Maximal Lp-Lq regularity for the two-phase Stokes equations; Model problems

Yoshihiro Shibata, Senjo Shimizu

    Research output: Contribution to journalArticle

    15 Citations (Scopus)

    Abstract

    In this paper we prove the generalized resolvent estimate and maximal Lp-Lq regularity of the Stokes equation with and without surface tension and gravity in the whole space with flat interface. We prove R boundedness of solution operators defined in a sector Σε,γ0={λεC\{0}||argλ| ≤ π-ε,|λ|≥γ0} with 0<ε<π/2 and γ0≥0, which combined with the Fourier multiplier theorem of S.G. Mihlin and the operator valued Fourier multiplier theorem of L. Weis yields the required generalized resolvent estimate and maximal Lp-Lq regularity at the same time. One of the character of the paper is to introduce special function spaces Eq(Ṙnε,γ0) and Ep,q,γ0(Ṙn×R) (cf. (1.7) and (1.8)), which is necessary to treat the situation that the normal component of velocity fields jumps across the interface. Such spaces never appear in the study of the Stokes equations with other boundary conditions like non-slip condition, Navier slip condition, Robin condition or pure Neumann condition appearing in the study of one phase problem (cf. Desch et al., 2001 [12], Farwig and Sohr, 1994 [13], Saal, 2003 [22], Shibata and Shimada, 2007 [23], Shibata and Shimizu 2008 [25], 2009 [26], in press [27]), because the normal component of the velocity fields vanishes at the boundary which is physical requirement that the flow does not go out and come in through the rigid boundary.

    Original languageEnglish
    Pages (from-to)373-419
    Number of pages47
    JournalJournal of Differential Equations
    Volume251
    Issue number2
    DOIs
    Publication statusPublished - 2011 Jul 15

    Fingerprint

    Resolvent Estimates
    Stokes Equations
    Velocity Field
    Operator-valued Fourier multipliers
    R-boundedness
    Regularity
    Fourier multipliers
    Slip Condition
    Neumann Condition
    Boundedness of Solutions
    Special Functions
    Theorem
    Surface Tension
    Function Space
    Surface tension
    Vanish
    Gravity
    Gravitation
    Sector
    Jump

    Keywords

    • Gravity
    • Maximal regularity
    • Resolvent estimate
    • Stokes equation
    • Surface tension
    • Two-phase problem

    ASJC Scopus subject areas

    • Analysis

    Cite this

    Maximal Lp-Lq regularity for the two-phase Stokes equations; Model problems. / Shibata, Yoshihiro; Shimizu, Senjo.

    In: Journal of Differential Equations, Vol. 251, No. 2, 15.07.2011, p. 373-419.

    Research output: Contribution to journalArticle

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