Finite time blow up of compressible Navier-Stokes equations on half space or outside a fixed ball

Abstract

In this paper, we consider the initial-boundary value problem to the compressible Navier-Stokes equations for ideal gases without heat conduction in the half space or outside a fixed ball in ${\mathbb{R}}^N$, with $N\ge 1$. We prove that any classical solutions $(\rho ,u,\theta )$, in the class $C^1([0,T];H^m(\Omega ))$, $m>[\frac{N}{2}]+2$, with bounded from below initial entropy and compactly supported initial density, which allows to touch the physical boundary, must blow-up in finite time, as long as the initial mass is positive. This paper extends the classical reault by Xin [CPAM, 1998], in which the Cauchy probelm is considered, to the case that with physical boundary.