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

Abstract

This paper is concerned with a coupled Navier-Stokes/Allen-Cahn system describinga diffuse interface model for two-phase flow of viscous incompressible fluidswith 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 finitetime in terms of the temporal integral of both the maximum norm of the deformation tensor ofvelocity 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.