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Global existence of small displacement solutions for Hookean incompressible viscoelasticity in 3D
1. | Raytheon Technologies, Goleta, CA 93117, USA |
2. | Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA |
3. | Department of Mathematics, University of California, Santa Barbara, CA 93016, USA |
The initial value problem for incompressible Hookean viscoelastic motion in three space dimensions has global strong solutions with small displacements.
References:
[1] |
Y. Cai, Z. Lei, F. Lin and N. Masmoudi,
Vanishing viscosity limit for incompressible viscoelasticity in two dimensions, Comm. Pure Appl. Math., 72 (2019), 2063-2120.
doi: 10.1002/cpa.21853. |
[2] |
J.-Y. Chemin and N. Masmoudi,
About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84-112.
doi: 10.1137/S0036141099359317. |
[3] |
P. Constantin, J. Wu, J. Zhao and Y. Zhu, High Reynolds number and high Weissenberg number Oldroyd-B model with dissipation, J. Evolution Equations, (2020).
doi: 10.1007/s00028-020-00616-8. |
[4] |
T. M. Elgindi and J. Liu,
Global wellposedness to the generalized Oldroyd type models in $\Bbb{R}^3$, J. Differential Equations, 259 (2015), 1958-1966.
doi: 10.1016/j.jde.2015.03.026. |
[5] |
T. M. Elgindi and F. Rousset,
Global regularity for some Oldroyd-B type models, Comm. Pure Appl. Math., 68 (2015), 2005-2021.
doi: 10.1002/cpa.21563. |
[6] |
B. Y. Jonov, Longtime Behavior of Small Solutions to Viscous Perturbations of Nonlinear Hyperbolic Systems in 3D, Thesis (Ph.D.)–University of California, Santa Barbara, 2014. URL http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:3645652. |
[7] |
B. Jonov and T. C. Sideris,
Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms, Commun. Pure Appl. Anal., 14 (2015), 1407-1442.
doi: 10.3934/cpaa.2015.14.1407. |
[8] |
P. Kessenich, Global Existence with Small Initial Data for Three-Dimensional Incompressible Isotropic Viscoelastic Materials, Thesis (Ph.D.)–University of California, Santa Barbara, 2008, URL http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3335016. |
[9] |
P. Kessenich, Global existence with small initial data for three-dimensional incompressible isotropic viscoelastic materials, arXiv e-prints, arXiv: 0903.2824. |
[10] |
A. A. Kiselev and O. A. Ladyženskaya,
On the existence and uniqueness of the solution of the nonstationary problem for a viscous, incompressible fluid, Izv. Akad. Nauk SSSR. Ser. Mat., 21 (1957), 655-680.
|
[11] |
Z. Lei, C. Liu and Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Commun. Math. Sci., 5 (2007), 595–616, URL http://projecteuclid.org/euclid.cms/1188405670.
doi: 10.4310/CMS.2007.v5.n3.a5. |
[12] |
Z. Lei, C. 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. |
[13] |
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. |
[14] |
P. L. Lions and N. Masmoudi,
Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B, 21 (2000), 131-146.
doi: 10.1142/S0252959900000170. |
[15] |
T. C. Sideris,
Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math., 151 (2000), 849-874.
doi: 10.2307/121050. |
[16] |
T. C. Sideris and B. Thomases,
Global existence for three-dimensional incompressible isotropic elastodynamics via the incompressible limit, Comm. Pure Appl. Math., 58 (2005), 750-788.
doi: 10.1002/cpa.20049. |
[17] |
T. C. Sideris and B. Thomases,
Global existence for three-dimensional incompressible isotropic elastodynamics, Comm. Pure Appl. Math., 60 (2007), 1707-1730.
doi: 10.1002/cpa.20196. |
show all references
References:
[1] |
Y. Cai, Z. Lei, F. Lin and N. Masmoudi,
Vanishing viscosity limit for incompressible viscoelasticity in two dimensions, Comm. Pure Appl. Math., 72 (2019), 2063-2120.
doi: 10.1002/cpa.21853. |
[2] |
J.-Y. Chemin and N. Masmoudi,
About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84-112.
doi: 10.1137/S0036141099359317. |
[3] |
P. Constantin, J. Wu, J. Zhao and Y. Zhu, High Reynolds number and high Weissenberg number Oldroyd-B model with dissipation, J. Evolution Equations, (2020).
doi: 10.1007/s00028-020-00616-8. |
[4] |
T. M. Elgindi and J. Liu,
Global wellposedness to the generalized Oldroyd type models in $\Bbb{R}^3$, J. Differential Equations, 259 (2015), 1958-1966.
doi: 10.1016/j.jde.2015.03.026. |
[5] |
T. M. Elgindi and F. Rousset,
Global regularity for some Oldroyd-B type models, Comm. Pure Appl. Math., 68 (2015), 2005-2021.
doi: 10.1002/cpa.21563. |
[6] |
B. Y. Jonov, Longtime Behavior of Small Solutions to Viscous Perturbations of Nonlinear Hyperbolic Systems in 3D, Thesis (Ph.D.)–University of California, Santa Barbara, 2014. URL http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:3645652. |
[7] |
B. Jonov and T. C. Sideris,
Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms, Commun. Pure Appl. Anal., 14 (2015), 1407-1442.
doi: 10.3934/cpaa.2015.14.1407. |
[8] |
P. Kessenich, Global Existence with Small Initial Data for Three-Dimensional Incompressible Isotropic Viscoelastic Materials, Thesis (Ph.D.)–University of California, Santa Barbara, 2008, URL http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3335016. |
[9] |
P. Kessenich, Global existence with small initial data for three-dimensional incompressible isotropic viscoelastic materials, arXiv e-prints, arXiv: 0903.2824. |
[10] |
A. A. Kiselev and O. A. Ladyženskaya,
On the existence and uniqueness of the solution of the nonstationary problem for a viscous, incompressible fluid, Izv. Akad. Nauk SSSR. Ser. Mat., 21 (1957), 655-680.
|
[11] |
Z. Lei, C. Liu and Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Commun. Math. Sci., 5 (2007), 595–616, URL http://projecteuclid.org/euclid.cms/1188405670.
doi: 10.4310/CMS.2007.v5.n3.a5. |
[12] |
Z. Lei, C. 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. |
[13] |
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. |
[14] |
P. L. Lions and N. Masmoudi,
Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B, 21 (2000), 131-146.
doi: 10.1142/S0252959900000170. |
[15] |
T. C. Sideris,
Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math., 151 (2000), 849-874.
doi: 10.2307/121050. |
[16] |
T. C. Sideris and B. Thomases,
Global existence for three-dimensional incompressible isotropic elastodynamics via the incompressible limit, Comm. Pure Appl. Math., 58 (2005), 750-788.
doi: 10.1002/cpa.20049. |
[17] |
T. C. Sideris and B. Thomases,
Global existence for three-dimensional incompressible isotropic elastodynamics, Comm. Pure Appl. Math., 60 (2007), 1707-1730.
doi: 10.1002/cpa.20196. |
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