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  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).
|Journal||Calculus of Variations and Partial Differential Equations|
|Publication status||Published - 2021 Oct|
ASJC Scopus subject areas
- Applied Mathematics