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

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 β_j 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 L^r-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).