The Muskat equation is well-posed on the critical Sobolev space

Abstract: This talk is about a series of papers with Thomas Alazard, devoted to the study of solutions with critical regularity for the two-dimensional Muskat equation. I will describe our main result, which states that the Cauchy problem is well-posed on the endpoint Sobolev space of L^2 functions with three-half derivative in L^2 (locally in time for large data, and globally for small enough data). This result is optimal with respect to the scaling of the equation. For the proof, we introduce weighted fractional laplacians and use these operators to estimate the solutions for a norm which depends on the initial data themselves. Another key ingredient of the proof is a null-type structure, allowing to compensate for the degeneracy of the parabolic behavior for large slopes.

Mathematics

Audience: researchers in the topic

ICCM 2020