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

We construct a divergence-free velocity field $u:[0,T] \times \mathbb{T}^2 \to \mathbb{R}^2$ satisfying $$u \in C^\infty([0,T];C^\alpha(\mathbb{T}^2)) \quad \forall \alpha \in [0,1)$$ 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.