The Obukhov--Corrsin spectrum of passive scalar turbulence through anomalous regularization

The Obukhov--Corrsin spectrum predicts the distribution of Fourier mass for a passive scalar field advected by a "turbulent" velocity field with spatial regularity $C^\alpha_x$ for $\alpha \in (0,1)$ and subject to a time-stationary forcing. We prove the Obukhov--Corrsin spectrum holds after summing over geometric annuli in Fourier space -- up to logarithmic corrections -- as a consequence of a sharp anomalous regularization result. We then prove this anomalous regularization for a broad class of Kraichnan-type models. The proof of anomalous regularization relies on a Fourier space $\ell^p$ energy equality and a weighted lattice Poincar\'e inequality.