Weak solutions of the Navier-Stokes equations with non-zero boundary values in an exterior domain satisfying the strong energy inequality

Reinhard Farwig, Hideo Kozono

    Research output: Contribution to journalArticle

    7 Citations (Scopus)

    Abstract

    In an exterior domain Ω⊂R3 and a time interval [0, T), 0<T≤∞, consider the instationary Navier-Stokes equations with initial value u0∈Lσ2(Ω) and external force f=divF, F∈L2(0, T;L2(Ω)). As is well-known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray-Hopf weak solutions to the case when u=g with non-zero time-dependent boundary values g. Although uniqueness for these solutions cannot be proved, we show the existence of at least one weak solution satisfying the strong energy inequality and a related energy estimate.

    Original languageEnglish
    Pages (from-to)2633-2658
    Number of pages26
    JournalJournal of Differential Equations
    Volume256
    Issue number7
    DOIs
    Publication statusPublished - 2014 Apr 1

    Fingerprint

    Energy Inequality
    Exterior Domain
    Boundary Value
    Navier Stokes equations
    Weak Solution
    Navier-Stokes Equations
    Energy Estimates
    Uniqueness
    Interval

    Keywords

    • Exterior domain
    • Instationary Navier-Stokes equations
    • Non-zero boundary values
    • Strong energy inequality
    • Time-dependent data
    • Weak solutions

    ASJC Scopus subject areas

    • Analysis

    Cite this

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    title = "Weak solutions of the Navier-Stokes equations with non-zero boundary values in an exterior domain satisfying the strong energy inequality",
    abstract = "In an exterior domain Ω⊂R3 and a time interval [0, T), 0<T≤∞, consider the instationary Navier-Stokes equations with initial value u0∈Lσ2(Ω) and external force f=divF, F∈L2(0, T;L2(Ω)). As is well-known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray-Hopf weak solutions to the case when u=g with non-zero time-dependent boundary values g. Although uniqueness for these solutions cannot be proved, we show the existence of at least one weak solution satisfying the strong energy inequality and a related energy estimate.",
    keywords = "Exterior domain, Instationary Navier-Stokes equations, Non-zero boundary values, Strong energy inequality, Time-dependent data, Weak solutions",
    author = "Reinhard Farwig and Hideo Kozono",
    year = "2014",
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    day = "1",
    doi = "10.1016/j.jde.2014.01.029",
    language = "English",
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    journal = "Journal of Differential Equations",
    issn = "0022-0396",
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    TY - JOUR

    T1 - Weak solutions of the Navier-Stokes equations with non-zero boundary values in an exterior domain satisfying the strong energy inequality

    AU - Farwig, Reinhard

    AU - Kozono, Hideo

    PY - 2014/4/1

    Y1 - 2014/4/1

    N2 - In an exterior domain Ω⊂R3 and a time interval [0, T), 0<T≤∞, consider the instationary Navier-Stokes equations with initial value u0∈Lσ2(Ω) and external force f=divF, F∈L2(0, T;L2(Ω)). As is well-known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray-Hopf weak solutions to the case when u=g with non-zero time-dependent boundary values g. Although uniqueness for these solutions cannot be proved, we show the existence of at least one weak solution satisfying the strong energy inequality and a related energy estimate.

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    KW - Exterior domain

    KW - Instationary Navier-Stokes equations

    KW - Non-zero boundary values

    KW - Strong energy inequality

    KW - Time-dependent data

    KW - Weak solutions

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