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arXiv · 1911.03271

Anomalous Dissipation in Passive Scalar Transport

Theodore D. Drivas·Tarek M. Elgindi·Gautam Iyer·In-Jee Jeong
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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 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.

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Theodore D. Drivas, Tarek M. Elgindi, Gautam Iyer, In-Jee Jeong. 2019-11-08. Anomalous Dissipation in Passive Scalar Transport. https://arxiv.org/abs/1911.03271

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