Dissipative Euler flows for vortex sheet initial data without distinguished sign

TL;DR

This paper constructs numerous weak solutions to 2D Euler equations with vortex sheet initial data lacking a sign, using convex integration, revealing turbulence growth and energy dissipation characteristics.

Contribution

It introduces a convex integration method to generate infinitely many solutions for vortex sheets without sign restrictions, advancing understanding of turbulence and dissipation in fluid dynamics.

Findings

Solutions are smooth outside a turbulence zone growing linearly in time.

The turbulence zone growth is controlled by the local energy inequality.

The maximal initial dissipation rate is measured by vortex sheet strength.

Abstract

We construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our initial datum has vorticity concentrated on a simple closed curve in a suitable H\"older space and the vorticity may not have a distinguished sign. Our solutions are obtained by means of convex integration; they are smooth outside a "turbulence" zone which grows linearly in time around the vortex sheet. As a byproduct, this approach shows how the growth of the turbulence zone is controlled by the local energy inequality and measures the maximal initial dissipation rate in terms of the vortex sheet strength.