Global energy conservation for distributional solutions to incompressible Hall-MHD equations without resistivity

Abstract

This paper concerns the global energy conservation for distributional solutions to incompressible Hall-MHD equations without resistivity. {\color{red} Motivated by the works of Tan and Wu in [arXiv:2111.13547v2] and Wu in [J. Math. Fluid Mech. 24, 111 (2022)],}we establish the energy balance for a distributional solution in whole spaces $\mathbb{R}^d(d\geq 2)$ provided that $u, b \in L^4L^4$ and $\nabla b\in L^{\frac{8}{3}}L^{\frac{8}{3}}$.
Moreover, as a corollary, we also obtain the energy conservation criterion for a Leray-Hopf weak solution.