Pointwise upper bounds for the solution of the Stokes equation on Lσ∞(Ω) and applications

Martin Bolkart, Matthias Georg Hieber

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Consider the Stokes semigroup T defined on Lσ∞(Ω) where Ω⊂Rn, n≥3, denotes an exterior domain with smooth boundary. It is shown that T(z)u0 for u0∈Lσ∞(Ω) and z∈σθ with θ∈(0, π/2) satisfies pointwise estimates similar to the ones known for G(z)u0 where G denotes the Gaussian semigroup on Rn. In particular, T extends to a bounded analytic semigroup on Lσ∞(Ω) of angle π/2. Moreover, T(t) allows Lσ∞(Ω)-C2+α(Ω-) smoothing for every t>0 and the Stokes semigroups Tp and Tq on Lσp(Ω) and Lσq(Ω) are consistent for all p, q∈(1, ∞].

Original languageEnglish
Pages (from-to)1678-1710
Number of pages33
JournalJournal of Functional Analysis
Volume268
Issue number7
DOIs
Publication statusPublished - 2015 Apr 1
Externally publishedYes

Keywords

  • Bounded analytic semigroups
  • Exterior domain
  • L estimates for Stokes equation
  • Pointwise bounds

ASJC Scopus subject areas

  • Analysis

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