New approach in the Besov space to the existence and uniqueness of solutions of the D-dimensional fractional magnetic Bénard system without thermal diffusion

Abstract

This work investigates the existence and uniqueness of local weak solutions for the d-dimensional $(d\ge 2)$ fractional magnetic Bénard system without thermal diffusion, integrating the Bénard equation and MHD system. For $\kappa =0$ and $1\le \alpha =\beta <1+\frac{d}{4}$, we establish that any starting conditions $(u_0,B_0)\in B_{2,1}^{1+\frac{d}{2}-2\alpha }({\mathbb{R}}^d)$ and ${\theta }_0\in B_{2,1}^{1+\frac{d}{2}-\alpha }({\mathbb{R}}^d)$.