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

الأصل · EN

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, (e₃,0). Compared with the the previous works 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 XLZMHD1, XZ15. By using Frobenius Theorem and anisotropic Littlewood-Paley theory for the Lagrangian formulation of the system, we achieve the global L¹ 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.