Remarks on the solution map for Yudovich solutions of the Euler equations

TL;DR

This paper provides a Lagrangian proof demonstrating the strong and weak-$*$ continuity of the solution map for Yudovich solutions of the 2D Euler equations in various domains, clarifying the mathematical properties of these solutions.

Contribution

It offers a novel purely Lagrangian proof establishing continuity properties of the solution map for Yudovich solutions, enhancing understanding of their stability and behavior.

Findings

Solution map is strongly continuous in $L^p$ for all $p eq \infty$

Solution map is weakly-$*$ continuous in $L^\infty$

Provides a new Lagrangian approach to analyze Euler solutions

Abstract

Consider Yudovich solutions to the incompressible Euler equations with bounded initial vorticity in bounded planar domains or in $\mathbb{R}^2$. We present a purely Lagrangian proof that the solution map is strongly continuous in $L^p$ for all $p\in [1, \infty)$ and is weakly-$*$ continuous in $L^\infty$.