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

## TL;DR

This paper proves that solutions to the 3-D incompressible MHD system with small perturbations near equilibrium decay over time, confirming a conjecture that energy dissipation is independent of magnetic resistivity.

## Contribution

It provides a rigorous mathematical justification for the energy dissipation conjecture in 3-D MHD systems without magnetic diffusion for small initial data.

## Key findings

- Global existence of solutions for small perturbations
- Decay of velocity and magnetic field difference at explicit rates
- Decay rate matches that of the linear system, indicating optimality

## 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.