2011/11/10 by Boris Haspot, Haspot, Boris
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP
paper · pdf · doi:10.48550/arxiv.1111.2464
arxiv created 2011/11/10 · arxiv updated 2011/11/11
Regularity and uniqueness of weak solutions of the compressible barotropic Navier-Stokes equations with constant viscosity coefficients 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 P(ρ)=aργ with γ>1 (in dimension three, additional conditions of size will be ask on γ). The second part of the paper is devoted to blow-up criteria for slightly subcritical initial data for the scaling of the equations when the viscosity coefficients (μ,λ) are assumed constant provided that their ratio is large enough (in particular 0<λ<(5/4)μ). More precisely we prove that under the condition ρ belongs to L∞((0,T)×\TN) then we can extend the unique solution beyond T>0. Finally, we prove that weak solutions in the torus \mathbbTN turn out to be smooth as long as the density remains bounded in L∞(0,T,L(N+1+\e)γ(\mathbbTN)) with \e>0 arbitrary small. This result may be considered as a Prodi-Serrin theorem (see \citeprodi and \citeserrin) for compressible Navier-Stokes system.