Leray’s inequality in general multi-connected domains in $${\mathbb{R}^n}$$

Abstract

Consider the stationary Navier–Stokes equations in a bounded domain \({\Omega \subset \mathbb{R}^n}\) whose boundary \({\partial\Omega}\) consists of L + 1 smooth (n − 1)-dimensional closed hypersurfaces Γ0, Γ1, . . . , Γ L , where Γ1, . . . , Γ L lie inside of Γ0 and outside of one another. The Leray inequality of the given boundary data β on \({\partial\Omega}\) plays an important role for the existence of solutions. It is known that if the flux \({\gamma_i \equiv \int_{\Gamma_i}\beta \cdot \nu dS = 0}\) on Γ i (ν: the unit outer normal to Γ i ) is zero for each i = 0, 1, . . . , L, then the Leray inequality holds. We prove that if there exists a sphere S in Ω separating \({\partial\Omega}\) in such a way that Γ1, . . . , Γ k (1 ≦ k ≦ L) are contained inside of S and that the others Γ k+1, . . . , Γ L are outside of S, then the Leray inequality necessarily implies that γ 1 + · · · + γ k = 0. In particular, suppose that there are L spheres S 1, . . . , S L in Ω lying outside of one another such that Γ i lies inside of S i for all i = 1, . . . , L. Then the Leray inequality holds if and only if γ 0 = γ 1 = · · · = γ L = 0.