Regularity of weak solutions of the compressible Navier-Stokes equations

Abstract

In this paper, we assume a density with integrability on the space $L^{\infty}(0,T; L^{q_0})$ for some $q_0$ and $T>0$. Under the assumption on the density, we obtain a regularity result for the weak solutions to the compressible Navier-Stokes equations. That is, the supremum of the density is finite and the infimum of the density is positive in the domain $ {\mathbf T}^3\times (0,T)$. Moreover, Moser type iteration scheme is developed for $L^\infty$ norm estimate for the velocity.