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

TL;DR

This paper proves the existence of solutions for the steady Navier-Stokes equations in complex domains under minimal flux conditions, extending Leray's classical problem using Bernoulli's law for weak Euler solutions.

Contribution

It establishes the existence of solutions in multiply connected and axially symmetric domains under zero flux, solving a long-standing problem posed by Leray.

Findings

Solution exists under zero total flux condition

Uses Bernoulli's law for weak Euler solutions

Addresses classical Leray problem in complex 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 axially-symmetric 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.