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

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发表日期：

2025年8月30日

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Marco Rehmeier,
Marco Romito

摘要

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 $L^1\cap L^p$ $L^{1} \cap L^{p}$ , $p \ge 2$ $p \geq 2$ , and related spaces such as the classical and uniformly localized Yudovich space. It is well-known that for $p <\infty $ $p < \infty$ weak solutions are in general not unique, which renders a suitable selection nontrivial.