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We present a rigorous analysis of the transient evolution of nearly circular bilayer interfaces evolving under the thin interface limit, ε ≪ 1, of the mass preserving L2-gradient flow of the strong scaling of the functionalized Cahn–Hilliard equation. For a domain Ω ⊂ R2 we construct a bilayer manifold with boundary comprised of quasi-equilibria of the flow and a projection onto the manifold that associates functions u in an H2 tubular neighborhood of the manifold with an interface Γ embedded in Ω. The linearization of the flow about the manifold does not present a clear spectral separation of modes normal and tangential to the manifold. The dimension of the parameterization of the interfaces and the bilayer manifold controls both the normal coercivity of the manifold and the coupling between normal and tangential modes, both of which increase with this dimension. The key step in the analysis is the identification of a range of dimensions in which coercivity dominates the coupling, permitting the closure of the nonlinear estimates that establish the asymptotic stability of the manifold. Orbits originating in a thin, forward invariant, tubular neighborhood ultimately converge to an equilibrium associated to a circular interface. Projections of these orbits yield interfacial evolution equivalent at leading order to the regularized curve-lengthening motion characterized by normal motion against mean curvature, regularized by a higher order Willmore expression. The curve lengthening is driven by absorption of excess mass from the regions of Ω away from the interface, leading to high dimensional dynamics that are ill-posed in the ε → 0+ limit. 

Experiments with diblock co-polymer melts display undulated bilayers that emanate from defects such as triple junctions and endcaps, [8]. Undulated bilayers are characterized by oscillatory perturbations of the bilayer width, which decay on a spatial length scale that is long compared to the bilayer width. We mimic defects within the functionalized Cahn-Hillard free energy by introducing spatially localized inhomogeneities within its parameters. For length parameter \begin{document}$ \varepsilon\ll1 $\end{document}, we show that this induces undulated bilayer solutions whose width perturbations decay on an \begin{document}$ O\!\left( \varepsilon^{-1/2}\right) $\end{document} inner length scale that is long in comparison to the \begin{document}$ O(1) $\end{document} scale that characterizes the bilayer width.

 

Abstract Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error $$\sigma $$ σ in the limit of small length scale parameter $$\epsilon \rightarrow 0$$ ϵ → 0 during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen–Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity.