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