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