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Rotation-strain decomposition for the incompressible viscoelasticity in two dimensions
1. | School of Mathematical Sciences, LMNS and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, China |
References:
[1] |
R. Agemi, Global existence of nonlinear elastic waves, Invent. Math., 142 (2000), 225-250.
doi: 10.1007/s002220000084. |
[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] |
Y. Chen and P. Zhang, The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions, Comm. Partial Differential Equations, 31 (2006), 1793-1810.
doi: 10.1080/03605300600858960. |
[4] |
Y. Du, C. Liu and Q. Zhang, A blow-up criterion for 3-D compressible viscoelasticity, arXiv:1202.3693. |
[5] |
J. Fan and T. Ozawa, Regularity criterion for the incompressible viscoelastic fluid system, Houston J. Math., 37 (2011), 627-636. |
[6] |
K. O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn's inequality, Ann. Math. (2), 48 (1947), 441-471.
doi: 10.2307/1969180. |
[7] |
M. E. Gurtin, An Introduction to Continuum Mechanics, Mathematics in Science and Engineering, 158, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. |
[8] |
L. He and L. Xu, Global well-posedness for viscoelastic fluid system in bounded domains, SIAM J. Math. Anal., 42 (2010), 2610-2625.
doi: 10.1137/10078503X. |
[9] |
X. Hu and D. Wang, Local strong solution to the compressible viscoelastic flow with large data, J. Differential Equations, 249 (2010), 1179-1198.
doi: 10.1016/j.jde.2010.03.027. |
[10] |
X. Hu and D. Wang, Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231.
doi: 10.1016/j.jde.2010.10.017. |
[11] |
F. John, Rotation and strain, Comm. Pure Appl. Math., 14 (1961), 391-413.
doi: 10.1002/cpa.3160140316. |
[12] |
F. John, Distance changes in deformations with small strain, in 1970 Studies and Essays (Presented to Yu-why Chen on his 60th Birthday, April 1, 1970), Math. Res. Center, Nat. Taiwan Univ., Taipei, 1970, 1-15. |
[13] |
P. Kessenich, Global Existence with Small Initial Data for Three-Dimensional Incompressible Isotropic Viscoelastic Materials, Ph.D Thesis, University of California, Santa Barbara, 2008. |
[14] |
Z. Lei, On 2D viscoelasticity with small strain, Archive Ration. Mech. Anal., 198 (2010), 13-37.
doi: 10.1007/s00205-010-0346-2. |
[15] |
Z. Lei, Global existence of classical solutions for some Oldroyd-B model via the incompressible limit, Chin. Ann. Math. Ser. B, 27 (2006), 565-580.
doi: 10.1007/s11401-005-0041-z. |
[16] |
Z. Lei, C. Liu and Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Comm. Math. Sci., 5 (2007), 595-616.
doi: 10.4310/CMS.2007.v5.n3.a5. |
[17] |
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. |
[18] |
Z. Lei and Y. Wang, Global solutions for micro-macro models of polymeric fluids, J. Differential Equations, 250 (2011), 3813-3830.
doi: 10.1016/j.jde.2011.01.005. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
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. |
[23] |
C. Liu and N. J. Walkington, An Eulerian description of fluids containing visco-elastic particles, Arch. Rat. Mech Anal., 159 (2001), 229-252.
doi: 10.1007/s002050100158. |
[24] |
N. Masmoudi, Global existence of weak solutions to macroscopic models of polymeric flows, J. Math. Pures Appl. (9), 96 (2011), 502-520.
doi: 10.1016/j.matpur.2011.04.008. |
[25] |
N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows, Invent. Math., 191 (2013), 427-500.
doi: 10.1007/s00222-012-0399-y. |
[26] |
J. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system, Nonlinear Anal., 72 (2010), 3222-3234.
doi: 10.1016/j.na.2009.12.022. |
[27] |
J. Qian, Initial boundary value problems for the compressible viscoelastic fluid, J. Differential Equations, 250 (2011), 848-865.
doi: 10.1016/j.jde.2010.07.026. |
[28] |
J. Qian and Z. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium, Arch. Ration. Mech. Anal., 198 (2010), 835-868.
doi: 10.1007/s00205-010-0351-5. |
[29] |
T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. (2), 151 (2000), 849-874.
doi: 10.2307/121050. |
[30] |
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. |
[31] |
T. C. Sideris and B. Thomases, Local energy decay for solutions of multi-dimensional isotropic symmetric hyperbolic systems, J. Hyperbolic Differ. Equ., 3 (2006), 673-690.
doi: 10.1142/S0219891606000975. |
[32] |
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. |
[33] |
Y. Sun and Z. Zhang, Global well-posedness for the 2D micro-macro models in the bounded domain, Comm. Math. Phys., 303 (2011), 361-383.
doi: 10.1007/s00220-010-1170-0. |
[34] |
T. Zhang and D. Fang, Global well-posedness for the incompressible viscoelastic fluids in the critical $L^p$ framework, arXiv:1101.5864. |
[35] |
T. Zhang and D. Fang, Global existence in critical spaces for incompressible viscoelastic fluids, arXiv:1101.5862. |
show all references
References:
[1] |
R. Agemi, Global existence of nonlinear elastic waves, Invent. Math., 142 (2000), 225-250.
doi: 10.1007/s002220000084. |
[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] |
Y. Chen and P. Zhang, The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions, Comm. Partial Differential Equations, 31 (2006), 1793-1810.
doi: 10.1080/03605300600858960. |
[4] |
Y. Du, C. Liu and Q. Zhang, A blow-up criterion for 3-D compressible viscoelasticity, arXiv:1202.3693. |
[5] |
J. Fan and T. Ozawa, Regularity criterion for the incompressible viscoelastic fluid system, Houston J. Math., 37 (2011), 627-636. |
[6] |
K. O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn's inequality, Ann. Math. (2), 48 (1947), 441-471.
doi: 10.2307/1969180. |
[7] |
M. E. Gurtin, An Introduction to Continuum Mechanics, Mathematics in Science and Engineering, 158, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. |
[8] |
L. He and L. Xu, Global well-posedness for viscoelastic fluid system in bounded domains, SIAM J. Math. Anal., 42 (2010), 2610-2625.
doi: 10.1137/10078503X. |
[9] |
X. Hu and D. Wang, Local strong solution to the compressible viscoelastic flow with large data, J. Differential Equations, 249 (2010), 1179-1198.
doi: 10.1016/j.jde.2010.03.027. |
[10] |
X. Hu and D. Wang, Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231.
doi: 10.1016/j.jde.2010.10.017. |
[11] |
F. John, Rotation and strain, Comm. Pure Appl. Math., 14 (1961), 391-413.
doi: 10.1002/cpa.3160140316. |
[12] |
F. John, Distance changes in deformations with small strain, in 1970 Studies and Essays (Presented to Yu-why Chen on his 60th Birthday, April 1, 1970), Math. Res. Center, Nat. Taiwan Univ., Taipei, 1970, 1-15. |
[13] |
P. Kessenich, Global Existence with Small Initial Data for Three-Dimensional Incompressible Isotropic Viscoelastic Materials, Ph.D Thesis, University of California, Santa Barbara, 2008. |
[14] |
Z. Lei, On 2D viscoelasticity with small strain, Archive Ration. Mech. Anal., 198 (2010), 13-37.
doi: 10.1007/s00205-010-0346-2. |
[15] |
Z. Lei, Global existence of classical solutions for some Oldroyd-B model via the incompressible limit, Chin. Ann. Math. Ser. B, 27 (2006), 565-580.
doi: 10.1007/s11401-005-0041-z. |
[16] |
Z. Lei, C. Liu and Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Comm. Math. Sci., 5 (2007), 595-616.
doi: 10.4310/CMS.2007.v5.n3.a5. |
[17] |
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. |
[18] |
Z. Lei and Y. Wang, Global solutions for micro-macro models of polymeric fluids, J. Differential Equations, 250 (2011), 3813-3830.
doi: 10.1016/j.jde.2011.01.005. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
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. |
[23] |
C. Liu and N. J. Walkington, An Eulerian description of fluids containing visco-elastic particles, Arch. Rat. Mech Anal., 159 (2001), 229-252.
doi: 10.1007/s002050100158. |
[24] |
N. Masmoudi, Global existence of weak solutions to macroscopic models of polymeric flows, J. Math. Pures Appl. (9), 96 (2011), 502-520.
doi: 10.1016/j.matpur.2011.04.008. |
[25] |
N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows, Invent. Math., 191 (2013), 427-500.
doi: 10.1007/s00222-012-0399-y. |
[26] |
J. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system, Nonlinear Anal., 72 (2010), 3222-3234.
doi: 10.1016/j.na.2009.12.022. |
[27] |
J. Qian, Initial boundary value problems for the compressible viscoelastic fluid, J. Differential Equations, 250 (2011), 848-865.
doi: 10.1016/j.jde.2010.07.026. |
[28] |
J. Qian and Z. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium, Arch. Ration. Mech. Anal., 198 (2010), 835-868.
doi: 10.1007/s00205-010-0351-5. |
[29] |
T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. (2), 151 (2000), 849-874.
doi: 10.2307/121050. |
[30] |
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. |
[31] |
T. C. Sideris and B. Thomases, Local energy decay for solutions of multi-dimensional isotropic symmetric hyperbolic systems, J. Hyperbolic Differ. Equ., 3 (2006), 673-690.
doi: 10.1142/S0219891606000975. |
[32] |
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. |
[33] |
Y. Sun and Z. Zhang, Global well-posedness for the 2D micro-macro models in the bounded domain, Comm. Math. Phys., 303 (2011), 361-383.
doi: 10.1007/s00220-010-1170-0. |
[34] |
T. Zhang and D. Fang, Global well-posedness for the incompressible viscoelastic fluids in the critical $L^p$ framework, arXiv:1101.5864. |
[35] |
T. Zhang and D. Fang, Global existence in critical spaces for incompressible viscoelastic fluids, arXiv:1101.5862. |
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