  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.
