On the decay of infinite energy solutions to the Navier-Stokes equations in the plane

► We study long-time behaviour of infinite energy solutions to a 2D Navier-Stokes equation. ► The finite energy part of these solutions satisfies a Navier-Stokes like equation. ► Asymptotic behaviour of these is that of the heat equation with the same initial data. ► This, in turn, determines the decay of infinite energy solutions towards specific solutions.