Anomalous dissipation and Euler flows

We show anomalous dissipation of scalars advected by weak solutions to the incompressible Euler equations with $C^{(\text{sfrac}13{)}^{-}}$ regularity, for an arbitrary initial datum in ${\dot{H}}^1({\text{T}}^3)$. 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.