# American Institute of Mathematical Sciences

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

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.
Citation: Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure & 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.  Google Scholar [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.  Google Scholar [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), 409-427.  Google Scholar [5] J. S. Fan and T. Ozawa, Regularity criterion for the incompressible viscoelastic fluid system, Houston J. Math., 37 (2011), 627-636.  Google Scholar [6] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193.  Google Scholar [7] M. E. Gurtin, "An Introduction to Continuum Mechanics, Mathematics in Science and Engineering," Academic Press, Vol. 158, 1981.  Google Scholar [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.  Google Scholar [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), 1179-1198.  Google Scholar [11] X. P. Hu and D. H. Wang, Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231.  Google Scholar [12] H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194.  Google Scholar [13] R. G. Larson, "The Structure and Rheology of Complex Fluids," Oxford University Press, New York, 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), 565-580.  Google Scholar [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), 371-398.  Google Scholar [17] Z. Lei, N. Masmoudi and Y. Zhou, Remarks on the blowup criteria for Oldroyd models, J. Differential Equations, 248 (2010), 328-341.  Google Scholar [18] Z. Lei and Y. Wang, Global solutions for micro-macro models of polymeric fluids, J. Differential Equations, 250 (2011), 3813-3830.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] F. H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Commun. Pure Appl. Math., 58 (2005), 1437-1471.  Google Scholar [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.  Google Scholar [23] C. Liu and N. G. Walkington, An Eulerian description of fluids containing viscohyperelastic particles, Arch. Rational Mech. Anal., 159 (2001), 229-252.  Google Scholar [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.  Google Scholar [25] N. Masmoudi, Global existence of weak solutions to macroscopic models of polymeric flows, J. Math. Pures Appl., 96 (2011), 502-520.  Google Scholar [26] N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows, Invent. Math., 191 (2013), 427-500.  Google Scholar [27] J. Z. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system, Nonlinear Anal., 72 (2010), 3222-3234.  Google Scholar [28] J. Z. Qian, Initial boundary value problems for the compressible viscoelastic fluid, J. Differential Equations, 250 (2011), 848-865.  Google Scholar [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.  Google Scholar [30] E. M. Stein, "Harmonic Analysis," Princeton Univ. Press, Princeton, 1993. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [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).  Google Scholar [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.  Google Scholar [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.  Google Scholar [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), 169-172.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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), 409-427.  Google Scholar [5] J. S. Fan and T. Ozawa, Regularity criterion for the incompressible viscoelastic fluid system, Houston J. Math., 37 (2011), 627-636.  Google Scholar [6] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193.  Google Scholar [7] M. E. Gurtin, "An Introduction to Continuum Mechanics, Mathematics in Science and Engineering," Academic Press, Vol. 158, 1981.  Google Scholar [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.  Google Scholar [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), 1179-1198.  Google Scholar [11] X. P. Hu and D. H. Wang, Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231.  Google Scholar [12] H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194.  Google Scholar [13] R. G. Larson, "The Structure and Rheology of Complex Fluids," Oxford University Press, New York, 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), 565-580.  Google Scholar [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), 371-398.  Google Scholar [17] Z. Lei, N. Masmoudi and Y. Zhou, Remarks on the blowup criteria for Oldroyd models, J. Differential Equations, 248 (2010), 328-341.  Google Scholar [18] Z. Lei and Y. Wang, Global solutions for micro-macro models of polymeric fluids, J. Differential Equations, 250 (2011), 3813-3830.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] F. H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Commun. Pure Appl. Math., 58 (2005), 1437-1471.  Google Scholar [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.  Google Scholar [23] C. Liu and N. G. Walkington, An Eulerian description of fluids containing viscohyperelastic particles, Arch. Rational Mech. Anal., 159 (2001), 229-252.  Google Scholar [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.  Google Scholar [25] N. Masmoudi, Global existence of weak solutions to macroscopic models of polymeric flows, J. Math. Pures Appl., 96 (2011), 502-520.  Google Scholar [26] N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows, Invent. Math., 191 (2013), 427-500.  Google Scholar [27] J. Z. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system, Nonlinear Anal., 72 (2010), 3222-3234.  Google Scholar [28] J. Z. Qian, Initial boundary value problems for the compressible viscoelastic fluid, J. Differential Equations, 250 (2011), 848-865.  Google Scholar [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.  Google Scholar [30] E. M. Stein, "Harmonic Analysis," Princeton Univ. Press, Princeton, 1993. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [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).  Google Scholar [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.  Google Scholar [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.  Google Scholar [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), 169-172.  Google Scholar
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