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August  2022, 15(4): 621-649. doi: 10.3934/krm.2021038

Global existence of small displacement solutions for Hookean incompressible viscoelasticity in 3D

Boyan Jonov 1, , Paul Kessenich 2, and  Thomas C. Sideris 3,,

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

*Corresponding author: Thomas C. Sideris

In memory of Bob Glassey, penetrating mathematician, inspiring teacher, patient mentor, and friend. -T.C.S.

Received  June 2021 Published  August 2022 Early access  November 2021

The initial value problem for incompressible Hookean viscoelastic motion in three space dimensions has global strong solutions with small displacements.

Keywords: Incompressible viscoelasticity, global existence.
Mathematics Subject Classification: Primary: 35Q74.
Citation: Boyan Jonov, Paul Kessenich, Thomas C. Sideris. Global existence of small displacement solutions for Hookean incompressible viscoelasticity in 3D. Kinetic and Related Models, 2022, 15 (4) : 621-649. doi: 10.3934/krm.2021038
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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