We study the Functionalized Cahn-Hilliard Energy (FCH), which is a higher-order reformulation of the Cahn-Hilliard energy, as a model for network formation in polymer-solvent mixtures. The model affords a finite interfacial width, accommodates merging and other topological reorganization, and couples naturally to momentum balance and other macroscopic mass transport equations.The corresponding constrained L2 gradient flow has a rich family of approximately steady-state solutions that include not only the single-layer heteroclinic front profile seen in gradient flows of the Cahn-Hilliard energy, but also a novel one parameter family of homoclinic bi-layer solutions. In this thesis we rigorously derive the geometric evolution of the single-layer polymer-solvent interface.We form a manifold of quasi-equilbria by "dressing" a large family of co-dimension one interfaces immersed in Rd with heteroclinic solutions of a one-dimensional equilibrium equation derived from the first variation of the FCH energy. We show that solutions of the gradient flow that start sufficiently close to the manifold remain close, and moreover the flow can be decomposed, at leading order, as a normal velocity for the underlying co-dimension one interface. Assuming the smoothness of the interface under this flow, we develop rigorous estimates on the proximity of the true solution to the manifold, in an appropriate norm, for long time.
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