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 nonuniqueness 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 ObukhovCorrsin monofractal theory of scalar turbulence along with other potential applications.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.03271
 Bibcode:
 2019arXiv191103271D
 Keywords:

 Mathematics  Analysis of PDEs;
 Physics  Fluid Dynamics