Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2014, Article ID 563084, 4 pages
http://dx.doi.org/10.1155/2014/563084
Research Article

An Osgood Type Regularity Criterion for the 3D Boussinesq Equations

1School of Resources and Safety Engineering, Central South University, Changsha, Hunan 410075, China
2Jiangxi University of Science and Technology, Ganzhou, Jiangxi 341000, China
3Department of Basic Teaching, Harbin Finance University, Harbin 150030, Heilongjiang, China

Received 2 November 2013; Accepted 2 February 2014; Published 11 March 2014

Academic Editors: D. Baleanu and H. Jafari

Copyright © 2014 Qiang Wu et al. 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 an Osgood type regularity criterion in terms of the velocity gradient.

1. Introduction

In this paper, we consider the following three-dimensional (3D) Boussinesq equations with the incompressibility condition: where is the fluid velocity, is a scalar pressure, and is the scalar temperature, while and are the prescribed initial velocity and temperature, respectively, with .

In case , (1) reduces to the incompressible Navier-Stokes equations. The regularity of its weak solutions and the existence of global strong solutions are important open problems; see [13]. Starting with [4, 5], there have been a lot of literatures devoted to finding sufficient conditions (which now are called regularity criteria) to ensure the smoothness of the solutions; see [616] and so forth. Since the convective terms are the same in the Navier-Stokes equations and Boussinesq equations, the authors also consider the regularity conditions for (1). In particular, Qiu et al. [17] obtained Serrin type regularity condition:

The extension to the multiplier spaces was established by the same authors in [18]. For the Besov-type regularity criterion, Fan and Zhou [19] and Ishimura and Morimoto [20] showed the following regularity conditions: Zhang [21, 22] then considers the regularity criterion in terms of the pressure or its gradient. The readers are also referred to [23] for generalized models.

Motivated by [2426], we will improve (3) as in the following.

Theorem 1. Let . Assume that is the smooth solution to (1) with the initial data for . If then the solution can be extended after time . Here, denotes the Fourier localization operator and .

Remark 2. The Osgood type condition (4) is weaker than (3). Notice that, for , we have

The rest of this paper is organized as follows. In Section 2, we recall the definition of Besov spaces and some interpolation inequalities. Section 3 is devoted to proving Theorem 1.

2. Preliminaries

Let be the Schwartz class of rapidly decreasing functions. For , its Fourier transform is defined by Let us choose a nonnegative radial function such that and let For , the Littlewood-Paley projection operators and are, respectively, defined by Observe that . Also, it is easy to check that if , then in the sense. By telescoping the series, we thus have the following Littlewood-Paley decomposition: for all , where the summation is the sense. Notice that then from Young's inequality, it readily follows that where and is an absolute constant independent of and .

Let , , ; the homogeneous Besov space is defined by the full-dyadic decomposition such that where and is the dual space of Also, it is well known that We refer to [27] for more detailed properties.

3. Proof of Theorem 1

This section is devoted to proving Theorem 1. From standard continuity arguments, we need to only provide the uniform bounds of the solution .

Taking the inner products of (1) with , (1) with , we obtain by adding together that For , we use Hölder’s inequality to get For , applying the Littilewood-Paley decomposition as in (11), we get where is positive integral to be determined later on. Plugging (20) into , we see that For , we dominate as For , we have Finally, for , we estimate as Gathering (22), (23), and (24) together and plugging them into (21), we deduce Substituting (19) and (25) into (18), we find Taking where is the largest integer smaller than and , then (26) implies that Applying Gronwall inequality three times, we deduce Recalling (4), we see the solution is uniformly bounded in . This completes the proof of Theorem 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the NSF of China (no. 11326238, no. 11326138, and no. 11101101), the Science Foundation of Jiangxi Provincial Department of Education (no. GJJ13374 and no. GJJ13658), and the Youth Natural Science Foundation of Jiangxi Province (20132BAB211007).

References

  1. E. Hopf, “Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen,” Mathematische Nachrichten, vol. 4, pp. 213–231, 1951. View at Google Scholar · View at MathSciNet
  2. P. G. Lemarié-Rieusset, Recent Developments in the Navier–Stokes Problem, vol. 431, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  3. 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
  4. 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
  5. J. Serrin, “The initial value problem for the Navier–Stokes equations,” in Nonlinear Problems, R. E. Langer, Ed., pp. 69–98, University of Wisconsin Press, Madison, Wis, USA, 1963. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. X. W. He and S. Gala, “Regularity criterion for weak solutions to the Navier–Stokes equations in terms of the pressure in the class L2(0,T;B,-13),” Nonlinear Analysis, Real World Applications, vol. 12, no. 6, pp. 3602–3607, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. 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,” in Topics in Mathematical Fluid Mechanics, vol. 10 of Quaderni di Matematica, pp. 163–183, Seconda Università degli Studi di Napoli, Caserta, Italy, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Z. J. 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
  9. Z. J. Zhang, “A remark on the regularity criterion for the 3D Navier–Stokes equations involving the gradient of one velocity component,” Journal of Mathematical Analysis and Applications, vol. 414, no. 1, pp. 472–479, 2014. View at Publisher · View at Google Scholar
  10. Z. J. 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,” 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
  11. 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, vol. 129, no. 1, pp. 175–181, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  12. Z. J. Zhang, Z. A. Yao, M. Lu, and L. D. 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
  13. Z. J. Zhang, P. Li, and D. X. Zhong, “Navier–Stokes equations with regularity in two entries of the velocity gradient tensor,” Applied Mathematics and Computation, vol. 228, pp. 546–551, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  14. 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
  15. 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, 11 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. 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
  17. H. Qiu, Y. Du, and Z. A. 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
  18. 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
  19. 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
  20. 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
  21. Z. J. Zhang, “A remark on the regularity criterion for the 3D Boussinesq equations via the pressure gradient,” Abstract and Applied Analysis, vol. 2014, Article ID 510924, 4 pages, 2014. View at Publisher · View at Google Scholar
  22. Z. J. Zhang, “A logarithmically improved regularity criterion for the 3D Boussinesq equations via the pressure,” Acta Aplicandae Mathematicae, 2013. View at Publisher · View at Google Scholar
  23. B. Mehdinejadiani, H. Jafari, and D. Baleanu, “Derivation of a fractional Boussinesq equation for modelling uncon-fined groundwater,” The European Physical Journal Special Topics, vol. 222, no. 8, pp. 1805–1812, 2013. View at Publisher · View at Google Scholar
  24. Q. Zhang, “Refined blow-up criterion for the 3D magnetohydrodynamics equations,” Applicable Analysis, vol. 92, no. 12, pp. 2590–2599, 2013. View at Publisher · View at Google Scholar
  25. Z. J. Zhang and S. Gala, “Osgood type regularity criterion for the 3D Newton-Boussinesq equation,” Electronic Journal of Differential Equations, vol. 2013, no. 223, pp. 1–6, 2013. View at Google Scholar · View at MathSciNet
  26. Z. J. Zhang, T. Tang, and L. H. Liu, “An Osgood type regularity criterion for the liquid crystal flows,” Nonlinear Differerntial Equations and Applications, 2013. View at Publisher · View at Google Scholar
  27. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18, North-Holland, Amsterdam, The Netherlands, 1978. View at MathSciNet