A Regularity Criterion for the 3D Incompressible Magnetohydrodynamics Equations in the Multiplier Spaces

Here u, b describe the flow velocity vector and the magnetic field vector, respectively, p is a scalar pressure, ] > 0 is the kinematic viscosity, and η > 0 is the magnetic diffusivity, while u0 and b0 are the given initial velocity and initial magnetic field, respectively, with ∇ ⋅ u0 = ∇ ⋅ b0 = 0. If ] = η = 0, (1) is called the ideal MHD equations. Employing the standard energy method, it is easy to prove that, for given initial data (u0, b0) ∈ Hs(R3) with s > 1/2, there exist a positive time T = T(‖(u0, b0)‖Hs) and a unique smooth solution (u(t, x), b(t, x)) on [0, T) to the MHD equations satisfying (u, b) ∈ C ([0, T) ;Hs) ∩ C1 ((0, T) ;Hs) ∩ C ((0, T) ;Hs+2) . (2)


Introduction
We consider the 3D incompressible magnetohydrodynamics (MHD) equations.( Here ,  describe the flow velocity vector and the magnetic field vector, respectively,  is a scalar pressure, ] > 0 is the kinematic viscosity, and  > 0 is the magnetic diffusivity, while  0 and  0 are the given initial velocity and initial magnetic field, respectively, with ∇ ⋅  0 = ∇ ⋅  0 = 0.If ] =  = 0, (1) is called the ideal MHD equations.
However, the solution regularity can be derived when certain growth conditions are satisfied.For regularity of the weak solutions to the 3D MHD equations (1), some numerical experiments [2,3] seem to indicate that the velocity field plays a more important role than the magnetic field in the regularity theory of solutions to the MHD equations.Recently, inspired by Constantin and Fefferman initial work [4] where the regularity condition of the direction of vorticity was used to describe the regularity criterion to the Navier-Stokes equations, He and Xin [5] extended it to the MHD equations and obtained some integrability condition of the magnitude of the only velocity  alone; that is, Later, Zhou and Gala [6] extended it to the multiplier spaces.
Other important studies such as the regularity criteria can be found in [7][8][9] and references therein.
Recently, some regularity criteria in terms of partial velocity components and magnetic components or partial derivative of the velocity components and magnetic components were established [10][11][12].However, the spaces used are not scaling invariant (in other words, not of Serrin's type).Many researchers were devoted to studying it along this direction.In 2010, Ji and Lee [13] obtained the following regularity: In 2012, Ni et al. [14] got some new regularity as follows: Our purpose in this paper is to obtain a new regularity criterion of weak solution for the 3D MHD equations in a sense of scaling invariant by employing a different decomposition for nonlinear terms.
Notation 1.Throughout the paper, we use the following notations for the simplicity.The planar components of (, ) will be denoted by ũ = ( 1 ,  2 ) and b = ( 1 ,  2 ) and ∇ will be used for ∇ = ( 1 ,  2 ).Now we state our result as follows.

Preliminaries
First, we recall the definition and some properties of the multiplier space Ẋ introduced recently by Lemarié-Rieusset [15] (see also [16]).The space Ẋ of pointwise multipliers which map  2 into Ḣ− is defined in the following way.
Definition 5.For 0 ≤  < 3/2, we define the homogeneous space Ẋ by where we denote by Ḣ (R  ) the completion of the space , where û() denotes the Fourier transform of .
The norm of Ẋ is given by the operator norm of pointwise multiplication          Ẋ = sup It is easy to check that Hence, for any function (, ) defined for both spatial and time variables, for any  > 0, with   (, ) = (,  2 ).Therefore, if (, ) solves the MHD equations, then so does (  ,   ).This is the so-called scaling dimension zero property.
Lemma 6. Assume 0 ≤  ≤ 1.Then the following inequality holds: The proof of this lemma can be easily obtained by Parseval's equality and Hölder's inequality, and we omit it.
Step 1 ( 2 -estimates).Multiplying the first equation and the second equation in (1) by , , respectively, and integrating the resulting equations by parts over R 3 , we obtain after adding them together from the incompressibility condition ∇ ⋅  = 0 and ∇ ⋅  = 0 where ⟨⋅, ⋅⟩ denotes the inner-product in  2 (R 3 ).Integrating from 0 to for the above inequality, we have Step 2 ( Ḣ1 -estimates).Differentiating the first equation and the second equation of ( 1) with respect to   (1 ≤  ≤ 3) and multiplying the first and the second equations of (1) by /  =    and /  =   , respectively, and then, by integrating by parts over R 3 , we get Noting the incompressibility conditions ∇⋅ = 0 and ∇⋅ = 0, since then ( 15) and ( 16) can be rewritten as First, we rewrite  1 into seven parts as follows: According to the definition of ∇ũ, the former four terms of the right-hand side of ( 20 Next, we will bound the term  2 .We can get, by splitting  2 into seven parts, Obviously, the former four terms of the right-hand side of (24) are bounded by Combining ( 14) with (32), we have ,  ∈  ∞ ([0, ) ;  1 ) .
By the standard arguments of continuation of local solutions [1], we can draw the conclusion that the smooth solution (, ) remains smooth at  = .This completes the proof of Theorem 2.