Research article

Export file:

Format

• RIS(for EndNote,Reference Manager,ProCite)
• BibTex
• Text

Content

• Citation Only
• Citation and Abstract

Osgood type blow-up criterion for the 3D Boussinesq equations with partial viscosity

School of Mathematical Science, Shanghai Jiao Tong University, Shanghai 200240, P. R. China

## Abstract    Full Text(HTML)    Figure/Table    Related pages

This paper is dedicated to studying the blow-up criterion of smooth solutions to the three-dimensional Boussinesq equations with partial viscosity. By means of the Littlewood-Paley decomposition, we give an improved logarithmic Sobolev inequality and through this, we obtain the corresponding blow-up criterion in a space larger than $\dot{B}^0_{\infty,\infty}$, which extends several previous works.
Figure/Table
Supplementary
Article Metrics

# References

1. D. Chae and H. S. Nam, Local existence and blow-up criterion for the Boussinesq equations, Proc. Roy. Soc. Edinburgh Sect. A., 127 (1997), 935--946.

2. D. Chae, S.K. Kim and H. S. Nam, Local existence and blow-up criterion of Hlder continuous solutions of the Boussinesq equations, Nagoya Math. J., 155 (1999), 55--80.

3. D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203 (2006), 497--513.

4. B. Dong, S. Jiang and W. Zhang, Blow-up criterion via pressure of three-dimensional Boussinesq equations with partial viscosity, Scientia Sinica Mathematica, 40 (2010), 1225--1236.

5. B. Dong, Y. Jia and X. Zhang, Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion, Commun. Pur. Appl. Anal., 12 (2012), 923--937.

6. M. Fu and C. Cai, Remarks on Pressure Blow-Up Criterion of the 3D Zero-Diffusion Boussinesq Equations in Margin Besov Spaces, Adv. Math. Phys., 2017 (2017), 1--7.

7. S. Gala, M. Mechdene and M. A. Ragusa, Logarithmically improved regularity criteria for the Boussinesq equations, AIMS Math., 2 (2017), 336--347.

8. T. Kato and G. Ponce, Commutator estimates and Euler and Navier-Stokes equations, Commun. Pur. Appl. Math., 41 (1988), 891--907.

9. J. Fan and T. Ozawa, Regularity criterion for 3D density-dependent Boussinesq equations, Nonlinearity, 22 (2009), 553--568.

10. J. Fan and Y. Zhou, A note on regularity criterion for the 3D Boussinesq systems with partial viscosity, Appl. Math. Lett., 22 (2009), 802--805.

11. J. Fan, H. Malaikah, S. Monaquel, et al. Global Cauchy problem of 2D generalized MHD equations, Monatsh. Math., 175 (2014), 127--131.

12. J. Fan, F. S. Alzahrani, T. Hayat, et al. Global regularity for the 2D liquid crystal model with mixed partial viscosity, Anal. Appl. (Singap.), 13 (2015), 185--200.

13. A. Majda, Introduction to PDEs and waves for the atmosphere and ocean, Courant lecture notes in mathematics, Vol. 9, New York (NY), AMS/CIMS, 2003.

14. A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Univ. Press, Cambridge, 2002.

15. M. Mechdene, S. Gala, Z. Guo and A. M. Ragusa, Logarithmical regularity criterion of the three-dimensional Boussinesq equations in terms of the pressure, Z. Angew. Math. Phys., 67 (2016), 120.

16. N. Ishihara and H. Morimoto, Remarks on the blow-up criterion for the 3D Boussinesq equations, Math. Mod. Meth. Appl. S., 9 (1999), 1323--1332.

17. T. Ogawa and Y. Taniuchi, On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain, J. Differ. Equations, 190 (2003), 39--63.

18. J. Pedlosky, Geophysical Fluid Dynamics, New York: Springer-Verlag, 1987.

19. H. Qiu, Y. Du and Z. Yao, A blow-up criterion for 3D Boussinesq equations in Besov spaces, Nonlinear Anal-Theor, 73 (2010), 806--815.

20. W. Ren, On the blow-up criterion for the 3D Boussinesq system with zero viscosity constant, Appl. Anal., 94 (2015), 856--862.

21. Q. Wu, H. Lin and G. Liu, An Osgood Type Regularity Criterion for the 3D Boussinesq Equations, The Scientific World J., 2014 (2014), 563084.

22. Z. Ye, Blow-up criterion of smooth solutions for the Boussinesq equations, Nonlinear Anal-Theor, 110 (2014), 97--103.

23. Z. Ye, On the regularity criteria of the 2D Boussinesq equations with partial dissipation, Comput. Math. Appl., 72 (2016), 1880--1895.

24. Z. Zhang and S. Gala, Osgood type regularity criterion for the 3D Newton-Boussinesq equation, Electron. J. Differ. Equ., 2013 (2013), 1--6.