Regularized Curve Lengthening from the Strong FCH Flow

Abstract

We present a rigorous analysis of the transient evolution of nearly circular bilayer interfaces evolving under the thin interface limit, $\varepsilon\ll1$, of the mass preserving $L^2$-gradient flow of the strong scaling of the functionalized Cahn-Hilliard equation. For a domain $\Omega\subset{\mathbb R}^2$ we construct a bilayer manifold with boundary comprised of quasi-equilibrium of the flow and a projection onto the manifold that associates functions $u$ in an $H^2$ tubular neighborhood of the manifold with an interface $\Gamma$ embedded in $\Omega$. These interfaces, and hence the bilayer manifold, are parameterized by a finite but asymptotically large number of degrees of freedom. The manifold contains a unique, up to translation and mass constraint, equilibrium of the gradient flow whose projected interface is circular up to exponentially small corrections. The thin tubular neighborhood is forward invariant under the flow with orbits that ultimately converge to the equilibrium. Projections of these orbits yield an interfacial evolution equivalent at leading order to the regularized curve-lengthening motion characterized by normal motion {\sl against} mean curvature, regularized by a higher order Willmore expression. The curve lengthening is driven by absorption of excess mass from the regions of $\Omega$ away from the interface, generically leading to nontrivial dynamics that are ill-posed in the $\varepsilon\to0$ limit.