[Paper Review] Regularized Curve Lengthening from the Strong FCH Flow

Yuan Chen, Keith Promislow|arXiv (Cornell University)|Jul 4, 2019

Fluid Dynamics and Turbulent Flows29 references4 citations

TL;DR

This paper rigorously analyzes the transient evolution of nearly circular bilayer interfaces under the strong scaling of the functionalized Cahn-Hilliard (FCH) equation in the thin interface limit (ε ≪ 1). It constructs a bilayer manifold parameterized by finite degrees of freedom, proves forward invariance of a tubular neighborhood, and shows that projected dynamics converge to equilibrium, with interfacial evolution equivalent to regularized curve-lengthening driven by mass absorption, regularized by a higher-order Willmore term.

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 $Ω\subset{\mathbb R}^2$ 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 $Γ$ embedded in $Ω$. 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 $Ω$ away from the interface, generically leading to nontrivial dynamics that are ill-posed in the $\varepsilon o0$ limit.

Motivation & Objective

• To rigorously characterize the transient dynamics of nearly circular bilayer interfaces evolving under the mass-preserving L²-gradient flow of the strong FCH functional.
• To construct a finite-dimensional bilayer manifold embedded in H²(Ω) that captures quasi-equilibrium configurations of the FCH energy.
• To establish the existence of a unique equilibrium on the manifold, up to translation and mass constraint, with a circular interface up to exponentially small corrections.
• To prove that orbits in a thin H² tubular neighborhood of the manifold converge to the equilibrium and induce interfacial evolution equivalent to regularized curve-lengthening.
• To identify the mechanism of curve lengthening as driven by absorption of excess mass from the bulk, leading to dynamics ill-posed in the ε → 0 limit.

Proposed method

• Constructs a bilayer manifold using quasi-equilibrium solutions ΦΓ ∈ H²(Ω) parameterized by a smooth embedded curve Γ ⊂ Ω and a bulk parameter σ.
• Uses the ε-scaled signed distance z(x) to Γ and the homoclinic solution φ₀ of the ODE ∂²zφ₀ = W′(φ₀) to define the leading-order bilayer profile.
• Imposes a mass constraint ∫(u − b₋)dx = εM₀ to relate σ and |Γ| via |Γ| = M₀/m₀ − |Ω|σ/(W″(b₋))²m₀, ensuring mass scaling consistent with ε.
• Reduces the FCH energy to a Canham-Helfrich-type energy E(Γ,σ) = (νₛ/2)∫Γ|κ|²ds + (ν_b/(2ε))(σ − σ₁*)², with νₛ, ν_b > 0 depending on system parameters.
• Defines a projection map from functions in an H² tubular neighborhood of the manifold to the interface Γ, enabling analysis of interfacial evolution.
• Derives the leading-order interfacial evolution as regularized curve-lengthening: normal velocity v = −κ + ε(νₛΔΓκ + νₛκ|κ|²), with regularization from a higher-order Willmore term.

Experimental results

Research questions

• RQ1 What is the structure of the quasi-equilibrium manifold of bilayer configurations in the strong FCH flow for ε ≪ 1?
• RQ2 How does the interfacial evolution behave under the mass-preserving L²-gradient flow, and what is its effective dynamics at leading order?
• RQ3 What drives the curve-lengthening phenomenon in the FCH system, and how is it regularized in the thin interface limit?
• RQ4 Why is the dynamics ill-posed in the ε → 0 limit, and how does the regularization prevent this breakdown?
• RQ5 Can the convergence of orbits in a tubular neighborhood to equilibrium be rigorously established, and what is the role of mass absorption in this process?

Key findings

• A unique equilibrium exists on the bilayer manifold, up to translation and mass constraint, with the projected interface circular up to exponentially small corrections.
• The H² tubular neighborhood of the bilayer manifold is forward invariant under the FCH flow, and all orbits converge to the equilibrium.
• The projected interfacial evolution is equivalent at leading order to regularized curve-lengthening: v = −κ + ε(νₛΔΓκ + νₛκ|κ|²), where νₛ = m₁² and m₁ = ‖φ₀′‖L²(ℝ).
• The curve lengthening is driven by absorption of excess surfactant mass from the bulk, which is not captured in the ε → 0 limit, rendering the dynamics ill-posed there.
• The regularization term arises from the Willmore-type residual in the strong scaling (p = 1), which stabilizes the dynamics and prevents blow-up in the limit.
• The equilibrium value σ₁* = −(η₁ + η₂)/(2m₀)m₁² balances surfactant absorption and ejection, corresponding to a high-energy state for dispersed surfactant.