Stability and inviscid limit of the 3D anisotropic MHD system near a background magnetic field with mixed fractional partial dissipation

A main result of this paper establishes the global stability of the three-dimensional MHD equations near a background magnetic field with mixed fractional partial dissipation with $\alpha ,\beta \in (\frac{1}{2},1]$. Namely, the velocity equations involve dissipation $({\mathrm{\Lambda }}_1^{2\alpha }+{\mathrm{\Lambda }}_2^{2\alpha }+\sigma {\mathrm{\Lambda }}_3^{2\alpha })u$ with the case $\sigma =1$ and $\sigma =0$. The magnetic equations without partial magnetic diffusion ${\mathrm{\Lambda }}_i^{2\beta }{b}_i$ but with the diffusion $(-\mathrm{\Delta }{)}^{\beta }b$, where ${\mathrm{\Lambda }}_i^s(s>0)$ with $i=1,2,3$ are the directional fractional operators. Then we focus on the vanishing vertical kinematic viscosity coefficient limit of the MHD system with the case $\sigma =1$ to the case $\sigma =0$. The convergent result is obtained in the sense of $H^1$-norm.