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August  2015, 35(8): 3437-3461. doi: 10.3934/dcds.2015.35.3437

Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows

Xianpeng Hu 1, and  Hao Wu 2,

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.
Keywords: long-time behavior, optimal decay rate, Viscoelastic flow, weak-strong uniqueness., Navier-Stokes equations.
Mathematics Subject Classification: Primary: 35B40, 35Q35; Secondary: 35L6.
Citation: Xianpeng Hu, Hao Wu. Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows. Discrete & 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.  Google Scholar

[2]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.  Google Scholar

[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.  Google Scholar

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A. C. Eringen, E. S. Suhubi, Elastodynamics. Vol. I. Finite motions, Academic Press, New York-London, 1974.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[7]

X.-P. Hu and F.-H. Lin, Global solution to two dimensional incompressible viscoelastic fluid with discontinuous data,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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T. Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in $\mathbbR^3$, J. Differential Equations, 184 (2002), 587-619. doi: 10.1006/jdeq.2002.4158.  Google Scholar

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R.-G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, New York, 1995. Google Scholar

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Z. Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-37. doi: 10.1007/s00205-010-0346-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

H.-L. Li, A. Matsumura and G.-J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbbR^3$, Arch. Ration. Mech. Anal., 196 (2010), 681-713. doi: 10.1007/s00205-009-0255-4.  Google Scholar

[23]

H.-L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in $\mathbbR^3$, Math. Methods Appl. Sci., 34 (2011), 670-682. doi: 10.1002/mma.1391.  Google Scholar

[24]

F.-H. Lin, Some analytical issues for elastic complex fluids, Comm. Pure Appl. Math., 65 (2012), 893-919. doi: 10.1002/cpa.21402.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

J. Serrin, On the interior regularity of weak solutions of Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195.  Google Scholar

[36]

Z. Tan and G.-C. Wu, Large time behavior of solutions for compressible Euler equations with damping in $\mathbbR^3$, J. Differential Equations, 252 (2012), 1546-1561. doi: 10.1016/j.jde.2011.09.003.  Google Scholar

[37]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2, North-Holland, Amsterdam, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.  Google Scholar

[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.  Google Scholar

[4]

A. C. Eringen, E. S. Suhubi, Elastodynamics. Vol. I. Finite motions, Academic Press, New York-London, 1974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

X.-P. Hu and F.-H. Lin, Global solution to two dimensional incompressible viscoelastic fluid with discontinuous data,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

T. Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in $\mathbbR^3$, J. Differential Equations, 184 (2002), 587-619. doi: 10.1006/jdeq.2002.4158.  Google Scholar

[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 $\mathbbR^3$, Comm. Math. Phys., 200 (1999), 621-659. doi: 10.1007/s002200050543.  Google Scholar

[17]

R.-G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, New York, 1995. Google Scholar

[18]

Z. Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-37. doi: 10.1007/s00205-010-0346-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

H.-L. Li, A. Matsumura and G.-J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbbR^3$, Arch. Ration. Mech. Anal., 196 (2010), 681-713. doi: 10.1007/s00205-009-0255-4.  Google Scholar

[23]

H.-L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in $\mathbbR^3$, Math. Methods Appl. Sci., 34 (2011), 670-682. doi: 10.1002/mma.1391.  Google Scholar

[24]

F.-H. Lin, Some analytical issues for elastic complex fluids, Comm. Pure Appl. Math., 65 (2012), 893-919. doi: 10.1002/cpa.21402.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

J. Serrin, On the interior regularity of weak solutions of Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195.  Google Scholar

[36]

Z. Tan and G.-C. Wu, Large time behavior of solutions for compressible Euler equations with damping in $\mathbbR^3$, J. Differential Equations, 252 (2012), 1546-1561. doi: 10.1016/j.jde.2011.09.003.  Google Scholar

[37]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2, North-Holland, Amsterdam, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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