On the global existence for the Muskat problem

TL;DR

This paper establishes new maximum principles and proves global existence of Lipschitz and strong solutions for the Muskat problem, with explicit bounds on initial data, advancing understanding of interface dynamics between fluids.

Contribution

It introduces a novel log conservation law, proves global existence for Lipschitz solutions with explicit initial bounds, and establishes uniqueness under small initial data conditions.

Findings

Proves an $L^2$ maximum principle via a new log conservation law.

Demonstrates global existence of Lipschitz solutions with bounded initial data.

Establishes global uniqueness for solutions with explicitly small initial data.

Abstract

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2(\R)$ maximum principle, in the form of a new ``log'' conservation law \eqref{ln} which is satisfied by the equation \eqref{ec1d} for the interface. Our second result is a proof of global existence of Lipschitz continuous solutions for initial data that satisfy $\|f_0\|_{L^\infty}<\infty$ and $\|\partial_x f_0\|_{L^\infty}<1$. We take advantage of the fact that the bound $\|\partial_x f_0\|_{L^\infty}<1$ is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law. Lastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance $\| f\|_1 \le 1/5$. Previous results of this sort used a small constant $\epsilon \ll1$ which was not explicit.