Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities

Yinghua  Li; Shijin Ding; Mingxia  Huang

doi:10.3934/dcdsb.2016009

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2016, Volume 21, Issue 5: 1507-1523. Doi: 10.3934/dcdsb.2016009

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Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities

1.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631
2.
Department of Mathematics, South China Normal University, Guangzhou, Guangdong 510631

Received: February 28, 2013

Revised: August 31, 2013

Published: June 2016

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Abstract

This paper is concerned with a coupled Navier-Stokes/Allen-Cahn system describing a diffuse interface model for two-phase flow of viscous incompressible fluids with different densities in a bounded domain $\Omega\subset\mathbb R^N$($N=2,3$). We establish a criterion for possible break down of such solutions at finite time in terms of the temporal integral of both the maximum norm of the deformation tensor of velocity gradient and the square of maximum norm of gradient of phase field variable in 2D. In 3D, the temporal integral of the square of maximum norm of velocity is also needed. Here, we suppose the initial density function $\rho_0$ has a positive lower bound.

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Mathematics Subject Classification: Primary: 76T10, 35Q30; Secondary: 35B44.

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