On the stationary Navier-Stokes equations in exterior domains

Hyunseok Kim, Hideo Kozono

    Research output: Contribution to journalArticle

    5 Citations (Scopus)

    Abstract

    This paper is concerned with the existence and uniqueness questions on weak solutions of the stationary Navier-Stokes equations in an exterior domain Ω in R3, where the external force is given by divF with F=F(x)=(Fji(x))i,j=1,2,3. First, we prove the existence and uniqueness of a weak solution for F∈L 3/2,∞(Ω)∩L p,q(Ω) with 3/2<p<3 and 1≤q≤∞ provided ||F||L3/2,∞(Ω) is sufficiently small. Here L p,q(Ω) denotes the well-known Lorentz space. We next show that weak solutions satisfying the energy inequality are unique for F∈L 3/2,∞(Ω)∩L 2(Ω) under the same smallness condition on ||F||L3/2,∞(Ω). This result provides a complete answer to the uniqueness question of weak solutions satisfying the energy inequality, the existence of which was proved by Leray in 1933. Finally, we establish the existence of weak solutions for data F in a very large class, for instance, in L 3/2(Ω)+L 2(Ω), which generalizes Leray's existence result.

    Original languageEnglish
    Pages (from-to)486-495
    Number of pages10
    JournalJournal of Mathematical Analysis and Applications
    Volume395
    Issue number2
    DOIs
    Publication statusPublished - 2012 Nov 15

    Fingerprint

    Stationary Navier-Stokes Equations
    Exterior Domain
    Navier Stokes equations
    Weak Solution
    Energy Inequality
    Existence and Uniqueness
    Lorentz Spaces
    Existence of Weak Solutions
    Existence Results
    Uniqueness
    Denote
    Generalise

    Keywords

    • Energy inequality
    • Exterior problem
    • Lorentz space
    • Navier-Stokes equations
    • Regularity
    • Uniqueness

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

    Cite this

    On the stationary Navier-Stokes equations in exterior domains. / Kim, Hyunseok; Kozono, Hideo.

    In: Journal of Mathematical Analysis and Applications, Vol. 395, No. 2, 15.11.2012, p. 486-495.

    Research output: Contribution to journalArticle

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