# Energy conditional measures and 2D turbulence

## TL;DR

This paper demonstrates the convergence of invariant measures of point vortices to enstrophy measures under energy conditioning and establishes the existence of solutions to 2D Euler equations with these measures as invariants.

## Contribution

It introduces a novel approach to connect point vortex measures with enstrophy measures through energy conditioning and proves existence of Euler solutions with these measures.

## Key findings

- Invariant measure of point vortices converges to enstrophy measure under Hamiltonian conditioning.
- Existence of 2D Euler solutions with energy-conditioned invariant measures.
- Numerical simulations support theoretical results.

## Abstract

We show that the invariant measure of point vortices, when conditioning the Hamiltonian to a finite interval, converges weakly to the enstrophy measure by conditioning the renormalized energy to the same interval. We also prove the existence of solutions to 2D Euler equations having the energy conditional measure as invariant measure. Some heuristic discussions and numerical simulations are presented in the last section.