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

Regularity of weak solutions of the compressible isentropic Navier-Stokes equations is proven for small time in dimension N = 2 of 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 and p(rho) = a rho(gamma). Moreover, weak solutions in T-2 turn out to be smooth as long as the density remains bounded in L-infinity(T-2).