On the R -boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer

Hirokazu Saito

    Research output: Contribution to journalArticle

    2 Citations (Scopus)

    Abstract

    In this paper, we prove the R-boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter λεΣ∈,γ0, where Σ∈,γ0={λεC||argλ|≤π-∈,|λ|≥γ0}(0<∈<πâ•2,γ0>0), and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 < ∈ < π / 2 and γ<inf>0</inf> > 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large γ<inf>0</inf> > 0 for given 0 < ∈ < π / 2. We also prove the maximal L<inf>p</inf> - L<inf>q</inf> regularity theorem of the nonstationary Stokes problem as an application of the R-boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable x′=(x1,...,xN-1).

    Original languageEnglish
    Pages (from-to)1888-1925
    Number of pages38
    JournalMathematical Methods in the Applied Sciences
    Volume38
    Issue number9
    DOIs
    Publication statusPublished - 2015 Jun 1

    Fingerprint

    R-boundedness
    Boundedness of Solutions
    Resolvent
    Stokes
    Operator
    Choose
    Fourier transforms
    Stokes Problem
    Boundary conditions
    Theorem
    Dirichlet
    Lemma
    Fourier transform
    Exact Solution
    Regularity
    Partial
    Family

    Keywords

    • infinite layer
    • maximal regularity
    • R -boundedness
    • Stokes equations

    ASJC Scopus subject areas

    • Mathematics(all)
    • Engineering(all)

    Cite this

    On the R -boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer. / Saito, Hirokazu.

    In: Mathematical Methods in the Applied Sciences, Vol. 38, No. 9, 01.06.2015, p. 1888-1925.

    Research output: Contribution to journalArticle

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    KW - Stokes equations

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