Global Classical Solutions To The Incompressible Allen-Cahn-Navier-Stokes Equations With Small Initial Data

We study the global well-posedness of Allen-Cahn-Navier-Stokes equations,a diffuse-interface model for two-phase incompressible flows with different densities.We first prove the local-in-time existence of classical solutions with finite initial energy.The key point is to carefully design an energy norm.The derivative f?(?) of the physical relevant energy density f(?) produces a damping effect near equilibrium ?=±1.We thereby establish the unique global classical solution near equilibrium under small size of initial energy.The paper consists of four parts.The first part gives a detailed introduction to Allen-Cahn-Navier-Stokes equations,we can understand the background and characteristics of the equations.Then we gather the all notations and conventions used throughout this paper,and state the main results of the paper,namely,Local well-posedness and global well-posedness of the Allen-Cahn-Navier-Stokes equations in the sense of classical solutions.At the same time,we also analyzed the difficulties encountered in the research process and determined the basic research methods.The second part gives the a priori estimate of Allen-Cahn-Navier-Stokes equations.In this section we gives the local energy estimate of the equations.It is worth mentioning here that we design the energy functional,which makes the calculation of the energy estimate a certain simplification.The third part of this paper is that we prove the local well-posedness of the Allen-Cahn-Navier-Stokes equations in the sense of classical solutions.We construct the linear approximate system by iteration.Then,we prove the existence of the uniform positive time lower bound to the iterative approximate system and thereby the uniform energy bound will hold.Finally,by compactness arguments,we justify the local existence results.The last part we proves the global well-posedness of the Allen-Cahn-Navier-Stokes equations in the sense of classical solution.In this section we consider the global well-posedness near the equilibrium(0,±1).Then we give the global energy estimation and the uniform energy bound,which proves the existence of the classical solution of the equations under small size of initial energy.