March  2013, 12(2): 923-937. doi: 10.3934/cpaa.2013.12.923

Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion

1. 

School of Mathematical Sciences, Anhui University, Hefei 230601, China, China

2. 

School of Mathematical Science, Anhui University, Hefei 230039

Received  January 2012 Revised  January 2012 Published  September 2012

This article is concerned with the blow-up criterion for smooth solutions of three-dimensional Boussinesq equations with zero diffusion. It is shown that if the velocity field $u(x,t)$ satisfies \begin{eqnarray*} u\in L^p(0,T_1;B^r_{q,\infty}(R^3)),\quad \frac{2}{p}+\frac{3}{q}=1+r,\quad \frac{3}{1+r}< q \leq \infty, \quad -1 < r \leq 1, \end{eqnarray*} then the solution can be continually extended to the interval $(0,T)$ for some $T>T_1$.
Citation: 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 & Applied Analysis, 2013, 12 (2) : 923-937. doi: 10.3934/cpaa.2013.12.923
References:
[1]

H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system,, J. Differential Equations, 233 (2007), 199. doi: 10.1016/j.jde.2006.10.008.

[2]

H. Abidi, T. Hmidi and K. Sahbi, On the global regularity of axisymmetric Navier-Stokes-Boussinesq system,, Discrete and Continuous Dynamical Systems, 29 (2011), 737. doi: 10.3934/dcds.2011.29.737.

[3]

J.-M. Bony, Calcul symbolique et propagation des singulariteś pour lesequations aux deŕiveés partielles non lineáires,, Ann. Sci. E\'cde Norm. Sup., 14 (1981), 209.

[4]

C. Cao and E. S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor,, Arch. Rational Mech. Anal., 202 (2011), 919. doi: 10.1007/s00205-011-0439-6.

[5]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations,, J. Differential Equations, 248 (2010), 2263. doi: 10.1016/j.jde.2009.09.020.

[6]

D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms,, Adv. Math., 203 (2006), 497. doi: 10.1016/j.aim.2005.05.001.

[7]

D. Chae and H.-S. Nam, Local existence and blow-up criterion for the Boussinesq equations,, Proc. Roy. Soc. Edinburgh, 127 (1997), 935. doi: 10.1017/S0308210500026810.

[8]

Q. Chen, C. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous Magneto-Hydrodynamics equations,, Comm. Math. Phys., 284 (2008), 919. doi: 10.1007/s00220-008-0545-y.

[9]

Q. Chen and Z. Zhang, Space-time estimates in the Besov spaces and the Navier-Stokes equations,, Methods Appl. Anal., 13 (2006), 107.

[10]

A. Cheskidov and R. Shvydkoy, On the regularity of weak solutions of the 3D Navier-Stokes equations in $B_{\infty ,\infty}^{-1}$, , Arch. Rational Mech. Anal., 195 (2010), 159. doi: s00205-009-0265-2.

[11]

R. Danchin and M. Paicu, Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces,, Phys. D, 237 (2008), 1444. doi: 10.1016/j.physd.2008.03.034.

[12]

R. Danchin and M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data,, Comm. Math. Phys., 290 (2009), 1. doi: 10.1007/s00220-009-0821-5.

[13]

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. doi: 10.1360/012010-567.

[14]

B.-Q. Dong and Z. Zhang, The BKM criterion for the 3D Navier-Stokes equations via two velocity components,, Nonlinear Analysis: Real World Applications \textbf{11} (2010), 11 (2010), 2415. doi: 10.1016/j.nonrwa.2009.07.013.

[15]

B.-Q. Dong, Y. Jia and Z.-M. Chen, Pressure regularity criteria of the three-dimensional micropolar fluid flows,, Math. Meth. Appl. Sci. \textbf{34} (2011), 34 (2011), 595. doi: 10.1002/mma.1383.

[16]

J. Fan and T. Ozawa, Regularity criteria for the 3D density Boussinesq equations,, Nonlinearity, 22 (2009), 553. doi: 10.1088/0951-7715/22/3/003.

[17]

J. Fan and Y. Zhou, A note on regularity criterion for the 3D Boussinesq system with partial viscosity,, Appl. Math. Lett., 22 (2009), 802. doi: 10.1016/j.aml.2008.06.041.

[18]

C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, J. Differential Equations, 213 (2005), 235. doi: 10.1016/j.jde.2004.07.002.

[19]

T. Hmidi and F. Rousset, Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data,, Ann. I. H. Poincar\'e-AN., 27 (2010), 1227. doi: 10.1016/j.anihpc.2010.06.001.

[20]

T.-Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations,, Discrete Contin. Dyn. Syst., 12 (2005), 1.

[21]

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

[22]

Y. Jia, W. Zhang and B.-Q. Dong, Remarks on the regularity criterion of the 3D micropolar fluid equations in terms of the pressure,, Appl. Math. Letters, 24 (2011), 199. doi: 10.1016/j.aml.2010.09.003.

[23]

T. Kato and G. Ponce, Commutator estimates and the Euler and the Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891. doi: 10.1002/cpa.3160410704.

[24]

P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem,", Chapman Hall/CRC, (2002). doi: 10.1201/9781420035674.

[25]

A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean,", in: Courant Lecture Notes in Mathematics, (2003).

[26]

J. Pedlosky, "Geophysical Fluid Dynamics,", Springer-Verlag, (1987).

[27]

H. Qiu, Y. Du and Z. Yao, A blow-up criterion for 3D Boussinesq equations in Besov spaces,, Nonlinear Analysis TMA, 73 (2010), 806. doi: 10.1016/j.na.2010.04.021.

[28]

H. Triebel, "Theory of Function Spaces,", Birkh\, (). doi: 10.1007/978-3-0346-0416-1.

[29]

X. Xu, Global regularity of solutions of 2D Boussinesq equations with fractional diffusion,, Nonlinear Analysis TMA, 72 (2010), 677. doi: 10.1016/j.na.2009.07.008.

[30]

Y. Zhou, Remarks on regularities for the 3D MHD equations,, Discrete Contin. Dyn. Syst., 12 (2005), 881. doi: 10.3934/dcds.2005.12.881.

show all references

References:
[1]

H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system,, J. Differential Equations, 233 (2007), 199. doi: 10.1016/j.jde.2006.10.008.

[2]

H. Abidi, T. Hmidi and K. Sahbi, On the global regularity of axisymmetric Navier-Stokes-Boussinesq system,, Discrete and Continuous Dynamical Systems, 29 (2011), 737. doi: 10.3934/dcds.2011.29.737.

[3]

J.-M. Bony, Calcul symbolique et propagation des singulariteś pour lesequations aux deŕiveés partielles non lineáires,, Ann. Sci. E\'cde Norm. Sup., 14 (1981), 209.

[4]

C. Cao and E. S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor,, Arch. Rational Mech. Anal., 202 (2011), 919. doi: 10.1007/s00205-011-0439-6.

[5]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations,, J. Differential Equations, 248 (2010), 2263. doi: 10.1016/j.jde.2009.09.020.

[6]

D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms,, Adv. Math., 203 (2006), 497. doi: 10.1016/j.aim.2005.05.001.

[7]

D. Chae and H.-S. Nam, Local existence and blow-up criterion for the Boussinesq equations,, Proc. Roy. Soc. Edinburgh, 127 (1997), 935. doi: 10.1017/S0308210500026810.

[8]

Q. Chen, C. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous Magneto-Hydrodynamics equations,, Comm. Math. Phys., 284 (2008), 919. doi: 10.1007/s00220-008-0545-y.

[9]

Q. Chen and Z. Zhang, Space-time estimates in the Besov spaces and the Navier-Stokes equations,, Methods Appl. Anal., 13 (2006), 107.

[10]

A. Cheskidov and R. Shvydkoy, On the regularity of weak solutions of the 3D Navier-Stokes equations in $B_{\infty ,\infty}^{-1}$, , Arch. Rational Mech. Anal., 195 (2010), 159. doi: s00205-009-0265-2.

[11]

R. Danchin and M. Paicu, Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces,, Phys. D, 237 (2008), 1444. doi: 10.1016/j.physd.2008.03.034.

[12]

R. Danchin and M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data,, Comm. Math. Phys., 290 (2009), 1. doi: 10.1007/s00220-009-0821-5.

[13]

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. doi: 10.1360/012010-567.

[14]

B.-Q. Dong and Z. Zhang, The BKM criterion for the 3D Navier-Stokes equations via two velocity components,, Nonlinear Analysis: Real World Applications \textbf{11} (2010), 11 (2010), 2415. doi: 10.1016/j.nonrwa.2009.07.013.

[15]

B.-Q. Dong, Y. Jia and Z.-M. Chen, Pressure regularity criteria of the three-dimensional micropolar fluid flows,, Math. Meth. Appl. Sci. \textbf{34} (2011), 34 (2011), 595. doi: 10.1002/mma.1383.

[16]

J. Fan and T. Ozawa, Regularity criteria for the 3D density Boussinesq equations,, Nonlinearity, 22 (2009), 553. doi: 10.1088/0951-7715/22/3/003.

[17]

J. Fan and Y. Zhou, A note on regularity criterion for the 3D Boussinesq system with partial viscosity,, Appl. Math. Lett., 22 (2009), 802. doi: 10.1016/j.aml.2008.06.041.

[18]

C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, J. Differential Equations, 213 (2005), 235. doi: 10.1016/j.jde.2004.07.002.

[19]

T. Hmidi and F. Rousset, Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data,, Ann. I. H. Poincar\'e-AN., 27 (2010), 1227. doi: 10.1016/j.anihpc.2010.06.001.

[20]

T.-Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations,, Discrete Contin. Dyn. Syst., 12 (2005), 1.

[21]

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

[22]

Y. Jia, W. Zhang and B.-Q. Dong, Remarks on the regularity criterion of the 3D micropolar fluid equations in terms of the pressure,, Appl. Math. Letters, 24 (2011), 199. doi: 10.1016/j.aml.2010.09.003.

[23]

T. Kato and G. Ponce, Commutator estimates and the Euler and the Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891. doi: 10.1002/cpa.3160410704.

[24]

P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem,", Chapman Hall/CRC, (2002). doi: 10.1201/9781420035674.

[25]

A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean,", in: Courant Lecture Notes in Mathematics, (2003).

[26]

J. Pedlosky, "Geophysical Fluid Dynamics,", Springer-Verlag, (1987).

[27]

H. Qiu, Y. Du and Z. Yao, A blow-up criterion for 3D Boussinesq equations in Besov spaces,, Nonlinear Analysis TMA, 73 (2010), 806. doi: 10.1016/j.na.2010.04.021.

[28]

H. Triebel, "Theory of Function Spaces,", Birkh\, (). doi: 10.1007/978-3-0346-0416-1.

[29]

X. Xu, Global regularity of solutions of 2D Boussinesq equations with fractional diffusion,, Nonlinear Analysis TMA, 72 (2010), 677. doi: 10.1016/j.na.2009.07.008.

[30]

Y. Zhou, Remarks on regularities for the 3D MHD equations,, Discrete Contin. Dyn. Syst., 12 (2005), 881. doi: 10.3934/dcds.2005.12.881.

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