Résumé en anglais

Regularity of weak solutions of the compressible isentropic Navier-Stokes equations is proven for small time in dimension N = 2 or 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 τ > 1 when N = 2 and p() = aτ. Moreover,weak solutions in T2turn out to be smooth as long as the density remains bounded in L∞( T2).