Entropy maximization in the two-dimensional Euler equations

TL;DR

This paper investigates entropy maximization in 2D Euler equations to understand vortex dynamics and stability, employing variational principles, optimal transport, and rearrangement inequalities.

Contribution

It introduces novel variational methods and analytical techniques to characterize entropy maximizers and analyze vortex stability in fluid dynamics.

Findings

Characterization of entropy maximizers in vortex dynamics

Application of optimal transport methods to Euler equations

Results on stability of the canonical Gibbs measure

Abstract

We consider variational problem related to entropy maximization in the two-dimensional Euler equations, in order to investigate the long-time dynamics of solutions with bounded vorticity. Using variations on the classical min-max principle and borrowing ideas from optimal transportation and quantitative rearrangement inequalities, we prove results on the structure of entropy maximizers arising in the investigation of the long-time behavior of vortex patches. We further show that the same techniques apply in the study of stability of the canonical Gibbs measure associated to a system of point vortices.