Global wellposedness for the 3D Muskat problem with medium size slope

2020/02/02 by Stephen Cameron, Cameron, Stephen · 3 citations
Mathematics · Computer Science · #Navier-Stokes equation solutions #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2002.00508

Abstract

We prove the existence and uniqueness of global, classical solutions to the 3D Muskat problem in the stable regime whenever the initial interface has sublinear growth and slope ||∇_x f₀||_L^∞< 5^-1/2. We show under these assumptions that the equation is fundamentally parabolic, satisfying a comparison principle. Applying the modulus of continuity technique, we show that rough initial data instantly becomes C^1,1 with the curvature decaying like O(t^-1).

Citations

• The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid

Cited by

• Global well-posedness for the one-phase Muskat problem
• The Muskat problem with C¹ data
• Global Results for the Inhomogeneous Muskat Problem

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