Navier–Stokes equations with external forces in Lorentz spaces and its application to the self-similar solutions

Hideo Kozono, Senjo Shimizu

    Research output: Contribution to journalArticle

    4 Citations (Scopus)

    Abstract

    We show existence theorem of global mild solutions with small initial data and external forces in Lorentz spaces with scaling invariant norms. If the initial data have more regularity in another scaling invariant class, then our mild solution is actually the strong solution. The result on local existence of solutions for large data is also discussed. Our method is based on the maximal regularity theorem on the Stokes equations in Lorentz spaces. Then we apply our theorem to prove existence of self-similar solutions provided both initial data and external forces are homogeneous functions. Since we construct the global solution by means of the implicit function theorem, as a byproduct, its stability with respect to the given data is necessarily obtained.

    Original languageEnglish
    Pages (from-to)1693-1708
    Number of pages16
    JournalJournal of Mathematical Analysis and Applications
    Volume458
    Issue number2
    DOIs
    Publication statusPublished - 2018 Feb 15

    Fingerprint

    Lorentz Spaces
    Self-similar Solutions
    Navier-Stokes Equations
    Mild Solution
    Scaling
    Maximal Regularity
    Homogeneous Function
    Implicit Function Theorem
    Invariant
    Byproducts
    Local Existence
    Stokes Equations
    Large Data
    Strong Solution
    Theorem
    Global Solution
    Existence Theorem
    Existence of Solutions
    Regularity
    Norm

    Keywords

    • Global solutions
    • Implicit function theorem
    • Lorentz space
    • Maximal regularity theorem
    • Navier–Stokes equations
    • Self-similar solutions

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

    Cite this

    Navier–Stokes equations with external forces in Lorentz spaces and its application to the self-similar solutions. / Kozono, Hideo; Shimizu, Senjo.

    In: Journal of Mathematical Analysis and Applications, Vol. 458, No. 2, 15.02.2018, p. 1693-1708.

    Research output: Contribution to journalArticle

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