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A Regularity Criterion for the 3D Incompressible Magnetohydrodynamics Equations in the Multiplier Spaces
We are concerned with the regularity criterion for weak solutions to the 3D incompressible MHD equations in this paper. We show that if some partial derivatives of the velocity components and magnetic components belong to the multiplier spaces, then the solution actually is smooth on .
We consider the 3D incompressible magnetohydrodynamics (MHD) equations.
(MHD)Here , describe the flow velocity vector and the magnetic field vector, respectively, is a scalar pressure, is the kinematic viscosity, and is the magnetic diffusivity, while and are the given initial velocity and initial magnetic field, respectively, with . If , (1) is called the ideal MHD equations.
Employing the standard energy method, it is easy to prove that, for given initial data with , there exist a positive time and a unique smooth solution on to the MHD equations satisfying Whether smooth solutions of (1) on will lead to a singularity at is an outstanding open problem; see Sermange and Temam .
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  where the regularity condition of the direction of vorticity was used to describe the regularity criterion to the Navier-Stokes equations, He and Xin  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  extended it to the multiplier spaces. Other important studies such as the regularity criteria can be found in [7–9] 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 [10–12]. 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  obtained the following regularity:In 2012, Ni et al.  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 and and will be used for .
Now we state our result as follows.
Theorem 2. Let with . If the weak solutions to (1) satisfy the following integrability conditions:then the solution remains smooth on .
Remark 3. Since the components of and without and are less than those of and , therefore, our result improves that in .
Remark 4. Sinceas , where denotes the weak -space, therefore, our result extends that in .
First, we recall the definition and some properties of the multiplier space introduced recently by Lemarié-Rieusset  (see also ). The space of pointwise multipliers which map into is defined in the following way.
Definition 5. For , we define the homogeneous space by where we denote by the completion of the space with respect to the norm , where denotes the Fourier transform of .
The norm of is given by the operator norm of pointwise multiplication
It is easy to check that Hence, for any function defined for both spatial and time variables, for any , with . Therefore, if solves the MHD equations, then so does . This is the so-called scaling dimension zero property.
Lemma 6. Assume . 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.
3. Proof of Theorem 2
Step 1 (-estimates). Multiplying the first equation and the second equation in (1) by , , respectively, and integrating the resulting equations by parts over , we obtain after adding them together from the incompressibility condition and where denotes the inner-product in . Integrating from to for the above inequality, we have
Step 2 (-estimates). Differentiating the first equation and the second equation of (1) with respect to and multiplying the first and the second equations of (1) by and , respectively, and then, by integrating by parts over , we getNoting the incompressibility conditions and , since then (15) and (16) can be rewritten as Since , adding up (18), we have by summing up over First, we rewrite into seven parts as follows: According to the definition of , the former four terms of the right-hand side of (20) are bounded by And the latter three terms can be estimated, noting that from the incompressible condition Thus, using Hölder’s inequality and Young’s inequality, we have Next, we will bound the term . We can get, by splitting into seven parts, Obviously, the former four terms of the right-hand side of (24) are bounded by Noting that and from the incompressible conditions, the latter three terms can be estimated as follows: So, by Hölder’s inequality and Young’s inequality, we obtainSimilar to the estimate of , for , we have Combining the above estimates (23), (27), and (28) into (19), we deduce Therefore, we have Due to Gronwall’s inequality, it follows from (30) that Thanks to , we haveCombining (14) with (32), we haveBy the standard arguments of continuation of local solutions , we can draw the conclusion that the smooth solution remains smooth at . This completes the proof of Theorem 2.
Conflicts of Interest
The author declares that they have no conflicts of interest.
The work was supported by the National Natural Science Foundation of China (11302102) and China Postdoctoral Science Foundation funded project (2014M561893).
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