A new blowup criterion for the 3D barotropic compressible Navier-Stokes equations with vacuum

TL;DR

This paper establishes new blowup criteria for 3D barotropic compressible Navier-Stokes equations with vacuum, extending classical incompressible results and covering various boundary conditions.

Contribution

It proves global existence of solutions under bounded vorticity and divergence norms, extending the Beale-Kato-Majda criterion to compressible flows with vacuum.

Findings

Global solutions exist if vorticity and divergence norms are bounded.

Extends Beale-Kato-Majda criterion to compressible Navier-Stokes equations.

Includes vacuum states and different boundary conditions.

Abstract

We investigate the blowup criterion of the barotropic compressible viscous fluids for the Cauchy problem, Dirichlet problem and Navier-slip boundary condition. The main novelty of this paper is two-fold: First, for the Cauchy problem and Dirichlet problem, we prove that a strong or smooth solution exists globally, provided that the vorticity of velocity satisfies Serrin's condition and the maximum norm of the divergence of the velocity is bounded. Second, for the Navier-slip boundary condition, we show that if both the maximum norm of the vorticity of velocity and the maximum norm of the divergence of velocity are bounded, then the solution exists globally. In particular, this criterion extends the well-known Beale-Kato-Majda's blowup criterion for the 3D incompressible Euler equations (Comm. Math. Phys. 94(1984):61-66) to the 3D barotropic compressible Navier-Stokes equations, and can be regarded as a complement for the work by Huang-Li-Xin (Comm. Math. Phys. 301(2011):23-35). The vacuum is allowed to exist here.