Non-unique $L^2$ solutions to the stationary Navier--Stokes equations on the whole plane

Abstract

We prove that, for a suitable small external force in ${\dot{H}}^{-2}({\mathbb{R}}^2)$, the stationary Navier--Stokes equations on ${\mathbb{R}}^2$ admit multiple solutions in $L^2({\mathbb{R}}^2)$.