International Journal of Partial Differential Equations
Volume 2014 (2014), Article ID 213083, 5 pages
http://dx.doi.org/10.1155/2014/213083

Research Article

MHD Equations with Regularity in One Direction

Zujin Zhang

School of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, China

Received 3 July 2014; Accepted 6 October 2014; Published 27 October 2014

Academic Editor: Antonin Novotny

Copyright © 2014 Zujin Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider the 3D MHD equations and prove that if one directional derivative of the fluid velocity, say, , with , , then the solution is in fact smooth.  This improves previous results greatly.

1. Introduction

We consider the following three-dimensional (3D) magnetohydrodynamic (MHD) equations:

Here, and are the fluid velocity and magnetic fields, respectively; and 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 ( 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 [4–18] and references cited therein.

For the Navier-Stokes equations, the authors have established that the regularity of the velocity in one direction (say, ), one component of the velocity (say, ), 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 [19–29] 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

shown in [30]. That is, we enlarge the scaling dimension from to (almost) . Precisely, we show that the condition

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.

2. Preliminaries

In this section, we first recall the weak formulation of (1).

Definition 1. Let satisfying , be given. A measurable pair on is said to be a weak solution to (1) provided that

(1);(2) (1)1,2 are satisfied in the sense of distributions;(3) the energy inequality is given as

 for all .

Then a fundamental Sobolev inequality is given.

Lemma 2. Suppose that ; then, one has

where is a generic constant independent of and ; , , and satisfy

Proof . Consider the following

where in the last inequality we have used Hölder inequality with

and the Gagliardo-Nirenberg inequality.

3. The Main Result and Its Proof

In this section, we state and prove our main regularity criterion.

Theorem 3. Let satisfying , be given. Assume that the measurable pair is a weak solution as in Definition 1 on . If

then .

Proof. For any , we can find a such that

since 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 and (1)₂ by to get

Integration by parts formula together with the divergence-free conditions yields

where we use the summation convention; that is, the repeated index (say, here) is automatically summed over .
Substituting (14) into (13) and using a simple Cauchy-Schwarz inequality, we obtain

where is to be determined later on.
Due to the Calderón-Zygmund inequality,

( being the Riesz transform) we may take that the in (15) equals to get

To further bound , , we introduce some notations. Denote

Then, by (10), we have

Invoking Lemma 2 with as in (10), as in (18), and as in (19), we may estimate as

where is as in (16). Applying Hölder inequality with

(21) becomes

when , or

when , or

when .
Similarly, can be dominated as

when , or

when , or

when .
Thus, if , then, combing (23) and (26), we deduce from (17) that

Applying Gronwall inequality and noting that as in Definition 1, we obtain (12) as desired.
If , we gather (24) and (27) into (17) to get

To deduce (12) by applying Gronwall inequality to (30), we need only to show that

This is indeed true. First, simple interpolation inequality together with the fact that

yields

Second, the integrability indices as in (31) satisfy

and , by (18) and (19).
If, however, , then , and, combining (25) and (28), we deduce from (17) that

where we recall from (10). Applying Gronwall inequality and noting that as in Definition 1, we obtain (12) as desired.
The proof of Theorem 3 is completed.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author was partially supported by the Youth Natural Science Foundation of Jiangxi Province (20132BAB211007) and the National Natural Science Foundation of China (11326138, 11361004).

References

G. Duvaut and J.-L. Lions, “Inéquations en thermoélasticité et magnétohydrodynamique,” Archive for Rational Mechanics and Analysis, vol. 46, pp. 241–279, 1972.

E. Hopf, “Üer die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen,” Mathematische Nachrichten, vol. 4, pp. 213–231, 1951.

J. Leray, “Sur le mouvement d'un liquide visqueux emplissant l'espace,” Acta Mathematica, vol. 63, no. 1, pp. 193–248, 1934.

C. He and Z. Xin, “On the regularity of weak solutions to the magnetohydrodynamic equations,” Journal of Differential Equations, vol. 213, no. 2, pp. 235–254, 2005.

Y. Zhou, “Remarks on regularities for the 3D MHD equations,” Discrete and Continuous Dynamical Systems Series A, vol. 12, no. 5, pp. 881–886, 2005.

J. T. Beale, T. Kato, and A. Majda, “Remarks on the breakdown of smooth solutions for the -D Euler equations,” Communications in Mathematical Physics, vol. 94, no. 1, pp. 61–66, 1984.

H. Beirão da Veiga, “A new regularity class for the Navier-Stokes equations in ,” Chinese Annals of Mathematics B, vol. 16, no. 4, pp. 407–412, 1995.

D. Chae and J. Lee, “Regularity criterion in terms of pressure for the Navier-Stokes equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 46, no. 5, pp. 727–735, 2001.

Q. Chen, C. Miao, and Z. Zhang, “On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations,” Communications in Mathematical Physics, vol. 284, no. 3, pp. 919–930, 2008.

L. Escauriaza, G. Seregin, and V. Šverák, “Backward uniqueness for parabolic equations,” Archive for Rational Mechanics and Analysis, vol. 169, no. 2, pp. 147–157, 2003.

C. He and Y. Wang, “On the regularity criteria for weak solutions to the magnetohydrodynamic equations,” Journal of Differential Equations, vol. 238, no. 1, pp. 1–17, 2007.

H. Kozono, T. Ogawa, and Y. Taniuchi, “The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations,” Mathematische Zeitschrift, vol. 242, no. 2, pp. 251–278, 2002.

J. Serrin, “On the interior regularity of weak solutions of the Navier-Stokes equations,” Archive for Rational Mechanics and Analysis, vol. 9, pp. 187–195, 1962.

J. Wu, “Regularity criteria for the generalized MHD equations,” Communications in Partial Differential Equations, vol. 33, no. 1–3, pp. 285–306, 2008.

J. Wu, “Regularity results for weak solutions of the 3D MHD equations,” Discrete and Continuous Dynamical Systems, vol. 10, no. 1-2, pp. 543–556, 2004.

Z. Zhang, Z.-a. Yao, M. Lu, and L. Ni, “Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations,” Journal of Mathematical Physics, vol. 52, no. 5, Article ID 053103, 7 pages, 2011.

Y. Zhou, “Regularity criteria for the generalized viscous {MHD} equations,” Annales de l'Institut Henri Poincaré. Analyse Non Lineairé, vol. 24, no. 3, pp. 491–505, 2007.

Y. Zhou, “Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain,” Mathematische Annalen, vol. 328, no. 1-2, pp. 173–192, 2004.

C. S. Cao, “Sufficient conditions for the regularity to the 3D Navier-Stokes equations,” Discrete and Continuous Dynamical Systems. Series A, vol. 26, no. 4, pp. 1141–1151, 2010.

C. Cao and E. S. Titi, “Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor,” Archive for Rational Mechanics and Analysis, vol. 202, no. 3, pp. 919–932, 2011.

C. S. Cao and E. S. Titi, “Regularity criteria for the three-dimensional Navier-Stokes equations,” Indiana University Mathematics Journal, vol. 57, no. 6, pp. 2643–2661, 2008.

Z. Zhang, D. Zhong, and L. Hu, “A new regularity criterion for the 3D Navier-Stokes equations via two entries of the velocity gradient tensor,” Acta Applicandae Mathematicae, vol. 129, pp. 175–181, 2014.

Z. Zhang, F. Alzahrani, T. Hayat, and Y. Zhou, “Two new regularity criteria for the Navier-Stokes equations via two entries of the velocity Hessian tensor,” Applied Mathematics Letters, vol. 37, pp. 124–130, 2014.

Z. Zhang, “A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component,” Communications on Pure and Applied Analysis, vol. 12, no. 1, pp. 117–124, 2013.

Z. L. Zhang, Z. A. Yao, P. Li, C. C. Guo, and M. Lu, “Two new regularity criteria for the 3D Navier-Stokes equations via two entries of the velocity gradient tensor,” Acta Applicandae Mathematicae, vol. 123, pp. 43–52, 2013.

Y. Zhou, “A new regularity criterion for the Navier-Stokes equations in terms of the gradient of one velocity component,” Methods and Applications of Analysis, vol. 9, no. 4, pp. 563–578, 2002.

Y. Zhou, “A new regularity criterion for weak solutions to the Navier-Stokes equations,” Journal de Mathématiques Pures et Appliquées, vol. 84, no. 11, pp. 1496–1514, 2005.

Y. Zhou and M. Pokorný, “On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component,” Journal of Mathematical Physics, vol. 50, no. 12, Article ID 123514, 2009.

Y. Zhou and M. Pokorný, “On the regularity of the solutions of the Navier-Stokes equations via one velocity component,” Nonlinearity, vol. 23, no. 5, pp. 1097–1107, 2010.

C. Cao and J. Wu, “Two regularity criteria for the 3D MHD equations,” Journal of Differential Equations, vol. 248, no. 9, pp. 2263–2274, 2010.

Z. J. Zhang, P. Li, and G. H. Yu, “Regularity criteria for the 3D MHD equations via one directional derivative of the pressure,” Journal of Mathematical Analysis and Applications, vol. 401, no. 1, pp. 66–71, 2013.

Z. Zhang, Z.-a. Yao, and X. Wang, “A regularity criterion for the 3D magneto-micropolar fluid equations in Triebel-Lizorkin spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 6, pp. 2220–2225, 2011.

Z. J. Zhang, “Magneto-micropolar fluid equations with regularity in one direction,” submitted