Note on the blowup criterion of smooth solution to the incompressible viscoelastic flow

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May 2013, 33(5): 2211-2219. doi: 10.3934/dcds.2013.33.2211

Note on the blowup criterion of smooth solution to the incompressible viscoelastic flow

Baoquan Yuan ^1,

1.	School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, Henan, China

Received December 2011 Revised March 2012 Published December 2012

Abstract
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We study the blowup criterion of smooth solution to the Oldroyd model. Let $(u(t,x), F(t,x)$ be a smooth solution in $[0,T)$, it is shown that the solution $(u(t,x), F(t,x)$ does not appear breakdown until $t=T$ provided $∇ u(t,x)∈ L^1([0,T]; L^∞(\mathbb{R}^n))$ for $n=2,3$.

Keywords: Oldroyd model, Incompressible viscoelastic fluids, blowup criterion of smooth solution..

Mathematics Subject Classification: 76A10, 76A0.

Citation: Baoquan Yuan. Note on the blowup criterion of smooth solution to the incompressible viscoelastic flow. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2211-2219. doi: 10.3934/dcds.2013.33.2211

References:

[1]	J. T. 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 fluid, SIAM J. Math. Anal., 33 (2001), 84-112. doi: 10.1137/S0036141099359317.
[3]	X. P. Hu and R. Hynd, A blowup criterion for ideal viscelastic flow, Preprint, arXiv:1102.1113v1.
[4]	T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.
[5]	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.
[6]	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.
[7]	F. H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471. doi: 10.1002/cpa.20074.
[8]	F. H. Lin and P. Zhang, On the initial-boundary value problem of the incompressible viscoelastic fluid system, Comm. Pure Appl. Math., 61 (2008), 539-558. doi: 10.1002/cpa.20219.
[9]	A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Univ. Press, 2002.
[10]	C. X. Miao, "Harmonic Analysis and Applications to Partial Differential Equations," $2^{nd}$ edition, Science Press, Beijing, 2004.
[11]	E. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton Univ. Press, 1971.
[12]	L. G. Zhao, B. L. Guo and H. Y. Huang, Blow-up solutions to a viscoelastic fluid system and a coupled Navier-Stokes/phase-field system in $\mathbb{R}^2$, Chin. Phys. Lett., 28 (2011), 1-3. 060206.

show all references

References:

[1]	J. T. 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 fluid, SIAM J. Math. Anal., 33 (2001), 84-112. doi: 10.1137/S0036141099359317.
[3]	X. P. Hu and R. Hynd, A blowup criterion for ideal viscelastic flow, Preprint, arXiv:1102.1113v1.
[4]	T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.
[5]	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.
[6]	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.
[7]	F. H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471. doi: 10.1002/cpa.20074.
[8]	F. H. Lin and P. Zhang, On the initial-boundary value problem of the incompressible viscoelastic fluid system, Comm. Pure Appl. Math., 61 (2008), 539-558. doi: 10.1002/cpa.20219.
[9]	A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Univ. Press, 2002.
[10]	C. X. Miao, "Harmonic Analysis and Applications to Partial Differential Equations," $2^{nd}$ edition, Science Press, Beijing, 2004.
[11]	E. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton Univ. Press, 1971.
[12]	L. G. Zhao, B. L. Guo and H. Y. Huang, Blow-up solutions to a viscoelastic fluid system and a coupled Navier-Stokes/phase-field system in $\mathbb{R}^2$, Chin. Phys. Lett., 28 (2011), 1-3. 060206.

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