Dynamical behavior for the solutions of the Navier-Stokes equation

Kuijie Li, Tohru Ozawa, Baoxiang Wang

    Research output: Contribution to journalArticle

    Abstract

    We study several quantitative properties of solutions to the incompressible Navier-Stokes equation in three and higher dimensions: (Equation Presented) More precisely, for the blow up mild solutions with initial data in L(ℝRd) and Hd/2-1(ℝd), we obtain a concentration phenomenon and blowup profile decomposition respectively. On the other hand, if the Fourier support has the form supp u0 ⊂ {ξ ∈ ℝn : ξ1 ≥ L} and ||u0|| ≪ L for some L > 0, then (1) has a unique global solution u ∈ C(ℝR+;L). In 3D, we show the compactness of the set consisting of minimal-Lp singularity-generating initial data with 3 < p < 1, furthermore, if the mild solution with data in Lp(ℝ3) blows up in a Type-I manner, we prove the existence of a blowup solution which is uniformly bounded in critical Besov spaces B-1+6/p p/2,∞ (ℝ3).

    Original languageEnglish
    Pages (from-to)1511-1560
    Number of pages50
    JournalCommunications on Pure and Applied Analysis
    Volume17
    Issue number4
    DOIs
    Publication statusPublished - 2018 Jul 1

    Fingerprint

    Dynamical Behavior
    Blow-up
    Navier Stokes equations
    Navier-Stokes Equations
    Mild Solution
    Concentration Phenomena
    Blow-up Solution
    Besov Spaces
    Incompressible Navier-Stokes Equations
    Global Solution
    Higher Dimensions
    Compactness
    Three-dimension
    Singularity
    Decompose
    Decomposition

    Keywords

    • Blowup profile
    • Concentration phenomena
    • L-minimal singularity-generating data
    • Navier-stokes equation
    • Type-I blowup solution

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

    Cite this

    Dynamical behavior for the solutions of the Navier-Stokes equation. / Li, Kuijie; Ozawa, Tohru; Wang, Baoxiang.

    In: Communications on Pure and Applied Analysis, Vol. 17, No. 4, 01.07.2018, p. 1511-1560.

    Research output: Contribution to journalArticle

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