Anomalous dissipation via spontaneous stochasticity with a two-dimensional autonomous velocity field

TL;DR

This paper constructs a specific 2D divergence-free velocity field demonstrating anomalous dissipation of passive scalars, linking spontaneous stochasticity to dissipation through a fluctuation-dissipation approach.

Contribution

It introduces a novel 2D autonomous velocity field in $C^eta$ that exhibits anomalous dissipation, connecting stochastic trajectory variance to dissipation in the zero noise limit.

Findings

Constructed a $C^eta$ velocity field with anomalous dissipation

Linked spontaneous stochasticity to dissipation via fluctuation-dissipation formula

Addressed open questions on anomalous dissipation in 3D Navier-Stokes

Abstract

We study anomalous dissipation in the context of passive scalars and we construct a two-dimensional autonomous divergence-free velocity field in $C^\alpha$ (with $\alpha \in (0,1)$ arbitrary but fixed) which exhibits anomalous dissipation. Our proof employs the fluctuation-dissipation formula, which links spontaneous stochasticity with anomalous dissipation. Therefore, we address the issue of anomalous dissipation by showing that the variance of stochastic trajectories, in the zero noise limit, remains positive. Based on this result, we answer Question 2.2 and Question 2.3 in [Bru\`{e} & De Lellis '22] regarding anomalous dissipation for the forced three-dimensional Navier-Stokes equations.