Abstract and Applied Analysis

Volume 2012 (2012), Article ID 682436, 7 pages

doi:10.1155/2012/682436

Research Article

A Regularity Criterion for the Navier-Stokes Equations in the Multiplier Spaces

Xiang'ou Zhu1,2

¹College of Physics and Electronic Information Engineering, Wenzhou University, Zhejiang, Wenzhou 325035, China

²The Key Laboratory of Low-voltage Apparatus Intellectual Technology of Zhejiang, Wenzhou 325035, China

Received 16 February 2012; Accepted 23 April 2012

Academic Editor: Benchawan Wiwatanapataphee

Copyright © 2012 Xiang'ou Zhu. 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 exhibit a regularity condition concerning the pressure gradient for the Navier-Stokes equations in a special class. It is shown that if the pressure gradient belongs to , where is the multipliers between Sobolev spaces whose definition is given later for , then the Leray-Hopf weak solution to the Navier-Stokes equations is actually regular.

1. Introduction

Consider the Navier-Stokes equations in :

where is the velocity field, is the scalar pressure, and with div in the sense of distribution is the initial velocity field. For simplicity, we assume that the external force has a scalar potential and is included into the pressure gradient.

In the famous paper, Leray [1] and Hopf [2] constructed a weak solution of (1.1) for arbitrary with . The solution is called the Leray-Hopf weak solution. Regularity of such Leray-Hopf weak solutions is one of the most significant open problems in mathematical fluid mechanics. We note here that there are partial regularity results from Scheffer and from Caffarelli et al., see [3, 4] and references therein. Besides, more work was pioneered by Serrin [5] and extended and improved by Giga [6], Struwe [7, 8], and Zhou [9]. Further results can be found in [10–16] and references therein.

Introducing the class , Serrin [5] showed that if we have a Leray-Hopf weak solution belonging to with the exponents and satisfying , , , then the solution , while the limit case was shown much later by Sohr [17] (see also [18]).

Regularity results including assumptions on the pressure gradient have been given by Zhou [15], and it was extended later by Struwe [8] to any dimension . It is shown that if the gradient of pressure with , then the corresponding weak solution is actually strong. For the recent work on the regularity problem containing the pressure, velocity field, and the quotient of pressure-velocity, we refer to [19–21] for details.

The purpose of this short paper is to establish a regularity criterion in terms of the pressure gradient for weak solutions to the Navier-Stokes equations in the class . This work is motivated by the recent results [22, 23] on the Navier-Stokes equations. It is an unusual, larger space considered in the current paper than (the following Lemma 2.3) and possesses more information. Obviously, the present result extends some previous ones. For more facts concerning regularity of weak solutions, we refer the readers to the celebrated papers [24–30].

2. Preliminaries

We recall the definition of the multiplier space, which was introduced in [31] (see also [32, 33]). The space of pointwise multipliers, which map into , is defined in the following way.

Definition 2.1. For , is a Banach space of all distributions on such that there exists a constant such that for all we have and

where we denote by the completion of the space with respect to the norm and denote by the Schwarz class.

The norm of is given by the operator norm of pointwise multiplication

Remark 2.2. Equivalently, we will say that if and only if the inequality

holds for all .

Lemma 2.3. Let . Then the following embedding:

holds.

Proof. Indeed, let . By using the following well-known Sobolev embedding:

with , we have by Hölder’s inequality

where . Then, it follows that

This completes the proof.

Example 2.4. Due to the well-known inequality

we see that .

Indeed, since the functions of class are dense in in the norm , suppose . Then by virtue of the Cauchy-Schwarz inequality we obtain

and thus for , and

3. Regularity Theorem

Now we state our result as following.

Theorem 3.1. Let for some and in the sense of distributions. Suppose that is a Leray-Hopf solution of (1.1) in . If the pressure gradient satisfies

then is a regular solution in the sense that

Proof. In order to prove this result, we have to do a priori estimates for the Navier-Stokes equations and then show that the solution satisfies the well-known Serrin regularity condition. Multiply both sides of the first equation of (1.1) by and integrate by parts to obtain (see, e.g., [30])

for . Then we have

where we have used

Let us estimate the integral

on the right-hand side of (3.4). By the Hölder inequality and the Young inequality, we have

where ; we have used the inequality

and the Young inequality with :

for . Hence by (3.4) and the above inequality, we derive

Now by Gronwall’s lemma (see for instance in [28, Lemma 2]), we have

Due to the integrability of the pressure gradient, it follows that

Consequently falls into the well-known Serrin’s regularity framework. Therefore, the smoothness of follows immediately. This completes the proof of Theorem 3.1.

Remark 3.2. By a strong solution we mean a weak solution of the Navier-Stokes equation such that

It is wellknown that strong solutions are regular (we say classical) and unique in the class of weak solutions.

Acknowledgment

The author thanks the anonymous referee for his/her comments on this paper.

References

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

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

L. Caffarelli, R. Kohn, and L. Nirenberg, “Partial regularity of suitable weak solutions of the Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 35, no. 6, pp. 771–831, 1982.

V. Scheffer, “Partial regularity of solutions to the Navier-Stokes equations,” Pacific Journal of Mathematics, vol. 66, no. 2, pp. 535–552, 1976.

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.

Y. Giga, “Solutions for semilinear parabolic equations in and regularity of weak solutions of the Navier-Stokes system,” Journal of Differential Equations, vol. 62, no. 2, pp. 186–212, 1986.

M. Struwe, “On partial regularity results for the Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 41, no. 4, pp. 437–458, 1988.

M. Struwe, “On a Serrin-type regularity criterion for the Navier-Stokes equations in terms of the pressure,” Journal of Mathematical Fluid Mechanics, vol. 9, no. 2, pp. 235–242, 2007.

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.

H. B. da Veiga, “A sufficient condition on the pressure for the regularity of weak solutions to the Navier-Stokes equations,” Journal of Mathematical Fluid Mechanics, vol. 2, no. 2, pp. 99–106, 2000.

J. Fan, S. Jiang, and G. Ni, “On regularity criteria for the n-dimensional Navier-Stokes equations in terms of the pressure,” Journal of Differential Equations, vol. 244, no. 11, pp. 2963–2979, 2008.

J. Fan, S. Jiang, G. Nakamura, and Y. Zhou, “Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations,” Journal of Mathematical Fluid Mechanics, vol. 13, no. 4, pp. 557–571, 2011.

H. Kozono and H. Sohr, “Regularity criterion of weak solutions to the Navier-Stokes equations,” Advances in Differential Equations, vol. 2, no. 4, pp. 535–554, 1997.

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.

Y. Zhou, “On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in ,” Zeitschrift für Angewandte Mathematik und Physik, vol. 57, no. 3, pp. 384–392, 2006.

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.

H. Sohr, The Navier-Stokes equations, An Elementary Functional Analytic Approach, Birkhäuser Advanced Texts, Springer Basel, Basel, Switzerland, 2001.

H. Sohr and W. von Wahl, “On the regularity of the pressure of weak solutions of Navier-Stokes equations,” Archiv der Mathematik, vol. 46, no. 5, pp. 428–439, 1986.

Z. Guo and S. Gala, “Remarks on logarithmical regularity criteria for the Navier-Stokes equations,” Journal of Mathematical Physics, vol. 52, no. 6, p. 063503, 9, 2011.

Z. Guo and S. Gala, “A note on the regularity criteria for the Navier-Stokes equations,” Applied Mathematics Letters, vol. 25, no. 3, pp. 305–309, 2012.

Z. Guo, P. Wittwer, and W. Wang, “Regularity issue of the Navier-Stokes equations involving the combination of pressure and velocity field,” Acta Applicandae Mathematicae. In press

Y. Zhou and S. Gala, “Regularity criteria in terms of the pressure for the Navier-Stokes equations in the critical Morrey-Campanato space,” Zeitschrift für Analysis und ihre Anwendungen, vol. 30, no. 1, pp. 83–93, 2011.

Y. Zhou and S. Gala, “Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces,” Journal of Mathematical Analysis and Applications, vol. 356, no. 2, pp. 498–501, 2009.

L. C. Berselli and G. P. Galdi, “Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations,” Proceedings of the American Mathematical Society, vol. 130, no. 12, pp. 3585–3595, 2002.

C. 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.

C. Cao, J. Qin, and E. S. Titi, “Regularity criterion for solutions of three-dimensional turbulent channel flows,” Communications in Partial Differential Equations, vol. 33, no. 1–3, pp. 419–428, 2008.

S. Gala, “Remark on a regularity criterion in terms of pressure for the Navier-Stokes equations,” Quarterly of Applied Mathematics, vol. 69, no. 1, pp. 147–155, 2011.

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 the Navier-Stokes equations in terms of the direction of vorticity,” Monatshefte für Mathematik, vol. 144, no. 3, pp. 251–257, 2005.

Y. Zhou, “On regularity criteria in terms of pressure for the Navier-Stokes equations in ,” Proceedings of the American Mathematical Society, vol. 134, no. 1, pp. 149–156, 2006.

P. G. Lemarié-Rieusset and S. Gala, “Multipliers between Sobolev spaces and fractional differentiation,” Journal of Mathematical Analysis and Applications, vol. 322, no. 2, pp. 1030–1054, 2006.

V. G. Maz'ya and I. E. Verbitsky, “The Schrödinger operator on the energy space: boundedness and compactness criteria,” Acta Mathematica, vol. 188, no. 2, pp. 263–302, 2002.

V. G. Maz'ya and I. E. Verbitsky, “The form boundedness criterion for the relativistic Schrödinger operator,” Annales de l'Institut Fourier, vol. 54, no. 2, pp. 317–339, 2004.