ON THE GLOBAL WELL-POSEDNESS TO THE 3-D NAVIER-STOKES-MAXWELL SYSTEM

. The present paper is devoted to the well-posedness issue of solutions of a full system of the 3- D incompressible magnetohydrodynamic(MHD) equations. By means of Littlewood-Paley analysis we prove the global wellposedness of solutions in the Besov spaces ˙ B 12 2 , 1 × B 32 2 , 1 × B 32 2 , 1 provided the norm of initial data is small enough in the sense that for some suﬃciently small constant c .

2,1 provided the norm of initial data is small enough in the sense that for some sufficiently small constant c 0 .
1. Introduction. In this paper, we study the global well-posedness of the following three-dimensional full Magnetohydrodynamics (MHD) system: where u = u(t, x), B = B(t, x) and E = E(t, x) are the fluid velocity, magnetic and electric fields, depending on the spatial position x and the time t. The scalar functions p = p(t, x) denote the pressure. j is the electric current which is given by Ohm's law. The positive constants ν is the viscosity coefficients and σ is the electric resistivity. Here, u, E, B, j are defined on R 3 and take their values in R 3 . The first equation in (1.1) is the Navier-Stokes equation for incompressible flows with a Lorentz force term j × B under a quasi-neutrality assumption of the net charge carried by the fluid. The second equation is the Ampere-Maxwell equation which includes here the displacement current ∂ t E for an electric field E. The third equation is the Faraday's law. The fifth equation is the Ohm's law which states that the electric current is proportional to the electric field measured in a frame moving with the local velocity of the conductor. One may refer to [6] for a physical review of the background to magnetohydrodynamics.
Similar to the Navier-Stokes equations, by using the standard L 2 energy estimate, we have the following formal energy identity: This identity shows that the L 2 energy is dissipated by the viscosity and the electric resistivity. However, a global finite energy weak solution with initial data lying in (L 2 (R d )) 3 remains an interesting open problem for the Navier-Stokes-Maxwell equations (1.1) in both d = 2, 3.
For the 2-D Navier-Stokes-Maxwell system, the global existence of regular solutions as well as an exponential growth estimate of the solution has been obtained by Masmoudi [11]. For the 3-D case, Ibrahim and Yoneda [10] proved the existence of local in time unique solution and the loss of smoothness via Fujita-Katos method, and the authors [9] investigated the existence and uniqueness of a global weak solution for the 2-D and 3-D Navier-Stokes-Maxwell equations by means of the Fourier localization technique and Bonys para-product decomposition. For the 2-D case, the authors introduce the space L 2 log and L 2 H to deal with low and high frequencies, repectively, which defined by In the framework of these spaces, they can obtain an extra logarithmic regularity on both B and E. Later on, Germain, Ibrahim and Masmoudi [7] proved the local and global well-posedness of bidimensional and three-dimensional full system of magneto-hydrodynamical(MHD) equations with small initial data by using the Picard's theorem. Our goal of this paper is to prove that the system (1.1) is globally well-posedness when the initial velocity field with the third component being large. We are going to relax the smallness condition in the following Theorem 1.1. The main difficulty of this work lies in the fact that we will have to deal with the nonlinear coupling between the Navier-Stokes equations with a forcing induced by the electric current and the magnetic field. Toward this, we will have to work in the framework of a weighted Chemin-Lerner type space introduced in [12] here. These spces have been used in the study of the global well-posedness to 3-D incompressible inhomogeneous Navier-Stokes equations [13].
Our first result can be stated as follows.
The rest of the paper unfolds as follows. In the next section, we introduce the main tool for the proof -the Littlewood-Paley decomposition-and some related functional spaces. In Section 3, we focus on the proof of the existence and uniqueness of a solution of (1.1). In section 4, we shall present the estimate to the pressure function. Finally in the last section, we shall complete the proof of Theorem 1.2. Let us end this section with the notations we are going to use in this paper.
• Throughout this paper, C represents some "harmless" constant, which can be understood from the context. In some places, we shall alternately use the notation A B instead of A ≤ CB, and A ≈ B means A B and B A. • If X is a Banach space, T > 0 and p ∈ [1, +∞] then L p T (X) stands for the set of Lebesgue measurable functions f from [0, T ) to X such that t → f (t) X belongs to L p ([0, T )). If T = +∞, then the space is merely denoted by L p (X).
• Throughout this paper, (d j ) j∈Z denotes a generic element of the space of 1 (Z) so that d j ≥ 0 and j∈Z d j = 1.

2.
Basic results on Besov spaces. In order to define Besov space, we need the following a dyadic decomposition of the Foueier space in the case x ∈ R 3 , see [1,4].
For a ∈ S (R 3 ), we set Then we have the formal decomposition where P[R 3 ] is the set of polynomials. Moreover, the dyadic operators satisfy the property of almost orthogonality: We recall now the definition of homogeneous Besov spaces from [1].
From the above definition we know that for s ∈ R, 1 ≤ p, r ≤ ∞, and u ∈ S (R 3 ), then u belongs to B s p,r (R 3 ) if and only if there exists {c j,r } j∈Z such that c j,r ≥ 0, c j,r r = 1 and ∆ j u L p ≤ Cc j,r 2 −js u B s p,r for all j ∈ Z.
Similarly, we can also define the inhomogeneous Besov spaces. Indeed, let χ ∈ [0, 1] be a smooth function supported in the ball {ξ ∈ R 3 , |ξ| ≤ 4 3 } so that For u ∈ S (R 3 ), we set Then for all u ∈ S (R 3 ), we have the inhomogeneous Littlewood-Paley decomposition u = j∈Z ∆ j u, and for (p, r) ∈ [1, +∞] 2 , s ∈ R and u ∈ S (R 3 ), we define the inhomogeneous Besov space B s p,r (R 3 ) as follows: We point out that if s > 0 then B s p,r (R 3 ) =Ḃ s p,r (R 3 ) ∩ L p (R 3 ) and u B s p,r ≈ u Ḃs p,r + u L p and B s p,r →Ḃ s p,r with p < ∞. We also define the following Chemin-Lerner type spaces, which were introduced by Chemin and Lerner [3]. [3].) Let s ≤ 3 p (or, in general, s ∈ R), (r, λ, p) ∈ [1, +∞] 3 , and T ∈ (0, +∞]. We define the L λ T (Ḃ s p,r (R 3 ))-norm by With the usual change if r = ∞. For short, we just denote this space by L λ T (Ḃ s p,r ). Similarly, we can define L λ T (B s p,r ), which will be used in the subsequent sections. By virtue of the Minkowski inequality, we have In particular, we have . As we shall repeatedly use the Littlewood-Paley theory in what follows, we list some basic facts here. Bernstein's inequality is fundamental in the analysis involving Besov spaces. Please see the details in [2,4]. Lemma 2.3. Let B be a ball and C a ring of R 3 . A constant C exists so that for any positive real number λ, any nonnegative integer k, any homogeneous function σ of degree m smooth outside of 0 and any couple of real numbers (a, b) with b ≥ a ≥ 1, there hold We shall use the following Bony's decomposition in the homogeneous context: We first present the following property of Besov space, which is crucial to obtain the a priori estimate of the solution of (1.1).
In order to deal with the nonlinear coupling between the Navier-Stokes equations with a forcing induced by the magnetic field and the induction equation, we recall the following form of weighted Chemin-Lerner type norm from [12].
, and with the standard modification for q = ∞ or r = ∞.
We now state a standard estimate on the parabolic regularity that we will use in proving the existence of solutions of Navier-Stokes-Maxwell system (1.1).

Local existence.
We define by induction a sequence Γ n = (u n , B n , E n ) of smooth functions by solving the following linear equations:

First Step. Unifrom Bounds
We claim that the following estimates hold for some T > 0, and u n+1 ≤ C 0 , then for any small γ 0 , there exists T 0 , such that the following estimate holds e νt∆ u 0 ≤ γ 0 , and e νtL E 0 Thus, we find that (3.1) and (3.2) hold for n = 0. Assume (3.1) and (3.2) hold for n−1. By using Lemma 2.4 and Lemma 2.6 we can choose Similarly, we have u n+1

Second Step. Convergence
We are going to show that According to Lemma 2.6, we need to estimate I i , i = 1, 2. ), and .
In addition, we perform a simple energy estimate for Maxwell's equations and obtian .

GAOCHENG YUE AND CHENGKUI ZHONG
Combining the above estimates with Lemma 2.6 and interpolation inequality a , by choosing t and ε 0 small enough. Hence, {Γ n } n∈N is a Cauchy sequence in 2,1 ) and converge to some limit func- . By using the procedure of the second step on small time interval, we can prove w 1 = w 2 = w 3 = 0 in this interval. By repeating the procedure, we can obtain w 1 = w 2 = w 3 = 0 on [0, T ]. We finish the proof of local existence.
3.2. Global existence. We construct approximate solutions of (1.1) which are smooth solutions. Let (u 0 , B 0 , E 0 ) = (0, 0, 0) and let the sequence Γ n = (u n , B n , E n ) be solutions of the following system by induction The sequence {θ n } ⊂ N is chosen so that Before stating our local existence result, we introduce the following function space: We are going to show that {(u n , B n , E n )} n∈N is a Cauchy sequence in E. According to Lemma 2.6, we have the following inequality for all n ∈ N : .

Applying Lemma 2.4 yields
, which together with L 2 (R 3 ) energy estimate implies that .

Combining (3.3) and (3.4), we obtain
Similarly, applying the above estimate to Γ n+1 − Γ n , we obtain It is now obvious that {Γ n } n∈N is Cauchy sequences in E, hence converge to some functions (u, B, E) ∈ E, which resolves the system (1.1).
Using a similar argument as that in [9], one can easily obtain the uniqueness of the solutions (u, B, E). Thus, we finish the proof of Theorem 1.1.

4.
The estimate of the pressure. The goal of this section is to provide the pressure estimates in the framework of weighted Chemin-Lerner type norms. We first get by taking div to the momentum equation of (1.1) that where, for a vector field u = (u h , u 3 ) we denote div h u h = ∂ 1 u 1 + ∂ 2 u 2 .
The following proposition concerning the estimate of the pressure will be the main ingredient used in the estimate of u h . 2,1 ). We denote 2,1 ) which decays to zero when |x| → ∞ so that for all t ∈ [0, T ], there holds .
The proof of this proposition will mainly be based on the following lemmas: .
Proof. Using the standard product law in Besov space and the interpolation inequality ensure that .

This proves the first inequality of Lemma 4.2.
Where again thanks to the standard product law in Besov space and the interpolation inequality we get that .
This completes the proof of the lemma. and .

NAVIER-STOKES-MAXWELL SYSTEM 5827
Proof. Using the standard product law in Besov space and the interpolation inequality ensure that .
This proves the first inequality of Lemma 4.2.
Where again thanks to the standard product law in Besov space and the interpolation inequality we get that .
This completes the proof of the lemma. and .
Proof. Using the standard product law in Besov space and the interpolation inequality ensure that .

This proves the first inequality of Lemma 4.4.
Where again thanks to the standard product law in Besov space and the interpolation inequality we get that .
This completes the proof of the lemma. Now we are in a position to complete the proof of Proposition 4.1.
Proof of Proposition 4.1. Again as both the proof of the existence and uniqueness of solutions to (4.1) is essentially followed by the estimates (4.3) for some appropriate approximate solutions of (4.1). For simplicity, we just prove (4.3) for smooth enough solutions of (4.1). Indeed, thanks to (4.1) and divu = 0, we have Applying∆ j to the above equation and using Lemma 2.3 leads to ). Applying Lemma 2.4 and standard product laws in Besov space gives rise to , which along with Lemmas 4.2-4.4 and (4.5) implied that .
This finishes the proof of Proposition 4.1.
To deal with the estimate of u 3 , we also need the following proposition: , for all t ∈ (0, T ).
Proof. The proof of this proposition follows exactly the same lines as that of Proposition 4.1. We first applying the standard product laws in Besov space gives rise to from which and Lemma 4.2-Lemma 4.4, we deduce that , for all t ≤ T, from which we conclude the proof of (4.6). 2,1 ). We denote T * to be the largest time so that there holds (5.1). Hence to prove Theorem 1.2, we only need to prove that T * = ∞ and there holds (1.4)-(1.5). Toward this and motivated by [8,12,14], we first rewrite (1.1) as follows: Applying the operator∆ j to the above equation and taking the L 2 inner product of the resulting equation with∆ j u h λ , we obtain 1 2 However thanks to [5], there exists a positive constant c such that from which, we deduce that

Applying Lemma 2.4 and Lemma 4.2, we obtain
.
While applying Lemma 2.4, Lemma 4.2 and Lemma 4.3 leads to .
, for all t ≤ T.

5.2.
The estimate of u 3 . We use that the equation on the vertical component of the velocity is a linear equation with coefficients depending on the horizontal components. Thanks to the u 3 equation of (5.2), we get by a similar derivation of (5.4) that ∆ j u 3

Applying Lemma 4.2 and Lemma 4.4 ensures that
.
While applying Lemma 2.4 and standard product laws in Besov space yields .
Then we get by substituting the above estimates and (4.6) into (5.6) that .

5.3.
The estimate of B λ and E λ . The existence and uniqueness of solutions to Maxwell's equations is essentially followed by the estimates (4.3) for some appropriate approximate solutions of (4.1). For simplicity, we just prove (4.3) for smooth enough solutions of Maxwell's equations. Indeed thanks to (1.1), we have Applying the operator∆ j to the above two equations and then taking the L 2 inner product of the resulting equation with∆ j E λ and∆ j B λ , respectively, we have which implies that Thus, we complete the proofs of (1.4) and (1.5).