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The Cahn--Hilliard equation is a fundamental model that describes phaseseparation processes of binary mixtures. In recent years, several types ofdynamic boundary conditions have been proposed in order to account for possibleshort-range interactions of the material with the solid wall. Our first aim inthis paper is to propose a new class of dynamic boundary conditions for theCahn--Hilliard equation in a rather general setting. The derivation is based onan energetic variational approach that combines the least action principle andOnsager's principle of maximum energy dissipation. One feature of our model isthat it naturally fulfills three important physical constraints such asconservation of mass, dissipation of energy and force balance relations. Next,we provide a comprehensive analysis of the resulting system of partialdifferential equations. Under suitable assumptions, we prove the existence anduniqueness of global weak/strong solutions to the initial boundary valueproblem with or without surface diffusion. Furthermore, we establish theuniqueness of asymptotic limit as $t\to+\infty$ and characterize the stabilityof local energy minimizers for the system.

An Energetic Variational Approach for the Cahn--Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis