\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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 $T>T_1$.
    Mathematics Subject Classification: Primary: 35Q35; Secondary: 76D05.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(106) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return