Anomalous Dissipation, Spontaneous Stochasticity & Onsager's Conjecture

Abstract

Turbulence displays a number of remarkable features. It is a super dissipator, able to efficiently deplete its energy without the direct aid of viscosity. Non-vanishing energy dissipation in the limit of zero viscosity is termed anomalous dissipation and it is so fundamental to our modern understanding that it is ofter referred to as the "zeroth law of turbulence". Turbulent fluids are also exceptionally strong mixers, capable of very rapidly separating nearby particles within the flow. This is related to the phenomenon of spontaneous stochasticity, or the non-uniqueness of Lagrangian particle trajectories at infinite Reynolds number. Though seemingly distinct, these features are conjectured to be closely connected:

"There seems to be a strong relation between the behavior of the Lagrangian trajectories and the basic hydrodynamic properties of developed turbulent flows: we expect the appearance of non-unique trajectories for Re→∞ to be responsible for the dissipative anomaly, the direct energy cascade, the dissipation of higher conserved quantities and the pertinence of weak solutions of hydrodynamical equations at Re = ∞. " -- K. Gawedzki & M. Vergassola (2000)

This dissertation contains detailed and mathematically rigorous investigations of these properties of turbulence for number of hydrodynamic models, with a particular focus on establishing the connections conjectured above.