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

In this paper, we are concerned with the conservation of energy and magnetic helicity of weak solutions for both the electron and Hall magnetohydrodynamic equations. Various energy and magnetic helicity conservation criteria in Onsager’s critical spaces ${\underline{B}}_{p,VMO}^{\alpha }$ and $B_{p,c\left(N\right)}^{\alpha }$ in these systems are established. Furthermore, we observe that the conservation criteria for energy and magnetic helicity in the EMHD equations correspond to the helicity and energy conservation principles in ideal incompressible Euler equations, respectively.