Many partial differential equations from fluid mechanics possess one or more conserved quantities, such as the energy or helicity of a fluid. These quantities are conserved by smooth solutions, but not necessarily by weak solutions. The aim of this project to study whether weak solutions of several fluid models conserve energy or related quantities. Examples of such models include:

1. The hydrostatic Euler equations (also known as the inviscid primitive equations of large-scale oceanic and atmospheric dynamics). These equations constitute a fundamental model in ocean and climate modelling.

2. The subgrid-scale α-models of turbulence. These models have been applied to model turbulent pipe flow. In particular, we are interested in proving sufficient conditions for weak solutions to conserve energy and showing that these thresholds are sharp by constructing non-energy conserving low regularity solutions.

Publications:

D.W. Boutros, S. Markfelder, E.S. Titi: On Energy Conservation for the Hydrostatic Euler Equations: An Onsager Conjecture. Preprint available at: https://doi.org/10.48550/arXiv.2208.08334

D.W. Boutros, E.S. Titi: Onsager’s Conjecture for Subgrid Scale α-Models of Turbulence. Preprint available at: https://doi.org/10.48550/arXiv.2207.03416