Characterization of initial data in the homogeneous Besov space for solutions in the Serrin class of the Navier-Stokes equations

Hideo Kozono, Akira Okada, Senjo Shimizu

Research output: Contribution to journalArticle

Abstract

Consider the Cauchy problem of the Navier-Stokes equations in Rn with initial data a in the homogeneous Besov space [Formula presented] for n<p<∞ and 1≦q≦∞. We show that the Stokes flow ea can be controlled in Lα,q(0,∞;B˙r,1 0(Rn)) for [Formula presented] with p≦r<∞, where Lα,q denotes the Lorentz space. As an application, the global existence theorem of mild solutions for the small initial data is established in the above class which is slightly stronger than Serrin's. Conversely, if the global solution belongs to the usual Serrin class Lα,q(0,∞;Lr(Rn)) for r and α as above with 1<q≦∞, then the initial data a necessarily belongs to B˙r,q −1+nr(Rn). Moreover, we prove that such solutions are analytic in the space variables. Our method for the proof of analyticity is based on a priori estimates of higher derivatives of solutions in Lp(Rn) with Hölder continuity in time.

Original languageEnglish
Article number108390
JournalJournal of Functional Analysis
Volume278
Issue number5
DOIs
Publication statusPublished - 2020 Mar 15

Keywords

  • Analyticity
  • Homogeneous Besov space
  • Navier-Stokes equations
  • Serrin class

ASJC Scopus subject areas

  • Analysis

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