Anomalous dissipation and lack of selection in the Obukhov-Corrsin theory of scalar turbulence

The Obukhov-Corrsin theory of scalar turbulence [Obu49, Cor51] advances quantitative predictions on passive-scalar advection in a turbulent regime and can be regarded as the analogue for passive scalars of Kolmogorov's K41 theory of fully developed turbulence [Kol41]. The scaling analysis of Obukhov and Corrsin from 1949-1951 identifies a critical regularity threshold for the advection-diffusion equation and predicts anomalous dissipation in the limit of vanishing diffusivity in the supercritical regime. In this paper we provide a fully rigorous mathematical validation of this prediction by constructing a velocity field such that the unique bounded solution of the advection-diffusion equation is bounded uniformly-in-diffusivity in the full supercritical Obukhov-Corrsin regularity regime and exhibits anomalous dissipation. We also show that for a velocity field in $C^{\alpha}$ of space and time (for an arbitrary $0 \leq {\alpha} < 1$) neither vanishing diffusivity nor regularization by convolution provide a selection criterion for bounded solutions of the advection equation.