Weak solutions of the Navier-Stokes equations with non-zero boundary values in an exterior domain satisfying the strong energy inequality

Reinhard Farwig, Hideo Kozono

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

In an exterior domain Ω⊂R3 and a time interval [0, T), 0<T≤∞, consider the instationary Navier-Stokes equations with initial value u0∈Lσ2(Ω) and external force f=divF, F∈L2(0, T;L2(Ω)). As is well-known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray-Hopf weak solutions to the case when u=g with non-zero time-dependent boundary values g. Although uniqueness for these solutions cannot be proved, we show the existence of at least one weak solution satisfying the strong energy inequality and a related energy estimate.

Original languageEnglish
Pages (from-to)2633-2658
Number of pages26
JournalJournal of Differential Equations
Volume256
Issue number7
DOIs
Publication statusPublished - 2014 Apr 1

Keywords

  • Exterior domain
  • Instationary Navier-Stokes equations
  • Non-zero boundary values
  • Strong energy inequality
  • Time-dependent data
  • Weak solutions

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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