Leray's plane stationary solutions at small Reynolds numbers

TL;DR

This paper proves that Leray's stationary solutions to the Navier-Stokes equations in two dimensions achieve the prescribed velocity at infinity for small Reynolds numbers, resolving a long-standing open problem using a novel blow-down technique.

Contribution

It establishes the asymptotic behavior of Leray's solutions at infinity for small Reynolds numbers, building on recent methods and solving a major open problem.

Findings

Leray's solutions attain the prescribed velocity at infinity at small Reynolds numbers.

The proof introduces a new blow-down argument rescaling domains to the unit disc.

The result confirms the expected behavior of stationary flows around obstacles in 2D.

Abstract

In the celebrated paper by Jean Leray, published in JMPA journal in 1933, the invading domains method was proposed to construct D-solutions for the stationary Navier-Stokes flow around obstacle problem. In two dimensions, whether Leray's D-solution achieves the prescribed limiting velocity at spatial infinity became a major open problem since then. In this paper, we solve this problem at small Reynolds numbers. The proof builds on a novel blow-down argument which rescales the invading domains to the unit disc, and the ideas developed in a recent paper [Korobkov-Pileckas-Russo2020], where the nontriviality of Leray solutions in the general case was proved, and [Korobkov-Ren-2021], where the uniqueness result for small Reynolds number was established.