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March  2014, 13(2): 823-833. doi: 10.3934/cpaa.2014.13.823

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

Hua Qiu 1, and  Shaomei Fang 1,

1. 

Department of Applied Mathematics, South China Agricultural University, Guangzhou 510642, China

Received  May 2013 Revised  August 2013 Published  October 2013

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 $n=2,3$.
Keywords: regularity criterion, Incompressible viscoelastic fluids, Oldroyd model, Besov space..
Mathematics Subject Classification: Primary: 76A10, 76A05; Secondary: 35B0.
Citation: Hua Qiu, Shaomei Fang. A BKM's criterion of smooth solution to the incompressible viscoelastic flow. Communications on Pure and Applied Analysis, 2014, 13 (2) : 823-833. doi: 10.3934/cpaa.2014.13.823
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, "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. Zhang, A 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. 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. 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. 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. 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. 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.

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, "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. Zhang, A 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. 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. 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. 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. 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. 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.

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