Anomalous Dissipation in Passive Scalar Transport

We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible C^∞([0 , T) × T^d) ∩ L¹([0 , T] ; C^{1 -}(T^d)) velocity field which explicitly exhibits anomalous dissipation. As a consequence, this example also shows the non-uniqueness of solutions to the transport equation with an incompressible L¹([0 , T] ; C^{1 -}(T^d)) drift, which is smooth except at one point in time. We also give a sufficient condition for anomalous dissipation based on solutions to the inviscid equation becoming singular in a controlled way. Finally, we discuss connections to the Obukhov-Corrsin monofractal theory of scalar turbulence along with other potential applications.