Global well-posedness of 2D incompressible Magnetohydrodynamic equations with horizontal dissipation

Abstract

<p style='text-indent:20px;'>This paper focuses on two-dimensional incompressible non-resistive MHD equations with only horizontal dissipation in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{T}\times\mathbb{R} $\end{document}</tex-math></inline-formula>. Invoking three Poincaré-type inequalities about the horizontal derivative, we study the global well-posedness of the system near a background magnetic via the structure of the perturbation MHD system and the symmetry condition imposed on the initial data. By a precise time-weighted energy estimate, we also establish the global well-posedness of the system with only horizontal magnetic damping. Here we overcome the difficulties brought by the absence of magnetic diffusion and the appearance of the boundary. We note that the stability of MHD equations with one-directional dissipation in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> or a bounded domain appears to be unknown.</p>