Decay rate of global solutions to three dimensional generalized MHD system

We investigate the initial value problem for the three dimensional generalized incompressible MHD system. Analyticity of global solutions was proved by energy method in the Fourier space and continuous argument. Then decay rate of global small solutions in the function space \begin{document}$ \mathcal {X}^{1-2\alpha}\bigcap \mathcal {X}^{1-2\beta} $\end{document} follows by constructing time weighted energy inequality.

1. Introduction. In this paper, we investigate decay rate of global solutions to the three dimensional generalized incompressible magnetohydrodynamics (GMHD) system        ∂ t u + u · ∇u − H · ∇H + µΛ 2α u + ∇P = 0, ∂ t H + u · ∇H − H · ∇u + νΛ 2β H = 0, Here u ∈ R 3 is the velocity, H ∈ R 3 is the magnetic field, p is the pressure of the flow and P = p + 1 2 |H| 2 . µ > 0 and ν > 0 are viscosity, magnetic diffusivity, respectively. The fractional Laplacian operator Λ 2α = (−∆) α is defined by Λ 2α f (ξ) = |ξ| 2α f (ξ), where f (ξ) denotes Fourier transform of f . When α = β = 1, (1) is reduced to the classical MHD system. Global wellposedness of MHD system is an important problem. We recall the global wellposedness and asymptotic decay of solutions in Lei-Lin function spaces for our purpose. Motivated by the argument of Lei and Lin [9] for the incompressible Navier-Stokes sytem, global existence of mild solutions in the critical function space was established in [17], proved that the norms of the initial data are bounded exactly by the minimal value of the viscosity coefficients. Later, the first author of this paper [19] proved asymptotic behavior and stability of global mild solution established in [17]. For a class of special large initial data with the critical norm of the initial velocity and magnetic field can be arbitrarily large, global existence of smooth solutions was obtained in [12]. We may refer to [4], [5], [6], [7], [8], [10], [13], [14], [15], [18], [24] [25], [26] for more results. When α = 1 and B = 0, (1) is reduced to the incompressible Navier-Stokes system, global existence of small solutions and long time behavior have been obtained in Lei-Lin type spaces(see [3], [2], [3], [9] and [23]). For the GMHD system (1) with α = β, Ye [21] proved the global well-posedness and decay results with small initial value. Global well-posedness and analyticity results of mild solutions with small initial data are established in [16]. Liu and Wang [11] obtained global large solutions with a class of large data. Very recently, Xiao, Yuan and Zhang [21] established temporal decay estimate of global small solutions obtained in [22]. For 1/2 ≤ α, β ≤ 1, Ye and Zhao [22] proved that global existence of small solutions. Moreover, the authors also proved that the norm of global solutions tends to zero when time tends to infinity.
Firstly, we state global solutions and long time behavior results that have been established by Ye and Zhao [22].
Then there exists δ 0 > 0 such that if δ ≤ δ 0 , then the problem (1), (2) has a global solution u ∈ C([0, ∞); with 1/2 ≤ α, β ≤ 1 be global solutions to the problem (1), (2) given by Theorem 1.1, then Our first result about analyticity of global small solutions, which will play a very important role in our decay rate.
Let (u, H) be the global solutions established in Theorem 1.1. Then there exists Remark 1. When α = β, the result in Theorem 1.3 has been proved in [16]. Therefore, Theorem 1.3 generalizes the corresponding analyticity of global solutions result obtained in [16].
Theorem 1.4. Assume that the conditions of Theorem 1.1 and 1.2 hold. Furthermore, we assume u 0 , H 0 ∈ L 2 . Then and Remark 2. On one hand, (8)- (11) implies that (5) holds. On the other hand, when α = β, the result in Theorem 1.4 is reduced to the corresponding one in [20]. Hence this paper can be viewed as complementary results of the works [20] and [22].
Notations For s ∈ R, the function space X s is defined by which is equipped with the norm 2. Proof of theorem 1.3. In this section, our main aim is to prove the analyticity of global solutions established in Theorem 1.1. The following basic inequality will play a very important role in the proof of Theorem 1.3.
Then for any ξ, η ∈ R 3 , the following inequality holds.
In what follows, we give the proof of Theorem 1.3.
then U (ξ, t) and H(ξ, t) satisfy Noting that (13), then J 1 can be written as Similarly, we have and Noting that the facts is uniformly bounded with respect to time t and are uniformly bounded independently t and τ . Therefore, combining (14)- (16) yields Similarly, from (14), (17) and (18), it holds that Let and Multiplying (19) by |ξ| 1−2α and then integrating the result over R 3 , using Lemma 2.1, we have Similarly, we can obtain from (19), (20) and Lemma 2.1 and Multiplying (19) by |ξ| 1−2α and then integrating the result over R 3 × (0, t), using Young inequality, Lemma 2.1, we obtain (27) The same procedure leads to Collecting (23)-(30) yields Thus we can complete the proof of Theorem 1.3 by continuous argument.
3. Proof of theorem 1.4. In this section, our main purpose is to prove decay rate of global small solutions. We need the following Lemma, which has been established in [3].