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

We construct a divergence-free velocity field \(u:[0,T] \times \mathbb {T}^2 \rightarrow \mathbb {R}^2\) $u:[0,T]\times {\mathbb{T}}^2\to {\mathbb{R}}^2$ satisfying \(u \in C^\infty ([0,T];C^\alpha (\mathbb {T}^2)) \quad \forall \alpha \in [0,1)\) $$u\in C^{\infty }([0,T];C^{\alpha }({\mathbb{T}}^2))\forall \alpha \in [0,1)$$ such that the corresponding drift-diffusion equation exhibits anomalous dissipation for all smooth initial data. We also show that, given any \(\alpha _0 < 1\) ${\alpha }_0<1$ , the flow can be modified such that it is uniformly bounded only in \(C^{\alpha _0}(\mathbb {T}^2)\) $C^{{\alpha }_0}({\mathbb{T}}^2)$ 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 \(H^1\) $H^1$ growth for all solutions to the transport equation, which may be of independent interest.