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 RN, with N≥1. We prove that any classical solutions (ρ,u,θ), in the class C1([0,T];Hm(Ω)), m>[N2]+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 result by Xin (1998) [19], in which the Cauchy problem is considered, to the case that with physical boundary.