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

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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

Yan Jia ^1, , Xingwei Zhang ^1, and Bo-Qing Dong ^2,

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

Abstract
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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 $T>T_1$.

Keywords: Boussinesq equations, blow-up criterion, Besov spaces., zero diffusion.

Mathematics Subject Classification: Primary: 35Q35; Secondary: 76D0.

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 and 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), 199-220. 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), 737-756. 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. Ećde Norm. Sup., 14 (1981), 209-246.
[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), 919-932. 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), 2263-2274. 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), 497-513. 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, Sect. A, 127 (1997), 935-946. 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), 919-930. 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), 107-122.
[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), 159-169. 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), 1444-1460. 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), 1-14. 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), 1225-1236. 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 11 (2010), 2415-2421. 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. 34 (2011), 595-606. doi: 10.1002/mma.1383.
[16]	J. Fan and T. Ozawa, Regularity criteria for the 3D density Boussinesq equations, Nonlinearity, 22 (2009), 553-568. 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), 802-805. 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), 235-254. 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. Poincaré-AN., 27 (2010), 1227-1246. 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), 1-12.
[21]	N. Ishimura and H. Morimoto, Remarks on the blow-up criterion for the 3D Boussinesq equations, Math. Methods Appl. Sci., 9 (1999), 1323-1332.
[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), 199-203. 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), 891-907. doi: 10.1002/cpa.3160410704.
[24]	P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," Chapman Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.
[25]	A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean," in: Courant Lecture Notes in Mathematics, vol. 9, AMS/CIMS, 2003.
[26]	J. Pedlosky, "Geophysical Fluid Dynamics," Springer-Verlag, New York, 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), 806-815. doi: 10.1016/j.na.2010.04.021.
[28]	H. Triebel, "Theory of Function Spaces," Birkhäuser Verlag, Basel-Boston,1983. 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), 677-681. 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), 881-886. 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), 199-220. 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), 737-756. 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. Ećde Norm. Sup., 14 (1981), 209-246.
[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), 919-932. 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), 2263-2274. 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), 497-513. 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, Sect. A, 127 (1997), 935-946. 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), 919-930. 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), 107-122.
[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), 159-169. 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), 1444-1460. 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), 1-14. 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), 1225-1236. 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 11 (2010), 2415-2421. 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. 34 (2011), 595-606. doi: 10.1002/mma.1383.
[16]	J. Fan and T. Ozawa, Regularity criteria for the 3D density Boussinesq equations, Nonlinearity, 22 (2009), 553-568. 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), 802-805. 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), 235-254. 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. Poincaré-AN., 27 (2010), 1227-1246. 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), 1-12.
[21]	N. Ishimura and H. Morimoto, Remarks on the blow-up criterion for the 3D Boussinesq equations, Math. Methods Appl. Sci., 9 (1999), 1323-1332.
[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), 199-203. 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), 891-907. doi: 10.1002/cpa.3160410704.
[24]	P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," Chapman Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.
[25]	A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean," in: Courant Lecture Notes in Mathematics, vol. 9, AMS/CIMS, 2003.
[26]	J. Pedlosky, "Geophysical Fluid Dynamics," Springer-Verlag, New York, 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), 806-815. doi: 10.1016/j.na.2010.04.021.
[28]	H. Triebel, "Theory of Function Spaces," Birkhäuser Verlag, Basel-Boston,1983. 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), 677-681. 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), 881-886. doi: 10.3934/dcds.2005.12.881.

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