Stationary Navier–Stokes equations under inhomogeneous boundary conditions in 3D exterior domains

Matthias Hieber, Hideo Kozono*, Anton Seyfert, Senjo Shimizu, Taku Yanagisawa

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

In an exterior domain Ω ⊂ R3 having compact boundary ∂Ω=⋃j=1LΓj with L disjoint smooth closed surfaces Γ 1, … , Γ L, we consider the problem on the existence of weak solutions v of the stationary Navier–Stokes equations in Ω satisfying v|Γj=βj, j= 1 , … , L and v→ 0 as | x| → ∞, where βj are the given data on the boundary component Γ j, j= 1 , … , L. Our first task is to find an appropriate solenoidal extension b into Ω , i.e., divb=0 satisfying b|Γj=βj, j= 1 , … , L. By our previous result [8] on the Lr-Helmholtz-Weyl decomposition, b is expressed as b=h+rotw, where h is a harmonic vector field depending only on the flux ∫Γjβj·νdS through Γ j, j= 1 , … , L. Next, we prove that if h is small in L3(Ω) , then there exists a weak solution v with ∇ v∈ L2(Ω).

Original languageEnglish
Article number180
JournalCalculus of Variations and Partial Differential Equations
Volume60
Issue number5
DOIs
Publication statusPublished - 2021 Oct

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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