Anomalous Dissipation in Passive Scalar Transport

Abstract

We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible \(C^\infty ([0,T)\times {\mathbb {T}}^d)\cap L^1([0,T]; C^{1-}({\mathbb {T}}^d))\) 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 \(L^1([0,T]; C^{1-}({\mathbb {T}}^d))\) 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.