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

Yoshihiro Shibata, Senjo Shimizu

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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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ASJC Scopus subject areas

  • Mathematics(all)

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