Long Time Dynamics Of Implicit Euler Method For The Cahn-Hilliard Equation

The Cahn-Hilliard equation,as an important class of fourth-order diffusion equa-tions,has become a major concern in the field of partial differential equations.It was originally proposed by Cahn and Hilliard in 1958 when studying the complex phase separation and coarsening phenomenon,the phase separation occurs when two kinds of mixtures are rapidly cooled from high temperature to low temperature(quenching).From the existing relevant literature,however,there are still a considerable number of theoretical and practical problems to be solved.In this paper,we consider the initial value problem for the Cahn-Hilliard equation with Nuemann boundary,use implicit Euler method to discretize the time variables,and get the following results,1.First,we proved two important properties of this equation which still preserves the property of mass conservation and energy decay in discrete sense,2.Next,the uniformly dissipative estimate of implicit Euler scheme on spatial Hl is obtained,where,the constant ?>?0k =(2C1(?)?12+?k/?)1/2,and is an positive integer n0 =no(R,?,?,k)?3.Again,the uniformly dissipative estimate of implicit Euler scheme on space H-1 is obtained,4.Finally,the uniformly dissipative estimate of implicit Euler scheme on space H2 is obtained,...