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 $L^{ \infty}(L^{ (N+1+\e)\gamma})$ with $\e>0$ arbitrary small.
Categories: math.AP
