Paper detail

Manifolds of Amphiphilic Bilayers: Stability up to the Boundary

We consider the mass preserving $L^2$-gradient flow of the strong scaling of the functionalized Cahn Hilliard gradient flow and establish the nonlinear stability of a manifold comprised of quasi-equilibrium bilayer \muckmucks up to the manifold's boundary. In the limit of thin but non-zero interfacial width, $\varepsilon\ll1,$ the bilayer manifold is parameterized by meandering modes that describe the interfacial evolution and "pearling" modes that control the structure of the profile near the interface. The pearling modes are weakly damped and can lead to the dynamic rupture of the interface. Amphiphilic interfaces can lengthen to decrease energy. We introduce an implicitly defined parameterization of the interfacial shape that uncouples this growth from the parameters describing the shape and introduce a nonlinear projection onto the manifold from a surrounding neighborhood. The bilayer manifold has asymptotically large but finite dimension tuned to maximize normal coercivity while preserving the wave-number gap between the meandering and the pearling modes. Modulo a pearling stability assumption, we show that the manifold attracts nearby orbits into a tubular neighborhood about itself so long as the interfacial shape remains sufficiently smooth and far from self-intersection. In a companion paper, arXiv:1907.02196, we identify open sets of initial data whose orbits converge to circular equilibrium after a significant transient, and derive a singularly perturbed interfacial evolution comprised of motion against curvature regularized by an asymptotically weak Willmore term.