Regularity And Boundary Layer Separation Of Solutions Of 2D Allen-Cahn-Navier-Stokes System

Allen-Cahn-Navier-Stokes system,the important model describing viscous fluid flow,consists of Allen-Cahn equations coupled with Navier-Stokes equations.In this thesis,regularity of steady-state solutions and global weak solutions,the conditions of boundary layer separation for 2D Allen-Cahn-Navier-Stokes system are studied.The main contents of this thesis include three parts as follows.In the first part,the existence of steady-state weak solutions of the Allen-Cahn-NavierStokes system is obtained by using acute angle principle of weakly continuous operators.Furthermore,ADN theory and linear elliptic operator theory are used to prove the regularity of the steady-state solutions.In the second part,the existence,uniqueness and regularity of global weak solutions of the Allen-Cahn-Navier-Stokes system are considered.First of all,the T-weakly continuous method is used to prove the existence of the global weak solutions of the Allen-Cahn-NavierStokes system.Secondly,the uniqueness and regularity of global weak solutions are obtained by means of weakly continuous operator theory,ADN theory and linear elliptic operator theory.In the third part,the conditions of boundary layer separation under flat and curved boundary of the Allen-Cahn-Navier-Stokes system are investigated.The conditions of boundary layer separation of the Allen-Cahn-Navier-Stokes system are gained by making use of the Taylor expansion and the geometric theory of incompressible flow.