Bounded weak solutions to the thin film Muskat problem via an infinite family of Liapunov functionals

2021/10/04 by Philippe Laurençot, Laurençot, Philippe, Bogdan–Vasile Matioc +2 · 2 citations
Engineering · Materials Science · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Thin Films #Nonlinear Partial Differential Equations #Solidification and crystal growth phenomena #math.AP

paper · pdf · doi:10.48550/arxiv.2110.01234

arxiv created 2021/10/04 · openalex publication_date 2021/10/04 · arxiv updated 2021/10/05 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

A countably infinite family of Liapunov functionals is constructed for the thin film Muskat problem, which is a second-order degenerate parabolic system featuring cross-diffusion. More precisely, for each n ≥ 2 we construct an homogeneous polynomial of degree n, which is convex on [0, ∞)² , with the property that its integral is a Liapunov functional for the problem. Existence of global bounded non-negative weak solutions is then shown in one space dimension.

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