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On the Weak Solutions to the Maxwell–Landau–Lifshitz Equations and to the Hall–Magneto–Hydrodynamic Equations

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Abstract

In this paper we deal with weak solutions to the Maxwell–Landau–Lifshitz equations and to the Hall–Magneto–Hydrodynamic equations. First we prove that these solutions satisfy some weak-strong uniqueness property. Then we investigate the validity of energy identities. In particular we give a sufficient condition on the regularity of weak solutions to rule out anomalous dissipation. In the case of the Hall–Magneto–Hydrodynamic equations we also give a sufficient condition to guarantee the magneto-helicity identity. Our conditions correspond to the same heuristic scaling as the one introduced by Onsager in hydrodynamic theory. Finally we examine the sign, locally, of the anomalous dissipations of weak solutions obtained by some natural approximation processes.

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Authors and Affiliations

  1. Institut Fourier, Université Grenoble 1, 100, rue des mathématiques, BP 74, 38402, Saint Martin d’Hères, France

    Eric Dumas

  2. Sorbonne Universités, Laboratoire Jacques-Louis Lions, UPMC Univ Paris 06, UMR 7598, 75005, Paris, France

    Franck Sueur

  3. Laboratoire Jacques-Louis Lions, CNRS, UMR 7598, 75005, Paris, France

    Franck Sueur

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  1. Eric Dumas
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  2. Franck Sueur
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Correspondence to Franck Sueur.

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Communicated by W. Schlag

E. Dumas support by The Nano-Science Foundation, Grenoble, Project HM-MAG, is acknowledged.

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Dumas, E., Sueur, F. On the Weak Solutions to the Maxwell–Landau–Lifshitz Equations and to the Hall–Magneto–Hydrodynamic Equations. Commun. Math. Phys. 330, 1179–1225 (2014). https://doi.org/10.1007/s00220-014-1924-1

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