Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence

R. Farwig, H. Kozono, H. Sohr

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Consider a smooth bounded domain Ω ⊆ ℝ3 with boundary ∂Ω, a time interval [0, T), 0<T ≤ ∞, and the Navier-Stokes system in [0, T) × Ω, with initial value u0 ∈ L2σ(Ω) and external force f = div F, F ∈ L2(0, T;L2(Ω)). Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying u{pipe}∂Ω = 0, div u = 0 to the more general class of Leray-Hopf type weak solutions u with general data u{pipe}∂Ω = g, div u = k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u = υ + E with some vector field E in [0, T) × Ω satisfying the (linear) Stokes system with f = 0 and nonhomogeneous data. This reduces the general system to a perturbed Navier-Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier-Stokes system we get the existence of global weak solutions for the more general system.

Original languageEnglish
Pages (from-to)51-70
Number of pages20
JournalRendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova
Volume125
DOIs
Publication statusPublished - 2011 Jan 1
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Mathematical Physics
  • Geometry and Topology

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