# 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 & Continuous Dynamical Systems - A, 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.  doi: 10.1007/s002220000084.  Google Scholar [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.  doi: 10.1137/S0036141099359317.  Google Scholar [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.  doi: 10.1080/03605300600858960.  Google Scholar [4] Y. Du, C. Liu and Q. Zhang, A blow-up criterion for 3-D compressible viscoelasticity,, , ().   Google Scholar [5] J. Fan and T. Ozawa, Regularity criterion for the incompressible viscoelastic fluid system,, Houston J. Math., 37 (2011), 627.   Google Scholar [6] K. O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn's inequality,, Ann. Math. (2), 48 (1947), 441.  doi: 10.2307/1969180.  Google Scholar [7] M. E. Gurtin, An Introduction to Continuum Mechanics,, Mathematics in Science and Engineering, 158 (1981).   Google Scholar [8] L. He and L. Xu, Global well-posedness for viscoelastic fluid system in bounded domains,, SIAM J. Math. Anal., 42 (2010), 2610.  doi: 10.1137/10078503X.  Google Scholar [9] X. Hu and D. Wang, Local strong solution to the compressible viscoelastic flow with large data,, J. Differential Equations, 249 (2010), 1179.  doi: 10.1016/j.jde.2010.03.027.  Google Scholar [10] X. Hu and D. Wang, Global existence for the multi-dimensional compressible viscoelastic flows,, J. Differential Equations, 250 (2011), 1200.  doi: 10.1016/j.jde.2010.10.017.  Google Scholar [11] F. John, Rotation and strain,, Comm. Pure Appl. Math., 14 (1961), 391.  doi: 10.1002/cpa.3160140316.  Google Scholar [12] F. John, Distance changes in deformations with small strain,, in 1970 Studies and Essays (Presented to Yu-why Chen on his 60th Birthday, (1970), 1.   Google Scholar [13] P. Kessenich, Global Existence with Small Initial Data for Three-Dimensional Incompressible Isotropic Viscoelastic Materials,, Ph.D Thesis, (2008).   Google Scholar [14] Z. Lei, On 2D viscoelasticity with small strain,, Archive Ration. Mech. Anal., 198 (2010), 13.  doi: 10.1007/s00205-010-0346-2.  Google Scholar [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.  doi: 10.1007/s11401-005-0041-z.  Google Scholar [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.  doi: 10.4310/CMS.2007.v5.n3.a5.  Google Scholar [17] Z. Lei, C. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids,, Arch. Ration. Mech. Anal., 188 (2008), 371.  doi: 10.1007/s00205-007-0089-x.  Google Scholar [18] Z. Lei and Y. Wang, Global solutions for micro-macro models of polymeric fluids,, J. Differential Equations, 250 (2011), 3813.  doi: 10.1016/j.jde.2011.01.005.  Google Scholar [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.  doi: 10.1137/040618813.  Google Scholar [20] F.-H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids,, Comm. Pure Appl. Math., 58 (2005), 1437.  doi: 10.1002/cpa.20074.  Google Scholar [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.  doi: 10.1002/cpa.20219.  Google Scholar [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.   Google Scholar [23] C. Liu and N. J. Walkington, An Eulerian description of fluids containing visco-elastic particles,, Arch. Rat. Mech Anal., 159 (2001), 229.  doi: 10.1007/s002050100158.  Google Scholar [24] N. Masmoudi, Global existence of weak solutions to macroscopic models of polymeric flows,, J. Math. Pures Appl. (9), 96 (2011), 502.  doi: 10.1016/j.matpur.2011.04.008.  Google Scholar [25] N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows,, Invent. Math., 191 (2013), 427.  doi: 10.1007/s00222-012-0399-y.  Google Scholar [26] J. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system,, Nonlinear Anal., 72 (2010), 3222.  doi: 10.1016/j.na.2009.12.022.  Google Scholar [27] J. Qian, Initial boundary value problems for the compressible viscoelastic fluid,, J. Differential Equations, 250 (2011), 848.  doi: 10.1016/j.jde.2010.07.026.  Google Scholar [28] J. Qian and Z. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium,, Arch. Ration. Mech. Anal., 198 (2010), 835.  doi: 10.1007/s00205-010-0351-5.  Google Scholar [29] T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves,, Ann. of Math. (2), 151 (2000), 849.  doi: 10.2307/121050.  Google Scholar [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.  doi: 10.1002/cpa.20049.  Google Scholar [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.  doi: 10.1142/S0219891606000975.  Google Scholar [32] T. C. Sideris and B. Thomases, Global existence for three-dimensional incompressible isotropic elastodynamics,, Comm. Pure Appl. Math., 60 (2007), 1707.  doi: 10.1002/cpa.20196.  Google Scholar [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.  doi: 10.1007/s00220-010-1170-0.  Google Scholar [34] T. Zhang and D. Fang, Global well-posedness for the incompressible viscoelastic fluids in the critical $L^p$ framework,, , ().   Google Scholar [35] T. Zhang and D. Fang, Global existence in critical spaces for incompressible viscoelastic fluids,, , ().   Google Scholar

show all references

##### References:
 [1] R. Agemi, Global existence of nonlinear elastic waves,, Invent. Math., 142 (2000), 225.  doi: 10.1007/s002220000084.  Google Scholar [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.  doi: 10.1137/S0036141099359317.  Google Scholar [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.  doi: 10.1080/03605300600858960.  Google Scholar [4] Y. Du, C. Liu and Q. Zhang, A blow-up criterion for 3-D compressible viscoelasticity,, , ().   Google Scholar [5] J. Fan and T. Ozawa, Regularity criterion for the incompressible viscoelastic fluid system,, Houston J. Math., 37 (2011), 627.   Google Scholar [6] K. O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn's inequality,, Ann. Math. (2), 48 (1947), 441.  doi: 10.2307/1969180.  Google Scholar [7] M. E. Gurtin, An Introduction to Continuum Mechanics,, Mathematics in Science and Engineering, 158 (1981).   Google Scholar [8] L. He and L. Xu, Global well-posedness for viscoelastic fluid system in bounded domains,, SIAM J. Math. Anal., 42 (2010), 2610.  doi: 10.1137/10078503X.  Google Scholar [9] X. Hu and D. Wang, Local strong solution to the compressible viscoelastic flow with large data,, J. Differential Equations, 249 (2010), 1179.  doi: 10.1016/j.jde.2010.03.027.  Google Scholar [10] X. Hu and D. Wang, Global existence for the multi-dimensional compressible viscoelastic flows,, J. Differential Equations, 250 (2011), 1200.  doi: 10.1016/j.jde.2010.10.017.  Google Scholar [11] F. John, Rotation and strain,, Comm. Pure Appl. Math., 14 (1961), 391.  doi: 10.1002/cpa.3160140316.  Google Scholar [12] F. John, Distance changes in deformations with small strain,, in 1970 Studies and Essays (Presented to Yu-why Chen on his 60th Birthday, (1970), 1.   Google Scholar [13] P. Kessenich, Global Existence with Small Initial Data for Three-Dimensional Incompressible Isotropic Viscoelastic Materials,, Ph.D Thesis, (2008).   Google Scholar [14] Z. Lei, On 2D viscoelasticity with small strain,, Archive Ration. Mech. Anal., 198 (2010), 13.  doi: 10.1007/s00205-010-0346-2.  Google Scholar [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.  doi: 10.1007/s11401-005-0041-z.  Google Scholar [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.  doi: 10.4310/CMS.2007.v5.n3.a5.  Google Scholar [17] Z. Lei, C. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids,, Arch. Ration. Mech. Anal., 188 (2008), 371.  doi: 10.1007/s00205-007-0089-x.  Google Scholar [18] Z. Lei and Y. Wang, Global solutions for micro-macro models of polymeric fluids,, J. Differential Equations, 250 (2011), 3813.  doi: 10.1016/j.jde.2011.01.005.  Google Scholar [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.  doi: 10.1137/040618813.  Google Scholar [20] F.-H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids,, Comm. Pure Appl. Math., 58 (2005), 1437.  doi: 10.1002/cpa.20074.  Google Scholar [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.  doi: 10.1002/cpa.20219.  Google Scholar [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.   Google Scholar [23] C. Liu and N. J. Walkington, An Eulerian description of fluids containing visco-elastic particles,, Arch. Rat. Mech Anal., 159 (2001), 229.  doi: 10.1007/s002050100158.  Google Scholar [24] N. Masmoudi, Global existence of weak solutions to macroscopic models of polymeric flows,, J. Math. Pures Appl. (9), 96 (2011), 502.  doi: 10.1016/j.matpur.2011.04.008.  Google Scholar [25] N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows,, Invent. Math., 191 (2013), 427.  doi: 10.1007/s00222-012-0399-y.  Google Scholar [26] J. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system,, Nonlinear Anal., 72 (2010), 3222.  doi: 10.1016/j.na.2009.12.022.  Google Scholar [27] J. Qian, Initial boundary value problems for the compressible viscoelastic fluid,, J. Differential Equations, 250 (2011), 848.  doi: 10.1016/j.jde.2010.07.026.  Google Scholar [28] J. Qian and Z. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium,, Arch. Ration. Mech. Anal., 198 (2010), 835.  doi: 10.1007/s00205-010-0351-5.  Google Scholar [29] T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves,, Ann. of Math. (2), 151 (2000), 849.  doi: 10.2307/121050.  Google Scholar [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.  doi: 10.1002/cpa.20049.  Google Scholar [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.  doi: 10.1142/S0219891606000975.  Google Scholar [32] T. C. Sideris and B. Thomases, Global existence for three-dimensional incompressible isotropic elastodynamics,, Comm. Pure Appl. Math., 60 (2007), 1707.  doi: 10.1002/cpa.20196.  Google Scholar [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.  doi: 10.1007/s00220-010-1170-0.  Google Scholar [34] T. Zhang and D. Fang, Global well-posedness for the incompressible viscoelastic fluids in the critical $L^p$ framework,, , ().   Google Scholar [35] T. Zhang and D. Fang, Global existence in critical spaces for incompressible viscoelastic fluids,, , ().   Google Scholar
 [1] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [2] Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268 [3] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [4] Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345 [5] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273 [6] Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 16: 331-348. doi: 10.3934/jmd.2020012 [7] Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073 [8] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [9] Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080 [10] José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376 [11] Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082 [12] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [13] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [14] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [15] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [16] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [17] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [18] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [19] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267 [20] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

2019 Impact Factor: 1.338