-
Previous Article
Weak solutions for generalized large-scale semigeostrophic equations
- CPAA Home
- This Issue
-
Next Article
Local well-posedness of quasi-linear systems generalizing KdV
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 |
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). 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. |
[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). 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. |
[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. |
[1] |
Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333 |
[2] |
Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225 |
[3] |
Ming Lu, Yi Du, Zheng-An Yao, Zujin Zhang. A blow-up criterion for the 3D compressible MHD equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1167-1183. doi: 10.3934/cpaa.2012.11.1167 |
[4] |
Baoquan Yuan, Xiao Li. Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2167-2179. doi: 10.3934/dcdss.2016090 |
[5] |
Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809 |
[6] |
Xin Zhong. A blow-up criterion for three-dimensional compressible magnetohydrodynamic equations with variable viscosity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3249-3264. doi: 10.3934/dcdsb.2018318 |
[7] |
Kyouhei Wakasa. Blow-up of solutions to semilinear wave equations with non-zero initial data. Conference Publications, 2015, 2015 (special) : 1105-1114. doi: 10.3934/proc.2015.1105 |
[8] |
Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639 |
[9] |
Houyu Jia, Xiaofeng Liu. Local existence and blowup criterion of the Lagrangian averaged Euler equations in Besov spaces. Communications on Pure & Applied Analysis, 2008, 7 (4) : 845-852. doi: 10.3934/cpaa.2008.7.845 |
[10] |
Qunyi Bie, Qiru Wang, Zheng-An Yao. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinetic & Related Models, 2015, 8 (3) : 395-411. doi: 10.3934/krm.2015.8.395 |
[11] |
Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399 |
[12] |
Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63 |
[13] |
Monica Marras, Stella Vernier Piro. Blow-up phenomena in reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4001-4014. doi: 10.3934/dcds.2012.32.4001 |
[14] |
Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182 |
[15] |
Yu-Zhu Wang, Weibing Zuo. On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1327-1336. doi: 10.3934/cpaa.2014.13.1327 |
[16] |
Anthony Suen. Corrigendum: A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1387-1390. doi: 10.3934/dcds.2015.35.1387 |
[17] |
Yinghua Li, Shijin Ding, Mingxia Huang. Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1507-1523. doi: 10.3934/dcdsb.2016009 |
[18] |
Anthony Suen. A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3791-3805. doi: 10.3934/dcds.2013.33.3791 |
[19] |
Bin Li. On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks. Kinetic & Related Models, 2019, 12 (5) : 1131-1162. doi: 10.3934/krm.2019043 |
[20] |
Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170 |
2018 Impact Factor: 0.925
Tools
Metrics
Other articles
by authors
[Back to Top]