The Maximum Estimate For Solutions Of Two Variants Of Cahn-Hilliard Equation

In this thesis,the existence and the maximum estimate for the solutions of the Cahn-Hilliard equation with a proliferation term is concerned.Meanwhile,the maximum estimate and the norm of^-1andestimate for solutions of the Viscous Cahn–Hilliard equation with a nonlocal term is also concerned.The thesis is divided into three chapters,and the details are as follows:In Chapter 1,the background and current research status of the two types of equations are introduced,and the related concepts and some main lemmas and inequalities used in this thesis are given.In Chapter 2,the Cahn-Hilliard equation with a proliferation term is studied.Firstly,the existence results of the solutions are obtained by using Leray-Schauder fixed point theorem.Then,the maximum estimate of the solutions are obtained by an iterative technique based on the Gagliarda-Nirenberg inequality which is developed by Bates P.W.in[Bates P W.On some nonlocal evolution equations arising in materials science[M]//.Fields Institute Commu-nications.Nonlinear dynamics and evolution equations.Providence:American Mathematical Society,2006:13-52.].In Chapter 3,some norm estimates for solutions of the viscous Cahn-Hilliard equation with a non-local term is obtained.Firstly,as in Chapter 2,the maximum estimate of the so-lutions are obtained by an iterative technique based on the Gagliarda-Nirenberg inequality.Then,the^-1norm estimate and thenorm estimate of the solutions are obtained by the uniform Gronwall inequality.