Linear stability analysis of 2D incompressible MHD equations with only magnetic diffusion

Although many physical experiments and numerical simulations show that the magnetic field can stabilize and inhibit electrically conducting fluids, whether 2D incompressible MHD equations with only magnetic diffusion develop finite time singularities or not is one of the most challenging problems and remains open. Therefore, this issue has always attracted a lot of attention of mathematicians. Due to its linearized system plays a crucial role, to deeper understand the aforesaid issue, in this paper, we make the first attempt to study its linear stability when the magnetic field close to the equilibrium state $e_2=(0,1)$ in the periodic domain and ultimately proposed the linear stability condition (1.4). To be more precise, we show that the solution of its linearized system will be time-asymptotically stable and converge to the equilibrium state in the algebraic rate via the method of spectral analysis, as long as the integrals in the vertical direction of initial perturbations are zeros.