Entropy maximization in the two-dimensional Euler equations

Coti Zelati, Michele; Delgadino, Matias G.

doi:10.2140/apde.2026.19.505

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Entropy maximization in the two-dimensional Euler equations

Michele Coti Zelati and Matias G. Delgadino

Vol. 19 (2026), No. 3, 505–538

DOI: 10.2140/apde.2026.19.505

Abstract

We consider a 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.

Keywords

Euler equations, entropy maximization, statistical hydrodynamics, rearrangement inequalities, optimal transport

Mathematical Subject Classification

Primary: 35Q31

Secondary: 37K58, 49Q22

Milestones

Received: 30 May 2024

Revised: 20 February 2025

Accepted: 14 April 2025

Published: 11 March 2026

Authors

Michele Coti Zelati
Department of Mathematics Imperial College London United Kingdom

Matias G. Delgadino
Department of Mathematics University of Texas at Austin United States

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Vol. 19, No. 3, 2026