Norm Growth, Non-uniqueness, and Anomalous Dissipation in Passive Scalars

Abstract

We construct a divergence-free velocity field $u:[0,T]\times {\mathbb{T}}^2\to {\mathbb{R}}^2$ satisfying $$u\in C^{\infty }([0,T];C^{\alpha }({\mathbb{T}}^2))\forall \alpha \in [0,1)$$ such that the corresponding drift-diffusion equation exhibits anomalous dissipation for every smooth initial data. We also show that, given any ${\alpha }_0<1$, the flow can be modified such that it is uniformly bounded only in $C^{{\alpha }_0}({\mathbb{T}}^2)$ and the regularity of solutions satisfy sharp (time-integrated) bounds predicted by the Obukhov-Corrsin theory. The proof is based on a general principle implying $H^1$ growth for all solutions to the transport equation, which may be of independent interest.

23 pages, 1 figure