Well-posedness of the Langmuir film problem

TL;DR

This paper establishes the local well-posedness of the inviscid Langmuir film problem by reformulating it as a boundary integral equation, analyzing the Dirichlet-to-Neumann operator, and introducing a finite-element scheme for simulations.

Contribution

It provides a rigorous mathematical analysis of the Langmuir film problem, including the explicit representation of the DtN operator, a boundary integral formulation, and a stable numerical scheme.

Findings

Perimeter decreases monotonically over time.

Explicit Fourier multiplier for the DtN operator identified.

Finite-element scheme captures relaxation dynamics.

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.