On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain

Yoshihiro Shibata, Senjo Shimizu

Research output: Contribution to journalArticle

45 Citations (Scopus)

Abstract

In this paper, we prove the Lp-Lq maximal regularity of solutions to the Neumann problem for the Stokes equations with non-homogeneous boundary condition and divergence condition in a bounded domain. The result was first stated by Solonnikov V. A. Solonnikov, On the transient motion of an isolated volume of viscous incompressible fluid, Math. USSR Izvest. 31 (1988), 381405., but he assumed that p = q > 3 and considered only the finite time interval case. In this paper, we consider not only the case: 1 < p, q < but also the infinite time interval case. Especially, we obtain the Lp-Lq maximal regularity theorem with exponential stability on the infinite time interval. Our method can be applied to any initial boundary value problem for the equation of parabolic type with suitable boundary condition which generates an analytic semigroup, for example the Stokes equation with non-slip, slip or Robin boundary conditions.

Original languageEnglish
Pages (from-to)157-209
Number of pages53
JournalJournal fur die Reine und Angewandte Mathematik
Issue number615
DOIs
Publication statusPublished - 2008 Feb 1
Externally publishedYes

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Maximal Regularity
Stokes Equations
Neumann Problem
Bounded Domain
Boundary conditions
Interval
Nonhomogeneous Boundary Conditions
Analytic Semigroup
Robin Boundary Conditions
Slip Boundary Condition
Regularity of Solutions
Exponential Stability
Asymptotic stability
Viscous Fluid
Incompressible Fluid
Initial-boundary-value Problem
Boundary value problems
Divergence
Fluids
Motion

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain. / Shibata, Yoshihiro; Shimizu, Senjo.

In: Journal fur die Reine und Angewandte Mathematik, No. 615, 01.02.2008, p. 157-209.

Research output: Contribution to journalArticle

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