Global Large Smooth Solutions for 3-D Hall-magnetohydrodynamics

In this paper, the global smooth solution of Cauchy's problem of incompressible, resistive, viscous Hall-magnetohydrodynamics (Hall-MHD) is studied. By exploring the nonlinear structure of Hall-MHD equations, a class of large initial data is constructed, which can be arbitrarily large in $H^3(\mathbb{R}^3)$. Our result may also be considered as the extension of work of Lei-Lin-Zhou from the second-order semi-linear equations to the second-order quasilinear equations, because the Hall term elevates the Hall-MHD system to the quasilinear level.

For Hall-MHD equations, some newly developments have been made. For example, Chae et al. in [3] proved global smooth solutions of three-dimensional Hall-MHD equation with small initial data in H 3 (R 3 ) 3 × H 3 (R 3 ) 3 . Chae and Lee improved their results under weaker smallness assumptions on the initial data, see [4] for details. There are other prominent works for small solutions for Hall-MHD equations, for examples, [1,11,27,26,6,7,8]. These results of the global well-posedness of the three-dimensional Hall-MHD system under the smallness condition on the initial data in the deterministic case requires positive diffusion on both velocity and magnetic fields equations. However, with noise, zero viscosity is allowed; Yamazaki and Moha in [26] proved the global well-posedness of the three-dimensional stochastic Hall-MHD system with zero viscosity under the smallness condition on the initial data. However, none of results are known for Hall-MHD equations for general initial data without smallness conditions. Under a class of large initial data, we found some results for incompressible Navier-Stokes equations and incompressible standard MHD equations, see [10,9,15,17,28] for details. Those motivate us to study the global well-posedness of Cauchy's problem of Hall-MHD equations with large inital data. But the Hall term heightens the level of nonlinearity of the standard MHD system from a second-order semilinear to a second-order quasilinear level, significantly making its qualitative analysis more difficult. To the author's knowledge, it's quite rare to prove the existence of large, smooth, global solutions for quasilnear system. Using the large initial data constructed in [15], the author is very fortunately to go through these difficulties by combining nonlinear structures and commutator energy estimates for resistive, viscous Hall-MHD equations. The aim of this paper is to prove the existence of a unique, global smooth solution of Hall-MHD equations with initial data being arbitrarily large in H 3 (R 3 ). Our result completely drops the smallness condition on the initial data.
Before we state our main results, we first give some notations. Let is a cut off function satisfying |χ(x)| 2 and Let v 0 be that constructed by Lei et al. [15], and it has the following properties where M 1 , M 2 are positive constants,v 0 is the Fourier transform of v 0 and the operator √ −∆ is defined through the Fourier transform √ −∆f (ξ) = |ξ|f (ξ).
Our main result is as follows.
Remark 1.2. In the limiting case δ = 0, ∇ × u 02 = u 02 , ∇ × b 02 = b 02 , and the flow, magnetic field are called Beltrami flow and force-free fields respectively. Let us also mention that the magnetic energy achieves the minimum value for force-free fields, one can refer [24] for details.
The proof of Theorem 1.1 is based on a perturbation argument along with a standard cut-off technique, and the perturbation is as large as the initial data. Compared with the standard MHD equations, a part of the nonlinearities may not be small for Hall-MHD equations (see (3.2) and (3.6)). Fortunately, by combining the nonlinear structure of the term and commutator estimates, these terms can be estimated carefully.
This paper is organized as follows: In section 2, we introduce commutator estimates and give some estimates of some quadratic terms. Section 3 is devoted to prove the global existence and uniqueness of large smooth solutions for Hall-MHD equations.

Preliminaries
In this section, we first give some notations.
. Let m be a positive integer and r > 0. We use the notation ||g|| H m (|x| r) to denote the Sobolev norm localized in bounded domain x ∈ R 3 |x| r , that is, Next, we introduce the commutator estimate.
The following commutator estimate holds.
3. The proof of Theorem 1.1 In this section, we will prove Theorem 1.1 using a perturbation argument along with a standard cut-off technique.
Proof of Theorem 1.1. Letf = χ M0 f,g = χ M0 g, and u = U +f , b = B +g. Then U, B satisfy In what follows, we will derive some energy estimates of U, B.
Step 1: Energy inequalities of B.
Operating Equation (3.2) with ∂ α , |α| 3, and taking L 2 inner product with ∂ α B, we get where Next, we need to estimate I 1 , I 2 , I 3 and I 4 . For using Lemma 2.1, we deduce that We calculate To estimate I 3 , we divide it into two parts Applying Hölder inequality, we easily infer Moreover, thus, Combining (3.10) and (3.11), we get As for I 4 , we can write We thus have Step 2: Energy inequalities of U .
Firstly, we have then we have Using Hölder inequality and Calderon-Zygmund estimate [