This paper examines the existence and uniqueness of weak solutions to the d‐dimensional magnetohydrodynamic (MHD) equations with fractional dissipation ${(-\mathrm{\Delta })}^{\alpha }u$ and fractional magnetic diffusion ${(-\mathrm{\Delta })}^{\beta }b$. The aim is at the uniqueness of weak solutions in the weakest possible inhomogeneous Besov spaces. We establish the local existence and uniqueness in the functional setting $u\in L^{\infty }(0,T;B_{2,1}^{d/2-2\alpha +1}({\mathbb{R}}^d))$ and $b\in L^{\infty }(0,T;B_{2,1}^{d/2}{\mathbb{R}}^d))$ when $\alpha >1/2$, $\beta \ge 0$ and $\alpha +\beta \ge 1$. The case when $\alpha =1$ with $\nu >0$ and $\eta =0$ has previously been studied in [7, 19]. However, their approaches can not be directly extended to the fractional case when $\alpha <1$ due to the breakdown of a bilinear estimate. By decomposing the bilinear term into different frequencies, we are able to obtain a suitable upper bound on the bilinear term for $\alpha <1$, which allows us to close the estimates in the aforementioned Besov spaces.

中文翻译：

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具有分数耗散的磁流体动力学方程的唯一弱解

本文研究了具有分数耗散的d维磁流体动力学（MHD）方程的弱解的存在性和唯一性${（-\mathrm{\Delta }）}^{\alpha }ü$ 和分数磁扩散 ${（-\mathrm{\Delta }）}^{\beta }b$。目的是在最弱的不均匀Besov空间中弱解的唯一性。我们在功能设置中建立局部存在性和唯一性$ü\in {\mathrm{大号}}^{\infty }（0，Ť;{乙}_{2，\mathrm{1个}}^{d/2-2\alpha +\mathrm{1个}}（{\mathbb{[R}}^d））$ 和 $b\in {\mathrm{大号}}^{\infty }（0，Ť;{乙}_{2，\mathrm{1个}}^{d/2}{\mathbb{[R}}^d））$ 什么时候 $\alpha >\mathrm{1个}/2$， $\beta \ge 0$ 和 $\alpha +\beta \ge \mathrm{1个}$。当时的情况$\alpha =\mathrm{1个}$ 与 $\nu >0$ 和 $\eta =0$先前已经在[7，19]中进行了研究。但是，它们的方法不能直接扩展到小数情况下$\alpha <\mathrm{1个}$由于双线性估计的细分。通过将双线性项分解为不同的频率，我们可以在双线性项上获得合适的上限$\alpha <\mathrm{1个}$，这使我们可以关闭上述Besov空间中的估计。