  We study anomalous dissipation in hydrodynamic turbulence in 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 anomalous dissipation. As a
consequence, this example also shows non-uniqueness of solutions to the
transport equation with an incompressible $L^1([0,T]; C^{1-}(\mathbb{T}^d))$
drift, which is smooth except at one point in time. We also provide three
sufficient conditions for anomalous dissipation provided solutions to the
inviscid equation become singular in a controlled way. Finally, we discuss
connections to the Obukhov-Corrsin monofractal theory of scalar turbulence
along with other potential applications.
