Global well-posedness and L2 decay estimate of smooth solutions for 3-D incompressible two-phase flows

Abstract

In this paper, we consider the following Cauchy problem for the 3-D incompressible two-phase flows model{ut+(u⋅∇)u+∇p=μΔu−∇χΔχ,x∈R3,t>0,divu=0,χt+u⋅∇χ−Δχ=0,u|t=0=u0,χ|t=0=χ0. We prove that the Cauchy problem has global weak solutions by means of Galerkin approximation method. Under the assumptions of small initial data, we prove that there exists a global smooth solution by applying the classical energy estimates methods. Moreover, we show the large time behavior of smooth solution to the Cauchy problem through using the Fourier-splitting method.