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

A BKM's criterion of smooth solution to the incompressible viscoelastic flow

Abstract / Introduction Related Papers Cited by
  • In this paper, we study the regularity criterion of smooth solution to the Oldroyd model in $R^n(n=2,3)$. We obtain a Beale-Kato-Majda-type criterion in terms of vorticity in two and three space dimensions, namely, the solution $(u(t,x),F(t,x))$ does not develop singularity until $t=T$ provided that $\nabla \times u \in L^1(0,T;\dot{B}_{\infty,\infty}^0(R^n))$ in the case $n=2,3$.
    Mathematics Subject Classification: Primary: 76A10, 76A05; Secondary: 35B05.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    J. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94 (1984), 61-66.

    [2]

    J. Y. Chemin, "Perfect Incompressible Fluids," Oxford Lecture Ser. Math. Appl., vol. 14, The Clarendon Press/Oxford Univ. Press, New York, 1998.

    [3]

    J. Y. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84-112.

    [4]

    Y. Du, C. Liu and Q. T. ZhangA blow-up criterion for 3-D compressible viscoelasticity, arXiv:1202.3693.

    [5]

    W. N. E, T. J. Li and P. W. Zhang, Well-posedness for the dumbbell model of polymeric fluids, Comm. Math. Phys., 248 (2004), 409-427.doi: 10.1007/s00220-004-1102-y.

    [6]

    J. S. Fan and T. Ozawa, Regularity criterion for the incompressible viscoelastic fluid system, Houston J. Math., 37 (2011), 627-636.

    [7]

    M. E. Gurtin, "An Introduction to Continuum Mechanics, Mathematics in Science and Engineering," Academic Press, Vol. 158, 1981.

    [8]

    L. B. He and L. Xu, Global well-posedness for viscoelastic fluid system in bounded domains, SIAM J. Math. Anal., 42 (2010), 2610-2625.doi: 10.1137/10078503X.

    [9]

    X. P. Hu and R. HyndA blowup criterion for ideal viscelastic flow, arXiv:1102.1113v1.

    [10]

    X. P. Hu and D. H. Wang, Local strong solution to the compressible viscoelastic flow with large data, J. Differential Equations, 249 (2010), 1179-1198.doi: 10.1016/j.jde.2010.03.027.

    [11]

    X. P. Hu and D. H. Wang, Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231.doi: 10.1016/j.jde.2010.10.017.

    [12]

    H. Kozono,T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002), 251-278.doi: 10.1007/s002090100332.

    [13]

    R. G. Larson, "The Structure and Rheology of Complex Fluids," Oxford University Press, New York, 1995.

    [14]

    Z. Lei, Global existence of classical solutions for some Oldroyd-B model via the incompressible limit, Chin. Ann. Math. Ser. B, 27 (2006), 565-580.doi: 10.1007/s11401-005-0041-z.

    [15]

    Z. LeiRotation-strain decomposition for the incompressible viscoelasticity in two dimensions, arXiv:1204.5763v1.

    [16]

    Z. Lei, C. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Rational Mech. Anal., 188 (2008), 371-398.doi: 10.1007/s00205-007-0089-x.

    [17]

    Z. Lei, N. Masmoudi and Y. Zhou, Remarks on the blowup criteria for Oldroyd models, J. Differential Equations, 248 (2010), 328-341.doi: 10.1016/j.jde.2009.07.011.

    [18]

    Z. Lei and Y. Wang, Global solutions for micro-macro models of polymeric fluids, J. Differential Equations, 250 (2011), 3813-3830.doi: 10.1016/j.jde.2011.01.005.

    [19]

    Z. Lei and Y. Zhou, Global existence of classical solutions for 2D Oldroyd model via the incompressible limit, SIAM J. Math. Anal., 37 (2005), 797-814.doi: 10.1137/040618813.

    [20]

    Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst., 25 (2009), 575-583.doi: 10.3934/dcds.2009.25.575.

    [21]

    F. H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Commun. Pure Appl. Math., 58 (2005), 1437-1471.doi: 10.1002/cpa.20074.

    [22]

    F. H. Lin and P. Zhang, On the initial-boundary value problem of the incompressible viscoelastic fluid system, Commun. Pure Appl. Math., 61 (2008), 539-558.doi: 10.1002/cpa.20219.

    [23]

    C. Liu and N. G. Walkington, An Eulerian description of fluids containing viscohyperelastic particles, Arch. Rational Mech. Anal., 159 (2001), 229-252.doi: 10.1007/s002050100158.

    [24]

    A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, Vol. 27, 2002.doi: 10.1007/978-1-4612-0873-0.

    [25]

    N. Masmoudi, Global existence of weak solutions to macroscopic models of polymeric flows, J. Math. Pures Appl., 96 (2011), 502-520.doi: 10.1016/j.matpur.2011.04.008.

    [26]

    N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows, Invent. Math., 191 (2013), 427-500.doi: 10.1007/s00222-012-0399-y.

    [27]

    J. Z. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system, Nonlinear Anal., 72 (2010), 3222-3234.doi: 10.1016/j.na.2009.12.022.

    [28]

    J. Z. Qian, Initial boundary value problems for the compressible viscoelastic fluid, J. Differential Equations, 250 (2011), 848-865.doi: 10.1016/j.jde.2010.07.026.

    [29]

    J. Z. Qian and Z. F. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium, Arch. Rational Mech. Anal., 198 (2010), 835-868.doi: 10.1007/s00205-010-0351-5.

    [30]

    H. Qiu, Regularity criteria of smooth solution to the incompressible viscoelastic flow, Comm. Pure Appl. Anal., 12 (2013), 2873-2888.doi: 10.3934/cpaa.2013.12.2873.

    [31]

    Y. Z. Sun and Z. F. Zhang, Global well-posedness for the 2D micro-macro models in the bounded domain, Comm. Math. Phys., 303 (2011), 361-383.doi: 10.1007/s00220-010-1170-0.

    [32]

    B. Q. Yuan, Note on the blowup criterion of smooth solution to the incompressible viscoelastic flow, Discrete Contin. Dyn. Syst., 33 (2013), 2211-2219.doi: 10.3934/dcds.2013.33.2211.

    [33]

    B. Q. Yuan and R. Li, The blowup criterion of a smooth solution to the incompressible viscoelastic flow, J. Math. Anal. Anal., 406 (2013), 158-164.doi: 10.1016/j.jmaa.2013.04.055.

    [34]

    T. Zhang and D. Y. FangGlobal well-posedness for the incompressible viscoelastic fluids in the critical $L^p$ framework, arXiv:1101.5864.

    [35]

    T. Zhang and D. Y. FangGlobal existence in critical spaces for incompressible viscoelastic fluids, arXiv:1101.5862.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(90) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return