Global well-posedness for the incompressible Navier–Stokes equations in the critical Besov space under the Lagrangian coordinates

Takayoshi Ogawa, Senjo Shimizu

Research output: Contribution to journalArticlepeer-review

Abstract

We consider global well-posedness of the Cauchy problem of the incompressible Navier–Stokes equations under the Lagrangian coordinates in scaling critical Besov spaces. We prove the system is globally well-posed in the homogeneous Besov space B˙p,1−1+n/p(Rn) with 1≤p<∞. The former result was restricted for 1≤p<2n and the main reason why the well-posedness space is enlarged is that the quasi-linear part of the system has a special feature called a multiple divergence structure and the bilinear estimate for the nonlinear terms are improved by such a structure. Our result indicates that the Navier–Stokes equations can be transferred from the Eulerian coordinates to the Lagrangian coordinates even for the solution in the limiting critical Besov spaces.

Original languageEnglish
Pages (from-to)613-651
Number of pages39
JournalJournal of Differential Equations
Volume274
DOIs
Publication statusPublished - 2021 Feb 15

Keywords

  • Critical Besov space
  • End-point
  • Global well-posedness
  • Lagrangian coordinates
  • Maximal L regularity
  • Navier–Stokes equations

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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