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

Abstract

In this paper, we establish a new blowup criterion for the strongsolutions in a smooth bounded domain $\Omega\subset\mathbb{R}^3$. In[13], Wen, Yao, and Zhu prove that if the strongsolutions blow up at finite time $T^*$, the mass in$L^\infty(\Omega)$ norm must concentrate at $T^*$. Here we extendWen, Yao, and Zhu's work in the sense of the concentration of massin $BMO(\Omega)$ norm at $T^*$. The method can be applied to studythe 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 allowedin the paper.