# Lagrangian solutions to the 2D Euler system with $L^1$ vorticity and infinite energy

created by crippa on 18 Aug 2015
modified by bohun on 16 Feb 2016

[BibTeX]

Published Paper

Inserted: 18 aug 2015
Last Updated: 16 feb 2016

Journal: Nonlinear Analysis: Theory, Methods & Applications
Volume: 132
Pages: 160-172
Year: 2016

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 the only assumption of L1 weak convergence of the initial vorticity. The existence of Lagrangian solutions to the Euler system follows for arbitrary L1 vorticity. Relations with previously known notions of solutions are established.