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

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

2010-01-11

arxiv.org/abs/1001.1581v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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