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 $C_x^{\alpha }$ 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 ${\mathrm{\ell }}^p$ energy equality and a weighted lattice Poincaré inequality.