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Communications in Mathematical Physics

, Volume 330, Issue 3, pp 1179–1225 | Cite as

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

  • Eric Dumas
  • Franck SueurEmail author
Article

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.

Keywords

Weak Solution Euler Equation Energy Inequality Global Weak Solution Energy Identity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut FourierUniversité Grenoble 1Saint Martin d’HèresFrance
  2. 2.Sorbonne UniversitésLaboratoire Jacques-Louis LionsParisFrance
  3. 3.Laboratoire Jacques-Louis LionsCNRS, UMR 7598ParisFrance

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