Scalar anomalous dissipation and optimal regularity via iterated homogenization

Abstract

For any $\beta_0<1/3$ we construct divergence free vector fields in $ C_{x,t}^{\beta_0}$ and a sequence of diffusivities $\kappa_q \searrow 0$ such that, for an arbitrary initial datum from a low regularity class, the classical solution $\rho_q$ to the advection-diffusion equation exhibits anomalous dissipation along the sequence $\kappa_q$. At the same time $\rho_q$ remains uniformly bounded in $C_t^{0} C_x^{\alpha_0}$, where $\beta_0 + 2\alpha_0<1$. Our result confirms a conjecture of Armstrong and Vicol \cite{ArmstrongVicol} and shows sharpness of the Obukhov-Corrsin threshold within the context of iterated homogenization. Our construction confirms time-homogeneity of the dissipation anomaly, as required in turbulence theory, and as a consequence we also obtain better time regularity for the scalar $\rho_q$ than the classical prediction of Yaglom.