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Anomalous Dissipation in Passive Scalar Transport

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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 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.

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  1. We remark that it is also unknown whether this double exponential lower bound above is attained for any flow. In discrete time [38] produce an example where is in fact attained. In continuous time, however, there are no examples exhibiting the double exponential decay. Moreover, Miles and Doering [49] provide numerical evidence and a heuristic argument that the Batchelor scale limits the effectiveness of mixing, suggesting that the \(L^2\) energy can only decay exponentially.

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Authors and Affiliations

  1. Department of Mathematics, Stony Brook University, Stony Brook, NY, USA

    Theodore D. Drivas

  2. Department of Mathematics, Duke University, Durham, USA

    Tarek M. Elgindi

  3. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, USA

    Gautam Iyer

  4. Department of Mathematical Sciences and RIM, Seoul National University, Seoul, South Korea

    In-Jee Jeong

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  1. Theodore D. Drivas
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  2. Tarek M. Elgindi
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  3. Gautam Iyer
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  4. In-Jee Jeong
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Correspondence to In-Jee Jeong.

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Communicated by P. Constantin.

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This work has been partially supported by the National Science Foundation under grants DMS-1703997 to TD, DMS-1817134 to TE, DMS-1814147 to GI, as well as by the Center for Nonlinear Analysis. IJ has been supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA2002-04. TD and TE would like to thank the Korean Institute for Advanced Study (KIAS) for its hospitality. TD would like to thank Navid Constantinou for useful discussions.

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Drivas, T.D., Elgindi, T.M., Iyer, G. et al. Anomalous Dissipation in Passive Scalar Transport. Arch Rational Mech Anal 243, 1151–1180 (2022). https://doi.org/10.1007/s00205-021-01736-2

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