AIMS Mathematics, 2017, 2(3): 451-457. doi: 10.3934/Math.2017.2.451

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A regularity criterion of weak solutions to the 3D Boussinesq equations

1 Department of Mathematical Science , Faculty of Applied Science, Umm Alqura University, P. O.Box 14035, Makkah 21955, Saudi Arabia
2 Department of Mathematics, University of Mostaganem, Algeria
3 Dipartimento di Mathematica e Informatica, Università di Catania, Viale Andrea Doria, 6, 95125Catania, Italy
4 RUDN University, 6 Miklukho - Maklay St, Moscow, 117198, Russia

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

In this note, a regularity criterion of weaksolutions to the 3D-Boussinesq equations with respect to Serrin type condition under the framework of Besov space $\overset{.}{B}_{\infty ,\infty}^{r}$. It is shown that the weak solution $(u,\theta )$ is regular on $%(0,T]$ if $u$ satisfies $\int\limits_{0}^{T}{\left\| u\left( \cdot ,t \right) \right\|_{\overset{\cdot R}{\mathop{{{B}_{\infty ,\infty }}}}\,}^{\frac{2}{1+r}}}\ \ dt < \infty ,$ for 0<r<1. This result improves some previous works.
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# References

1. J. R. Cannon and E. Dibenedetto, The initial problem for the Boussinesq equation with data in Lp,Lecture Notes in Mathematics, Springer, Berlin, 771 (1980), 129-144.

2. L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the navierstokesequations Comm. Pure Appl. Math., 35 (1982), 771-831.

3. 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.

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

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

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

7. S. Gala, On the regularity criterion of strong solutions to the 3D Boussinesq equations, ApplicableAnalysis, 90 (2011), 1829-1835.

8. S. Gala and M.A. Ragusa, Logarithmically improved regularity criterion for the Boussinesq equationsin Besov spaces with negative indices, Applicable Analysis, 95 (2016), 1271-1279.

9. S. Gala, Z. Guo, and M. A. Ragusa, A remark on the regularity criterion of Boussinesq equationswith zero heat conductivity, Appl. Math. Lett., 27 (2014), 70-73.

10. Z. Guo and S. Gala, Regularity criterion of the Newton-Boussinesq equations in R3, Commun. PureAppl. Anal., 11 (2012), 443-451.

11. J. Geng and J. Fan, A note on regularity criterion for the 3D Boussinesq system with zero thermalconductivity, Appl. Math. Lett., 25 (2012), 63-66.

12. E. Hopf, Über die Anfangswertaufgabe für die hydrodynamichen Grundgleichungen, Math. Nach.,4 (1950/1951), 213-231.

13. Y. Jia, X. Zhang, and B. Dong, Remarks on the blow-up criterion for smooth solutions of theBoussinesq equations with zero diffusion, C.P.A.A., 12 (2013), 923-937.

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

15. C. Kenig, G. Ponce, and L. Vega, Well-posedness of the initial value problem for the Korteweg-de-Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.

16. J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta. Math., 63 (1934),183-248.

17. A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notesin Mathematics, AMS/CIMS, 9 (2003).

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

19. G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959),173-182.

20. J. Serrin, The initial value problem for the Navier-Stokes equations, In Nonlinear Problems (Proc.Sympos., Madison, Wis.), Univ. of Wisconsin Press, Madison, Wis., 1963, 69-98.

21. M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math.,41 (1988), 437-458.

22. N. Ishimura and H. Morimoto, Remarks on the blow-up criterion for the 3D Boussinesq equations,Math. Meth. Appl. Sci., 9 (1999), 1323-1332.

23. H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, Basel, 1983.

24. H. Qiu, Y. Du, and Z. Yao, Blow-up criteria for 3D Boussinesq equations in the multiplier space,Comm. Nonlinear Sci. Num. Simulation, 16 (2011), 1820-1824.

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

26. Z. Xiang, The regularity criterion of the weak solution to the 3D viscous Boussinesq equations inBesov spaces, Math. Methods Appl. Sci., 34 (2011), 360-372.

27. F. Xu, Q. Zhang, and X. Zheng, Regularity Criteria of the 3D Boussinesq Equations in the Morrey-Campanato Space, Acta Appl. Math., 121 (2012), 231-240.

28. Z. Ye, A Logarithmically improved regularity criterion of smooth solutions for the 3D Boussinesqequations, Osaka J. Math., 53 (2016), 417-423.

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