Well-posedness of the Langmuir film problem

TL;DR

This work provides a rigorous mathematical framework for the inviscid Langmuir film coupled to a Stokes subfluid (ILLSS). It recasts the bulk–surface system as a boundary integral equation on the Langmuir film by identifying the Dirichlet-to-Neumann operator with the fractional Laplacian $(-\Delta)^{1/2}$ and constructing a fundamental solution tensor $E(x)=\frac{x\otimes x}{2\pi|x|^3}$. The authors establish a curve-shortening identity, prove local well-posedness via maximal $L^2$-regularity aided by a DeTurck-type reparametrization, and show instantaneous smoothing and equivalence with the original ILLSS model. They also develop a linearly implicit parametric finite-element scheme that captures experimentally observed relaxation dynamics, including bola-shaped to circular domain evolution with area preservation. The results provide a rigorous analytical foundation and a robust numerical framework for understanding interfacial dynamics in Langmuir films and related hierarchical fluid systems.

Abstract

We analyze the inviscid Langmuir layer--Stokesian subfluid (ILLSS) model for two-phase Langmuir monolayers coupled to a Stokes flow in the underlying subfluid. Eliminating the bulk variables, we reformulate the coupled three-dimensional system as an evolution on the film involving the Dirichlet-to-Neumann (DtN) operator. We identify the Fourier symbol of the DtN operator and show it coincides with that of the fractional Laplacian, which yields an explicit Fourier-multiplier representation and allows construction of the corresponding fundamental solution. Using this representation we express the surface velocity as a convolution of the fundamental solution with the interfacial curvature forcing and analyze its normal limit to derive a boundary integral equation for the moving curve. Independently, exploiting the DtN representation we establish a curve-shortening identity: the interfacial perimeter decreases monotonically and its time derivative is controlled by $\dot{H}^{1/2}(\mathbb{R}^2)$-norm of the surface velocity. Building on the boundary integral equation, we prove local well-posedness via maximal $L^2$-regularity for quasilinear parabolic systems, employing a DeTurck-type reparametrization, and show equivalence with the original ILLSS system. Finally, we introduce a linearly implicit parametric finite-element scheme which captures experimentally observed relaxation dynamics.

Figures (2)

• Figure 1: A schematic diagram of the ILLSS model.
• Figure 2: Numerical result with initial curve \ref{['eq:bola']}.

Theorems & Definitions (55)

• Definition 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Proposition 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof