Necessary and sufficient condition on initial data in the Besov space for solutions in the Serrin class of the Navier–Stokes equations

Hideo Kozono, Akira Okada, Senjo Shimizu*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The Cauchy problem of the Navier–Stokes equations in Rn with the initial data a in the Besov space Bp,q-1+np(Rn) for n< p< ∞ and 1 ≤ q≤ ∞ is considered. We construct the local solution in Lα,q(0,T;Br,10(Rn)) for p≤ r< ∞ satisfying 2α+nr=1 with the initial data a∈Bp,q-1+np(Rn), where Lα,q denotes the Lorentz space. Conversely, if the solution belongs to Lα,q(0 , T; Lr(Rn)) with 2α+nr=1, then the initial data a necessarily belong to Br,q-1+nr(Rn). It implies that the initial data in the Besov space Bp,q-1+np(Rn) are a necessary and sufficient condition for the existence of solutions in the Serrin class.

Original languageEnglish
Pages (from-to)3015-3033
Number of pages19
JournalJournal of Evolution Equations
Volume21
Issue number3
DOIs
Publication statusPublished - 2021 Sep

Keywords

  • Inhomogeneous Besov space
  • Navier–Stokes equations
  • Serrin class

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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