Abstract and Applied Analysis

Volume 2014, Article ID 510924, 4 pages

http://dx.doi.org/10.1155/2014/510924

## A Remark on the Regularity Criterion for the 3D Boussinesq Equations Involving the Pressure Gradient

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

Received 10 November 2013; Accepted 28 December 2013; Published 21 January 2014

Academic Editor: Giovanni P. Galdi

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 three-dimensional Boussinesq equations and obtain a regularity criterion involving the pressure gradient in the Morrey-Companato space . This extends and improves the result of Gala (Gala 2013) for the Navier-Stokes equations.

#### 1. Introduction

This paper concerns itself with the following three-dimensional (3D) Boussinesq equations: where is given time, is the fluid velocity, is a scalar pressure, and is the temperature, while and are the prescribed initial velocity field and temperature, respectively.

When , (1) reduces to the incompressible Navier-Stokes equations. The regularity of its weak solutions and the existence of global strong solutions are challenging open problems; see [1–3]. Starting with [4, 5], there have been a lot of literature devoted to finding sufficient conditions to ensure the smoothness of the solutions; see [6–15] and the references cited therein. Since the convective terms are similar in the Navier-Stokes equations and Boussinesq equations, the authors also consider the regularity conditions for (1); see [16–20] and so forth.

In [6], Gala uses intricate decomposition technique to obtain the following regularity criterion for the Navier-Stokes equations: Here, is the point-wise multiplier space from to , which is strictly larger than (see [6, Lemma 1.2]).

In this paper, we will extend and improve the regularity condition (2) to the Boussineq equations (1).

Before stating the precise result, let us recall the weak formulation of (1).

*Definition 1. *Let , . A measurable pair is said to be a weak solution of (1) in , provided that(1), ;(2)(1)_{1,2,3} are satisfied in the sense of distributions;(3)the energy inequality
for all .

Now, our main result reads the following.

Theorem 2. *Let with in the sense of distributions, . Supposing that is a weak solution of (1) in , and the pressure gradient satisfies
**
then the solution .*

Here, is the Morrey-Campanato space, which will be introduced in Section 2. And Section 3 is devoted to the proof of Theorem 2.

*Remark 3. *Noticing that for (see (10)), we indeed improve the result of [6] for the Navier-Stokes equations.

#### 2. Preliminaries

In this section, we will introduce the definition of Morrey-Campanato space and recall its fundamental properties. The space plays an important role in studying the regularity of solutions to partial differential equations (see [21–23], e.g.).

*Definition 4. *For , the Morrey-Campanato space is defined as
where is the ball with center and radius .

One sees readily that is a Banach space under the norm and contains the classical Lebesgue space as a subspace:

Moreover, the following scaling property holds:

Due to the following characterization in [24].

Lemma 5. *For , the space is defined as the space of all functions such that
**Then if and only if with equivalent norm.*

*And with the fact that
we have
*

*Here is the Besov space, which is intermediate between and (see [25]):
*

*3. Proof of Theorem 2*

*In this section, we will prove Theorem 2.*

*Due to the Serrin type regularity criterion
in [19], we need only to prove
*

*We just do a priori estimates, with the justification being from passage to limits for the Galerkin approximated solutions.*

*Taking the inner product of (1) _{2} with in , we find
*

*Thus,
*

*One can also take the inner product of (1) _{2} with () in to derive the estimate of in -norm and invoke the maximum principle to bound the -norm of , as stated in Definition 1.*

*Taking the divergence of (1) _{1}, we get
*

*Consequently,
*

*Taking the inner product of (1) _{1} with in , we get
*

*For , we estimate as*

*
The term can be dominated as
*

*Plugging (19) and (20) into (18), we deduce that
*

*Applying Gronwall inequality, we see that
for every . Recalling (13), we complete the proof of Theorem 2.*

*Conflict of Interests*

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

*Acknowledgments*

*Zujin Zhang was partially supported by the Youth Natural Science Foundation of Jiangxi Province (20132BAB211007), the Science Foundation of Jiangxi Provincial Department of Education (GJJ13658, GJJ13659), and the National Natural Science Foundation of China (11326138, 11361004, and 11326238).*

*References*

- E. Hopf, “Über die anfangswertaufgabe für die hydrodynamischen Grundgleichungen,”
*Mathematische Nachrichten*, vol. 4, pp. 213–231, 1951. View at Google Scholar · View at MathSciNet - P. G. Lemarié-Rieusset,
*Recent Developments in the Navier-Stokes Problem*, Chapman & Hall, Boca Raton, Fla, USA, 2002. View at MathSciNet - J. Leray, “Sur le mouvement d'un liquide visqueux emplissant l'espace,”
*Acta Mathematica*, vol. 63, no. 1, pp. 193–248, 1934. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Serrin, “On the interior regularity of weak solutions of the Navier-Stokes equations,”
*Archive for Rational Mechanics and Analysis*, vol. 9, no. 1, pp. 187–195, 1962. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Serrin, “The initial value problem for the Navier-Stokes equations,” in
*Nonlinear Problems*, R. E. Langer, Ed., University of Wisconsin Press, Madison, Wis, USA, 1963. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Gala, “Remarks on regularity criterion for weak solutions to the Navier-Stokes equations in terms of the gradient of the pressure,”
*Applicable Analysis*, vol. 92, no. 1, pp. 96–103, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. He and S. Gala, “Regularity criterion for weak solutions to the Navier-Stokes equations in terms of the pressure in the class ${L}^{2}(0,T;{\stackrel{\u0307}{B}}_{\infty ,\infty}^{-1}({\mathbb{R}}^{3})$,”
*Nonlinear Analysis: Real World Applications*, vol. 12, no. 6, pp. 3602–3607, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Neustupa, A. Novotný, and P. Penel, “An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity, topics in mathematical uid mechanics,”
*Quaderni di Matematica*, vol. 10, pp. 163–183, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zhang, Z.-A. Yao, P. Li, 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, no. 1, pp. 43–52, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Z. J. Zhang, D. X. 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*, 2013. View at Publisher · View at Google Scholar - 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, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Zhou, “A new regularity criterion for weak solutions to the Navier-Stokes equations,”
*Journal des Mathematiques Pures et Appliquees*, vol. 84, no. 11, pp. 1496–1514, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. S. Fan and Y. Zhou, “A note on regularity criterion for the 3D Boussinesq system with partial viscosity,”
*Applied Mathematics Letters*, vol. 22, no. 5, pp. 802–805, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - N. Ishimura and H. Morimoto, “Remarks on the blow-up criterion for the 3-D Boussinesq equations,”
*Mathematical Models and Methods in Applied Sciences*, vol. 9, no. 9, pp. 1323–1332, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - H. Qiu, Y. Du, and Z. Yao, “Blow-up criteria for 3D Boussinesq equations in the multiplier space,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 4, pp. 1820–1824, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - H. Qiu, Y. Du, and Z. Yao, “Serrin-type blow-up criteria for 3D Boussinesq equations,”
*Applicable Analysis*, vol. 89, no. 10, pp. 1603–1613, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Z. J. Zhang, “A logarithmically improved regularity criterion for the 3D Boussinesq equations via the pressure,”
*Acta Applicandae Mathematicae*, 2013. View at Publisher · View at Google Scholar - T. Kato, “Strong solutions of the Navier-Stokes equation in Morrey spaces,”
*Boletim da Sociedade Brasileira de Matemática*, vol. 22, no. 2, pp. 127–155, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - M. E. Taylor, “Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations,”
*Communications in Partial Differential Equations*, vol. 17, no. 9-10, pp. 1407–1456, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zhang, “Remarks on the regularity criteria for generalized MHD equations,”
*Journal of Mathematical Analysis and Applications*, vol. 375, no. 2, pp. 799–802, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - P. G. Lemarié-Rieusset,
*Recent Developments in the Navier-Stokes Problem*, Chapman & Hall, London, UK, 2002. View at MathSciNet - Y. Zhou and S. Gala, “A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 9-10, pp. 3643–3648, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus

*
*