Finite energy of generalized suitable weak solutions to the Navier–Stokes equations and Liouville-type theorems in two dimensional domains

Hideo Kozono, Yutaka Terasawa, Yuta Wakasugi

    Research output: Contribution to journalArticle

    Abstract

    Introducing a new notion of generalized suitable weak solutions, we first prove validity of the energy inequality for such a class of weak solutions to the Navier–Stokes equations in the whole space Rn. Although we need certain growth condition on the pressure, we may treat the class even with infinite energy quantity except for the initial velocity. We next handle the equation for vorticity in 2D unbounded domains. Under a certain condition on the asymptotic behavior at infinity, we prove that the vorticity and its gradient of solutions are both globally square integrable. As their applications, Loiuville-type theorems are obtained.

    Original languageEnglish
    JournalJournal of Differential Equations
    DOIs
    Publication statusAccepted/In press - 2018 Jan 1

    Fingerprint

    Suitable Weak Solutions
    Liouville Type Theorem
    Vorticity
    Navier-Stokes Equations
    Energy Inequality
    Unbounded Domain
    Growth Conditions
    Energy
    Weak Solution
    Asymptotic Behavior
    Infinity
    Gradient
    Theorem
    Class

    Keywords

    • Energy inequalities
    • Liouville-type theorems
    • Navier–Stokes equations

    ASJC Scopus subject areas

    • Analysis

    Cite this

    @article{b776f9864629492b800304d01ccccadf,
    title = "Finite energy of generalized suitable weak solutions to the Navier–Stokes equations and Liouville-type theorems in two dimensional domains",
    abstract = "Introducing a new notion of generalized suitable weak solutions, we first prove validity of the energy inequality for such a class of weak solutions to the Navier–Stokes equations in the whole space Rn. Although we need certain growth condition on the pressure, we may treat the class even with infinite energy quantity except for the initial velocity. We next handle the equation for vorticity in 2D unbounded domains. Under a certain condition on the asymptotic behavior at infinity, we prove that the vorticity and its gradient of solutions are both globally square integrable. As their applications, Loiuville-type theorems are obtained.",
    keywords = "Energy inequalities, Liouville-type theorems, Navier–Stokes equations",
    author = "Hideo Kozono and Yutaka Terasawa and Yuta Wakasugi",
    year = "2018",
    month = "1",
    day = "1",
    doi = "10.1016/j.jde.2018.03.027",
    language = "English",
    journal = "Journal of Differential Equations",
    issn = "0022-0396",
    publisher = "Academic Press Inc.",

    }

    TY - JOUR

    T1 - Finite energy of generalized suitable weak solutions to the Navier–Stokes equations and Liouville-type theorems in two dimensional domains

    AU - Kozono, Hideo

    AU - Terasawa, Yutaka

    AU - Wakasugi, Yuta

    PY - 2018/1/1

    Y1 - 2018/1/1

    N2 - Introducing a new notion of generalized suitable weak solutions, we first prove validity of the energy inequality for such a class of weak solutions to the Navier–Stokes equations in the whole space Rn. Although we need certain growth condition on the pressure, we may treat the class even with infinite energy quantity except for the initial velocity. We next handle the equation for vorticity in 2D unbounded domains. Under a certain condition on the asymptotic behavior at infinity, we prove that the vorticity and its gradient of solutions are both globally square integrable. As their applications, Loiuville-type theorems are obtained.

    AB - Introducing a new notion of generalized suitable weak solutions, we first prove validity of the energy inequality for such a class of weak solutions to the Navier–Stokes equations in the whole space Rn. Although we need certain growth condition on the pressure, we may treat the class even with infinite energy quantity except for the initial velocity. We next handle the equation for vorticity in 2D unbounded domains. Under a certain condition on the asymptotic behavior at infinity, we prove that the vorticity and its gradient of solutions are both globally square integrable. As their applications, Loiuville-type theorems are obtained.

    KW - Energy inequalities

    KW - Liouville-type theorems

    KW - Navier–Stokes equations

    UR - http://www.scopus.com/inward/record.url?scp=85045097951&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=85045097951&partnerID=8YFLogxK

    U2 - 10.1016/j.jde.2018.03.027

    DO - 10.1016/j.jde.2018.03.027

    M3 - Article

    JO - Journal of Differential Equations

    JF - Journal of Differential Equations

    SN - 0022-0396

    ER -