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

TL;DR

This paper studies weak solutions to the Maxwell-Landau-Lifshitz and Hall-Magneto-Hydrodynamic equations, establishing weak-strong uniqueness, energy identities, and conditions to prevent anomalous dissipation, inspired by Onsager's scaling.

Contribution

It provides new conditions on regularity that ensure energy conservation and magneto-helicity identity, extending understanding of weak solutions in these complex systems.

Findings

Weak-strong uniqueness property established.

Conditions to rule out anomalous dissipation derived.

Sufficient criteria for magneto-helicity identity confirmed.

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.