Well-posedness of the Muskat problem with H² initial data

2014/12/24 by C. H. Arthur Cheng, Cheng, C. H. Arthur, Rafael Granero-Belinchón +3 · 2 citations
Mathematics · #35Q35 #35R35 #35S10 #76B03 #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.1412.7737

openalex publication_date 2014/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the dynamics of the interface between two incompressible fluids in a two-dimensional porous medium whose flow is modeled by the Muskat equations. For the two-phase Muskat problem, we establish global well-posedness and decay to equilibrium for small H² perturbations of the rest state. For the one-phase Muskat problem, we prove local well-posedness for H² initial data of arbitrary size. Finally, we show that solutions to the Muskat equations instantaneously become infinitely smooth.

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