  We develop the concept of an infinite-energy statistical solution to the
Navier-Stokes and Euler equations in the whole plane. We use a velocity
formulation with enough generality to encompass initial velocities having
bounded vorticity, which includes the important special case of vortex patch
initial data. Our approach is to use well-studied properties of statistical
solutions in a ball of radius R to construct, in the limit as R goes to
infinity, an infinite-energy solution to the Navier-Stokes equations. We then
construct an infinite-energy statistical solution to the Euler equations by
making a vanishing viscosity argument.
