Infinite-energy 2D statistical solutions to the equations of incompressible fluids

TL;DR

This paper introduces the concept of infinite-energy statistical solutions for 2D incompressible fluid equations, extending the analysis to the entire plane and including vortex patch initial data, using a limiting approach from finite domains.

Contribution

It develops a framework for infinite-energy statistical solutions to Navier-Stokes and Euler equations in the whole plane, accommodating bounded vorticity initial data.

Findings

Constructed infinite-energy solutions via limits of finite domain solutions.

Extended statistical solution concepts to Euler equations with vanishing viscosity.

Applicable to vortex patch initial data.

Abstract

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.