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Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows
1. | Department of Mathematics, City University of Hong Kong, Hong Kong, China |
2. | School of Mathematical Sciences, Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Han Dan Road 220, 200433 Shanghai |
References:
[1] |
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. |
[2] |
R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.
doi: 10.1007/s002220000078. |
[3] |
Y. Du, C. Liu and Q.-T. Zhang, Blow-up criterion for compressible visco-elasticity equations in three dimensional space, Comm. Math. Sci., 12 (2014), 473-484.
doi: 10.4310/CMS.2014.v12.n3.a4. |
[4] |
A. C. Eringen, E. S. Suhubi, Elastodynamics. Vol. I. Finite motions, Academic Press, New York-London, 1974. |
[5] |
Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212.
doi: 10.1016/0022-0396(86)90096-3. |
[6] |
X.-P. Hu and R. Hynd, A blowup criterion for ideal viscoelastic flow, J. Math. Fluid Mech., 15 (2013), 431-437.
doi: 10.1007/s00021-012-0124-z. |
[7] |
X.-P. Hu and F.-H. Lin, Global solution to two dimensional incompressible viscoelastic fluid with discontinuous data, preprint, arXiv:1312.6749. |
[8] |
X.-P. Hu and D.-H. 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. |
[9] |
X.-P. Hu and D.-H. 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. |
[10] |
X.-P. Hu and D.-H. Wang, Strong solutions to the three-dimensional compressible viscoelastic fluids, J. Differential Equations, 252 (2012), 4027-4067.
doi: 10.1016/j.jde.2011.11.021. |
[11] |
X.-P. Hu and G.-C. Wu, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal., 45 (2013), 2815-2833.
doi: 10.1137/120892350. |
[12] |
R. Kajikiya and T. Miyakawa, On $L^2$ decay of weak solutions of the Navier-Stokes Equations in $R^n$, Math. Z., 192 (1986), 135-148.
doi: 10.1007/BF01162027. |
[13] |
T. Kato, Strong $L^p$ solutions of Navier-Stokes equation in $R^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[14] |
H. Kozono and H. Sohr, Remark on uniqueness of weak solutions to the Navier-Stokes equations, Analysis, 16 (1996), 255-271.
doi: 10.1524/anly.1996.16.3.255. |
[15] |
T. Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in $\mathbb{R}^3$, J. Differential Equations, 184 (2002), 587-619.
doi: 10.1006/jdeq.2002.4158. |
[16] |
T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in $\mathbb{R}^3$, Comm. Math. Phys., 200 (1999), 621-659.
doi: 10.1007/s002200050543. |
[17] |
R.-G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, New York, 1995. |
[18] |
Z. Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-37.
doi: 10.1007/s00205-010-0346-2. |
[19] |
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. |
[20] |
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. |
[21] |
Z. Lei and Y. Zhou, Global existence of classical solutions for 2D Oldroyd model via the incompressible limit, SIAM J. Math. Anal., 37 (2005), 797-814.
doi: 10.1137/040618813. |
[22] |
H.-L. Li, A. Matsumura and G.-J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 196 (2010), 681-713.
doi: 10.1007/s00205-009-0255-4. |
[23] |
H.-L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in $\mathbb{R}^3$, Math. Methods Appl. Sci., 34 (2011), 670-682.
doi: 10.1002/mma.1391. |
[24] |
F.-H. Lin, Some analytical issues for elastic complex fluids, Comm. Pure Appl. Math., 65 (2012), 893-919.
doi: 10.1002/cpa.21402. |
[25] |
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. |
[26] |
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. |
[27] |
P.-L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chin. Ann. Math. Ser. B, 21 (2000), 131-146.
doi: 10.1142/S0252959900000170. |
[28] |
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. |
[29] |
G. Prodi, Un teorema di unicitá per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182.
doi: 10.1007/BF02410664. |
[30] |
J.-Z. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluids system, Nonlinear Anal., 72 (2010), 3222-3234.
doi: 10.1016/j.na.2009.12.022. |
[31] |
J.-Z. 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. |
[32] |
J.-Z. Qian and Z.-F. 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. |
[33] |
M. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 88 (1985), 209-222.
doi: 10.1007/BF00752111. |
[34] |
M. Schonbek, Large time behavior of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763.
doi: 10.1080/03605308608820443. |
[35] |
J. Serrin, On the interior regularity of weak solutions of Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195. |
[36] |
Z. Tan and G.-C. Wu, Large time behavior of solutions for compressible Euler equations with damping in $\mathbb{R}^3$, J. Differential Equations, 252 (2012), 1546-1561.
doi: 10.1016/j.jde.2011.09.003. |
[37] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2, North-Holland, Amsterdam, 1977. |
[38] |
T. Zhang and D.-Y. Fang, Global existence of strong solution for equations related to the incompressible viscoelastic fluids in the critical $L^p$ framework, SIAM J. Math. Anal., 44 (2012), 2266-2288.
doi: 10.1137/110851742. |
[39] |
S. Zheng, Nonlinear Evolution Equations, Pitman series Monographs and Survey in Pure and Applied Mathematics, 133, Chapman & Hall/CRC, Boca Raton, Florida, 2004.
doi: 10.1201/9780203492222. |
show all references
References:
[1] |
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. |
[2] |
R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.
doi: 10.1007/s002220000078. |
[3] |
Y. Du, C. Liu and Q.-T. Zhang, Blow-up criterion for compressible visco-elasticity equations in three dimensional space, Comm. Math. Sci., 12 (2014), 473-484.
doi: 10.4310/CMS.2014.v12.n3.a4. |
[4] |
A. C. Eringen, E. S. Suhubi, Elastodynamics. Vol. I. Finite motions, Academic Press, New York-London, 1974. |
[5] |
Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212.
doi: 10.1016/0022-0396(86)90096-3. |
[6] |
X.-P. Hu and R. Hynd, A blowup criterion for ideal viscoelastic flow, J. Math. Fluid Mech., 15 (2013), 431-437.
doi: 10.1007/s00021-012-0124-z. |
[7] |
X.-P. Hu and F.-H. Lin, Global solution to two dimensional incompressible viscoelastic fluid with discontinuous data, preprint, arXiv:1312.6749. |
[8] |
X.-P. Hu and D.-H. 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. |
[9] |
X.-P. Hu and D.-H. 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. |
[10] |
X.-P. Hu and D.-H. Wang, Strong solutions to the three-dimensional compressible viscoelastic fluids, J. Differential Equations, 252 (2012), 4027-4067.
doi: 10.1016/j.jde.2011.11.021. |
[11] |
X.-P. Hu and G.-C. Wu, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal., 45 (2013), 2815-2833.
doi: 10.1137/120892350. |
[12] |
R. Kajikiya and T. Miyakawa, On $L^2$ decay of weak solutions of the Navier-Stokes Equations in $R^n$, Math. Z., 192 (1986), 135-148.
doi: 10.1007/BF01162027. |
[13] |
T. Kato, Strong $L^p$ solutions of Navier-Stokes equation in $R^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[14] |
H. Kozono and H. Sohr, Remark on uniqueness of weak solutions to the Navier-Stokes equations, Analysis, 16 (1996), 255-271.
doi: 10.1524/anly.1996.16.3.255. |
[15] |
T. Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in $\mathbb{R}^3$, J. Differential Equations, 184 (2002), 587-619.
doi: 10.1006/jdeq.2002.4158. |
[16] |
T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in $\mathbb{R}^3$, Comm. Math. Phys., 200 (1999), 621-659.
doi: 10.1007/s002200050543. |
[17] |
R.-G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, New York, 1995. |
[18] |
Z. Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-37.
doi: 10.1007/s00205-010-0346-2. |
[19] |
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. |
[20] |
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. |
[21] |
Z. Lei and Y. Zhou, Global existence of classical solutions for 2D Oldroyd model via the incompressible limit, SIAM J. Math. Anal., 37 (2005), 797-814.
doi: 10.1137/040618813. |
[22] |
H.-L. Li, A. Matsumura and G.-J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 196 (2010), 681-713.
doi: 10.1007/s00205-009-0255-4. |
[23] |
H.-L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in $\mathbb{R}^3$, Math. Methods Appl. Sci., 34 (2011), 670-682.
doi: 10.1002/mma.1391. |
[24] |
F.-H. Lin, Some analytical issues for elastic complex fluids, Comm. Pure Appl. Math., 65 (2012), 893-919.
doi: 10.1002/cpa.21402. |
[25] |
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. |
[26] |
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. |
[27] |
P.-L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chin. Ann. Math. Ser. B, 21 (2000), 131-146.
doi: 10.1142/S0252959900000170. |
[28] |
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. |
[29] |
G. Prodi, Un teorema di unicitá per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182.
doi: 10.1007/BF02410664. |
[30] |
J.-Z. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluids system, Nonlinear Anal., 72 (2010), 3222-3234.
doi: 10.1016/j.na.2009.12.022. |
[31] |
J.-Z. 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. |
[32] |
J.-Z. Qian and Z.-F. 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. |
[33] |
M. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 88 (1985), 209-222.
doi: 10.1007/BF00752111. |
[34] |
M. Schonbek, Large time behavior of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763.
doi: 10.1080/03605308608820443. |
[35] |
J. Serrin, On the interior regularity of weak solutions of Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195. |
[36] |
Z. Tan and G.-C. Wu, Large time behavior of solutions for compressible Euler equations with damping in $\mathbb{R}^3$, J. Differential Equations, 252 (2012), 1546-1561.
doi: 10.1016/j.jde.2011.09.003. |
[37] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2, North-Holland, Amsterdam, 1977. |
[38] |
T. Zhang and D.-Y. Fang, Global existence of strong solution for equations related to the incompressible viscoelastic fluids in the critical $L^p$ framework, SIAM J. Math. Anal., 44 (2012), 2266-2288.
doi: 10.1137/110851742. |
[39] |
S. Zheng, Nonlinear Evolution Equations, Pitman series Monographs and Survey in Pure and Applied Mathematics, 133, Chapman & Hall/CRC, Boca Raton, Florida, 2004.
doi: 10.1201/9780203492222. |
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