MHD Equations with Regularity in One Direction

p is a scalar pressure. The MHD system (1) is a mathematical model for electronically conducting fluids such as plasma and salted water, which governs the dynamics of the fluid velocity and the magnetic fields. There have been extensive studies on (1). In particular, Duvaut and Lions [1] constructed a class of global weak solutions with finite energy, which is similar to the Leray-Hopf weak solutions (see [2, 3]) for the Navier-Stokes equations (b = 0 in (1)). However, the issue of uniqueness and regularity for a given weak solution remains a challenging open problem. Initiated by He and Xin [4] and Zhou [5], a lot of literatures have been devoted to the study of conditionswhich would ensure the smoothness of the solutions to (1) and which involve only the fluid velocity field. Such conditions are called regularity criteria. The readers, who are interested in the regularity criteria for the Navier-Stokes equations, are referred to [4–18] and references cited therein. For the Navier-Stokes equations, the authors have established that the regularity of the velocity in one direction (say, ∂ 3 u), one component of the velocity (say, u 3 ), or some other partial components of the velocity, velocity gradient, velocity Hessian, vorticity, pressure, and so forth, would guarantee the regularity of theweak solutions; see [19–29] and references therein.Many of these regularity criteria have been proved to be enjoyed by the MHD equations (1); see [30–33]. However, due to the strong coupling of the fluid velocity and the magnetic fields, the scaling dimensions for the MHD equations are not as good (large) as that for the Navier-Stokes equations. In this paper, we would like to improve the regularity criterion


Introduction
We consider the following three-dimensional (3D) magnetohydrodynamic (MHD) equations: Here, u and b are the fluid velocity and magnetic fields, respectively; u 0 and b 0 are the corresponding initial data satisfying the compatibility conditions is a scalar pressure.The MHD system (1) is a mathematical model for electronically conducting fluids such as plasma and salted water, which governs the dynamics of the fluid velocity and the magnetic fields.
There have been extensive studies on (1).In particular, Duvaut and Lions [1] constructed a class of global weak solutions with finite energy, which is similar to the Leray-Hopf weak solutions (see [2,3]) for the Navier-Stokes equations (b = 0 in (1)).However, the issue of uniqueness and regularity for a given weak solution remains a challenging open problem.Initiated by He and Xin [4] and Zhou [5], a lot of literatures have been devoted to the study of conditions which would ensure the smoothness of the solutions to (1) and which involve only the fluid velocity field.Such conditions are called regularity criteria.The readers, who are interested in the regularity criteria for the Navier-Stokes equations, are referred to [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references cited therein.
For the Navier-Stokes equations, the authors have established that the regularity of the velocity in one direction (say,  3 u), one component of the velocity (say,  3 ), or some other partial components of the velocity, velocity gradient, velocity Hessian, vorticity, pressure, and so forth, would guarantee the regularity of the weak solutions; see [19][20][21][22][23][24][25][26][27][28][29] and references therein.Many of these regularity criteria have been proved to be enjoyed by the MHD equations (1); see [30][31][32][33].However, due to the strong coupling of the fluid velocity and the magnetic fields, the scaling dimensions for the MHD equations are not as good (large) as that for the Navier-Stokes equations.
In this paper, we would like to improve the regularity criterion shown in [30].That is, we enlarge the scaling dimension from 1 to (almost) 3/2.Precisely, we show that the condition International Journal of Partial Differential Equations is enough to ensure the smoothness of the solution.The key idea is a multiplicative Sobolev inequality, which is in spirit similar to that in [20]; see Lemma 2. The rest of this paper is organized as follows.In Section 2, we recall the weak formulation of (1) and establish the fundamental Sobolev inequality.Section 3 is devoted to stating and proving the main result.
Then a fundamental Sobolev inequality is given.
where  is a generic constant independent of  and ; 1 ≤ ,  ≤ ∞, and 1 <  ≤ ∞ satisfy Proof .Consider the following where in the last inequality we have used Hölder inequality with and the Gagliardo-Nirenberg inequality.

The Main Result and Its Proof
In this section, we state and prove our main regularity criterion.
Proof.For any  ∈ (0, ), we can find a  ∈ (0, ) such that since (u, b) ∈  2 (0, ;  1 (R 3 )) as in Definition 1.Our strategy is to show that under condition (10) the weak solution is in fact strong; that is, which would imply the smoothness of the solution via standard energy estimates and Sobolev embeddings.Due to the arbitrariness of , we complete the proof.
To prove (12), we multiply (1) 1 by −Δu and (1) 2 by −Δb to get International Journal of Partial Differential Equations 3 Integration by parts formula together with the divergence-free conditions ∇ ⋅ u = ∇ ⋅ b = 0 yields where we use the summation convention; that is, the repeated index (say,  here) is automatically summed over {1, 2, 3}.