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The initial-boundary value problem for the compressible viscoelastic flows
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 |
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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