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Dyadic models for ideal MHD

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Abstract

We study two dyadic models for incompressible ideal magnetohydrodynamics, one with a uni-directional energy cascade and the other one with both forward and backward energy cascades. Global existence of weak solutions and local well-posedness are established for both models. In addition, solutions to the model with uni-directional energy cascade associated with positive initial data are shown to develop blow-up at a finite time. Moreover, a set of fixed points is found for each model. Linear instability about some particular fixed points is proved.

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Acknowledgements

The work of M. Dai is partially supported by NSF Grants DMS-1815069 and DMS-2009422; the work of S. Friedlander is partially supported by NSF Grant DMS-1613135. S. Friedlander is grateful to IAS for its hospitality in 2020-2021.

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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 G. P. Galdi.

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Dai, M., Friedlander, S. Dyadic models for ideal MHD. J. Math. Fluid Mech. 24, 21 (2022). https://doi.org/10.1007/s00021-021-00640-9

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