# American Institute of Mathematical Sciences

July  2014, 34(7): 2861-2871. doi: 10.3934/dcds.2014.34.2861

## 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

Received  April 2013 Revised  August 2013 Published  December 2013

In this paper, we revisit the 2D rotation-strain model which was derived in [14] for the motion of incompressible viscoelastic materials and prove its global well-posedness theory without making use of the equation of the rotation angle. The proof relies on a new identity satisfied by the strain matrix. The smallness assumptions are only imposed on the $H^2$ norm of initial velocity field and the initial strain matrix, which implies that the deformation tensor is allowed being away from the equilibrium of 2 in the maximum norm.
Citation: Zhen Lei. Rotation-strain decomposition for the incompressible viscoelasticity in two dimensions. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2861-2871. doi: 10.3934/dcds.2014.34.2861
##### 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,, , (). [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,, , (). [35] T. Zhang and D. Fang, Global existence in critical spaces for incompressible viscoelastic fluids,, , ().

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##### 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,, , (). [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,, , (). [35] T. Zhang and D. Fang, Global existence in critical spaces for incompressible viscoelastic fluids,, , ().
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