Remarks on uniqueness and energy conservation for electron-MHD system

This paper is concerned with the uniqueness and energy conservation of weak solutions for Electron-MHD system. Under suitable assumptions, we first show that the Electron-MHD system has a unique weak solution. In addition, we show that weak solution conserves energy if \(\nabla \times b\in L^2(0, T; L^4({\mathbb {R}}^d))(d\ge 2)\) $\mathrm{\nabla }\times b\in L^2(0,T;L^4({\mathbb{R}}^d))(d\ge 2)$ or \( \nabla \times b \in L^{\frac{4d+8}{d+4}}\left( 0, T; L^{\frac{4d+8}{d+4}}({\mathbb {R}}^{d})\right) (d=2, 3, 4)\) $\mathrm{\nabla }\times b\in L^{\frac{4d+8}{d+4}}\left(0,,,T,;,L^{\frac{4d+8}{d+4}},({\mathbb{R}}^d)\right)(d=2,3,4)$ .