On energy and magnetic helicity equality in the electron magnetohydrodynamic equations

In this paper, we are concerned with the conservation of energy and magnetic helicity of weak solutions for the three-dimensional electron magnetohydrodynamic (EMHD) equations. Firstly, we establish sufficient conditions to guarantee the energy (magnetic helicity) balance of weak solutions for the EMHD equations based on the magnetic field, which can be viewed as an analogue of famous Lions’ energy balance criterion of the Navier–Stokes equations for the EMHD equations. Secondly, in the spirit of recent works due to Berselli and Chiodaroli (Nonlinear Anal 192: 111704, 2020), as reported by Berselli (Three-Dimensional Navier–Stokes Equations for Turbulence. Academic Press, London, 2021), Berselli (Mathematics 11(4): 1–16, 2023), Berselli (J Differ Equ 368: 350–375, 2023), Berselli and Georgiadis (Nonlinear Differ Equ Appl 31(33): 1–14, 2024), we present energy (magnetic helicity) preservation criteria in terms of the current density in this system for both the whole space and the torus cases.