Global existence for the confined Muskat problem

2013/03/07 by Rafael Granero-Belinchón, Granero-Belinchón, Rafael
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1303.1769

openalex publication_date 2013/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we show global existence of Lipschitz continuous solution for the stable Muskat problem with finite depth (confined) and initial data satisfying some smallness conditions relating the amplitude, the slope and the depth. The cornerstone of the argument is that, for these small initial data, both the amplitude and the slope remain uniformly bounded for all positive times. We notice that, for some of these solutions, the slope can grow but it remains bounded. This is very different from the infinite deep case, where the slope of the solutions satisfy a maximum principle. Our work generalizes a previous result where the depth is infinite.

Citations

• Darcy's flow with prescribed contact angle -- Well-posedness and\n lubrication approximation
• Local solvability and turning for the inhomogeneous Muskat problem

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• Global well-posedness of arbitrarily large Lipschitz solutions for the Muskat problem with surface tension
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