Please use this identifier to cite or link to this item:
http://doi.org/10.25358/openscience-5962
Authors: | Tolksdorf, Patrick |
Title: | The Stokes resolvent problem : optimal pressure estimates and remarks on resolvent estimates in convex domains |
Online publication date: | 31-May-2021 |
Year of first publication: | 2020 |
Language: | english |
Abstract: | The Stokes resolvent problem ππ’βΞπ’+βπ=π Ξ» u β Ξ u + β Ο = f with div(π’)=0 div ( u ) = 0 subject to homogeneous Dirichlet or homogeneous Neumann-type boundary conditions is investigated. In the first part of the paper we show that for Neumann-type boundary conditions the operator norm of L2π(Ξ©)βπβ¦πβL2(Ξ©) L Ο 2 ( Ξ© ) β f β¦ Ο β L 2 ( Ξ© ) decays like |π|β1/2 | Ξ» | β 1 / 2 which agrees exactly with the scaling of the equation. In comparison to that, the operator norm of this mapping under Dirichlet boundary conditions decays like |π|βπΌ | Ξ» | β Ξ± for 0β€πΌβ€1/4 0 β€ Ξ± β€ 1 / 4 and we show optimality of this rate, thereby, violating the natural scaling of the equation. In the second part of this article, we investigate the Stokes resolvent problem subject to homogeneous Neumann-type boundary conditions if the underlying domain Ξ© Ξ© is convex. Invoking a famous result of Grisvard (Elliptic problems in nonsmooth domains. Monographs and studies in mathematics, Pitman, 1985), we show that weak solutions u with right-hand side πβL2(Ξ©;βπ) f β L 2 ( Ξ© ; C d ) admit H2 H 2 -regularity and further prove localized H2 H 2 -estimates for the Stokes resolvent problem. By a generalized version of Shenβs Lπ L p -extrapolation theorem (Shen in Ann Inst Fourier (Grenoble) 55(1):173β197, 2005) we establish optimal resolvent estimates and gradient estimates in Lπ(Ξ©;βπ) L p ( Ξ© ; C d ) for 2π/(π+2)<π<2π/(πβ2) 2 d / ( d + 2 ) < p < 2 d / ( d β 2 ) (with 1<π<β 1 < p < β if π=2 d = 2 ). This interval is larger than the known interval for resolvent estimates subject to Dirichlet boundary conditions (Shen in Arch Ration Mech Anal 205(2):395β424, 2012) on general Lipschitz domains. |
DDC: | 510 Mathematik 510 Mathematics |
Institution: | Johannes Gutenberg-UniversitΓ€t Mainz |
Department: | FB 08 Physik, Mathematik u. Informatik |
Place: | Mainz |
ROR: | https://ror.org/023b0x485 |
DOI: | http://doi.org/10.25358/openscience-5962 |
Version: | Published version |
Publication type: | Zeitschriftenaufsatz |
License: | CC BY |
Information on rights of use: | https://creativecommons.org/licenses/by/4.0/ |
Journal: | Calculus of variations and partial differential equations 59 |
Pages or article number: | 154 |
Publisher: | Springer |
Publisher place: | Berlin u.a. |
Issue date: | 2020 |
ISSN: | 1432-0835 |
Publisher URL: | https://doi.org/10.1007/s00526-020-01811-8 |
Publisher DOI: | 10.1007/s00526-020-01811-8 |
Appears in collections: | JGU-Publikationen |
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