On the weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-Magneto-Hydrodynamic equations

Eric Dumas (1) , Franck Sueur (2)

1 IF - Institut Fourier
2 LJLL - Laboratoire Jacques-Louis Lions

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.

Mots clés: weak-strong uniqueness; suitable weak solutions.; suitable weak solutions; Maxwell-Landau-Lifshitz equation; Hall-Magneto-Hydrodynamic equation; anomalous dissipation

Domaines: Equations aux dérivées partielles [math.AP]

Eric Dumas, Franck Sueur. On the weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-Magneto-Hydrodynamic equations. Communications in Mathematical Physics, 2014, 330 (3), pp.1179-1225. ⟨hal-00816222⟩