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March  2015, 35(3): 917-934. doi: 10.3934/dcds.2015.35.917

The initial-boundary value problem for the compressible viscoelastic flows

Xianpeng Hu 1, and  Dehua Wang 2,

1. 

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, United States
2. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260

Received  March 2014 Revised  March 2014 Published  October 2014

The initial-boundary value problem for the equations of compressible viscoelastic flows is considered in a bounded domain of three-dimensional spatial dimensions. The global existence of strong solution near equilibrium is established. Uniform estimates in $W^{1,q}$ with $q>3$ on the density and deformation gradient are also obtained.
Keywords: strong solution, Compressible viscoelastic flows, existence., initial-boundary value problem.
Mathematics Subject Classification: 35A05, 76A10, 76D0.
Citation: Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917
References:
[1]

J. 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.

[2]

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.

[3]

R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8 (2006), 333-381. doi: 10.1007/s00021-004-0147-1.

[4]

M. E. Gurtin, An introduction to Continuum Mechanics, Mathematics in Science and Engineering,158. Academic Press, New York-London, 1981.

[5]

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.

[6]

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.

[7]

D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Applied Mathematical Sciences, 84. Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-4462-2.

[8]

K. Kunisch and M. Marduel, Optimal control of non-isothermal viscoelastic fluid flow, J. Non-Newtonian Fluid Mechanics, 88 (2000), 261-301.

[9]

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. doi: 10.4310/CMS.2007.v5.n3.a5.

[10]

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.

[11]

Z. Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-17. doi: 10.1007/s00205-010-0346-2.

[12]

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.

[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]

F. 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.

[15]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996.

[16]

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.

[17]

C. Liu and N. J. Walkington, An Eulerian description of fluids containing visco-elastic particles, Arch. Ration. Mech. Anal., 159 (2001), 229-252. doi: 10.1007/s002050100158.

[18]

A. Matsumura and T. Nishida, The initial-value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.

[19]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. doi: 10.1007/BF01214738.

[20]

A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Series in Mathematics and its Applications, 27. Oxford University Press, Oxford, 2004.

[21]

J. G. Oldroyd, On the formation of rheological equations of state, Proc. Roy. Soc. London, Series A, 200 (1950), 523-541. doi: 10.1098/rspa.1950.0035.

[22]

J. G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids, Proc. Roy. Soc. London, Series A, 245 (1958), 278-297. doi: 10.1098/rspa.1958.0083.

[23]

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.

[24]

M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Scientic and Technicaland copublished in the US with John Wiley, New York, 1987.

[25]

T. C. Sideris, Nonlinear hyperbolic systems and elastodynamics, Phase space analysis of partial differential equations, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, II (2004), 451-485.

[26]

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.

show all references

References:
[1]

J. 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.

[2]

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.

[3]

R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8 (2006), 333-381. doi: 10.1007/s00021-004-0147-1.

[4]

M. E. Gurtin, An introduction to Continuum Mechanics, Mathematics in Science and Engineering,158. Academic Press, New York-London, 1981.

[5]

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.

[6]

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.

[7]

D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Applied Mathematical Sciences, 84. Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-4462-2.

[8]

K. Kunisch and M. Marduel, Optimal control of non-isothermal viscoelastic fluid flow, J. Non-Newtonian Fluid Mechanics, 88 (2000), 261-301.

[9]

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. doi: 10.4310/CMS.2007.v5.n3.a5.

[10]

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.

[11]

Z. Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-17. doi: 10.1007/s00205-010-0346-2.

[12]

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.

[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]

F. 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.

[15]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996.

[16]

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.

[17]

C. Liu and N. J. Walkington, An Eulerian description of fluids containing visco-elastic particles, Arch. Ration. Mech. Anal., 159 (2001), 229-252. doi: 10.1007/s002050100158.

[18]

A. Matsumura and T. Nishida, The initial-value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.

[19]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. doi: 10.1007/BF01214738.

[20]

A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Series in Mathematics and its Applications, 27. Oxford University Press, Oxford, 2004.

[21]

J. G. Oldroyd, On the formation of rheological equations of state, Proc. Roy. Soc. London, Series A, 200 (1950), 523-541. doi: 10.1098/rspa.1950.0035.

[22]

J. G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids, Proc. Roy. Soc. London, Series A, 245 (1958), 278-297. doi: 10.1098/rspa.1958.0083.

[23]

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.

[24]

M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Scientic and Technicaland copublished in the US with John Wiley, New York, 1987.

[25]

T. C. Sideris, Nonlinear hyperbolic systems and elastodynamics, Phase space analysis of partial differential equations, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, II (2004), 451-485.

[26]

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.

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