American Institute of Mathematical Sciences

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

 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 $n=2,3$.
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
References:
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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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