Lp-Theory of the Stokes equation in a half space

Wolfgang Desch, Matthias Georg Hieber, Jan Prüss

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75 Citations (Scopus)

Abstract

In this paper, we investigate Lp-estimates for the solution of the Stokes equation in a half space H where 1 <p <∞. It is shown that the solution of the Stokes equation is governed by an analytic semigroup on B U Cσ(H), C0,σ(H) or Lσ (H). From the operatortheoretical point of view it is a surprising fact that the corresponding result for Lσ 1(H) does not hold true. In fact, there exists an L1 -function f satisfying div f = 0 such that the solution of the corresponding resolvent equation with right hand side f does not belong to L1 . Taking into account however a recent result of Kozono on the nonlinear Navier-Stokes equation, the L1 -result is not surprising and even natural. We also show that the Stokes operator admits a R-bounded H-calculus on Lp for 1 <p <∞ and obtain as a consequence maximal Lp-Lq-regularity for the solution of the Stokes equation.

Original languageEnglish
Pages (from-to)115-142
Number of pages28
JournalJournal of Evolution Equations
Volume1
Issue number1
Publication statusPublished - 2001
Externally publishedYes

Keywords

  • Analytic semigroups
  • H-calculus
  • Maximal L-regularity
  • R-boundedness
  • Resolvent estimates
  • Stokes system

ASJC Scopus subject areas

  • Ecology, Evolution, Behavior and Systematics

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    Desch, W., Hieber, M. G., & Prüss, J. (2001). Lp-Theory of the Stokes equation in a half space. Journal of Evolution Equations, 1(1), 115-142.