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 describing a diffuse interface model for two-phase flow of viscous incompressible fluids with different densities in a bounded domain Ω C RopfN (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 ρo has a positive lower bound.