On the maximal L p-L q regularity of the Stokes problem with first order boundary condition; model problems

Yoshihiro Shibata, Senjo Shimizu

Research output: Contribution to journalArticlepeer-review

25 Citations (Scopus)

Abstract

In this paper, we proved the generalized resolvent estimate and the maximal L p-L q regularity of the Stokes equation with first order boundary condition in the half-space, which arises in the mathematical study of the motion of a viscous incompressible one phase fluid flow with free surface. The core of our approach is to prove the R boundedness of solution operators defined in a sector Σ ∈,γ0 = {λ ∈ C \ {0} | | arg λ| ≤ π - ∈, |λ| > γ o} with 0 < ∈ < π/2 and γ o ≥0. This R boundedness implies the resolvent estimate of the Stokes operator and the combination of this R boundedness with the operator valued Fourier multiplier theorem of L. Weis implies the maximal L p-L q regularity of the non-stationary Stokes. For a densely defined closed operator A, we know that what A has maximal L p regularity implies that the resolvent estimate of A in λ ∈ Σ ∈,γ0, but the opposite direction is not true in general (cf. Kalton and Lancien[19]). However, in this paper using the R boundedness of the operator family in the sector Σ ∈ λ0, we derive a systematic way to prove the resolvent estimate and the maximal L p regularity at the same time.

Original languageEnglish
Pages (from-to)561-626
Number of pages66
JournalJournal of the Mathematical Society of Japan
Volume64
Issue number2
DOIs
Publication statusPublished - 2012

Keywords

  • Gravity force
  • Half space problem
  • Maximal regularity
  • Resolvent estimate
  • Stokes equation
  • Surface tension

ASJC Scopus subject areas

  • Mathematics(all)

Fingerprint Dive into the research topics of 'On the maximal L <sub>p</sub>-L <sub>q</sub> regularity of the Stokes problem with first order boundary condition; model problems'. Together they form a unique fingerprint.

Cite this