On the global solution of 3-D MHD system with initial data near equilibrium

Abstract

In this paper, we prove the global existence of smooth solutions to the three-dimensional incompressible magneto-hydrodynamical system with initial data close enough to the equilibrium state, $(e_3,0).$ Compared with the the previous works \cite{XLZMHD1, XZ15}, here we present a new Lagrangian formulation of the system, which is a damped wave equation and which is non-degenerate only in the direction of the initial magnetic field. Furthermore, we remove the admissible condition on the initial magnetic field, which was required in \cite{XLZMHD1, XZ15}. By using Frobenius Theorem and anisotropic Littlewood-Paley theory for the Lagrangian formulation of the system, we achieve the global $L^1$ in time Lipschwitz estimate of the velocity field, which allows us to conclude the global existence of solutions to this system. In the case when the initial magnetic field is a constant vector, the large time decay rate of the solution is also obtained.