Regularity thresholds for anomalous dissipation and related phenomena in passive scalars

TL;DR

This paper proves the absence of anomalous dissipation in passive scalars driven by certain random divergence-free vector fields, under specific regularity and geometric conditions, using dimension-theoretic methods.

Contribution

It establishes new regularity thresholds preventing anomalous dissipation and related turbulent phenomena for passive scalars in various dimensions.

Findings

No anomalous dissipation occurs under the specified conditions.

The results apply to passive scalars with minimal initial regularity.

Certain turbulent flow properties are also prevented by these assumptions.

Abstract

We prove the absence of anomalous dissipation for passive scalars driven by some random autonomous divergence-free vector fields in $\mathbb T^d$. In dimension $d=2$ we just need continuity almost surely and a mild nondegeneracy condition on the randomness. In dimension $d\geq 3$ we assume a special geometric structure and almost sure H\"older regularity with a H\"older exponent bigger than $\frac{1}{8}$. No regularity is assumed on the passive scalar except for boundedness in the initial data. The proof relies on dimension-theoretic arguments, as opposed to commutator estimates. A consequence of these results is that the same assumptions prevent (almost surely) many other expected properties of turbulent flows, such as anomalous regularization, the Yaglom-Obukhov-Corrsin law, and Richardson diffusion.