Statistical hydrodynamics and geophysical flows

In his seminal 1949 article, Onsager presented a groundbreaking theory that delves into the irreversible behaviour of fluids, particularly those operating far from equilibrium.

This theory's profound impact is evident in its influence on various conjectures concerning the long-term dynamics of 2D incompressible Euler solutions.

Notably, these conjectures are often grounded in the idea that solutions should maximize vorticity mixing while still adhering to the preservation of conserved quantities.

Despite its apparent simplicity in certain scenarios, such as solutions generated by a patch in a disc, the exploration of entropy maximisers at specific energy levels remains vastly uncharted territory.

Furthermore, while faint parallels to classical statistical mechanics have been drawn, the full extent of their connection remains elusive.

Therefore, the primary objective of this project is to investigate these intriguing aspects of the 2D Euler equations by employing optimization techniques, rearrangement inequalities, and min-max principles.

By doing so, we aim to shed new light on these central questions and provide a fresh perspective on this fascinating subject matter.