Stability and boundary layer properties of solutions of Cahn-Hilliard equations.

摘要

This dissertation consists of two parts. Both contain results concerning properties of solutions of Cahn-Hilliard equations.; In the first part, under some conditions on the initial data {dollar}usb0(x),{dollar} the solution u of the singularly perturbed (with small parameter {dollar}varepsilon>0{dollar}) Cahn-Hilliard equation with constant mobility converges to the solution of the nonlinear Cahn-Allen equation as {dollar}varepsilonrightarrow0.{dollar} The technique used here is based on a construction of the approximate solutions and on estimates on various Sobolev norms.; In the second part, the problems of solvability, uniqueness and asymptotic behavior as {dollar}trightarrowinfty{dollar} of the solution of the Cahn-Hilliard equation with general variable mobility are considered. Approximate solutions are first constructed by using a Galerkin method, and then, a priori bounds for these approximate solutions are obtained by applying appropriate embedding theorems and interpolation inequalities. Finally, compactness properties are used to pass to the limit.