Lagrangian solutions to the 2D euler system with L^1 vorticity and infinite energy

TL;DR

This paper develops a theory of Lagrangian solutions for 2D Euler equations with integrable vorticity, allowing for locally infinite energy, and proves their stability and existence under minimal initial conditions.

Contribution

It introduces a new framework for Lagrangian solutions with L^1 vorticity and establishes their stability and existence, extending previous solution concepts.

Findings

Proves strong stability of Lagrangian solutions under L^1 initial vorticity convergence.

Establishes existence of solutions for arbitrary L^1 vorticity.

Connects new solutions with previously known notions.

Abstract

We consider solutions to the two-dimensional incompressible Euler system with only integrable vorticity, thus with possibly locally infinite energy. With such regularity, we use the recently developed theory of Lagrangian flows associated to vector fields with gradient given by a singular integral in order to define Lagrangian solutions, for which the vorticity is transported by the flow. We prove strong stability of these solutions via strong convergence of the flow, under only the assumption of L^1 weak convergence of the initial vorticity. The existence of Lagrangian solutions to the Euler system follows for arbitrary L^1 vorticity. Relations with previously known notions of solutions are established.