AIMS Mathematics, 2017, 2(2): 336-347. doi: 10.3934/Math.2017.2.336.

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Logarithmically improved regularity criteria for the Boussinesq equations

1 Department of Mathematics, University of Mostaganem, Box 227, Mostaganem, 27000, Algeria
2 Dipartimento di Mathematicae Informatica, Universit `a di Catania Viale Andrea Doria, 6, 95125 Catania, Italy

In this paper, logarithmically improved regularity criteria for the Boussinesq equations are established under the framework of Besov space $\overset{.}{B}_{\infty ,\infty }^{-r}$. We prove the solution $(u,\theta )$ is smooth up to time $T>0$ provided that \begin{equation} \int_{0}^{T}\frac{\left\Vert u(\cdot ,t)\right\Vert _{\overset{.}{B} _{\infty ,\infty }^{-r}}^{\frac{2}{1-r}}}{\log (e+\left\Vert u(t,.)\right\Vert _{\overset{.}{B}_{\infty ,\infty }^{-r}})}dt<\infty \end{equation} for some $0\leq r<1$ or \begin{equation} \left\Vert u(\cdot ,t)\right\Vert _{L^{\infty }(0,T;\overset{.}{B}_{\infty ,\infty }^{-1}(\mathbb{R}^{3}))}<<1. \end{equation} This result improves some previous works.
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Keywords Regularity criterion; Boussinesq equations; A priori estimates

Citation: Sadek Gala, Mohamed Mechdene, Maria Alessandra Ragusa. Logarithmically improved regularity criteria for the Boussinesq equations. AIMS Mathematics, 2017, 2(2): 336-347. doi: 10.3934/Math.2017.2.336


  • 1. J. R. Cannon and E. Dibenedetto, The initial problem for the Boussinesq equation with data in Lp, in: Lecture Notes in Mathematics, Springer, Berlin, 771 (1980), 129-144.    
  • 2. 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.    
  • 3. B. Q. Dong, J. Song and W. Zhang, Blow-up criterion via pressure of three-dimensional Boussinesq equations with partial viscosity (in Chinese), Sci. Sin. Math., 40 (2010), 1225-1236.
  • 4. 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.    
  • 5. J. Fan and T. Ozawa, Regularity criteria for the 3D density-dependent Boussinesq equations, Nonlinearity, 22 (2009), 553-568.    
  • 6. S. Gala, On the regularity criterion of strong solutions to the 3D Boussinesq equations, Applicable Analysis, 90 (2011), 1829-1835.    
  • 7. S. Gala and M.A. Ragusa, Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces with negative indices, Applicable Analysis, 95 (2016), 1271-1279.    
  • 8. S. Gala, Z. Guo and M. A. Ragusa, A remark on the regularity criterion of Boussinesq equations with zero heat conductivity, Appl. Math. Lett., 27 (2014), 70-73.    
  • 9. Z. Guo and S. Gala, Regularity criterion of the Newton-Boussinesq equations in R3, Commun. Pure Appl. Anal., 11 (2012), 443-451.
  • 10. J. Geng and J. Fan, A note on regularity criterion for the 3D Boussinesq system with zero thermal conductivity, Appl. Math. Lett., 25 (2012), 63-66.    
  • 11. Y. Jia, X. Zhang and B. Dong, Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion, C.P.A.A., 12 (2013), 923-937.
  • 12. T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl. Math., 41 (1988), 891-907.    
  • 13. 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.    
  • 14. A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, 9 (2003).
  • 15. M. Mechdene, S. Gala, Z. Guo and M.A. Ragusa, Logarithmical regularity criterion of the threedimensional Boussinesq equations in terms of the pressure, Z. Angew. Math. Phys., 67 (2016), 67-120.    
  • 16. Y. Meyer, P. Gerard and F. Oru, Inégalités de Sobolev précisées, Séminaire équations aux dérivées partielles (Polytechnique), 4, 1996-1997.
  • 17. N. Ishimura and H. Morimoto, Remarks on the blow-up criterion for the 3D Boussinesq equations, Math. Meth. Appl. Sci., 9 (1999), 1323-1332.    
  • 18. H. Triebel, Theory of Function Spaces, Birkh¨auser Verlag, Basel, 1983.
  • 19. 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, 16 (2011), 1820-1824.    
  • 20. 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.    
  • 21. Z. Xiang, The regularity criterion of the weak solution to the 3D viscous Boussinesq equations in Besov spaces, Mathematical Methods in the Applied Sciences, 34 (2011), 360-372.    
  • 22. 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.    
  • 23. Z. Ye, A Logarithmically improved regularity criterion of smooth solutions for the 3D Boussinesq equations, Osaka J. Math., 53 (2016), 417-423.


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