Statistical solutions of the incompressible Euler equations

TL;DR

This paper investigates statistical solutions of the 2D incompressible Euler equations with various vorticity conditions, establishing existence, connection to Navier-Stokes solutions, and uniqueness in the Yudovich class.

Contribution

It introduces a framework for statistical solutions of the Euler equations, proves their existence via approximation, and links them to Navier-Stokes solutions in the inviscid limit, with uniqueness results in specific classes.

Findings

Existence of statistical solutions for 2D Euler equations with vorticity in L^p.

Statistical solutions can be obtained as limits of Navier-Stokes solutions.

Uniqueness of trajectory statistical solutions in the Yudovich class.

Abstract

We study statistical solutions of the incompressible Euler equations in two dimensions with vorticity in $L^p$, $1\leq p \leq \infty$, and in the class of vortex-sheets with a distinguished sign. Our notion of statistical solution is based on the framework due to Bronzi, Mondaini and Rosa. Existence in this setting is shown by approximation with discrete measures, concentrated on deterministic solutions of the Euler equations. Additionally, we provide arguments to show that the statistical solutions of the Euler equations may be obtained in the inviscid limit of statistical solutions of the incompressible Navier-Stokes equations. Uniqueness of trajectory statistical solutions is shown in the Yudovich class.