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

We construct a divergence-free velocity field u:[0,T]×T2→R2 satisfying (Formula presented.) such that the corresponding drift-diffusion equation exhibits anomalous dissipation for all smooth initial data. We also show that, given any α0<1, the flow can be modified such that it is uniformly bounded only in Cα0(T2) 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 H1 growth for all solutions to the transport equation, which may be of independent interest.