Global Existence for 3D Anisotropic MHD system with Horizontal Dissipation and Small Horizontal Variations

Abstract

This paper establishes the global well-posedness for the 3D anisotropic MHD system with partial dissipation: $\hat{I}{”}_{\mathrm{h}}u$ for velocity and ${\partial }_1^{2}b$ for magnetic field, near background field $(0,1,0)$. Crucially, only horizontal components $(u_0^{\mathrm{h}},b_0^{\mathrm{h}})$ need to be small in $H^2({\mathbb{R}}^3)$, while $(u_0^3,b_0^3)$ can be arbitrarily large. Our analysis develops novel techniques including component-decoupled energies and iterative control of dangerous nonlinearities using the background field structure. This establishes the global result for anisotropic MHD equations allowing large vertical data, breaking the full-smallness requirement of previous works.