October  2018, 38(10): 5261-5295. doi: 10.3934/dcds.2018233

Dispersive effects of the incompressible viscoelastic fluids

1. 

Department of Mathematics, Zhejiang University, Hangzhou 310027, China

2. 

School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China

* Corresponding author: Ruizhao Zi

Received  March 2018 Published  July 2018

We consider the Cauchy problem of the $N$-dimensional incompressible viscoelastic fluids with $N≥2$. It is shown that, in the low frequency part, this system possesses some dispersive properties derived from the one parameter group $e^{± it\Lambda}$. Based on this dispersive effect, we construct global solutions with large initial velocity concentrating on the low frequency part. Such kind of solution has never been seen before in the literature even for the classical incompressible Navier-Stokes equations.

Citation: Daoyuan Fang, Ting Zhang, Ruizhao Zi. Dispersive effects of the incompressible viscoelastic fluids. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5261-5295. doi: 10.3934/dcds.2018233
References:
[1]

H. Abidi, Équation de Navier-Stokes avec densité et viscosité variables dans l'espace critique, Rev. Mat. Iberoam., 23 (2007), 537-586. doi: 10.4171/RMI/505. Google Scholar

[2]

H. Amann, On the strong solvability of the Navier-Stokes equations, J. Math. Fluid Mech., 2 (2000), 16-98. doi: 10.1007/s000210050018. Google Scholar

[3]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar

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M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, 13 (1997), 515-541. doi: 10.4171/RMI/229. Google Scholar

[7]

J.-Y. Chemin, Localization in Fourier space and Navier-Stokes system, Phase Space Analysis of Partial Differential Equations, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 1 (2004), 53-135. Google Scholar

[8]

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328. doi: 10.1006/jdeq.1995.1131. Google Scholar

[9]

F. Charve and R. Danchin, A global existence result for the compressible Navier- Stokes Navier-Stokes equations in the critical Lp framework, Arch. Rational Mech. Anal., 198 (2010), 233-271. doi: 10.1007/s00205-010-0306-x. Google Scholar

[10]

Q. L. ChenC. X. Miao and Z. F. Zhang, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224. doi: 10.1002/cpa.20325. Google Scholar

[11]

Y. M. Chen and P. Zhang, The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions, Comm. Partial Differerntial Equations, 31 (2006), 1793-1810. doi: 10.1080/03605300600858960. Google Scholar

[12]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078. Google Scholar

[13]

R. Danchin, Zero mach numer limit in critical spaces for compresible Navier-Stokes equations, Ann. Scitent. Éc. Norm. Sup., 35 (2002), 27-75. doi: 10.1016/S0012-9593(01)01085-0. Google Scholar

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R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh, 133 (2003), 1311-1334. doi: 10.1017/S030821050000295X. Google Scholar

[15]

R. Danchin, On the uniqueness in critical spaces for compressible Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 111-128. doi: 10.1007/s00030-004-2032-2. Google Scholar

[16]

D. Y. Fang, T. Zhang and R. Z. Zi, Global solutions to the isentropic compressible Navier-Stokes equations with a class of large initial data, arXiv: 1608.06447.Google Scholar

[17]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem Ⅰ, Arch. Ration. Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188. Google Scholar

[18]

Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. Differ. Equ., 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3. Google Scholar

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B. Haspot, Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces, Journal of Differential Equations, 251 (2011), 2262-2295. doi: 10.1016/j.jde.2011.06.013. Google Scholar

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B. Haspot, Existence of global strong solutions in critical spaces for barotropic viscous fluids, Arch. Rational Mech. Anal., 202 (2011), 427-460. doi: 10.1007/s00205-011-0430-2. Google Scholar

[21]

D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676. doi: 10.1512/iumj.1995.44.2003. Google Scholar

[22]

E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231. doi: 10.1002/mana.3210040121. Google Scholar

[23]

X. P. Hu and F. H. Lin, On the Cauchy problem for two dimensional incompressible viscoelastic flows, arXiv: 1601.03497.Google Scholar

[24]

T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182. Google Scholar

[25]

A. A. Kiselev and O. A. Ladyzenskaya, On the existence of uniqueness of solutions of the non-stationary problems for flows of non-compressible fluids, Am. Math. Soc. Transl. Ser. 2., 24 (1963), 79-106. doi: 10.1090/trans2/024/05. Google Scholar

[26]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937. Google Scholar

[27]

Z. Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-37. doi: 10.1007/s00205-010-0346-2. Google Scholar

[28]

Z. LeiC. Liu and Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Commun. Math. Sci., 5 (2007), 595-616. doi: 10.4310/CMS.2007.v5.n3.a5. Google Scholar

[29]

Z. LeiC. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188 (2008), 371-398. doi: 10.1007/s00205-007-0089-x. Google Scholar

[30]

Z. Lei and Y. Zhou, Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit, SIAM J. Math. Anal., 37 (2005), 797-814. doi: 10.1137/040618813. Google Scholar

[31]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354. Google Scholar

[32]

F. H. LinC. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471. doi: 10.1002/cpa.20074. Google Scholar

[33]

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. Google Scholar

[34]

J. Z. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system, Nonlinear Analysis, 72 (2010), 3222-3234. doi: 10.1016/j.na.2009.12.022. Google Scholar

[35]

T. C. Sideris and B. Thomases, Global existense for 3D incompressible isotropic elastodynamics via the incompressible limit, Comm. Pure Appl. Math., 58 (2005), 750-788. doi: 10.1002/cpa.20049. Google Scholar

[36]

T. Zhang and D. Y. Fang, Global existence of strong solution for equations related to the incompressible viscoelastic fluids in the critical $L^p$ framework, SIAM J. Math. Anal., 44 (2012), 2266-2288. doi: 10.1137/110851742. Google Scholar

[37]

T. Zhang, Global strong solutions for equations related to the incompressible viscoelastic fluids with a class of large initial data, Nonlinear Analysis, 100 (2014), 59-77. doi: 10.1016/j.na.2014.01.014. Google Scholar

show all references

References:
[1]

H. Abidi, Équation de Navier-Stokes avec densité et viscosité variables dans l'espace critique, Rev. Mat. Iberoam., 23 (2007), 537-586. doi: 10.4171/RMI/505. Google Scholar

[2]

H. Amann, On the strong solvability of the Navier-Stokes equations, J. Math. Fluid Mech., 2 (2000), 16-98. doi: 10.1007/s000210050018. Google Scholar

[3]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar

[4]

J.-M. Bony, Calcul symbolique et propagation des singularités pour équations aux dérivées partielles nonlinéaires, Annales Scinentifiques de l'école Normale Supérieure, 14 (1981), 209-246. doi: 10.24033/asens.1404. Google Scholar

[5]

M. Cannone, Y. Meyer and F. Planchon, Solutions autosimilaires des équations de Navier-Stokes, Séminaire "Équations aux Dérivées Partielles" de l'École Polytechnique, 1993–1994, Exp. No. Ⅷ, 12 pp., école Polytech., Palaiseau, 1994. doi: 10.1108/09533239410052824. Google Scholar

[6]

M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, 13 (1997), 515-541. doi: 10.4171/RMI/229. Google Scholar

[7]

J.-Y. Chemin, Localization in Fourier space and Navier-Stokes system, Phase Space Analysis of Partial Differential Equations, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 1 (2004), 53-135. Google Scholar

[8]

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328. doi: 10.1006/jdeq.1995.1131. Google Scholar

[9]

F. Charve and R. Danchin, A global existence result for the compressible Navier- Stokes Navier-Stokes equations in the critical Lp framework, Arch. Rational Mech. Anal., 198 (2010), 233-271. doi: 10.1007/s00205-010-0306-x. Google Scholar

[10]

Q. L. ChenC. X. Miao and Z. F. Zhang, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224. doi: 10.1002/cpa.20325. Google Scholar

[11]

Y. M. Chen and P. Zhang, The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions, Comm. Partial Differerntial Equations, 31 (2006), 1793-1810. doi: 10.1080/03605300600858960. Google Scholar

[12]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078. Google Scholar

[13]

R. Danchin, Zero mach numer limit in critical spaces for compresible Navier-Stokes equations, Ann. Scitent. Éc. Norm. Sup., 35 (2002), 27-75. doi: 10.1016/S0012-9593(01)01085-0. Google Scholar

[14]

R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh, 133 (2003), 1311-1334. doi: 10.1017/S030821050000295X. Google Scholar

[15]

R. Danchin, On the uniqueness in critical spaces for compressible Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 111-128. doi: 10.1007/s00030-004-2032-2. Google Scholar

[16]

D. Y. Fang, T. Zhang and R. Z. Zi, Global solutions to the isentropic compressible Navier-Stokes equations with a class of large initial data, arXiv: 1608.06447.Google Scholar

[17]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem Ⅰ, Arch. Ration. Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188. Google Scholar

[18]

Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. Differ. Equ., 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3. Google Scholar

[19]

B. Haspot, Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces, Journal of Differential Equations, 251 (2011), 2262-2295. doi: 10.1016/j.jde.2011.06.013. Google Scholar

[20]

B. Haspot, Existence of global strong solutions in critical spaces for barotropic viscous fluids, Arch. Rational Mech. Anal., 202 (2011), 427-460. doi: 10.1007/s00205-011-0430-2. Google Scholar

[21]

D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676. doi: 10.1512/iumj.1995.44.2003. Google Scholar

[22]

E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231. doi: 10.1002/mana.3210040121. Google Scholar

[23]

X. P. Hu and F. H. Lin, On the Cauchy problem for two dimensional incompressible viscoelastic flows, arXiv: 1601.03497.Google Scholar

[24]

T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182. Google Scholar

[25]

A. A. Kiselev and O. A. Ladyzenskaya, On the existence of uniqueness of solutions of the non-stationary problems for flows of non-compressible fluids, Am. Math. Soc. Transl. Ser. 2., 24 (1963), 79-106. doi: 10.1090/trans2/024/05. Google Scholar

[26]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937. Google Scholar

[27]

Z. Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-37. doi: 10.1007/s00205-010-0346-2. Google Scholar

[28]

Z. LeiC. Liu and Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Commun. Math. Sci., 5 (2007), 595-616. doi: 10.4310/CMS.2007.v5.n3.a5. Google Scholar

[29]

Z. LeiC. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188 (2008), 371-398. doi: 10.1007/s00205-007-0089-x. Google Scholar

[30]

Z. Lei and Y. Zhou, Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit, SIAM J. Math. Anal., 37 (2005), 797-814. doi: 10.1137/040618813. Google Scholar

[31]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354. Google Scholar

[32]

F. H. LinC. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471. doi: 10.1002/cpa.20074. Google Scholar

[33]

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. Google Scholar

[34]

J. Z. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system, Nonlinear Analysis, 72 (2010), 3222-3234. doi: 10.1016/j.na.2009.12.022. Google Scholar

[35]

T. C. Sideris and B. Thomases, Global existense for 3D incompressible isotropic elastodynamics via the incompressible limit, Comm. Pure Appl. Math., 58 (2005), 750-788. doi: 10.1002/cpa.20049. Google Scholar

[36]

T. Zhang and D. Y. Fang, Global existence of strong solution for equations related to the incompressible viscoelastic fluids in the critical $L^p$ framework, SIAM J. Math. Anal., 44 (2012), 2266-2288. doi: 10.1137/110851742. Google Scholar

[37]

T. Zhang, Global strong solutions for equations related to the incompressible viscoelastic fluids with a class of large initial data, Nonlinear Analysis, 100 (2014), 59-77. doi: 10.1016/j.na.2014.01.014. Google Scholar

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