Solution of Leray's problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains

Abstract

We study the nonhomogeneous boundary value problem for the Navier-Stokes equations of steady motion of a viscous incompressible fluid in arbitrary bounded multiply connected plane or axiallysymmetric spatial domains. We prove that this problem has a solution under the sole necessary condition of zero total flux through the boundary. The problem was formulated by Jean Leray 80 years ago. The proof of the main result uses Bernoulli's law for a weak solution to the Euler equations.

The flux problem for the Navier-Stokes equations

“…Here we collect results on the properties obtained in papers before [22] for the limiting solution ( w, p) of (3.5).…”

“…Some results on the Euler equations. Here we have collected results from papers before [22] for the limiting solution ( w, p) of the system (4.2), (4.3).…”

“…Results from [18]- [24] are central in this survey. The main result was proved by three of the authors in the planar case [22], and it can be stated as follows.…”

On the steady Navier–Stokes equations in 2D exterior domains

“…Since diam C > 0 for every C ∈ [E j , E 0 ], we obtain, by [18,Lemma 3.6], that the function Φ| [Ej ,E0] has the following analog of Luzin's N -property. Below we will say that a value t ∈ (0, δ) is regular if it satisfies the assertion of Corollary 4.3.…”

“…We say that a set Z ⊂ [E j , E 0 ] has T -measure zero if H 1 ({ψ(K) : K ∈ Z}) = 0. 9 See also the proof of Lemma 3.5 in [18]. 10 Recall, that by Lemma 4.2, the set [E j , E 0 ] is homeomorphic to the segment of a real line, i.e.…”