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Regularity criteria of smooth solution to the incompressible viscoelastic flow
1. | Department of Applied Mathematics, South China Agricultural University, Guangzhou 510642, China |
References:
[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.
|
[2] |
J. Y. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids,, SIAM J. Math. Anal., 33 (2001), 84.
|
[3] |
Y. Du, C. Liu and Q. T. Zhang, A blow-up criterion for 3-D compressible viscoelasticity,, \arXiv{1202.3693}., (). Google Scholar |
[4] |
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.
|
[5] |
J. S. Fan and T. Ozawa, Regularity criterion for the incompressible viscoelastic fluid system,, Houston J. Math., 37 (2011), 627.
|
[6] |
C. Fefferman and E. M. Stein, $H^p$ spaces of several variables,, Acta Math., 129 (1972), 137.
|
[7] |
M. E. Gurtin, "An Introduction to Continuum Mechanics, Mathematics in Science and Engineering,", Academic Press, (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.
|
[9] |
X. P. Hu and R. Hynd, A blowup criterion for ideal viscelastic flow,, \arXiv{1102.1113v1}., (). Google Scholar |
[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.
|
[11] |
X. P. Hu and D. H. Wang, Global existence for the multi-dimensional compressible viscoelastic flows,, J. Differential Equations, 250 (2011), 1200.
|
[12] |
H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations,, Math. Z., 235 (2000), 173.
|
[13] |
R. G. Larson, "The Structure and Rheology of Complex Fluids,", Oxford University Press, (1995). Google Scholar |
[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.
|
[15] |
Z. Lei, Rotation-strain decomposition for the incompressible viscoelasticity in two dimensions,, \arXiv{1204.5763v1}., (). Google Scholar |
[16] |
Z. Lei, C. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids,, Arch. Rational Mech. Anal., 188 (2008), 371.
|
[17] |
Z. Lei, N. Masmoudi and Y. Zhou, Remarks on the blowup criteria for Oldroyd models,, J. Differential Equations, 248 (2010), 328.
|
[18] |
Z. Lei and Y. Wang, Global solutions for micro-macro models of polymeric fluids,, J. Differential Equations, 250 (2011), 3813.
|
[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.
|
[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.
|
[21] |
F. H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids,, Commun. Pure Appl. Math., 58 (2005), 1437.
|
[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.
|
[23] |
C. Liu and N. G. Walkington, An Eulerian description of fluids containing viscohyperelastic particles,, Arch. Rational Mech. Anal., 159 (2001), 229.
|
[24] |
A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,", Cambridge Texts in Applied Mathematics, (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.
|
[26] |
N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows,, Invent. Math., 191 (2013), 427.
|
[27] |
J. Z. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system,, Nonlinear Anal., 72 (2010), 3222.
|
[28] |
J. Z. Qian, Initial boundary value problems for the compressible viscoelastic fluid,, J. Differential Equations, 250 (2011), 848.
|
[29] |
J. Z. Qian and Z. F. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium,, Arch. Rational Mech. Anal., 198 (2010), 835.
|
[30] |
E. M. Stein, "Harmonic Analysis,", Princeton Univ. Press, (1993).
doi: 10.1007/978-1-4612-0873-0. |
[31] |
V. A. Solonnikov, Estimates of the solutions of the nonstationary Navier-Stokes system,, in, (1973), 153.
|
[32] |
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.
|
[33] |
B. Q. Yuan, Note On the blowup criterion of smooth solution to the incompressible viscoelastic flow,, Discrete Contin. Dyn. Syst., 33 (2013), 2211.
|
[34] |
T. Zhang and D. Y. Fang, Global well-posedness for the incompressible viscoelastic fluids in the critical $L^p$ framework,, \arXiv{1101.5864}., (). Google Scholar |
[35] |
T. Zhang and D. Y. Fang, Global existence in critical spaces for incompressible viscoelastic fluids,, \arXiv{1101.5862}., (). Google Scholar |
[36] |
Y. Zhou and J. S. Fan, A regularity criterion for the 2D MHD system with zero magnetic diffusivity,, J. Math. Anal. Appl., 378 (2011), 169.
|
show all references
References:
[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.
|
[2] |
J. Y. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids,, SIAM J. Math. Anal., 33 (2001), 84.
|
[3] |
Y. Du, C. Liu and Q. T. Zhang, A blow-up criterion for 3-D compressible viscoelasticity,, \arXiv{1202.3693}., (). Google Scholar |
[4] |
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.
|
[5] |
J. S. Fan and T. Ozawa, Regularity criterion for the incompressible viscoelastic fluid system,, Houston J. Math., 37 (2011), 627.
|
[6] |
C. Fefferman and E. M. Stein, $H^p$ spaces of several variables,, Acta Math., 129 (1972), 137.
|
[7] |
M. E. Gurtin, "An Introduction to Continuum Mechanics, Mathematics in Science and Engineering,", Academic Press, (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.
|
[9] |
X. P. Hu and R. Hynd, A blowup criterion for ideal viscelastic flow,, \arXiv{1102.1113v1}., (). Google Scholar |
[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.
|
[11] |
X. P. Hu and D. H. Wang, Global existence for the multi-dimensional compressible viscoelastic flows,, J. Differential Equations, 250 (2011), 1200.
|
[12] |
H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations,, Math. Z., 235 (2000), 173.
|
[13] |
R. G. Larson, "The Structure and Rheology of Complex Fluids,", Oxford University Press, (1995). Google Scholar |
[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.
|
[15] |
Z. Lei, Rotation-strain decomposition for the incompressible viscoelasticity in two dimensions,, \arXiv{1204.5763v1}., (). Google Scholar |
[16] |
Z. Lei, C. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids,, Arch. Rational Mech. Anal., 188 (2008), 371.
|
[17] |
Z. Lei, N. Masmoudi and Y. Zhou, Remarks on the blowup criteria for Oldroyd models,, J. Differential Equations, 248 (2010), 328.
|
[18] |
Z. Lei and Y. Wang, Global solutions for micro-macro models of polymeric fluids,, J. Differential Equations, 250 (2011), 3813.
|
[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.
|
[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.
|
[21] |
F. H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids,, Commun. Pure Appl. Math., 58 (2005), 1437.
|
[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.
|
[23] |
C. Liu and N. G. Walkington, An Eulerian description of fluids containing viscohyperelastic particles,, Arch. Rational Mech. Anal., 159 (2001), 229.
|
[24] |
A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,", Cambridge Texts in Applied Mathematics, (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.
|
[26] |
N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows,, Invent. Math., 191 (2013), 427.
|
[27] |
J. Z. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system,, Nonlinear Anal., 72 (2010), 3222.
|
[28] |
J. Z. Qian, Initial boundary value problems for the compressible viscoelastic fluid,, J. Differential Equations, 250 (2011), 848.
|
[29] |
J. Z. Qian and Z. F. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium,, Arch. Rational Mech. Anal., 198 (2010), 835.
|
[30] |
E. M. Stein, "Harmonic Analysis,", Princeton Univ. Press, (1993).
doi: 10.1007/978-1-4612-0873-0. |
[31] |
V. A. Solonnikov, Estimates of the solutions of the nonstationary Navier-Stokes system,, in, (1973), 153.
|
[32] |
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.
|
[33] |
B. Q. Yuan, Note On the blowup criterion of smooth solution to the incompressible viscoelastic flow,, Discrete Contin. Dyn. Syst., 33 (2013), 2211.
|
[34] |
T. Zhang and D. Y. Fang, Global well-posedness for the incompressible viscoelastic fluids in the critical $L^p$ framework,, \arXiv{1101.5864}., (). Google Scholar |
[35] |
T. Zhang and D. Y. Fang, Global existence in critical spaces for incompressible viscoelastic fluids,, \arXiv{1101.5862}., (). Google Scholar |
[36] |
Y. Zhou and J. S. Fan, A regularity criterion for the 2D MHD system with zero magnetic diffusivity,, J. Math. Anal. Appl., 378 (2011), 169.
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