The 2 D MHD Systems with Vertical Dissipation and Vertical Dissipation Magnetic Diffusion

In this paper, we study the global regularity of the classical solution of the 2D incompressible magnetohydrodynamic equation with vertical dissipation and vertical magnetic dissipation. We show that any solution of the second component ( ) 2 2 , u b has a global 2 r L -bound, where r satisfies 1 r ≤ < ∞ and the boundary does not grow faster than log r r as r increases.


Introduction
The generalized MHD system is where , , , 0 ν κ α β > , ( ) 1 2 Λ = −∆ , u denotes the velocity field and b denotes the magnetic field. The magnetohydrodynamic (MHD) systems [1] control the dynamics of velocity and magnetic fields in conductive fluids such as plasma and reflect the basic laws of physical conservation.
In recent years, the MHD equations with partial dissipation regularity problem have attracted considerable interests. For example, the n-dimensional MHD Equation (1), when the coefficient satisfies 1 , 0, 1 , 2 4 2 n n α β α β ≥ + > + ≥ + it has been proved that the solution has global regularity [2]. Wu [3] has been  And it is also proved that the condition satisfying 0, 1 ν β = > has a global smooth solution with the direction of the magnetic field that remains sufficiently smooth. Cao, Regmi and Wu [4] have been proved that the 2D MHD with horizontal dissipation and horizontal magnetic diffusion in horizontal component of any solutions has a global regularity. The global regularity of the class solution of the MHD equation with magnetic diffusion and mixed partial dissipation is established by Wu [5]. In [6], the global existence and uniqueness of the smooth solution of 2D micropolar fluid flow with zero angular viscosity have been proved. Other related articles can be seen in [7] [8] [9], etc.
In this paper, we study the 2D MHD systems with vertical dissipation and vertical dissipation magnetic diffusion, namely In this case, we only get the global 2r L -bound of the solution in the y-direction, and the global regularity problem for the complete directional solution has not been achieved. In the following article, let w u b ± = ± , this will provide us with convenience.
We have a symmetric equation by (2) The new Equation (3) consists of two vectors, which is more complicated in the calculation process, therefore, we use fractionally derivative triple product estimation [4] to solve this difficulty. This paper takes Cao and Wu recent study of two-dimensional partially dissipated Boussinesq equation [8] as an example to discuss the influence of known vertical component ( ) 2 2 , u b Lebesgue norm on global regularity. And in Section 4, we obtain the main Theorem 3, which proves that ( ) In fact, in Section 2 we get Theorem 1, which is about the solution of Equation (2) bounded by Lebesgue in the y-direction. The sameness of Theorem 1 and Theorem 3 is that boundedness is related to the r, but in Theorem 1, we get the case of 1 r = , and Theorem 3 has a slower bounded change with the increase of r.
The rest of this article is divided into four parts. In Section 2, we prove the global bounded for ( ) , and the boundedness depends on the index of r.
In Section 3, we show the global bounded for . In Section 4, we prove that the solution of (2) in y-direction has a global Lebesgue bound. In Section 5, we prove the bounded condition of ( )

A Global Bound in the Lebesgue Spaces
In this section, we prove the classical solution of (2) at the y-direction exists globally bounded in 2r L norm. The boundedness obtained here depends on the index of r. We have the following theorem.
Here we omit the proof of Lemma 1 and now begin to prove Theorem 1. Consequently, Based on the above estimates, we get Following the Gronwall's inequality, we obtain

Global Bounds for the Pressure
In this section, we show the solution of the first components ( ) 1 1 , u b has a 2 L -bound with 2 r = or 3 r = , and establish the pressure has a global bound.
The results can be stated as follows.
where 1 3 q < ≤ and ( ) 0,1 s ∈ , and C is a constant related to T and initial value.
Here we use two calculus inequalities of the following lemma.

Lemma 2. [4] Assume that
( ) Proof. We use the symmetric Equation (3) to prove the case of 2 r = in Theorem 2. Take the inner product of the first Equation (3) Using 0 w + ∇ ⋅ = and integrate by parts, we get

I
pw w w x p w w w C p w w w According to (7), Therefore, by Young's inequality, .
According to Lemma 2 and 0 According to (7), we get The same can be proved that by Hölder's inequality and (6), we get We now proved the inequality (9), taking the divergence of the first two equations in (3)

An Improved Global Lebesgue Bound
From the conclusions of Sections 2 and 3, we have the main theorem of this paper.
Theorem 3. Assume that ( ) ( ) be the corresponding solution of (2). Let 2 r < < ∞ , then Before proving the Theorem 3, we first describe the lemma that will be used.

Lemma 3. [4] Let
where ρ and γ are given by where  and 2) For any 2 q ≤ < ∞ satisfying where C is again bounded uniformly as 1 s − → , and we make ( ) ( )( ) For further estimation, we spilt into two parts and bound one of them by Lemma 4. Moreover, we get any 0 1 β ≤ ≤ , The condition in (28) with a constant 0 C is independent in s and C is bounded uniformly as Using (32), (35) and (37) to simplify this index and get

Conditional Global Regularity
This section estimates the global boundedness of the vertical component 2 u  We divide the proof of the theorem into two parts. be the corresponding solution of (2). Then, for any Proof. Taking the inner product of the first equation in (3)