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Well-posedness of the Muskat problem in subcritical Lp-Sobolev spaces

Published online by Cambridge University Press:  18 January 2021

H. ABELS
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040Regensburg, Germany, emails: [email protected]; [email protected]
B.-V. MATIOC
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040Regensburg, Germany, emails: [email protected]; [email protected]

Abstract

We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an Lp-setting with p ∈ (1, ∞). The Sobolev space $W_p^s(\mathbb R)$ with s = 1+1/p is a critical space for this problem. We prove, for each s ∈ (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.

Type
Papers
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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