Onsager-type conjectures for boundary-driven and geophysical flowsOberseminar Analysis

by Daniel Boutros (Cambridge University)

Thursday Nov 20, 2025, 2:15 PM → 3:15 PM Europe/Berlin

Endenicher Allee 60/1-016 - Lipschitzsaal (Mathezentrum)

Onsager's conjecture states that 1/3 is the critical spatial (Hölder) regularity threshold for energy conservation by weak solutions of the incompressible Euler equations. We consider Onsager’s conjecture for the Euler equations in the case of a bounded domain, as boundary effects play a crucial role in hydrodynamic turbulence. We present a regularity result for the pressure in the Euler equations, which is fundamental for the proof of the conservation part of the Onsager conjecture (in the presence of boundaries). As an essential part of the proof, we introduce a new weaker notion of boundary condition which we show to be necessary by means of an explicit example. Moreover, we derive this new boundary condition rigorously from the weak formulation of the Euler equations.

 

In addition, we consider an analogue of Onsager’s conjecture for the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics). The anisotropic nature of these equations allows us to introduce new types of weak solutions and prove a range of independent sufficient criteria for energy conservation. Therefore there probably is a ‘family’ of Onsager conjectures for these equations. Furthermore, we employ the method of convex integration to show the nonuniqueness of weak solutions to the inviscid and viscous primitive equations (and also the Prandtl equations), and to construct examples of solutions that do not conserve energy in the inviscid case. If time allows, I will also present some recent results on a model from sea-ice dynamics, which is the elastic-viscous-plastic sea-ice model. These are joint works with Claude Bardos, Xin Liu, Simon Markfelder, Marita Thomas and Edriss S. Titi.