Well-Posedness of MHD Equations in Sobolev-Gevery Space

Abstract

This paper is devoted to the study of the 3D incompressible magnetohydrodynamic system. We prove the local in time well-posedness for any large initial data in $\dot{H}^1_{a,1}(\mathbb{R}^3)$ or $H^1_{a,1}(\mathbb{R}^3).$ Furthermore, the global well-posedness of a strong solution in $\tilde{L}^∞(0, T; H^1_{ a,1}(\mathbb{R}^3)) ∩ L^2 (0, T; \dot{H}^1_{a,1}(\mathbb{R}^3) ∩ \dot{H}^2_{a,1}(\mathbb{R}^3))$ with initial data satisfying a smallness condition is established.