Dynamic Transitions for Quasilinear Systems and Cahn-Hilliard equation with Onsager mobility

TL;DR

This paper investigates how nonlinear Onsager mobility influences phase transitions and well-posedness in the Cahn-Hilliard equation, introducing a systematic approach for quasilinear PDEs based on dynamic transition theory.

Contribution

It demonstrates the independence of dynamic transition from mobility nonlinearity and develops a new method for analyzing phase transitions in quasilinear systems.

Findings

Dynamic transition is unaffected by mobility nonlinearity.

Nonlinearity complicates well-posedness and transition analysis.

Introduces a systematic approach for quasilinear phase transition problems.

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 equation, following the ideas of the dynamic transition theory developed recently by Ma and Wang.