Stability of hydrostatic equilibrium of the 2D Boussinesq-MHD equations without magnetic diffusion in two kinds of periodic domains

In this paper, we investigate the nonlinear stability of 2D Boussinesq-MHD equations around the hydrostatic equilibrium state \((\overline{U},\overline{B},\overline{\Theta }, \overline{P})=(0,e_{1},y,\frac{1}{2}y^{2})\) $(\overline{U},\overline{B},\overline{\mathrm{\Theta }},\overline{P})=(0,e_1,y,\frac{1}{2}{y}^2)$ in two kinds of periodic domains: \(\mathbb {T}\times (0,1)\) $\mathbb{T}\times (0,1)$ and \(\mathbb {T}\times \mathbb {R}\) $\mathbb{T}\times \mathbb{R}$ . Due to the absence of dissipation of magnetic field, it brings great difficulties to obtain the uniformly a priori estimate of \(\partial _{1}b\) ${\partial }_{1}b$ . The main technique is utilizing the couple of equations between u and b to obtain the anisotropic estimates of ( \(u,b,\theta \) $u,b,\theta$ ) and then we obtain the expected stability results.