On bounded two-dimensional globally dissipative Euler flows

TL;DR

This paper studies the two-dimensional Euler equations with local energy considerations, providing criteria for the existence of multiple solutions and analyzing their dissipation properties, especially in the context of vortex sheets and Kelvin-Helmholtz instability.

Contribution

It introduces a relaxation framework for bounded solutions with local energy balance and establishes conditions for infinitely many solutions with the same initial data and dissipation.

Findings

Relaxation reduces to known Euler relaxation without local energy balance.

Sufficient criterion for multiple solutions from a given subsolution.

Existence of initial data leading to a wide range of dissipation measures.

Abstract

We examine the two-dimensional Euler equations including the local energy (in)equality as a differential inclusion and show that the associated relaxation essentially reduces to the known relaxation for the Euler equations considered without local energy (im)balance. Concerning bounded solutions we provide a sufficient criterion for a globally dissipative subsolution to induce infinitely many globally dissipative solutions having the same initial data, pressure and dissipation measure as the subsolution. The criterion can easily be verified in the case of a flat vortex sheet giving rise to the Kelvin-Helmholtz instability. As another application we show that there exists initial data, for which associated globally dissipative solutions realize every dissipation measure from an open set in $\mathcal{C}^0(\mathbb{T}^2\times[0,T])$. In fact the set of such initial data is dense in the space of solenoidal $L^2(\mathbb{T}^2;\mathbb{R}^2)$ vector fields.