TY - JOUR

T1 - Stationary Navier–Stokes equations under inhomogeneous boundary conditions in 3D exterior domains

AU - Hieber, Matthias

AU - Kozono, Hideo

AU - Seyfert, Anton

AU - Shimizu, Senjo

AU - Yanagisawa, Taku

N1 - Funding Information:
The research of the project was partially supported by JSPS Fostering Joint Research Program (B)-18KK0072. The research of H. Kozono was partially supported by JSPS Grant-in-Aid for Scientific Research (S) 16H06339. The research of S. Shimizu was partially supported by JSPS Grant-in-Aid for Scientific Research (B) -16H03945, MEXT.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2021/10

Y1 - 2021/10

N2 - 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).

AB - 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).

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U2 - 10.1007/s00526-021-02050-1

DO - 10.1007/s00526-021-02050-1

M3 - Article

AN - SCOPUS:85111495143

VL - 60

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 5

M1 - 180

ER -