Abstract (EN)

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 γ > 1 when N = 2 , 3 and P (ρ ) = a ρ γ. In a second part we prove a condition of blow-up in slightly subcritical initial data when ρ ∈ L ∞. We finish by proving that weak solutions in TN turn out to be smooth as long as the density remains bounded in L ∞(L (N +1+ǫ )γ ) with ǫ > 0 arbitrary small.