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Uniqueness and Non-Uniqueness Results for Forced Dyadic MHD Models

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Abstract

We construct non-unique Leray–Hopf solutions for some forced dyadic models for magnetohydrodynamics (MHD) when the intermittency dimension \(\delta \) is less than 1. Conventionally, the interaction of the velocity and magnetic fields is a major challenge in the context of MHD. However, in the dyadic MHD model scenario, we exploit to our benefit certain symmetries in the interactions of the fields to obtain a non-uniqueness result. In contrast, uniqueness of the Leray–Hopf solution to the dyadic MHD models is established in the case of \(\delta \ge 1\). Analogous results on uniqueness and non-uniqueness of Leray–Hopf solution are also obtained for dyadic models of MHD with fractional diffusion.

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Acknowledgements

M. Dai is partially supported by the NSF Grants DMS-1815069 and DMS-2009422. S. Friedlander is partially supported by the NSF Grant DMS-1613135. S. Friedlander is grateful to IAS for its hospitality in 2020–2021. M. Dai is also grateful to IAS for its hospitality in 2021–2022. The authors are very grateful to the anonymous referees for their constructive suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Mimi Dai

  2. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    Susan Friedlander

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  1. Mimi Dai
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  2. Susan Friedlander
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Correspondence to Mimi Dai.

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Communicated by Paul Newton.

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Dai, M., Friedlander, S. Uniqueness and Non-Uniqueness Results for Forced Dyadic MHD Models. J Nonlinear Sci 33, 10 (2023). https://doi.org/10.1007/s00332-022-09868-9

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