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

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J. Fan and T. Ozawa, Regularity criterion for the incompressible viscoelastic fluid system,, Houston J. Math., 37 (2011), 627.   Google Scholar

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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

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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

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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

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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

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