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

TL;DR

This paper proves the global existence and decay of smooth solutions for the 3D incompressible MHD system near equilibrium, introducing a new Lagrangian formulation and removing previous restrictions on initial magnetic fields.

Contribution

It presents a novel Lagrangian formulation of the 3D MHD system, removes initial magnetic field restrictions, and establishes global solutions with decay rates.

Findings

Global existence of smooth solutions near equilibrium.

New Lagrangian formulation as a damped wave equation.

Decay rates for solutions with constant initial magnetic field.

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.