$2D$ vorticity Euler equations: Superposition solutions and nonlinear Markov processes

TL;DR

This paper advances the understanding of 2D Euler equations in vorticity form by providing a generalized Lagrangian representation for weak solutions and constructing nonlinear Markov processes that select unique solutions from initial data.

Contribution

It introduces a generalized Lagrangian framework for weak solutions and develops nonlinear Markov processes for solution selection, extending classical theories.

Findings

Established a generalized Lagrangian representation for weak solutions.

Constructed nonlinear Markov processes for solution selection.

Extended classical results to broader function spaces.

Abstract

In this note we contribute two results to the theory of the $2D$ Euler equations in vorticity form on the full plane. First, we establish a generalized Lagrangian representation of weak (in general measure-valued) solutions, which includes and extends classical results on the Lagrangianity of weak solutions. Second, we construct nonlinear Markov processes which are uniquely determined by a selection of weak solutions from initial data in $L^1\cap L^p$, $p \geq 2$, and related spaces such as the classical and uniformly localized Yudovich space. It is well-known that for $p <\infty$ weak solutions are in general not unique, which renders a suitable selection nontrivial.