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

References:

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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[9]	Q. Chen and Z. Zhang, Space-time estimates in the Besov spaces and the Navier-Stokes equations,, Methods Appl. Anal., 13 (2006), 107. Google Scholar
[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
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[20]	T.-Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations,, Discrete Contin. Dyn. Syst., 12 (2005), 1. Google Scholar
[21]	N. Ishimura and H. Morimoto, Remarks on the blow-up criterion for the 3D Boussinesq equations,, Math. Methods Appl. Sci., 9 (1999), 1323. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem,", Chapman Hall/CRC, (2002). doi: 10.1201/9781420035674. Google Scholar
[25]	A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean,", in: Courant Lecture Notes in Mathematics, (2003). Google Scholar
[26]	J. Pedlosky, "Geophysical Fluid Dynamics,", Springer-Verlag, (1987). Google Scholar
[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. Google Scholar
[28]	H. Triebel, "Theory of Function Spaces,", Birkh\, (). doi: 10.1007/978-3-0346-0416-1. Google Scholar
[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. Google Scholar
[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. Google Scholar

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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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[9]	Q. Chen and Z. Zhang, Space-time estimates in the Besov spaces and the Navier-Stokes equations,, Methods Appl. Anal., 13 (2006), 107. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	T.-Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations,, Discrete Contin. Dyn. Syst., 12 (2005), 1. Google Scholar
[21]	N. Ishimura and H. Morimoto, Remarks on the blow-up criterion for the 3D Boussinesq equations,, Math. Methods Appl. Sci., 9 (1999), 1323. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem,", Chapman Hall/CRC, (2002). doi: 10.1201/9781420035674. Google Scholar
[25]	A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean,", in: Courant Lecture Notes in Mathematics, (2003). Google Scholar
[26]	J. Pedlosky, "Geophysical Fluid Dynamics,", Springer-Verlag, (1987). Google Scholar
[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. Google Scholar
[28]	H. Triebel, "Theory of Function Spaces,", Birkh\, (). doi: 10.1007/978-3-0346-0416-1. Google Scholar
[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. Google Scholar
[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. Google Scholar

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