Global Leray-Hopf weak solutions of the Navier-Stokes equations with Nonzero time-dependent boundary values

R. Farwig, H. Kozono, H. Sohr

Research output: Chapter in Book/Report/Conference proceedingChapter

4 Citations (Scopus)

Abstract

In a bounded smooth domain Ω ⸦ ℝ3 and a time interval [0, T), 0 < T ≤ ∞, consider the instationary Navier-Stokes equations with initial value U0 ∈ L2 σ(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|∂Ω= g with non-zero time-dependent boundary values g. Although there is no uniqueness result for these solutions, they satisfy a strong energy inequality and an energy estimate. In particular, the long-time behavior of energies will be analyzed.

Original languageEnglish
Title of host publicationProgress in Nonlinear Differential Equations and Their Application
PublisherSpringer US
Pages211-232
Number of pages22
DOIs
Publication statusPublished - 2011 Jan 1
Externally publishedYes

Publication series

NameProgress in Nonlinear Differential Equations and Their Application
Volume80
ISSN (Print)1421-1750
ISSN (Electronic)2374-0280

Keywords

  • Energy inequality
  • Instationary Navier-Stokes equations
  • Long-time behavior
  • Non-zero boundary values
  • Time-dependent data
  • Weak solutions

ASJC Scopus subject areas

  • Analysis
  • Computational Mechanics
  • Mathematical Physics
  • Control and Optimization
  • Applied Mathematics

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  • Cite this

    Farwig, R., Kozono, H., & Sohr, H. (2011). Global Leray-Hopf weak solutions of the Navier-Stokes equations with Nonzero time-dependent boundary values. In Progress in Nonlinear Differential Equations and Their Application (pp. 211-232). (Progress in Nonlinear Differential Equations and Their Application; Vol. 80). Springer US. https://doi.org/10.1007/978-3-0348-0075-4_11