Scalar anomalous dissipation and optimal regularity via iterated homogenization

Abstract

For any ${\hat{I}}_0^2<1/3$ we construct divergence free vector fields in $C_{x,t}^{{\hat{I}}_0^2}$ and a sequence of diffusivities $\hat{I}{º}_q\searrow 0$ such that, for an arbitrary initial datum from a low regularity class, the classical solution $\ddot{I}{}_q$ to the advection-diffusion equation exhibits anomalous dissipation along the sequence $\hat{I}{º}_q$. At the same time $\ddot{I}{}_q$ remains uniformly bounded in $C_t^0{C}_x^{\hat{I}{\pm }_0}$, where ${\hat{I}}_0^2+2\hat{I}{\pm }_0<1$. 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 $\ddot{I}{}_q$ than the classical prediction of Yaglom.

157 pages