Non-existence of global classical solutions to barotropic compressible Navier-Stokes equations with degenerate viscosity and vacuum

TL;DR

This paper proves that classical solutions to certain barotropic compressible Navier-Stokes equations with degenerate viscosity and vacuum conditions will blow up, highlighting limitations in solution existence under these circumstances.

Contribution

Introduces a new viscosity condition and demonstrates finite-time blow-up of classical solutions with initial vacuum and isolated mass groups.

Findings

Classical solutions blow up under specified conditions.

A new viscosity condition is proposed.

Results apply to bounded domains with initial vacuum.

Abstract

We are concerned about the barotropic compressible Navier-Stokes equations with density-dependent viscosities which may degenerate in vacuum. We show that any classical solution to barotropic compressible Navier-Stokes equations in bounded domains will blow up, when the initial density admits an isolated mass group and the viscousity coefficients satisfy some conditions. A new condition on viscosities is first put forward in this paper.