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

References:
[1]

Comm. Math. Phys., 94 (1984), 61-66.  Google Scholar

[2]

SIAM J. Math. Anal., 33 (2001), 84-112. doi: 10.1137/S0036141099359317.  Google Scholar

[3]

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

[4]

Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar

[5]

Arch. Rational Mech. Anal., 188 (2008), 371-398. doi: 10.1007/s00205-007-0089-x.  Google Scholar

[6]

J. Differential Equations, 248 (2010), 328-341. doi: 10.1016/j.jde.2009.07.011.  Google Scholar

[7]

Comm. Pure Appl. Math., 58 (2005), 1437-1471. doi: 10.1002/cpa.20074.  Google Scholar

[8]

Comm. Pure Appl. Math., 61 (2008), 539-558. doi: 10.1002/cpa.20219.  Google Scholar

[9]

Cambridge Univ. Press, 2002.  Google Scholar

[10]

$2^{nd}$ edition, Science Press, Beijing, 2004. Google Scholar

[11]

Princeton Univ. Press, 1971.  Google Scholar

[12]

Chin. Phys. Lett., 28 (2011), 1-3. 060206. Google Scholar

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