Large time behavior of solutions to 3-D MHD system with initial data near equilibrium

Abstract

In \cite{ChCa}, Califano and Chiuderi conjectured that the energy of incompressible Magnetic hydrodynamical system is dissipated at a rate that is independent of the ohmic resistivity. The goal of this paper is to mathematically justify this conjecture in three space dimension provided that the initial magnetic field and velocity is a small perturbation of the equilibrium state $(e_3,0).$ In particular, we prove that for such data, 3-D incompressible MHD system without magnetic diffusion has a unique global solution. Furthermore, the velocity field and the difference between the magnetic field and $e_3$ decay to zero in both $L^{\infty }$ and $L^2$ norms with explicit rates. We point out that the decay rate in the $L^2$ norm is optimal in sense that this rate coincides with that of the linear system. The main idea of the proof is to exploit H$\ddot{o}$rmander's version of Nash-Moser iteration scheme, which is very much motivated by the seminar papers \cite{Kl80, Kl82, Kl84} by Klainerman on the long time behavior to the evolution equations.

References in corpus (4)

• Large time behavior of solutions to 3-D MHD system with initial data near equilibrium
• Global well-posedness of the MHD equations in a homogeneous magnetic field
• Global Small Solutions to a Complex Fluid Model in 3D
• On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of Alfvén waves

Cited by in corpus (6)

• Large time behavior of solutions to 3-D MHD system with initial data near equilibrium
• Long time behavior of Alfvén waves in a flowing plasma: mathematical analysis on the generation of the magnetic island
• On asymptotic stability of the 3D Boussinesq equations without thermal conduction
• Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium
• Low regularity ill-posedness and shock formation for 3D ideal compressible MHD
• Asymptotic Behaviors of Global Solutions to the Two-Dimensional Non-resistive MHD Equations with Large Initial Perturbations