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

Abstract

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.

On Current Developments in Partial Differential Equations

“…Starting with [85] for a modified model of (2.26), Lin et al proved the global existence of solutions to (2.26) with initial velocity being sufficiently small and the initial magnetic field being sufficiently close to non-zero constant vector. The same result for (2.26) was proved in [1] and an optimal decay rates for such solutions was obtained in [41]. He et al proved the vanishing viscosity limit of the viscous MHD system in [58] (see also [16,118]).…”

Asymptotic Behaviors of Global Solutions to the Two-Dimensional Non-resistive MHD Equations with Large Initial Perturbations

“…where b 0 = 0 and v 0 ∈ H s with s ∈ (3/2, 3]. Very recently, under the assumption that the initial data are sufficiently smooth, Deng-Zhang further established the following faster time-decay than (1.5) [7]:…”

Global Well-Posedness for the 2-D MHD Equations with Magnetic Diffusion

“…Initiated by Lin and Zhang [11], there are many papers [1,2,6,7,10,12,14,15] devoted to the global well-posedness of the MHD system with dissipation but without magnetic resistivity or the MHD system with small full diffusion under a strong magnetic field. Then a natural question is whether the MHD system with partial diffusion has a global smooth solution for small data.…”