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

TL;DR

This paper introduces a novel approach within the Besov space framework to prove the existence and uniqueness of local weak solutions for a fractional magnetic Bénard system in multiple dimensions, without thermal diffusion.

Contribution

It develops a new method in Besov spaces to analyze the fractional magnetic Bénard system, extending previous results to higher dimensions and fractional orders.

Findings

Existence of local weak solutions under specified initial conditions.

Uniqueness of solutions in the considered functional framework.

Extension of results to fractional orders and higher dimensions.

Abstract

This work investigates the existence and uniqueness of local weak solutions for the d-dimensional $(d \geq 2)$ fractional magnetic B\'enard system without thermal diffusion, integrating the B\'enard equation and MHD system. For $\kappa = 0$ and $1 \leq \alpha=\beta < 1 + \dfrac{d}{4}$, we establish that any starting conditions $(u_{0},B_{0})\in B_{2,1}^{1+\frac{d}{2}-2\alpha}(\mathbb{R}^{d})$ and $\theta_{0}\in B_{2,1}^{1+\frac{d}{2}-\alpha}(\mathbb{R}^{d})$.