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

We construct a divergence-free velocity field \(u:[0,T] \times \mathbb {T}^2 \rightarrow \mathbb {R}^2\) 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 all 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.