Stability and sharp decay for 3D incompressible MHD system with fractional horizontal dissipation and magnetic diffusion

This paper aims as the stability and large-time behavior of 3D incompressible magnetohydrodynamic (MHD) equations with fractional horizontal dissipation and magnetic diffusion. By using the energy methods, we obtain that if the initial data are small enough in H3(R3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^3(\mathbb {R}^3)$$\end{document}, then this system possesses a global solution, and whose horizontal derivatives decay at least at the rate of (1+t)-12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+t)^{-\frac{1}{2}}$$\end{document}. Moreover, if we control the initial data further small in H3(R3)∩Hh-1(R3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^3(\mathbb {R}^3)\cap H_h^{-1}(\mathbb {R}^3)$$\end{document}, the sharp decay of this solution and its first-order derivatives is established.


Introduction
The standard incompressible MHD system can be written as where u represents the velocity field, P the total pressure and b the magnetic field, μ > 0 the kinematic viscosity and η > 0 the magnetic diffusivity. Because of their extensive application in physical (see, e.g., [2,4,9]) and stimulating in mathematics, MHD system has attracted considerable number of scholars, and significant progress has been made (see, e.g., [1,8,11,20,22,24]). In certain physical regimes and under suitable scaling, the full Laplacian dissipation is reduced to a partial dissipation [3]. For example, according to Pedlosky's book [12], the Navier-Stokes equations with only horizontal dissipation can model the anisotropic geophysical fluids when the vertical diffusion is much smaller than the horizontal one. Moreover, it can also model the turbulent diffusion of rotating fluids in Ekman layers. The anisotropic magnetic diffusion is related to the resistivity of electrically conducting fluids such as certain plasmas and liquid metal (see, e.g., [13]). Hittmeir and Merino-Aceituno [5] demonstrated that Boltzmann equation gives rise to fractional diffusion Navier-Stokes equation under appropriate rescaling. In mathematics, fractional dissipation allows us to study a family of equations simultaneously and gives us a broad view on how behavior of solutions changes as the dissipation power varies.
In this paper, we consider the 3D MHD system with fractional horizontal dissipation and magnetic diffusion: (1.2) where α > 0, μ = η = 1. And we have written Δ h = ∂ 2 1 + ∂ 2 2 . For notational convenience, we will use ∇ h := (∂ 1 , ∂ 2 ) for the horizontal gradient. The fractional operator (−Δ h ) α is defined via Fourier transform where ξ h := (ξ 1 , ξ 2 ) andf denotes the Fourier transform System (1.2) with its special structure and features has attracted numerous interests, and a rich array of results have been established. To place our results into the context of the existing research, we briefly describe some of the related work. We begin with the case α 5 4 , where the global existence has been proved. The study of 3D Navier-Stokes equations with fractional partial dissipation was initiated by Yang et al. [23], and a unique global solution in L ∞ ([0, ∞]), H 1 (R 3 )) was established. Their results were improved to L ∞ ([0, ∞]), H s (R 3 )) by Li and Yuan in [7]. Moreover, Yang, Jiu and Wu in [22] study the 3D MHD system resembling (1.2) with α = 5 4 and proved the existence of global solution. When 1 2 α < 5 4 , the global existence of (1.2) with general large initial data remains an open problem. However, any sufficiently small initial data with suitable regularity always leads to a unique global solution. Significant progress has been made on the small data global well-posedness. Wu and Zhu [21] established the small data global well-posedness for the 3D MHD equations with only horizontal velocity dissipation and vertical magnetic diffusion. Shang and Zhai [17] proved the global stability of solution for system (1.2) with α = 1 by energy estimate and further studied the large-time behavior of solution. In this paper, we confirm the global well-posedness for system (1.2) in the case 1 2 α < 1. In fact, we focus on the stability problem of the system (1.2) and prove the existence of the global solution.
In addition, we further analyze the large-time behavior of the solution. With a suitable initial data, we establish the sharp decay of this solution and its first-order derivatives.
Our main results can be stated as follows. The first one establishes a small data global well-posedness for system (1.2).
then, (1.2) has a global classical solution (u, b) satisfying, for any t > 0, A natural idea about the proof of Theorem 1.1 is to bound u H 3 + b H 3 via the energy estimate. However, system (1.2) has only fractional horizontal dissipation and magnetic diffusion. To bound the nonlinear terms, a more efficient use of the fractional horizontal dissipation is required. Therefore, we propose some new fractional anisotropic bounds in the following lemma, which are of great help in bounding the nonlinear terms by the dissipative parts when performing energy estimate for system (1.2). Incidentally, this lemma is also helpful for other systems with fractional anisotropic dissipation. A brief proof will be given in Sect. 2. Lemma 1.1. If the right-hand side norms are all bounded, and 0 < α < 1, then The second result assesses that the horizontal gradient of the solution in Theorem 1.1 decays at the rate of (1 + t) − 1 2 .
In addition, (u 0 , b 0 ) satisfy the smallness requirement of Theorem 1.1, namely for a small constant > 0. Then, the corresponding solution (u, b) satisfies the decay estimate, for any t 0, The decay result in Theorem 1.2 follows from the time integrability of Λ α h u 2 Details about the proof will be presented in Sect. 3. Theorem 1.2 does not provide decay rate on the L 2 -norm of (u, b). This is not surprising, we notice that even when we consider heat equation, the L 2 -norm of the solution is not known to decay in time if we only set initial data in L 2 -norm. Therefore, in order to study the decay rate for L 2 -norm of (u, b), we need to make extra assumptions on the initial data (u 0 , b 0 ). Recently, Ji, Wu and Yang [6] studied the 3D Navier-Stokes equations with only horizontal dissipation and offered an efficient method to establish decay rates. In addition, Shang and Zhai [17] established the optimal decay rates of these global solutions to the three-dimensional MHD equations with horizontal velocity dissipation and magnetic diffusion. Inspired by these, we set an extra condition in H −1 h (R 3 ). This kind of conditions are usually required when deal with large-time behavior of dissipative PDEs (see, e.g., [15,16]). Here, H −1 h (R 3 ), a kind of Sobolev space with negative index, denotes the space of distributions f satisfying In this way, the optimal decay rates for u and b and their first derivatives in L 2 -norm can be established.
Then, there exists a sufficiently small parameter > 0 such that, if then the global solution (u, b) of system (1.2) satisfies the following decay properties: (1.5) Remark 1.1. The decay rates established in Theorem 1.3 are optimal. Actually, we consider the corresponding heat equation For u 0 satisfy (1.4), then the solution of (1.6) satisfies the following sharp decay rate Which is exactly corresponding to (1.5). This implies that the decay rates are sharp.
The rest of this paper is divided into three sections. Section 2 devotes to prove Lemma 1.1 and Theorem 1.1. Sections 3 and 4 prove Theorems 1.2 and 1.3, respectively.

Proof of Lemma 1.1 and Theorem 1.1
This section is going to prove Lemma 1.1 and Theorem 1.1. We start with stating the following Minkowski's inequality by which we can estimate the Lebesgue norm with larger index firstly while the smaller index later. Precisely, for 1 p q ∞, where f is a nonnegative measurable function over R m × R n . More details and its proof can be found in [10].
Proof of Lemma 1.1. By Holder's inequality and Minkowski's inequality, we obtain when 0 < α < 1, where we wrote By Sobolev's inequalities, we obtain The second inequality can be proved similarly. Actually, we notice that These complete the proof. Proof of Theorem 1.1. The key part is a prior estimate of (u, b) in H 3 (R 3 ). Thanks to the norm equivalence we can estimate L 2 -norms andḢ 3 -norms, respectively. By a simple energy estimate to (1.2), we obtain Now, we focus on theḢ 3 -norms. Applying ∂ 3 i (i = 1, 2, 3) to equation (1.2) and then dotting by ( M 1 can be divided into three parts further, namely By Holder's inequality and Sobolev's inequality, we can bound M 11 , namely we can bound M 13 in the same way as M 12 , namely (2.5) We turn to M 2 which can be decomposed into three pieces, By Holder's inequality, we can bound M 21 , namely Similar procedure as M 12 , we obtain (2.7) Because of ∇ · b = 0, M 23 can be bound as M 22 , namely (2.8) Estimate on M 3 , we decomposed it into three pieces By Holder's inequalities, we can bound M 31 , namely By Lemma 1.1, we obtain where we need 1 2 α < 1. M 33 can be bound as M 13 , namely The estimation of M 4 is similar as M 1 and M 2 , so we just state the result.
Inserting inequalities (2.3-2.12) in (2.2), we have (2.13) Integrating in time and adding (2.1), we obtain (2.14) If we set Then, (2.14) implies that This inequality with a suitable initial data combined with the bootstrapping argument, proves Theorem 1.1.

Proof of Theorem 1.2
We first state a Lemma to help us estimate the decay rate, and more examples and applications can be found in [3,19]. (2) Generalized degression: for any 0 s < t, then let a 2 = max{2a 1 f (0), 2a 0 a 1 } and for any t > 0, we have

Lemma 3.1 implies that Theorem 1.2 is equivalent to verifying
which satisfies the two conditions in Lemma 3.1. The first condition has been established in Theorem 1.1. The second one, namely for any 0 s < t, is what we strive to establish. Proof of Theorem 1.2. We first focus on ∇ h u 2 Applying ∇ h to equation (1.2) and then dotting by (∇ h u, ∇ h b), we get Using Lemma 1.1, we can estimate N 1 , namely We should notice that α 2 3 is required in the last inequality above to establish Similar estimates hold for N 2 , N 3 and N 4 ; we omit the calculation and just state the result.
What's more, It remains to treat theḢ 2 -norm parts. Applying ∂ 2 i ∇ h to (1.2) and then dotting ( where To bound K 1 , we split it into three parts By Holder's inequalities, we can bound K 11 , namely As for K 12 , we further decompose it in two pieces, namely K 121 is easy to estimate by Lemma 1.1, (3.9) By Lemma 1.1, we get (3.10) K 13 is difficult; we further decompose it into two pieces, namely We first bound K 132 . Thanks to ∇ · u = 0, we get (3.11) As for K 131 , by Lemma 1.1, we obtain where the last inequality holds from 1 2 α 2 3 . Precisely, Adding (3.8)-(3.12), we have Similar procedure as K 1 , we decompose K 2 , namely By Holder's inequalities, we can easily get (3.14) Further decompose K 23 , namely Just like the process of K 121 , we can bound K 231 and K 232 , namely By Lemma 1.1, we can deal with K 233 , namely where we required 1 2 α 2 3 . We turn to estimate K 24 , which can be divided into two parts, By Lemma 1.1, we obtain (3.17) Similar as K 132 , we can bound K 242 , Adding (3.14)-(3.18) together, we get To bound K 3 , we decompose it into three parts, (3.20) By Holder's inequalities, we obtain (3.21) By Lemma 1.1, we obtain (3.22) Just like K 13 , we can estimate K 33 , Inserting (3.21)-(3.23) into (3.20), we obtain (3.24) Similar procedure as K 2 , we obtain (3.25) Inserting (3.3)-(3.6) in (3.2) and inserting (3.13, 3.19, 3.24, 3.25) in (3.7), and then adding two inequalities together, we obtain for any η > 0

Proof of Theorem 1.3
This section completes the proof of Theorem 3. We need some preparations which will be stated in the following two important lemmas.
The first lemma provides an exact decay estimate for the heat operator associated with the fractional Laplacian. A proof can be found in [14] and [19].
The second one provides an upper bound on a convolution integral, which can be proved similarly to Lemma 2.4 in [18] Lemma 4.2. if 0 < s 1 < 1 and 0 < s 2 , then Remark 4.1. The condition 0 < s 1 < 1 is necessary, because when s 1 1 and 0 < t < 1, the convolution integral is unbounded. But it is not necessary when we replace (t−s) −s1 with (t−s+1) −s1 , to be specific, Proof of Theorem 1.3. To study the decay rate of the solution for system (1.2), we first set up the following time-weighted norms: (4.1) We claim that Take the initial data (u 0 , b 0 ) to be sufficiently small. Precisely speaking, (u 0 , b 0 ) satisfy that for a suitable small constant > 0, Therefore, (4.2) implies This inequality combined with bootstrapping argument leads to which is exactly the statement of Theorem 3.
To verify (4.2), we rewrite equation (1.2) in integral form, where P denotes the Leray projection. For clarity of expression, we divide the proof into four subsections in each of which we dedicate to estimating one of the norms in the X(t) and Y (t). In addition, we verify (4.2) and complete the proof in the last subsection.

The estimate of u(t) L 2 + b(t) L 2
Taking the L 2 -norm of (4.3), we obtain  To bound I 1 , we consider it in two cases. When 0 < t 1, by Plancherel's theorem, we obtain when t > 1, by Lemma 4.1 we have Similarly, we get Thanks to the boundedness of P on L 2 -norm, we can bound I 3 by using Lemma 4.1.
By Holder's inequalities and Sobolev's inequalities, we get To bound J 32 , we first deal with ∇ h (b · ∇b) L 2 , namely