  Infinite energy solutions to the Navier-Stokes equations in $\mathbb{R}^2$
may be constructed by decomposing the initial data into a finite energy piece
and an infinite energy piece, which are then treated separately. We prove that
the finite energy part of such solutions is bounded for all time and decays
algebraically in time when the same can be said of heat energy starting from
the same data. As a consequence, we describe the asymptotic behavior of the
infinite energy solutions. Specifically, we consider the solutions of Gallagher
and Planchon [5] as well as solutions constructed from a "radial energy
decomposition". Our proof uses the Fourier Splitting technique of M. E.
Schonbek.
