Anomalous Dissipation in Passive Scalar Transport

We study anomalous dissipation in hydrodynamic turbulence the context of passive scalars. Our main result produces an incompressible $$C^\infty ([0,T)\times {\mathbb {T}}^d)\cap L^1([0,T]; C^{1-}({\mathbb {T}}^d))$$ velocity field which explicitly exhibits dissipation. As a consequence, this example also shows non-uniqueness solutions to transport equation with $$L^1([0,T]; drift, is smooth except at one point time. give sufficient condition for based on inviscid becoming singular controlled way. Finally, we discuss connections Obukhov-Corrsin monofractal theory scalar along other potential applications.