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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), 61-66. |
[2] |
J. Y. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84-112. |
[3] |
Y. Du, C. Liu and Q. T. Zhang, A blow-up criterion for 3-D compressible viscoelasticity,, \arXiv{1202.3693}., ().
|
[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), 409-427. |
[5] |
J. S. Fan and T. Ozawa, Regularity criterion for the incompressible viscoelastic fluid system, Houston J. Math., 37 (2011), 627-636. |
[6] |
C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193. |
[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. |
[9] |
X. P. Hu and R. Hynd, A 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. |
[11] |
X. P. Hu and D. H. Wang, Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231. |
[12] |
H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194. |
[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. |
[15] |
Z. Lei, Rotation-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. |
[17] |
Z. Lei, N. Masmoudi and Y. Zhou, Remarks on the blowup criteria for Oldroyd models, J. Differential Equations, 248 (2010), 328-341. |
[18] |
Z. Lei and Y. Wang, Global solutions for micro-macro models of polymeric fluids, J. Differential Equations, 250 (2011), 3813-3830. |
[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. |
[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. |
[21] |
F. H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Commun. Pure Appl. Math., 58 (2005), 1437-1471. |
[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. |
[23] |
C. Liu and N. G. Walkington, An Eulerian description of fluids containing viscohyperelastic particles, Arch. Rational Mech. Anal., 159 (2001), 229-252. |
[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. |
[26] |
N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows, Invent. Math., 191 (2013), 427-500. |
[27] |
J. Z. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system, Nonlinear Anal., 72 (2010), 3222-3234. |
[28] |
J. Z. Qian, Initial boundary value problems for the compressible viscoelastic fluid, J. Differential Equations, 250 (2011), 848-865. |
[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. |
[30] |
E. M. Stein, "Harmonic Analysis," Princeton Univ. Press, Princeton, 1993.
doi: 10.1007/978-1-4612-0873-0. |
[31] |
V. A. Solonnikov, Estimates of the solutions of the nonstationary Navier-Stokes system, in "Boundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions, 7," Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973) 153-231 (in Russian). |
[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), 361-383. |
[33] |
B. Q. Yuan, Note On the blowup criterion of smooth solution to the incompressible viscoelastic flow, Discrete Contin. Dyn. Syst., 33 (2013), 2211-2219. |
[34] |
T. Zhang and D. Y. Fang, Global well-posedness for the incompressible viscoelastic fluids in the critical $L^p$ framework,, \arXiv{1101.5864}., ().
|
[35] |
T. Zhang and D. Y. Fang, Global existence in critical spaces for incompressible viscoelastic fluids,, \arXiv{1101.5862}., ().
|
[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), 169-172. |
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), 61-66. |
[2] |
J. Y. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84-112. |
[3] |
Y. Du, C. Liu and Q. T. Zhang, A blow-up criterion for 3-D compressible viscoelasticity,, \arXiv{1202.3693}., ().
|
[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), 409-427. |
[5] |
J. S. Fan and T. Ozawa, Regularity criterion for the incompressible viscoelastic fluid system, Houston J. Math., 37 (2011), 627-636. |
[6] |
C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193. |
[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. |
[9] |
X. P. Hu and R. Hynd, A 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. |
[11] |
X. P. Hu and D. H. Wang, Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231. |
[12] |
H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194. |
[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. |
[15] |
Z. Lei, Rotation-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. |
[17] |
Z. Lei, N. Masmoudi and Y. Zhou, Remarks on the blowup criteria for Oldroyd models, J. Differential Equations, 248 (2010), 328-341. |
[18] |
Z. Lei and Y. Wang, Global solutions for micro-macro models of polymeric fluids, J. Differential Equations, 250 (2011), 3813-3830. |
[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. |
[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. |
[21] |
F. H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Commun. Pure Appl. Math., 58 (2005), 1437-1471. |
[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. |
[23] |
C. Liu and N. G. Walkington, An Eulerian description of fluids containing viscohyperelastic particles, Arch. Rational Mech. Anal., 159 (2001), 229-252. |
[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. |
[26] |
N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows, Invent. Math., 191 (2013), 427-500. |
[27] |
J. Z. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system, Nonlinear Anal., 72 (2010), 3222-3234. |
[28] |
J. Z. Qian, Initial boundary value problems for the compressible viscoelastic fluid, J. Differential Equations, 250 (2011), 848-865. |
[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. |
[30] |
E. M. Stein, "Harmonic Analysis," Princeton Univ. Press, Princeton, 1993.
doi: 10.1007/978-1-4612-0873-0. |
[31] |
V. A. Solonnikov, Estimates of the solutions of the nonstationary Navier-Stokes system, in "Boundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions, 7," Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973) 153-231 (in Russian). |
[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), 361-383. |
[33] |
B. Q. Yuan, Note On the blowup criterion of smooth solution to the incompressible viscoelastic flow, Discrete Contin. Dyn. Syst., 33 (2013), 2211-2219. |
[34] |
T. Zhang and D. Y. Fang, Global well-posedness for the incompressible viscoelastic fluids in the critical $L^p$ framework,, \arXiv{1101.5864}., ().
|
[35] |
T. Zhang and D. Y. Fang, Global existence in critical spaces for incompressible viscoelastic fluids,, \arXiv{1101.5862}., ().
|
[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), 169-172. |
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