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

Abstract

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 $L^{\infty}(L^{(N+1+\e)γ})$ with $\e>0$ arbitrary small.

References in corpus (3)

• On the isentropic compressible Navier-Stokes equation
• Local well-posedness results for density-dependent incompressible fluids
• Well-posedness in critical spaces for barotropic viscous fluids

Cited by in corpus (1)

• A blow-up criterion for compressible viscous heat-conductive flows