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

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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)=aργ. 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 ⁿ turn out to be smooth as long as the density remains bounded in L∞(L⁽ⁿ⁺¹⁺⁾γ) with >0 arbitrary small.