A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions

Yingshan  Chen; Mei  Zhang

doi:10.3934/krm.2016001

Article Contents

2016, Volume 9, Issue 3: 429-441. Doi: 10.3934/krm.2016001

This issue Previous Article Next Article Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain

A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions

Yingshan Chen^1, and
Mei Zhang^1,

1.
Department of Mathematics, South China University of Technology, Guangzhou 510641

Received: October 31, 2015

Revised: November 30, 2015

Published: August 2016

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Abstract

In this paper, we establish a new blowup criterion for the strong solutions in a smooth bounded domain $\Omega\subset\mathbb{R}^3$. In [13], Wen, Yao, and Zhu prove that if the strong solutions blow up at finite time $T^*$, the mass in $L^\infty(\Omega)$ norm must concentrate at $T^*$. Here we extend Wen, Yao, and Zhu's work in the sense of the concentration of mass in $BMO(\Omega)$ norm at $T^*$. The method can be applied to study the blow-up criterion in terms of the concentration of density in $BMO(\Omega)$ norm for the strong solutions to compressible Navier-Stokes equations in smooth bounded domains. Therefore, as a byproduct, we can also improves the corresponding result about Navier-Stokes equations in [11]. Moreover, the appearance of vacuum is allowed in the paper.

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Mathematics Subject Classification: 76T10, 76N10, 35L65.

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