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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

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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[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.  doi: 10.1137/S0036141099359317.  Google Scholar

[3]

X. P. Hu and R. Hynd, A blowup criterion for ideal viscelastic flow,, Preprint, ().   Google Scholar

[4]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891.  doi: 10.1002/cpa.3160410704.  Google Scholar

[5]

Z. Lei, C. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids,, Arch. Rational Mech. Anal., 188 (2008), 371.  doi: 10.1007/s00205-007-0089-x.  Google Scholar

[6]

Z. Lei, N. Masmoudi and Y. Zhou, Remarks on the blowup criteria for Oldroyd models,, J. Differential Equations, 248 (2010), 328.  doi: 10.1016/j.jde.2009.07.011.  Google Scholar

[7]

F. H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids,, Comm. Pure Appl. Math., 58 (2005), 1437.  doi: 10.1002/cpa.20074.  Google Scholar

[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.  doi: 10.1002/cpa.20219.  Google Scholar

[9]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,", Cambridge Univ. Press, (2002).   Google Scholar

[10]

C. X. Miao, "Harmonic Analysis and Applications to Partial Differential Equations,", $2^{nd}$ edition, (2004).   Google Scholar

[11]

E. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton Univ. Press, (1971).   Google Scholar

[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 $\mathbbR^2$,, Chin. Phys. Lett., 28 (2011), 1.   Google Scholar

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.   Google Scholar

[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.  doi: 10.1137/S0036141099359317.  Google Scholar

[3]

X. P. Hu and R. Hynd, A blowup criterion for ideal viscelastic flow,, Preprint, ().   Google Scholar

[4]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891.  doi: 10.1002/cpa.3160410704.  Google Scholar

[5]

Z. Lei, C. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids,, Arch. Rational Mech. Anal., 188 (2008), 371.  doi: 10.1007/s00205-007-0089-x.  Google Scholar

[6]

Z. Lei, N. Masmoudi and Y. Zhou, Remarks on the blowup criteria for Oldroyd models,, J. Differential Equations, 248 (2010), 328.  doi: 10.1016/j.jde.2009.07.011.  Google Scholar

[7]

F. H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids,, Comm. Pure Appl. Math., 58 (2005), 1437.  doi: 10.1002/cpa.20074.  Google Scholar

[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.  doi: 10.1002/cpa.20219.  Google Scholar

[9]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,", Cambridge Univ. Press, (2002).   Google Scholar

[10]

C. X. Miao, "Harmonic Analysis and Applications to Partial Differential Equations,", $2^{nd}$ edition, (2004).   Google Scholar

[11]

E. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton Univ. Press, (1971).   Google Scholar

[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 $\mathbbR^2$,, Chin. Phys. Lett., 28 (2011), 1.   Google Scholar

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