Stokes resolvent estimates in spaces of bounded functions

Ken Abe, Yoshikazu Giga, Matthias Georg Hieber

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

The Stokes equation on a domain Ω Rn is well understood in the Lp-setting for a large class of domains including bounded and exterior domains with smooth boundaries pro- vided 1 <p <∞. The situation is very different for the case p = ∞ since in this case the Helmholtz projection does not act as a bounded operator anymore. Nevertheless it was recently proved by the first and the second author of this paper by a contradiction argument that the Stokes operator generates an analytic semigroup on spaces of bounded functions for a large class of domains. This paper presents a new approach as well as new a priori L-type estimates to the Stokes equation. They imply in par- ticular that the Stokes operator generates a C0-analytic semigroup of angle π/2 on C0,α(Ω), or a non- C0-analytic semigroup on Lα (Ω) for a large class of domains. The approach presented is inspired by the so called Masuda-Stewart technique for elliptic operators. It is shown furthermore that the method presented applies also to different types of boundary conditions as, e.g., Robin boundary conditions.

Original languageEnglish
Pages (from-to)537-559
Number of pages23
JournalAnnales Scientifiques de l'Ecole Normale Superieure
Volume48
Issue number3
Publication statusPublished - 2015 May 1
Externally publishedYes

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Resolvent Estimates
Stokes
Analytic Semigroup
Stokes Operator
Stokes Equations
Robin Boundary Conditions
Exterior Domain
Hermann Von Helmholtz
Bounded Operator
Elliptic Operator
Bounded Domain
Projection
Boundary conditions
Imply
Angle
Estimate
Class

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Stokes resolvent estimates in spaces of bounded functions. / Abe, Ken; Giga, Yoshikazu; Hieber, Matthias Georg.

In: Annales Scientifiques de l'Ecole Normale Superieure, Vol. 48, No. 3, 01.05.2015, p. 537-559.

Research output: Contribution to journalArticle

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