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

Part of: Harmonic analysis in several variables Miscellaneous topics - Partial differential equations Parabolic equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  18 January 2021

H. ABELS Open the ORCID record for H. ABELS [Opens in a new window]  and
B.-V. MATIOC Open the ORCID record for B.-V. MATIOC [Opens in a new window]
H. ABELS
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040Regensburg, Germany, emails: helmut.abels@ur.de; bogdan.matioc@ur.de
B.-V. MATIOC
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040Regensburg, Germany, emails: helmut.abels@ur.de; bogdan.matioc@ur.de
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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.

Keywords

Muskat problemRayleigh–Taylor conditionsubcritical spacesingular integral

MSC classification

Primary: 35R37: Moving boundary problems 35K55: Nonlinear parabolic equations
Secondary: 35Q35: PDEs in connection with fluid mechanics 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 2 , April 2022 , pp. 224 - 266
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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