• Previous Article
    Solutions for polyharmonic elliptic problems with critical nonlinearities in symmetric domains
  • CPAA Home
  • This Issue
  • Next Article
    Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows
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 and 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.

[2]

S. Chen, Symmetry analysis of convection patterns, Commun. Theor. Phys., 1 (1982), 413-426.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

B. Guo, Spectral method for solving two-dimensional Newton-Boussinesq equation, Acta. Math. Appl. Sin., 5 (1989), 201-218.

[10]

B. Guo, Nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations, Chin. Ann. Math., 16 B (1995), 379-390.

[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.

[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.

[13]

T. Kato, Strong $L^p$ solutions of the Navier-Stokes equations in Morrey spaces, Bol. Soc. Bras. Mat., 22 (1992), 127-155.

[14]

P. G. Lemarié-Rieusset, The Navier-Stokes equations in the critical Morrey-Campanato space, Rev. Mat. Iberoam., 23 (2007), 897-930.

[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.

[16]

H. Triebel, "Theory of Function Spaces II," Birkhäuser, Basel, 1992.

[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.

[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.

[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.

[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.

[21]

Y. Zhou and S. Gala, On the existence of global solutions for the magneto-hydrodynamic equations, Preprint (2010).

show all references

References:
[1]

J. Bergh and J. Löfström, "Interpolation Spaces," Springer-Verlag, 1976.

[2]

S. Chen, Symmetry analysis of convection patterns, Commun. Theor. Phys., 1 (1982), 413-426.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

B. Guo, Spectral method for solving two-dimensional Newton-Boussinesq equation, Acta. Math. Appl. Sin., 5 (1989), 201-218.

[10]

B. Guo, Nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations, Chin. Ann. Math., 16 B (1995), 379-390.

[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.

[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.

[13]

T. Kato, Strong $L^p$ solutions of the Navier-Stokes equations in Morrey spaces, Bol. Soc. Bras. Mat., 22 (1992), 127-155.

[14]

P. G. Lemarié-Rieusset, The Navier-Stokes equations in the critical Morrey-Campanato space, Rev. Mat. Iberoam., 23 (2007), 897-930.

[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.

[16]

H. Triebel, "Theory of Function Spaces II," Birkhäuser, Basel, 1992.

[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.

[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.

[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.

[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.

[21]

Y. Zhou and S. Gala, On the existence of global solutions for the magneto-hydrodynamic equations, Preprint (2010).

[1]

Xue-Li Song, Yan-Ren Hou. Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 991-1009. doi: 10.3934/dcds.2012.32.991

[2]

Xueli Song, Jianhua Wu. Non-autonomous 2D Newton-Boussinesq equation with oscillating external forces and its uniform attractor. Evolution Equations and Control Theory, 2022, 11 (1) : 41-65. doi: 10.3934/eect.2020102

[3]

Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357

[4]

Sadek Gala. A new regularity criterion for the 3D MHD equations in $R^3$. Communications on Pure and Applied Analysis, 2012, 11 (3) : 973-980. doi: 10.3934/cpaa.2012.11.973

[5]

Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299

[6]

Ahmad Mohammad Alghamdi, Sadek Gala, Chenyin Qian, Maria Alessandra Ragusa. The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations. Electronic Research Archive, 2020, 28 (1) : 183-193. doi: 10.3934/era.2020012

[7]

Yan Jia, Xingwei Zhang, Bo-Qing Dong. Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion. Communications on Pure and Applied Analysis, 2013, 12 (2) : 923-937. doi: 10.3934/cpaa.2013.12.923

[8]

Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333

[9]

Daoyuan Fang, Chenyin Qian. Regularity criterion for 3D Navier-Stokes equations in Besov spaces. Communications on Pure and Applied Analysis, 2014, 13 (2) : 585-603. doi: 10.3934/cpaa.2014.13.585

[10]

Zujin Zhang. A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component. Communications on Pure and Applied Analysis, 2013, 12 (1) : 117-124. doi: 10.3934/cpaa.2013.12.117

[11]

Jishan Fan, Fucai Li, Gen Nakamura. A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1757-1766. doi: 10.3934/dcdsb.2018079

[12]

Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks and Heterogeneous Media, 2021, 16 (2) : 283-315. doi: 10.3934/nhm.2021007

[13]

B. S. Goh, W. J. Leong, Z. Siri. Partial Newton methods for a system of equations. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 463-469. doi: 10.3934/naco.2013.3.463

[14]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure and Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637

[15]

Siran Li, Jiahong Wu, Kun Zhao. On the degenerate boussinesq equations on surfaces. Journal of Geometric Mechanics, 2020, 12 (1) : 107-140. doi: 10.3934/jgm.2020006

[16]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems and Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025

[17]

Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933

[18]

Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934/proc.2011.2011.1001

[19]

Mimi Dai, Han Liu. Low modes regularity criterion for a chemotaxis-Navier-Stokes system. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2713-2735. doi: 10.3934/cpaa.2020118

[20]

Jianhua Huang, Tianlong Shen, Yuhong Li. Dynamics of stochastic fractional Boussinesq equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2051-2067. doi: 10.3934/dcdsb.2015.20.2051

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (136)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]