Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem

2021/03/03 by Thomas Alazard, Quoc-Hung Nguyen, Alazard, Thomas +1 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.2103.02474

38 pages. This article is based on and continues the authors' works from arXiv:2009.08442 and arXiv:2010.06915

arxiv created 2021/03/03 · openalex publication_date 2021/03/03 · arxiv updated 2021/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We exhibit a new decomposition of the nonlinearity for the Muskat equation and use it to commute Fourier multipliers with the equation. This allows to study solutions with critical regularity. As a corollary, we obtain the first well-posedness result for arbitrary large data in the critical space H²(ℝ²)∩ W^1,∞(ℝ²). Moreover, we prove the existence of solutions for initial data which are not Lipschitz.

Citations

• On the Cauchy problem for the Muskat equation. II: Critical initial data
• Endpoint Sobolev theory for the Muskat equation

Cited by

• Self-similar solutions of the three-dimensional Muskat problem with surface tension

Related

• On the Cauchy problem for the Muskat equation. II: Critical initial data
• Endpoint Sobolev theory for the Muskat equation
• On the Cauchy problem for the Muskat equation with non-Lipschitz initial data
• The Muskat problem with a large slope
• Global wellposedness for the 3D Muskat problem with medium size slope