On the Cauchy problem for the Muskat equation with non-Lipschitz initial data

2020/09/09 by Thomas Alazard, Quoc-Hung Nguyen, Alazard, Thomas +2 · 2 citations
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.2009.04343

41 pages

arxiv created 2020/09/09 · openalex publication_date 2020/09/09 · arxiv updated 2020/09/10 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

This article is devoted to the study of the Cauchy problem for the Muskat equation. We consider initial data belonging to the critical Sobolev space of functions with three-half derivative in L², up to a fractional logarithmic correction. As a corollary, we obtain the first local and global well-posedness results for initial free surface which are not Lipschitz.

Citations

• Regularity for a special case of two-phase Hele-Shaw flow via parabolic integro-differential equations

Cited by

• Self-similar solutions of the three-dimensional Muskat problem with surface tension
• On the Cauchy problem for the Muskat equation. II: Critical initial data

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