The Cahn-Hilliard Equation as a Gradient Flow
Craig Cowan · Summit (Simon Fraser University) · 2005
Some evolution equations can be interpreted as gradient flows. Mathematically this is subtle as the flow depends on the choice of a functional and an inner product (different functionals or inner products give rise to different dynamics). The Cahn-Hilliard equation is a simple model for the process of phase separation of a binary alloy at a fixed temperature. This equation was first derived using physical principles but can also be obtained as a specific gradient flow of a free energy. Having these two viewpoints is quite common in physics and often one prefers to work with the variational formulation. For example, a variational formulation allows one to obtain many possible evolutionary models for the system. For a gradient flow, the basic idea is to start with an energy functional (F) defined on a Hilbert space. One then writes out the gradient flow associated with the functional and the Hilbert space: The above becomes an evolution equation which will be dependent on the Hilbert space. Whether it is a good model for the dynamics of the system is another question as it is not based upon any dynamic physical law (eg. a force balance law). In this thesis we will examine the above ideas focusing on the Cahn-Hilliard equation. We will develop the necessary tools from functional analysis and PDE theory.