Endpoint Sobolev theory for the Muskat equation

2020/10/14 by Thomas Alazard, Quoc-Hung Nguyen, Alazard, Thomas +1 · 3 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.2010.06915

57 pages

arxiv created 2020/10/14 · arxiv updated 2020/10/15

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 L² functions with three-half derivative in L². 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.

Cited by

• Self-similar solutions of the three-dimensional Muskat problem with surface tension
• Existence, asymptotic behaviour and convergence of a generalised 3D Muskat problem in stable regime
• Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem

Related

• On the Cauchy problem for the Muskat equation with non-Lipschitz initial data
• Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem
• On the Cauchy problem for the Muskat equation. II: Critical initial data
• The Muskat problem with a large slope
• Well-posedness and stability results for some periodic Muskat problems