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

TL;DR

This paper constructs an infinite family of Lyapunov functionals for the thin film Muskat problem, enabling the proof of global bounded solutions in one dimension for a complex degenerate parabolic system.

Contribution

It introduces a novel infinite family of convex polynomial Lyapunov functionals tailored for the thin film Muskat problem, facilitating the analysis of solutions.

Findings

Existence of global bounded non-negative weak solutions in one dimension.

Construction of an infinite family of Lyapunov functionals for the problem.

Establishment of convex polynomial Lyapunov functionals for each degree n ≥ 2.

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 $\ge$ 2 we construct an homogeneous polynomial of degree n, which is convex on [0, $\infty$)^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.