No abstract

The three-dimensional incompressible magnetohydrodynamic (MHD) system with only vertical dissipation arises in the study of reconnecting plasmas. When the spatial domain is the whole space $\mathbb R^3$, the small data global well-posedness remains an extremely challenging open problem. The one-directional dissipation is simply not sufficient to control the nonlinearity in $\mathbb R^3$. This paper solves this open problem when the spatial domain is the strip $\Omega := \mathbb R^2\times [0,1]$ with Dirichlet boundary conditions. By invoking suitable Poincaré type inequalities and designing a multi-step scheme to separate the estimates of the horizontal and the vertical derivatives, we are able to establish the global well-posedness in the Sobolev setting $H^3$ as long as the initial horizontal derivatives are small. We impose no smallness condition on the vertical derivatives of the initial data. Furthermore, the $H^3$-norm of the solution is shown to decay exponentially in time. This exponential decay is surprising for a system with no horizontal dissipation. This large-time behavior reflects the smoothing and stabilizing phenomenon due to the interaction within the MHD system and with the boundary.

<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>

This paper examines the uniqueness of weak solutions to the ddimensional magnetohydrodynamic (MHD) equations with the fractional dissipation (−∆) α u and without the magnetic diffusion. Important progress has been made on the standard Laplacian dissipation case α = 1. This paper discovers that there are new phenomena with the case α < 1. The approach for α = 1 can not be directly extended to α < 1. We establish that, for α < 1, any initial data(R d ) guarantees the existence and uniqueness. These regularity requirements appear to be optimal.