Large Time Behavior of Solutions to 3-D MHD System with Initial Data Near Equilibrium

Califano and Chiuderi (Phys Rev E 60 (PartB):4701-4707, 1999) conjectured that the energy of an 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 dimensions 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, a 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 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 Hormander's version of the Nash-Moser iteration scheme, which is very much motivated by the seminar papers by Klainerman (Commun Pure Appl Math 33:43-101, 1980, Arch Ration Mech Anal 78:73-98, 1982, Long time behaviour of solutions to nonlinear wave equations. PWN, Warsaw, pp 1209-1215, 1984) on the long time behavior to the evolution equations.