Endpoint Sobolev theory for the Muskat equation

TL;DR

This paper establishes the well-posedness of the 2D Muskat equation at a critical Sobolev regularity, using novel weighted fractional Laplacian techniques and identifying a null structure to handle degeneracy.

Contribution

It proves the optimal well-posedness in the endpoint Sobolev space for the Muskat equation and introduces a new approach using weighted fractional Laplacians and null structures.

Findings

Well-posedness at critical Sobolev regularity $L^2$ with three-half derivatives.

Introduction of weighted fractional Laplacians for solution estimates.

First identification of a null structure in the Muskat equation.

Abstract

This paper is devoted to the study of solutions with critical regularity for the two-dimensional Muskat equation. We prove that the Cauchy problem is well-posed on the endpoint Sobolev space of $L^2$ functions with three-half derivative in $L^2$. This result is optimal with respect to the scaling of the equation. One well-known difficulty is that one cannot define a flow map such that the lifespan is bounded from below on bounded subsets of this critical Sobolev space. To overcome this, we estimate the solutions for a norm which depends on the initial data themselves, using the weighted fractional Laplacians introduced in our previous works. Our proof is the first in which a null-type structure is identified for the Muskat equation, allowing to compensate for the degeneracy of the parabolic behavior for large slopes.