Global Small Solutions to Three-Dimensional Incompressible Magnetohydrodynamical System

In this paper, we consider the global wellposedness of three-dimensional incompressible magneto-hydrodynamical (MHD) system with small and smooth initial data. The main difficulty of the proof lies in establishing the global in time L¹ estimate for gradient of the velocity field due to the strong degeneracy and anisotropic spectral properties of the linearized system. To achieve this and to avoid the difficulty of propagating anisotropic regularity for the transport equation, we first write our system in the Lagrangian formulation. Then we employ anisotropic Littlewood--Paley analysis to establish the key L¹ in time estimates to the velocity and the gradient of the pressure in the Lagrangian coordinate. With those estimates, we prove the global wellposedness of the Lagrangian formulation with smooth and small initial data by using the energy method. Toward this, we will have to use the algebraic structure of the Lagrangian formulation in a rather crucial way. The global wellposedness of the original system then follows by a suitable change of variables together with a continuous argument. We should point out that compared with the linearized systems of two-dimensional MHD equations of F. Lin, L. Xu, and P. Zhang Global Small Solutions to 2-D MHD System, arXiv: 1302. 5877 and that of the three-dimensional modified MHD equations of F. Lin and P. Zhang Comm. Pure Appl. Math. , 67 (2014), pp. 531--580 our linearized system here is much more degenerate and moreover, the formulation of the initial data for the Lagrangian formulation is more subtle than that of F. Lin, L. Xu, and P. Zhang.