Gutenberg Open Science: The Stokes resolvent problem : optimal pressure estimates and remarks on resolvent estimates in convex domains

Please use this identifier to cite or link to this item: http://doi.org/10.25358/openscience-5962

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DC Field	Value	Language
dc.contributor.author	Tolksdorf, Patrick	-
dc.date.accessioned	2021-05-31T10:34:24Z	-
dc.date.available	2021-05-31T10:34:24Z	-
dc.date.issued	2020	-
dc.identifier.uri	https://openscience.ub.uni-mainz.de/handle/20.500.12030/5971	-
dc.description.abstract	The Stokes resolvent problem 𝜆𝑢−Δ𝑢+∇𝜙=𝑓 λ u − Δ u + ∇ ϕ = f with div(𝑢)=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 L2𝜎(Ω)∋𝑓↦𝜙∈L2(Ω) L σ 2 ( Ω ) ∋ f ↦ ϕ ∈ L 2 ( Ω ) decays like \|𝜆\|−1/2 \| λ \| − 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 \|𝜆\|−𝛼 \| λ \| − α for 0≤𝛼≤1/4 0 ≤ α ≤ 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 Ω Ω 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 𝑓∈L2(Ω;ℂ𝑑) f ∈ L 2 ( Ω ; 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 Shen’s L𝑝 L p -extrapolation theorem (Shen in Ann Inst Fourier (Grenoble) 55(1):173–197, 2005) we establish optimal resolvent estimates and gradient estimates in L𝑝(Ω;ℂ𝑑) L p ( Ω ; C d ) for 2𝑑/(𝑑+2)<𝑝<2𝑑/(𝑑−2) 2 d / ( d + 2 ) < p < 2 d / ( d − 2 ) (with 1<𝑝<∞ 1 < p < ∞ if 𝑑=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):395–424, 2012) on general Lipschitz domains.	en_GB
dc.language.iso	eng	de
dc.rights	CC BY	*
dc.rights.uri	https://creativecommons.org/licenses/by/4.0/	*
dc.subject.ddc	510 Mathematik	de_DE
dc.subject.ddc	510 Mathematics	en_GB
dc.title	The Stokes resolvent problem : optimal pressure estimates and remarks on resolvent estimates in convex domains	en_GB
dc.type	Zeitschriftenaufsatz	de
dc.identifier.doi	http://doi.org/10.25358/openscience-5962	-
jgu.type.dinitype	article	en_GB
jgu.type.version	Published version	de
jgu.type.resource	Text	de
jgu.organisation.department	FB 08 Physik, Mathematik u. Informatik	de
jgu.organisation.number	7940	-
jgu.organisation.name	Johannes Gutenberg-Universität Mainz	-
jgu.rights.accessrights	openAccess	-
jgu.journal.title	Calculus of variations and partial differential equations	de
jgu.journal.volume	59	de
jgu.pages.alternative	154	de
jgu.publisher.year	2020	-
jgu.publisher.name	Springer	de
jgu.publisher.place	Berlin u.a.	de
jgu.publisher.uri	https://doi.org/10.1007/s00526-020-01811-8	de
jgu.publisher.issn	1432-0835	de
jgu.organisation.place	Mainz	-
jgu.subject.ddccode	510	de
jgu.publisher.doi	10.1007/s00526-020-01811-8
jgu.organisation.ror	https://ror.org/023b0x485
Appears in collections:	JGU-Publikationen

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