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

## TL;DR

This paper proves that classical solutions to the compressible Navier-Stokes equations with certain initial conditions on a half space or outside a ball must blow up in finite time, extending previous results to cases with physical boundaries.

## Contribution

It extends the finite-time blow-up results for compressible Navier-Stokes equations from the Cauchy problem to bounded domains with physical boundaries.

## Key findings

- Classical solutions blow up in finite time under specified conditions.
- Extension of Xin's results to bounded domains with boundary.
- Conditions include positive initial mass and bounded initial entropy.

## 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 $N\geq1$. We prove that any classical solutions $(\rho, u, \theta)$, in the class $C^1([0,T]; H^m(\Omega))$, $m>[\frac N2]+2$, 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 reault by Xin [CPAM, 1998], in which the Cauchy probelm is considered, to the case that with physical boundary.