Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces

Abstract

We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an $L_p$-setting with $p\in (1,\infty )$. The Sobolev space $W_p^s(\mathbb{R})$ with $s=1+1/p$ is a critical space for this problem. We prove, for $s\in (1+1/p,2),$ that the Rayleigh-Taylor condition identifies an open subset of $W_p^s(\mathbb{R})$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.

43 pages, to appear in European Journal of Applied Mathematics

References in corpus (2)

• A paradifferential approach for well-posedness of the Muskat problem
• Paralinearization of the Muskat equation and application to the Cauchy problem

Cited by in corpus (3)

• Regularity of solutions to the Muskat equation
• Entropy solutions to macroscopic IPM
• The Second Iterate of the Muskat Equation in Supercritical Spaces