In [7], 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 (e3, 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 e3 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 Hörmander's version of Nash-Moser iteration scheme, which is very much motivated by the seminar papers [18,19,20] by Klainerman on the long time behavior to the evolution equations. (2000): 35Q30, 76D03 for g given by (2.4). Since ∂ z 3 b 0 (y(w(z))) in the source term is a time independent function, we now
AMS Subject Classificationand a correction termỸ so that Y =Ỹ +Ȳ andThen in view of (2.23), (2.24) and (2.30) of [1],Ȳ solves(2.18) 2.2. The proof of Theorem 1.1. Before presenting the main result for the system (2.17-2.18), let us first introduce notations of the norms: For f : R 3 y → R, u : R + × R 3 y → R, and p ∈ [1, +∞], N ∈ N, we denote f W N,p def = |α|≤N D α y f L p and u L p ;k,N def = sup t>0(1 + t) k u(t) W N,p .In particular, when p = 1, p = 2 and p = ∞, we simplify the notations as(2.19)Theorem 2.1. There exist an integer L 0 and small constants η, ε 0 > 0 such that ifThen the system (2.17) has a unique global solutionȲ ∈ C 2 ([0, ∞); C N 1 −4 (R 3 )), where N 1 = [(L 0 − 12)/2]. Furthermore, for any κ > 0, there hold (2.21) |∂ 3Ȳ | 3 4 −κ,2 + |Ȳ t | 5 4 −κ,2 + |Ȳ | 1 4 −κ,2 ≤ C κ η, and |D| −1 (∂ 3Ȳ ,Ȳ t ) 0,N 1 +2 + ∇Ȳ 0,N 1 +1 + (Ȳ t , ∂ 3Ȳ ) 1 2 ,N 1 +1 + ∇Ȳ t 1,N 1 −1 + Ȳ t L 2 t (H N 1 +2 ) + (∂ 3Ȳ , t 1 2 ∇Ȳ t ) L 2 t (H N 1 +1 ) + Ȳ tt 1 2 ,N 1 −2 ≤ C.(2.22)Admitting Theorem 2.1 for the time being, let us now turn to the proof of Theorem 1.1.