Stability of hydrostatic equilibrium for the 2D BMHD system with partial dissipation

<p style='text-indent:20px;'>In this paper, we establish the nonlinear stability and large time behavior of hydrostatic equilibrium in a uniform magnetic field for Boussinesq system with magnetohydrodynamics convection whole space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^{2} $\end{document}</tex-math></inline-formula> mixed partial dissipation, motivated by Lai, Wu, Zhong [<xref ref-type="bibr" rid="b18">18</xref>] Lin, Ji, Wu Yan rid="b22">22</xref>]. Due to lack horizontal dissipation vertical second component velocity, natural energy is not easy be closed, which overcome introducing an additional functional derivative velocity. This shows that temperature have stabilizing effect on fluid. Large linear decay rate solution are also obtained.</p>