Skip to main content
Log in

Logarithmical regularity criterion of the three-dimensional Boussinesq equations in terms of the pressure

Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

This work establishes a sufficient condition for the regularity criterion of the Boussinesq equation in terms of the derivative of the pressure in one direction. It is shown that if the partial derivative of the pressure \({\partial _{3}\pi }\) satisfies the logarithmical Serrin-type condition

$$\int_{0}^{T}\frac{\left\Vert \partial _{3}\pi (s)\right\Vert_{L^{\lambda }}^{q}}{1+\ln (1+\left\Vert \theta \right\Vert_{L^{4}})} {d}s < \infty \quad \text{with}\quad\frac{2}{q}+\frac{3}{\lambda }=\frac{7}{4}\quad \text{and}\quad\frac{12}{7} < \lambda \leq \infty,$$

then the solution \({(u,\theta )}\) remains smooth on \({\left[0,T\right]}\). Compared to the Navier–Stokes result, there is a logarithmic correction involving \({\theta}\) in the denominator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, R.A., Fournier, J.F.: Sobolev Spaces, 2nd ed. Elsevier, Amsterdam (2003)

  2. Berselli L.C., Fan J.: Logarithmic and improved regularity criteria for the nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Commun. Pure Appl. Anal. 14, 637–655 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berselli L.C., Galdi G.P.: Regularity criteria involving the pressure for the weak solutions to the Navier–Stokes equations. Proc. Am. Math. Soc. 130, 3585–3595 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cannon, J.R., Dibenedetto, E.: The initial problem for the Boussinesq equation with data in L p. In: Lecture Notes in Mathematics, vol. 771, pp. 129–144. Springer, Berlin (1980)

  5. Cao C.: Sufficient conditions for the regularity to the 3D Navier–Stokes equations. Discrete Contin. Dyn. Syst. 26, 1141–1151 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cao C., Titi E.: Regularity criteria for the three-dimensional Navier–Stokes equations. Indiana Univ. Math. J. 57, 2643–2661 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cao C., Wu J.: Two regularity criteria for the 3D MHD equations. J. Differ. Equ. 248, 2263–2274 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen Q., Zhang Z.: Regularity criterion via the pressure on weak solutions to the 3D Navier–Stokes equations. Proc. Am. Math. Soc. 135, 1829–1837 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Constantin, P., Foias, C.: Navier–Stokes Equations. The University of Chicago Press, Chicago (1988)

  10. Dong, B.Q., Song, J., Zhang, W.: Blow-up criterion via pressure of three-dimensional Boussinesq equations with partial viscosity. Sci. Sin. Math. 40, 1225–1236 (2010). (in Chinese)

  11. Fan J., Jiang S., Ni G.: On regularity criteria for the n-dimensional Navier–Stokes equations in terms of the pressure. J. Differ. Equ. 244, 2963–2979 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fan, J., Jiang, S., Nakamura, G., Zhou, Y.: Logarithmically improved regularity criteria for the Navier–Stokes and MHD equations. J. Math. Fluid Mech. 13, 557–71 (2011)

  13. Fan J., Ozawa T.: Regularity criterion for weak solutions to the Navier–Stokes equations in terms of the gradient of the pressure. J. Inequal. Appl. 2008, 412678 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fan J., Ozawa T.: Regularity criteria for the 3D density-dependent Boussinesq equations. Nonlinearity 22, 553–568 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fan J., Zhou Y.: A note on regularity criterion for the 3D Boussinesq system with partial viscosity. Appl. Math. Lett. 22, 802–805 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gala S.: On the regularity criterion of strong solutions to the 3D Boussinesq equations. Appl. Anal. 90, 1829–1835 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gala S.: A remark on the logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure. Math. Methods Appl. Sci. 34, 1945–1953 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gala, S., Ragusa, M.A.: Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces with negative indices. Appl. Anal. (2015). doi:10.1080/00036811.2015.1061122

  19. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. I & II. Springer, Berlin (1994)

  20. Geng J., Fan J.: A note on regularity criterion for the 3D Boussinesq system with zero thermal conductivity. Appl. Math. Lett. 25, 63–66 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guo Z., Gala S.: Remarks on logarithmical regularity criteria for the Navier–Stokes equations. J. Math. Phys. 52, 063503 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. He, C.: New sufficient conditions for regularity of solutions to the Navier–Stokes equations. Adv. Math. Sci. Appl. 12, 535–48 (2002)

  23. Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–31 (1950)

  24. Ishimura, N., Morimoto, H.: Remarks on the blow-up criterion for the 3-D Boussinesq equations. Math. Models Methods Appl. Sci. 9, 1323–1332 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jia Y., Zhang X., Dong B.: Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion. Commun. Pure Appl. Anal. 12, 923–937 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jia, Y., Zhang, X., Dong, B.: Logarithmical regularity criteria of the three-dimensional micropolar fluid equations in terms of the Pressure. In: Abstract and Applied Analysis, vol. 2012, Article ID 395420 (2012). doi:10.1155/2012/395420

  27. Kozono H., Yatsu N.: Extension criterion via two components of vorticity on strong solution to the 3D Navier–Stokes equations. Math. Z. 246, 55–68 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kukavica I., Ziane M.: Navier–Stokes equations with regularity in one direction. J. Math. Phys. 48, 065203 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ladyzhenskaya, O.A.: The Boundary Value Problems of Mathematical Physics. Springer, Berlin (1985)

  30. Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 183–48 (1934)

  31. Majda, A.: Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics, vol. 9. AMS/CIMS (2003)

  32. Qiu, H., Du, Y., Yao, Z.: Blow-up criteria for 3D Boussinesq equations in the multiplier space. Commun. Nonlinear Sci. Numer. Simul. 16, 1820–1824 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Qiu, H., Du, Y., Yao, Z.: Serrin-type blow-up criteria for 3D Boussinesq equations. Appl. Anal. 89, 1603–1613 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Serrin, J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 9, 187–95 (1962)

  35. Struwe, M.: On partial regularity results for the Navier–Stokes equations. Commun. Pure Appl. Math. 41, 437–58 (1988)

  36. Zhang Z., Chen Q.: Regularity criterion via two components of vorticity on weak solutions to the Navier–Stokes equations in \({\mathbb{R}^{3}}\). J. Differ. Equ. 216, 470–481 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhang, Z.: Some regularity criteria for the 3D Boussinesq equations in the class \({L^{2}(0,T;\dot{B}_{\infty,\infty }^{-1})}\). ISRN Appl. Math. 2014, Article ID 564758 (2014). doi:10.1155/2014/564758

  38. Zhang Z.: A logarithmically improved regularity criterion for the 3D Boussinesq equations via the pressure. Acta Appl. Math. 131, 213–219 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, Z.: A remark on the regularity criterion for the 3D Boussinesq equations involving the pressure gradient. In: Abstract and Applied Analysis, vol. 2014, Article ID 510924 (2014)

  40. Zhou Y., Fan J.: Logarithmically improved regularity criteria for the 3D viscous MHD equations. Forum Math. 24, 691–708 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhou Y.: On regularity criteria in terms of pressure for the Navier–Stokes equations in \({\mathbb{R}^{3}}\). Proc. Am. Math. Soc. 134, 149–156 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhou Y.: On a regularity criterion in terms of the gradient of pressure for the Navier–Stokes equations in \({\mathbb{R}^{N}}\). Z. Angew. Math. Phys. 57, 384–392 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhou Y., Gala S.: Logarithmically improved regularity criteria for the Navier–Stokes equations in multiplier spaces. J. Math. Anal. Appl. 356, 498–501 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhou Y.: Regularity criteria for the 3D MHD equations in terms of the pressure. Int. J. Non Linear Mech. 41, 1174–1180 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhou Y., Fan J.: On the Cauchy problems for certain Boussinesq-\({\alpha }\) equations. Proc. R. Soc. Edinb. Sect. A 140(2), 319–327 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Xu F., Zhang Q., Zheng X.: Regularity Criteria of the 3D Boussinesq equations in the Morrey–Campanato Space. Acta Appl. Math. 121, 231–240 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sadek Gala.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mechdene, M., Gala, S., Guo, Z. et al. Logarithmical regularity criterion of the three-dimensional Boussinesq equations in terms of the pressure. Z. Angew. Math. Phys. 67, 120 (2016). https://doi.org/10.1007/s00033-016-0715-2

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-016-0715-2

Mathematics Subject Classification

Keywords

Navigation