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

TL;DR

This paper proves regularity and uniqueness of weak solutions to the compressible barotropic Navier-Stokes equations in 2D and 3D for small times, with new conditions on initial density and pressure law, and establishes blow-up criteria and smoothness results.

Contribution

It introduces new regularity results for weak solutions without requiring positive lower bounds on initial density and extends classical criteria to the compressible case.

Findings

Weak solutions are regular and unique for small times under certain conditions.

Blow-up criteria are established based on boundedness of density.

Weak solutions become smooth if density remains bounded in a specific Lebesgue space.

Abstract

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.