Anomalous Dissipation in Passive Scalar Transport (1911.03271v3)
Abstract: We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible $C\infty([0,T)\times \mathbb{T}d)\cap L1([0,T]; C{1-}(\mathbb{T}d))$ 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 $L1([0,T]; C{1-}(\mathbb{T}d))$ 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.