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

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

*この研究の対応する著者

## 抄録

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).

本文言語 English 180 Calculus of Variations and Partial Differential Equations 60 5 https://doi.org/10.1007/s00526-021-02050-1 Published - 2021 10

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