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

Marco Rehmeier, Marco Romito

TL;DR

This work addresses the 2D Euler equations in vorticity form on $\mathbb{R}^2$ by casting them as a first-order nonlinear Fokker–Planck equation and developing a probabilistic representation of solutions. It proves a generalized Lagrangian representation: every nonnegative finite measure-valued weak solution with mild integrability can be realized as a superposition of solution trajectories, extending classical Lagrangian results. It then constructs nonlinear Markov processes that are uniquely determined by selecting appropriate weak solutions from initial data in $L^1\cap L^p$ (with $p\ge 2$) and related Yudovich spaces, via flow selections and a nonlinear martingale framework. The paper further develops the theory in regularity-persistent spaces, showing that, under suitable modulus and Osgood conditions, there exist unique nonlinear Markov processes with persistent regularity, linking deterministic transport along the flow to a probabilistic pathwise representation. Together, these results provide a rigorous, flow-based, probabilistic framework for understanding non-unique weak solutions to the 2D vorticity Euler equation and for selecting physically meaningful evolutions through nonlinear Markov dynamics.

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.

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

TL;DR

This work addresses the 2D Euler equations in vorticity form on by casting them as a first-order nonlinear Fokker–Planck equation and developing a probabilistic representation of solutions. It proves a generalized Lagrangian representation: every nonnegative finite measure-valued weak solution with mild integrability can be realized as a superposition of solution trajectories, extending classical Lagrangian results. It then constructs nonlinear Markov processes that are uniquely determined by selecting appropriate weak solutions from initial data in (with ) and related Yudovich spaces, via flow selections and a nonlinear martingale framework. The paper further develops the theory in regularity-persistent spaces, showing that, under suitable modulus and Osgood conditions, there exist unique nonlinear Markov processes with persistent regularity, linking deterministic transport along the flow to a probabilistic pathwise representation. Together, these results provide a rigorous, flow-based, probabilistic framework for understanding non-unique weak solutions to the 2D vorticity Euler equation and for selecting physically meaningful evolutions through nonlinear Markov dynamics.

Abstract

In this note we contribute two results to the theory of the 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 , , and related spaces such as the classical and uniformly localized Yudovich space. It is well-known that for weak solutions are in general not unique, which renders a suitable selection nontrivial.
Paper Structure (11 sections, 17 theorems, 50 equations)

This paper contains 11 sections, 17 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Notation.
  3. Nonlinear Fokker--Planck equations
  4. Superposition solutions
  5. Nonlinear Markov processes associated with solutions to 2D vorticity Euler equation
  6. Nonlinear Markov processes
  7. Nonlinear Markov process associated with the 2D vorticity Euler equation
  8. Flow selections.
  9. Nonlinear Markov processes in regularity persistent spaces
  10. Appendix
  11. Acknowledgment.

Key Result

Theorem 1

Any nonnegative finite measure-valued weak solution $\omega$ to eq1, satisfying is a superposition solution, i.e. there is a nonnegative finite Borel measure $\eta$ on $C([0,\infty),\mathbb{R}^2)$ (the space of continuous functions from $[0,\infty)$ to $\mathbb{R}^2$) concentrated on solution curves to eq_intro such that where $\pi_t: C([0,\infty),\mathbb{R}^2)\to \mathbb{R}^2$, $\pi_t(f )=f(t)$

Theorems & Definitions (39)

  • Theorem 1: see Thm. \ref{['thm1']} for details
  • Theorem 2: see Thm. \ref{['thm:main-appl-selection']} and Prop. \ref{['prop:well-posed-case']} for details
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • Remark 3.5
  • ...and 29 more