Regularity criteria of smooth solution to the incompressible viscoelastic flow

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November 2013, 12(6): 2873-2888. doi: 10.3934/cpaa.2013.12.2873

Regularity criteria of smooth solution to the incompressible viscoelastic flow

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

Received December 2012 Revised January 2013 Published May 2013

Abstract
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In this paper, we study the regularity criterion of smooth solution to the Oldroyd model in $R^n(n=2,3)$. Firstly, we establish a regularity criterion in terms of the $BMO$ norm of the gradient of columns of the deformation tensor in two space dimensions; secondly, we obtain a Beale-Kato-Majda-type criterion in terms of vorticity with the $BMO$ norm in two and three space dimensions.

Keywords: Oldroyd model; $BMO$., Incompressible viscoelastic fluids, regularity criterion.

Mathematics Subject Classification: Primary: 76A10, 76A05; Secondary: 35B0.

Citation: Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2873-2888. doi: 10.3934/cpaa.2013.12.2873

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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