Leray's plane steady state solutions are nontrivial

TL;DR

This paper proves that classical solutions to the obstacle problem for stationary Navier-Stokes flows past a body in 2D are always nontrivial, extending previous results that required symmetry assumptions.

Contribution

It establishes the nontriviality of Leray solutions without symmetry or smallness conditions, broadening the understanding of stationary Navier-Stokes solutions.

Findings

Leray solutions are always nontrivial in the obstacle problem

No symmetry or smallness assumptions are needed for nontriviality

Extends classical results by removing additional constraints

Abstract

We study solutions to the obstacle problem for the stationary Navier--Stokes system in a~two dimensional exterior domain (flow past a prescribed body). We prove that the classical Leray solution to this problem is always nontrivial. No additional condition (on symmetry or smallness, etc.) is assumed. This is a complete extension of a~classical result of C.J. Amick (Acta Math. 1988) where nontriviality was proved under symmetry assumption.