• Previous Article
    Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains
  • DCDS Home
  • This Issue
  • Next Article
    Emergence of phase-locked states for the Winfree model in a large coupling regime
August  2015, 35(8): 3437-3461. doi: 10.3934/dcds.2015.35.3437

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

Received  September 2014 Revised  November 2014 Published  February 2015

We consider the Cauchy problem for incompressible viscoelastic fluids in the whole space $\mathbb{R}^d$ ($d=2,3$). By introducing a new decomposition via Helmholtz's projections, we first provide an alternative proof on the existence of global smooth solutions near equilibrium. Then under additional assumptions that the initial data belong to $L^1$ and their Fourier modes do not degenerate at low frequencies, we obtain the optimal $L^2$ decay rates for the global smooth solutions and their spatial derivatives. At last, we establish the weak-strong uniqueness property in the class of finite energy weak solutions for the incompressible viscoelastic system.
Citation: Xianpeng Hu, Hao Wu. Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3437-3461. doi: 10.3934/dcds.2015.35.3437
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.

[1]

Wenjing Zhao. Weak-strong uniqueness of incompressible magneto-viscoelastic flows. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2907-2917. doi: 10.3934/cpaa.2020127

[2]

Yang Liu. Long-time behavior of a class of viscoelastic plate equations. Electronic Research Archive, 2020, 28 (1) : 311-326. doi: 10.3934/era.2020018

[3]

Oscar Jarrín, Manuel Fernando Cortez. On the long-time behavior for a damped Navier-Stokes-Bardina model. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3661-3707. doi: 10.3934/dcds.2022028

[4]

Hongjun Gao, Šárka Nečasová, Tong Tang. On weak-strong uniqueness and singular limit for the compressible Primitive Equations. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4287-4305. doi: 10.3934/dcds.2020181

[5]

Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks and Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625

[6]

Takeshi Taniguchi. The exponential behavior of Navier-Stokes equations with time delay external force. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 997-1018. doi: 10.3934/dcds.2005.12.997

[7]

Chao Yang, Yanbing Yang. Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4643-4658. doi: 10.3934/dcdss.2021110

[8]

Xin Zhong. Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3563-3578. doi: 10.3934/dcdsb.2020246

[9]

Tongtong Liang. The stability with the general decay rate of the solution for stochastic functional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022127

[10]

Etienne Emmrich, Robert Lasarzik. Weak-strong uniqueness for the general Ericksen—Leslie system in three dimensions. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4617-4635. doi: 10.3934/dcds.2018202

[11]

Anhui Gu, Boling Guo, Bixiang Wang. Long term behavior of random Navier-Stokes equations driven by colored noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2495-2532. doi: 10.3934/dcdsb.2020020

[12]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123

[13]

Wojciech M. Zajączkowski. Long time existence of regular solutions to non-homogeneous Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1427-1455. doi: 10.3934/dcdss.2013.6.1427

[14]

Donatella Donatelli, Nóra Juhász. The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2859-2892. doi: 10.3934/dcds.2022002

[15]

Shan Ma, Chunyou Sun. Long-time behavior for a class of weighted equations with degeneracy. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1889-1902. doi: 10.3934/dcds.2020098

[16]

Chang Zhang, Fang Li, Jinqiao Duan. Long-time behavior of a class of nonlocal partial differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 749-763. doi: 10.3934/dcdsb.2018041

[17]

Hongtao Li, Shan Ma, Chengkui Zhong. Long-time behavior for a class of degenerate parabolic equations. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2873-2892. doi: 10.3934/dcds.2014.34.2873

[18]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163

[19]

Reinhard Farwig, Yasushi Taniuchi. Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1215-1224. doi: 10.3934/dcdss.2013.6.1215

[20]

Giovanni P. Galdi. Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1237-1257. doi: 10.3934/dcdss.2013.6.1237

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (132)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]