The Attractor For Viscous Cahn-Hilliard Equation

In the paper, the existence problem of attractor of the viscous Cahn-Hilliard equation is studied. Long time behavior of the solution of viscous Cahn-Hilliard equation is explored. The existence of global attractor for viscous Cahn-Hilliard equation is proved in the case of autonomous systems. While, uniform attractor of Cahn-Hilliard equation and viscous Cahn-Hilliard equation is proved in the case of nonautonomous systems respectively.The results of research in this paper are as follows. In the first part, the existence of global attractor for viscous Cahn-Hilliard equation in the space of L2 is proved by the new methods of compactness, the dimension of the global attractor is estimated, and the global attractor has finite dimension. In the second part, the global attractor of the viscous Cahn-Hilliard equation in space of H1 and H2 is studied, First of all, bounded absorbing set in the space of H1 and H2 is obtained respectively, whereby global attractor in the space of H1 is proved, The method obtaining the compactness is applied to prove the existence of global attractor in the space of H2. In the third part, the long time behavior of dynamic for non-autonomous Cahn-Hilliard equation is discussed, the existence of uniform attractor for the equation is proved. The process family with two parameters is adopted to describe the uniform attractor, by the method of constructing uniform attractor in extension space, with the skills of energy inequality, uniform attractor for Cahn-Hilliard equation is proved in the space of L2. In the fourth part, the existence of uniform attractor for Cahn-Hilliard equation is studied. Uniform attractor of the viscous Cahn-Hilliard equation is proved in the space of L2×H1.