Dynamical behavior for the solutions of the Navier-Stokes equation

Kuijie Li, Tohru Ozawa, Baoxiang Wang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

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/pp/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

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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