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Application of the subharmonic Melnikov method to piecewise-smooth systems
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 |
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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