Regularity of weak solutions of the compressible barotropic Navier-Stokes equations

Regularity and uniqueness of weak solutions of the compressible barotropic Navier-Stokes equations with constant viscosity coefficients 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 $P(\rho)=a\rho^{\gamma}$ with $\gamma>1$ (in dimension three, additional conditions of size will be ask on $\gamma$). The second part of the paper is devoted to blow-up criteria for slightly subcritical initial data for the scaling of the equations when the viscosity coefficients $(\mu,\lambda)$ are assumed constant provided that their ratio is large enough (in particular $0<\lambda<(5/4)\mu$). More precisely we prove that under the condition $\rho$ belongs to $L^{\infty}((0,T)\times\T^{N})$ then we can extend the unique solution beyond $T>0$. Finally, we prove that weak solutions in the torus $\mathbb{T}^{N}$ turn out to be smooth as long as the density remains bounded in $L^{\infty}(0,T,L^{(N+1+\e)\gamma}(\mathbb{T}^{N}))$ with $\e>0$ arbitrary small. This result may be considered as a Prodi-Serrin theorem (see \cite{prodi} and \cite{serrin}) for compressible Navier-Stokes system.