Dynamic transitions for quasilinear systems and Cahn-Hilliard equation with Onsager mobility

Abstract

The main objectives of this article are two-fold. First, we study the effect of the nonlinear Onsager mobility on the phase transition and on the well-posedness of the Cahn-Hilliard equation modeling a binary system. It is shown in particular that the dynamic transition is essentially independent of the nonlinearity of the Onsager mobility. However, the nonlinearity of the mobility does cause substantial technical difficulty for the well-posedness and for carrying out the dynamic transition analysis. For this reason, as a second objective, we introduce a systematic approach to deal with phase transition problems modeled by quasilinear partial differential equations, following the ideas of the dynamic transition theory developed in Ma and Wang [Phase Transition Dynamics in Nonlinear Sciences (Springer) (to be published); Bifurcation Theory and Applications, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises Vol. 53 (World Scientific, Hackensack, NJ, 2005)].