A uniqueness result for the d-dimensional magnetohydrodynamics equations with fractional dissipation in Besov spaces

In this paper, we consider the Cauchy problem of d-dimensional magnetohydrodynamics equations ($d\geq 2$) with fractional dissipation $(-\Delta)^{\alpha}u$ and fractional magnetic diffusion $(-\Delta)^{\beta}b$. The aim of this paper is to establish the uniqueness of weak solutions under the $L^p$ framework in sense of the weakest possible inhomogeneous Besov spaces. We obtain the local existence and uniqueness in the functional setting $u\in L_T^{\infty}(B_{p,1}^{\frac{d}{p}+1-2\alpha}(\mathbb{R}^d))$ and $b\in L_T^{\infty}(B_{p,1}^{\frac{d}{p}}(\mathbb{R}^d))$ when $\alpha$ and $\beta$ satisfy certain conditions by using the iterative scheme and compactness arguments.