Thin-Layer Dichotomy in Three-Phase Mullins-Sekerka Flow: Gap Dynamics and Composite Interfaces
Tokuhiro Eto
math.AP
Oct 20, 2026 · v1
TL;DR
Selected results on the fixed-mode linearized Mullins–Sekerka system were formalized in Lean 4, with AI assistants helping develop the formalization.
Abstract
We study the thin-layer limit of the Mullins-Sekerka flow in concentric circular geometries at fixed wave number $k$, contrasting a two-phase annulus with a genuine three-phase layer. A dichotomy emerges: for $k\ge2$ the rates blow up like the inverse thickness when the same phase lies on both sides of the layer, whereas between two different phases the matrix governing the linearized Fourier mode extends analytically to zero thickness. Both cases share the singular Dirichlet-to-Neumann factor of the thin annulus and differ only in whether the Gibbs-Thomson law forces an order-one potential difference across it. In the three-phase case the pure thickness-modulation rate stays below the logarithmic growth rate of the mean thickness, for every $k\ge1$ and all positive tensions, and, after projection onto the mean displacement and the outer interface, the collapsed pair moves as one interface carrying the sum of the two tensions. All statements concern the instantaneous fixed-mode linearization. The singularity is a property of the phase coupling, not of the thin geometry.
Problem
The paper studies the thin-layer limit of the linearized Mullins–Sekerka flow in concentric circular geometries at fixed wave number k. It asks whether the singular thin-annulus interaction comes from the thin geometry or from the phase coupling.
Approach
The concentric three-phase configuration is solved exactly. The flow is then linearized in a single Fourier mode, using Dirichlet-to-Neumann maps to derive an explicit 3x3 mode matrix. A resistor-form decomposition of the thin-annulus map isolates the δ^{-1} factor, and the result is compared with an exact two-phase annulus matrix. Selected results for the fixed-mode linearized system were formalized in Lean 4.
Results
For k≥2, rates blow up like δ^{-1} when the same phase lies on both sides of the layer (two-phase case). Between distinct phases (three-phase case), the mode matrix extends analytically to zero thickness. In the three-phase case the thickness-modulation rate stays below the mean-thickness growth rate, and the collapsed pair behaves as one interface carrying the summed tensions.
