On two conserved quantities in the inviscid electron and Hall magnetohydrodynamic equations

In this paper, we are concerned with the energy and magnetic helicity conservation of weak solutions for both the electron and Hall magnetohydrodynamic equations. Various sufficient criteria to ensure the energy and magnetic helicity conservation in Onsager’s critical spaces ${\underset{‾}{B}}_{p,VMO}^{\alpha }$ and $B_{p,c(\mathbb{N})}^{\alpha }$ in these systems are established. Moreover, for the E-MHD equations, we observe that the conservation criteria of energy and magnetic helicity to the E-MHD equations correspond to the helicity and energy to the ideal incompressible Euler equations, respectively.