On the stationary Navier-Stokes equations in exterior domains

Hyunseok Kim, Hideo Kozono

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

This paper is concerned with the existence and uniqueness questions on weak solutions of the stationary Navier-Stokes equations in an exterior domain Ω in R3, where the external force is given by divF with F=F(x)=(Fji(x))i,j=1,2,3. First, we prove the existence and uniqueness of a weak solution for F∈L 3/2,∞(Ω)∩L p,q(Ω) with 3/2<p<3 and 1≤q≤∞ provided ||F||L3/2,∞(Ω) is sufficiently small. Here L p,q(Ω) denotes the well-known Lorentz space. We next show that weak solutions satisfying the energy inequality are unique for F∈L 3/2,∞(Ω)∩L 2(Ω) under the same smallness condition on ||F||L3/2,∞(Ω). This result provides a complete answer to the uniqueness question of weak solutions satisfying the energy inequality, the existence of which was proved by Leray in 1933. Finally, we establish the existence of weak solutions for data F in a very large class, for instance, in L 3/2(Ω)+L 2(Ω), which generalizes Leray's existence result.

Original languageEnglish
Pages (from-to)486-495
Number of pages10
JournalJournal of Mathematical Analysis and Applications
Volume395
Issue number2
DOIs
Publication statusPublished - 2012 Nov 15

Keywords

  • Energy inequality
  • Exterior problem
  • Lorentz space
  • Navier-Stokes equations
  • Regularity
  • Uniqueness

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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