Local solvability and turning for the inhomogeneous Muskat problem

2013/11/09 by Luigi C. Berselli, Diego Córdoba, Berselli, Luigi +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1311.2194

openalex publication_date 2013/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this work we study the evolution of the free boundary between two different fluids in a porous medium where the permeability is a two dimensional step function. The medium can fill the whole plane ℝ² or a bounded strip S=ℝ×(-π/2,π/2). The system is in the stable regime if the denser fluid is below the lighter one. First, we show local existence in Sobolev spaces by means of energy method when the system is in the stable regime. Then we prove the existence of curves such that they start in the stable regime and in finite time they reach the unstable one. This change of regime (turning) was first proven in \citeccfgl for the homogeneus Muskat problem with infinite depth.

Citations

• Darcy's flow with prescribed contact angle -- Well-posedness and\n lubrication approximation

Cited by

• Global existence for the confined Muskat problem

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• The 2D Muskat Problem I: Local Regularity on the Half-plane, Plane, and Strips
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