Geometric evolution of bilayers under the functionalized Cahn—Hilliard equation

We use a multi-scale analysis to derive a sharp interface limit for the dynamics of bilayer structures of the functionalized Cahn—Hilliard equation. In contrast to analysis based on single-layer interfaces, we show that the Stefan and Mullins—Sekerka problems derived for the evolution of single-layer interfaces for the Cahn—Hilliard equation are trivial in this context, and the sharp interface limit yields a quenched mean-curvature-driven normal velocity at O(ε -1 ), whereas on the longer O(ε -2 ) time scale, it leads to a total surface area preserving Willmore flow. In particular, for space dimension n = 2, the constrained Willmore flow drives collections of spherically symmetric vesicles to a common radius, whereas for n = 3, the radii are constant, and for n ≥ 4 the largest vesicle dominates.