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March  2012, 11(2): 443-451. doi: 10.3934/cpaa.2012.11.443

## Regularity criterion of the Newton-Boussinesq equations in $R^3$

 1 College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, Zhejiang, China 2 Department of Mathematics, College of Science, Jazan University, Jazan, Kazakhstan

Received  January 2010 Revised  March 2011 Published  October 2011

In this paper, we consider the regularity problem under the critical condition to the Newton-Boussinesq equations. The Serrin type regularity criteria are established in terms of the critical Morrey-Campanato spaces and Besov spaces.
Citation: Zhengguang Guo, Sadek Gala. Regularity criterion of the Newton-Boussinesq equations in $R^3$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 443-451. doi: 10.3934/cpaa.2012.11.443
##### References:
 [1] J. Bergh and J. Löfström, "Interpolation Spaces," Springer-Verlag, 1976.  Google Scholar [2] S. Chen, Symmetry analysis of convection patterns, Commun. Theor. Phys., 1 (1982), 413-426.  Google Scholar [3] D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203 (2006), 497-513. doi: 10.1016/j.aim.2005.05.001.  Google Scholar [4] J. R. Cannon and E. Dibenedetto, The initial problem for the Boussinesq equation with data in $L^p$, in "Lecture Notes in Mathematics," vol. 771, Springer, Berlin, 1980, pp. 129-144.  Google Scholar [5] X. Chen, S. Gala and Z. Guo, A new regularity criterion in terms of the direction of the velocity for the MHD equations, Acta Appl. Math., 113 (2011), 207-213. doi: 10.1007/s10440-010-9594-2.  Google Scholar [6] 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. doi: 10.1016/j.aml.2008.06.041.  Google Scholar [7] G. Fucci, B. Wang and P. Singh, Asymptotic behavior of the Newton-Boussinesq equations in a two-dimensional channel, Nonlinear Anal., 70 (2009), 2000-2013. doi: 10.1016/j.na.2008.02.098.  Google Scholar [8] J. Geng, X. Chen and S. Gala, On regularity criteria for the 3D micropolar fluid equations in the critical Morrey-Campanato space, Comm. Pure Appl. Anal., 10 (2011), 583-592. doi: 10.3934/cpaa.2011.10.583.  Google Scholar [9] B. Guo, Spectral method for solving two-dimensional Newton-Boussinesq equation, Acta. Math. Appl. Sin., 5 (1989), 201-218.  Google Scholar [10] B. Guo, Nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations, Chin. Ann. Math., 16 B (1995), 379-390.  Google Scholar [11] Z. Guo and S. Gala, Remarks on logarithmical regularity criteria for the Navier-Stokes equations, J. Math. Phys., 52 (2011), 063503. doi: 10.1063/1.3569967.  Google Scholar [12] N. Ishimura and H. Morimoto, Remarks on the blow-up criterion for the 3D Boussinesq equations, Math. Models Methods Appl. Sci., 9 (1999), 1323-1332. Google Scholar [13] T. Kato, Strong $L^p$ solutions of the Navier-Stokes equations in Morrey spaces, Bol. Soc. Bras. Mat., 22 (1992), 127-155.  Google Scholar [14] P. G. Lemarié-Rieusset, The Navier-Stokes equations in the critical Morrey-Campanato space, Rev. Mat. Iberoam., 23 (2007), 897-930.  Google Scholar [15] M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes equations and other evolutions equations, Comm. Partial Differential Equations, 17 (1992), 1407-1456. doi: 10.1080/03605309208820892.  Google Scholar [16] H. Triebel, "Theory of Function Spaces II," Birkhäuser, Basel, 1992.  Google Scholar [17] Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys., 61 (2010), 193-199. Google Scholar [18] Y. Zhou and S. Gala, Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces, J. Math. Anal. Appl., 356 (2009), 498-501. Google Scholar [19] Y. Zhou and J. Fan, On the Cauchy problems for certain Boussinesq-$\alpha$ equations, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 319-327. doi: 10.1017/S0308210509000122.  Google Scholar [20] Y. Zhou and S. Gala, Regularity criteria in terms of the pressure for the Navier-Stokes equations in the critical Morrey-Campanato space, Z. Anal. Anwendungen, 30 (2011), 83-93. Google Scholar [21] Y. Zhou and S. Gala, On the existence of global solutions for the magneto-hydrodynamic equations, Preprint (2010). Google Scholar

show all references

##### References:
 [1] J. Bergh and J. Löfström, "Interpolation Spaces," Springer-Verlag, 1976.  Google Scholar [2] S. Chen, Symmetry analysis of convection patterns, Commun. Theor. Phys., 1 (1982), 413-426.  Google Scholar [3] D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203 (2006), 497-513. doi: 10.1016/j.aim.2005.05.001.  Google Scholar [4] J. R. Cannon and E. Dibenedetto, The initial problem for the Boussinesq equation with data in $L^p$, in "Lecture Notes in Mathematics," vol. 771, Springer, Berlin, 1980, pp. 129-144.  Google Scholar [5] X. Chen, S. Gala and Z. Guo, A new regularity criterion in terms of the direction of the velocity for the MHD equations, Acta Appl. Math., 113 (2011), 207-213. doi: 10.1007/s10440-010-9594-2.  Google Scholar [6] 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. doi: 10.1016/j.aml.2008.06.041.  Google Scholar [7] G. Fucci, B. Wang and P. Singh, Asymptotic behavior of the Newton-Boussinesq equations in a two-dimensional channel, Nonlinear Anal., 70 (2009), 2000-2013. doi: 10.1016/j.na.2008.02.098.  Google Scholar [8] J. Geng, X. Chen and S. Gala, On regularity criteria for the 3D micropolar fluid equations in the critical Morrey-Campanato space, Comm. Pure Appl. Anal., 10 (2011), 583-592. doi: 10.3934/cpaa.2011.10.583.  Google Scholar [9] B. Guo, Spectral method for solving two-dimensional Newton-Boussinesq equation, Acta. Math. Appl. Sin., 5 (1989), 201-218.  Google Scholar [10] B. Guo, Nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations, Chin. Ann. Math., 16 B (1995), 379-390.  Google Scholar [11] Z. Guo and S. Gala, Remarks on logarithmical regularity criteria for the Navier-Stokes equations, J. Math. Phys., 52 (2011), 063503. doi: 10.1063/1.3569967.  Google Scholar [12] N. Ishimura and H. Morimoto, Remarks on the blow-up criterion for the 3D Boussinesq equations, Math. Models Methods Appl. Sci., 9 (1999), 1323-1332. Google Scholar [13] T. Kato, Strong $L^p$ solutions of the Navier-Stokes equations in Morrey spaces, Bol. Soc. Bras. Mat., 22 (1992), 127-155.  Google Scholar [14] P. G. Lemarié-Rieusset, The Navier-Stokes equations in the critical Morrey-Campanato space, Rev. Mat. Iberoam., 23 (2007), 897-930.  Google Scholar [15] M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes equations and other evolutions equations, Comm. Partial Differential Equations, 17 (1992), 1407-1456. doi: 10.1080/03605309208820892.  Google Scholar [16] H. Triebel, "Theory of Function Spaces II," Birkhäuser, Basel, 1992.  Google Scholar [17] Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys., 61 (2010), 193-199. Google Scholar [18] Y. Zhou and S. Gala, Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces, J. Math. Anal. Appl., 356 (2009), 498-501. Google Scholar [19] Y. Zhou and J. Fan, On the Cauchy problems for certain Boussinesq-$\alpha$ equations, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 319-327. doi: 10.1017/S0308210509000122.  Google Scholar [20] Y. Zhou and S. Gala, Regularity criteria in terms of the pressure for the Navier-Stokes equations in the critical Morrey-Campanato space, Z. Anal. Anwendungen, 30 (2011), 83-93. Google Scholar [21] Y. Zhou and S. Gala, On the existence of global solutions for the magneto-hydrodynamic equations, Preprint (2010). Google Scholar
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