Anomalous dissipation and Euler flows

TL;DR

This paper proves the first rigorous case of anomalous scalar dissipation in weak solutions of the incompressible Euler equations with specific regularity, advancing the understanding of scalar turbulence and the zeroth law.

Contribution

It provides the first rigorous derivation of the zeroth law of scalar turbulence for Euler flows with weak solutions and scalar advection.

Findings

Demonstrates anomalous scalar dissipation in Euler flows with $C^{(1/3)^-}$ regularity

Establishes a lower bound on scalar variance consistent with turbulence theories

Provides a typicality statement for the drift in scalar advection

Abstract

We show anomalous dissipation of scalars advected by weak solutions to the incompressible Euler equations with $C^{(\sfrac{1}{3})^-}$ regularity, for an arbitrary initial datum in $\dot H^1 (\T^3)$. This is the first rigorous derivation of zeroth law of scalar turbulence, where the scalar is advected by solution to an equation of hydrodynamics (unforced and deterministic). As a byproduct of our method, we provide a typicality statement for the drift, and recover certain desired properties of turbulence, including a lower bound on scalar variance commensurate with the Richardson pair dispersion hypothesis.