On the stokes and Navier-Stokes equations in a perturbed half-space

Takayuki Kubo, Yoshihiro Shibata

    Research output: Contribution to journalArticle

    11 Citations (Scopus)

    Abstract

    We discuss Lp-Lq type estimates of the Stokes semigroup and their application to the Navier-Stokes equation in a perturbed halfspace. Especially, we have the Lp-Lq type estimate of the gradient of the Stokes semigroup for any p and q with 1 < p ≤ q < ∞, while the same estimate holds only for the exponents p and q with 1 < p ≤ q ≤ n in the exterior domain case, where n denotes the space dimension, and, therefore, we can get better results concerning the asymptotic behaviour of solutions to the Navier-Stokes equations compared with the exterior domain case. Our proof of the Lp-Lq type estimate of the Stokes semigroup is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the Stokes resolvent near the origin. The order of asymptotic expansion of the Stokes resolvent near the origin is one half better compared with the exterior domain case, because we have the reflection principle on the boundary in the half-space case unlike the whole space case, and, then, such better asymptotics near the boundary are also obtained in the perturbed half-space by a perturbation argument. This is one of the reasons why the result in the perturbed half-space case is essentially better compared with the exterior domain case.

    Original languageEnglish
    Pages (from-to)695-720
    Number of pages26
    JournalAdvances in Differential Equations
    Volume10
    Issue number6
    Publication statusPublished - 2005

    Fingerprint

    Stokes
    Half-space
    Navier Stokes equations
    Exterior Domain
    Navier-Stokes Equations
    Semigroup
    Resolvent
    Estimate
    Local Energy Decay
    Reflection Principle
    Decay Estimates
    Energy Estimates
    Asymptotic Behavior of Solutions
    Asymptotic Expansion
    Asymptotic Behavior
    Exponent
    Gradient
    Denote
    Perturbation

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

    Cite this

    On the stokes and Navier-Stokes equations in a perturbed half-space. / Kubo, Takayuki; Shibata, Yoshihiro.

    In: Advances in Differential Equations, Vol. 10, No. 6, 2005, p. 695-720.

    Research output: Contribution to journalArticle

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