We investigate regularity and well-posedness for a fluid evolution model in the presence of a three-phase contact point. We consider a fluid evolution governed by Darcy's Law. After linearization, we obtain a nonlocal third order operator which contains the Dirichlet-Neumann operator on the wedge with opening angle ∈ > 0. We show well-posedness and regularity for this linear evolution equation. In the limit of vanishing opening angle, we show the convergence of solutions to a fourth order degenerate parabolic operator, related to the thin-film equation. In the course of the analysis, we introduce and characterize a new type of sum of weighted Sobolev spaces which are suitable to capture the singular limit as ∈ → 0. In particular, the nature of the problem requires the use of techniques that are adapted to the problem in the singular domain as well as the degenerate limit problem.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics