation of \cite{BD} appears naturally. More precisely the system reads in a very surprising way as two parabolic equation on the density $\rho$ and the vorticity ${\rm curl}v$, and as a transport equation on the divergence ${\rm div}v$. We show the existence of strong solution with large initial data in finite time when (\rho_0-1)\in B^{\NN}_{p,1}. A remarkable feature of this solution is the regularizing effects on the density. We extend this result to the case of global strong solution with small initial data.Comment: 38 pages. arXiv admin note: text overlap with arXiv:1107.2332; and with arXiv:1302.2617 by other author

Regularity of weak solutions of the compressible isentropic Navier-Stokes equation

Regularity and uniqueness of weak solution of the compressible isentropic Navier-Stokes equations is proven for small time in dimension $N=2,3$ under periodic boundary conditions. In this paper, the initial density is not required to have a positive lower bound and the pressure law is assumed to satisfy a condition that reduces to $\gamma>1$ when $N=2,3$ and $P(\rho)=a\rho^{\gamma}$. In a second part we prove a condition of blow-up in slightly subcritical initial data when $\rho\in L^{\infty}$. We finish by proving that weak solutions in \T^{N} turn out to be smooth as long as the density remains bounded in L^{\infty}(L^{(N+1+\e)\gamma}) with \e>0 arbitrary small