Local in time regularity properties of the Navier-Stokes equations

Reinhard Farwig, Hideo Kozono, Hermann Sohr

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

Let u be a weak solution of the Navier-Stokes equations in a smooth domain Ω ⊆ ℝ and a time interval [0, T), 0 < T < ∞, with initial value u0, and vanishing external force. As is well known, global regularity of u for general u0 is an unsolved problem unless we pose additional assumptions on u0 or on the solution u itself such as Serrin's condition ||u||Ls(0,T;Lq(Ω)) < ∞ where 2/s + 3/q = 1. In the present paper we prove several new local and global regularity properties by using assumptions beyond Serrin's condition e.g. as follows: If Ω is bounded and the norm ||u||L1(0, T;Lq(Ω)), with Serrin's number 2/1 + 3/q strictly larger than 1, is sufficiently small, then u is regular in (0, T). Further local regularity conditions for general smooth domains are based on energy quantities such as ||u||L(T0,T1L2(Ω)) |w||i»(r0,Ti;Z.2(i))) and || ▽ u|| ||u||L2(T 0,T1L2(Ω)). Indiana University Mathematics Journal

Original languageEnglish
Pages (from-to)2111-2131
Number of pages21
JournalIndiana University Mathematics Journal
Volume56
Issue number5
DOIs
Publication statusPublished - 2007 Dec 1
Externally publishedYes

Keywords

  • Instationary Navier-Stokes equations
  • Local in time regularity
  • Serrin's condition

ASJC Scopus subject areas

  • Mathematics(all)

Fingerprint Dive into the research topics of 'Local in time regularity properties of the Navier-Stokes equations'. Together they form a unique fingerprint.

  • Cite this