A universal total anomalous dissipator

A fundamental feature of turbulence, first predicted by Richardson, is the explosive dispersion of advected particles: at sufficiently late times, the separation of initially close particles grows independently of their initial separation. This implies that turbulent flows exhibit spontaneous stochasticity, or the non-uniqueness of Lagrangian trajectories in the zero-noise limit. Spontaneous stochasticity is mathematically equivalent to another well-documented phenomenon in turbulence: the anomalous dissipation of passive scalars. However, a rigorous mathematical demonstration of these effects in fluid models remains elusive. To investigate anomalous dissipation in a more tractable setting, one seeks to construct incompressible vector fields that explicitly exhibit this behaviour. Only recently have explicit deterministic vector fields with anomalous dissipation been constructed.

In this talk, I will provide an overview of spontaneous stochasticity/anomalous dissipation and discuss my recent work with Keefer Rowan, where we construct a universal total anomalous dissipator—a vector field that completely dissipates any initial data in unit time in the vanishing noise limit. Specifically, we construct a vector field such that the laws of the associated SDEs remain diffuse even as the noise vanishes. In fact, as the noise approaches zero, the laws converge to the uniform distribution on the torus.