Energy conservation for the nonhomogeneous incompressible ideal Hall-MHD equations

In this paper, we study the energy conservation for the nonhomogeneous incompressible ideal Hall-magnetohydrodynamic system. Three types of sufficient conditions are obtained. Precisely, the first one provides ρ, u, P, and b with sufficient regularity to ensure the local energy conservation. The second one removes the regularity condition on P while requires Lp regularity on the spatial gradient of the density ∇ρ and Lr regularity on ρt. The last one removes the regularity condition on ρt while requires certain time regularity on the velocity field u. Our main strategy relies on commutator estimates in the work of Constantin et al. [Commun. Math. Phys. 165, 207–209 (1994)].