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December 2018, 7(4): 617-637. doi: 10.3934/eect.2018030

Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Yangrong Li

Received  August 2017 Revised  May 2018 Published  September 2018

Fund Project: Y. Li and R. Wang are supported by National Natural Science Foundation of China grant 11571283 and L. She is supported by the Natural Science Foundation of Guizhou Province:KY[2016]103

This paper analyzes the time-dependence and backward controllability of pullback attractors for the trajectory space generated by a non-autonomous evolution equation without uniqueness. A pullback trajectory attractor is called backward controllable if the norm of its union over the past is controlled by a continuous function, and backward compact if it is backward controllable and pre-compact in the past on the underlying space. We then establish two existence theorems for such a backward compact trajectory attractor, which leads to the existence of a pullback attractor with the backward compactness and backward boundedness in two original phase spaces respectively. An essential criterion is the existence of an increasing, compact and absorbing brochette. Applying to the non-autonomous Jeffreys-Oldroyd equations with a backward controllable force, we obtain a backward compact trajectory attractor, and also a pullback attractor with backward compactness in the negative-exponent Sobolev space and backward boundedness in the Lebesgue space.

Citation: Yangrong Li, Renhai Wang, Lianbing She. Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations. Evolution Equations & Control Theory, 2018, 7 (4) : 617-637. doi: 10.3934/eect.2018030
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A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractor for Infinite-dimensional Nonautonomous Dynamical Systems, Appl. Math. Sciences, 182, Springer, 2013.

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V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964. doi: 10.1016/S0021-7824(97)89978-3.

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V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, C. R. Acad. Sci. Paris, 321 (1995), 1309-1314. doi: 10.1016/S0021-7824(97)89978-3.

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I. Chueshov and A. Shcherbina, Semi-weak well-posedness and attractors for 2D Schrodinger-Boussinesq equations, Evolution Equ. Control Theory, 1 (2012), 57-80. doi: 10.3934/eect.2012.1.57.

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H. CuiJ. A. Langa and Y. Li, Regularity and structure of pullback attractors for reactiondiffusion type systems without uniqueness, Nonlinear Anal., 140 (2016), 208-235. doi: 10.1016/j.na.2016.03.012.

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H. CuiJ. A. LangaY. Li and J. Valero, Attractors for multi-valued non-autonomous dynamical systems: relationship, characterization and robustness, Set-Valued Var. Anal., 26 (2018), 493-530. doi: 10.1007/s11228-016-0395-2.

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H. CuiY. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 304-324. doi: 10.1016/j.na.2015.08.009.

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T. A. Cung and M. T. Vu, Local exact controllability to trajectories of the magneto-micropolar fluid equations, Evolution Equ. Control Theory, 6 (2017), 357-379. doi: 10.3934/eect.2017019.

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D. S. JorgeA. Marcio and V. Narciso, Long-time Dynamics for a class of extensible beams with nonlocal nonlinear damping, Evolution Equ. Control Theory, 6 (2017), 437-470. doi: 10.3934/eect.2017023.

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P. E. Kloeden and P. Marin-Rubio, Negatively invariant sets and entire trajectories of set-valued dynamical systems, Set-Valued Anal., 19 (2011), 43-57. doi: 10.1007/s11228-009-0123-2.

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P. E. KloedenP. Marin-Rubio and J. Valero, The envelope attractor of non-strict multivalued dynamical systems with application to the 3D Navier-Stokes and reaction-diffusion equations, Set-Valued Var. Anal., 21 (2013), 517-540. doi: 10.1007/s11228-012-0228-x.

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Y. LiA. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021.

[15]

Y. LiH. Cui and J. Li, Upper semi-continuity and regularity of random attractors on p-times integrable spaces and applications, Nonlinear Anal., 109 (2014), 33-44. doi: 10.1016/j.na.2014.06.013.

[16]

Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differ. Equ., 245 (2008), 1775-1800. doi: 10.1016/j.jde.2008.06.031.

[17]

Y. LiR. Wang and J. Yin, Backward compact attractors for non-autonomous BenjaminBona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. B, 22 (2017), 2569-2586. doi: 10.3934/dcdsb.2017092.

[18]

M. PaicuG. Raugel and A. Rekalo, Regularity of the global attractor and finite-dimensional behavior for the second grade fluid equations, J. Differ. Equ., 252 (2012), 3695-3751. doi: 10.1016/j.jde.2011.10.015.

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M. I. Vishik and V. V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Russian Math. Surveys, 66 (2011), 637-731. doi: 10.1070/RM2011v066n04ABEH004753.

[21]

M. I. Vishik and V. V. Chepyzhov, Trajectory and global attractors of the three-dimensional Navier-Stokes system, Mat. Zametki, 71 (2002), 194-213. doi: 10.1023/A:1014190629738.

[22]

D. A. Vorotnikov, Asymptotic behaviour of the non-autonomous 3D Navier-Stokes problem with coercive force, J. Differ. Equ., 251 (2011), 2209-2225. doi: 10.1016/j.jde.2011.07.008.

[23]

D. A. Vorotnikov and V. G. Zvyagin, Trajectory and global attractors of the boundary value problem for autonomous motion equations of viscoelastic medium, J. Math. Fluid Mech., 10 (2008), 19-44. doi: 10.1007/s00021-005-0215-1.

[24]

D. A. Vorotnikov and V. G. Zvyagin, Uniform attractors for non-autonomous motion equations of viscoelastic medium, J. Math. Anal. Appl., 325 (2007), 438-458. doi: 10.1016/j.jmaa.2006.01.078.

[25]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.

[26]

Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differ. Equ., 259 (2015), 728-776. doi: 10.1016/j.jde.2015.02.026.

[27]

J. YinA. Gu and Y. Li, Backwards compact attractors for non-autonomous damped 3D Navier-Stokes equations, Dynamics of PDE, 14 (2017), 201-218. doi: 10.4310/DPDE.2017.v14.n2.a4.

[28]

J. YinY. Li and H. Cui, Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations on an unbounded domain, J. Math. Anal. Appl., 450 (2017), 1180-1207. doi: 10.1016/j.jmaa.2017.01.064.

[29]

J. YinY. Li and H. Zhao, Random attractors for stochastic semi-linear degenerate parabolic equations with additive noise in Lq, Appl. Math. Comput., 225 (2013), 526-540. doi: 10.1016/j.amc.2013.09.051.

[30]

M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM J. Math. Anal., 47 (2015), 1530-1561. doi: 10.1137/140978995.

[31]

V. Zvyagin and S. Kondratyev, Pullback attractor of Jeffreys-Oldroyd equations, J. Differ. Equ., 260 (2016), 5020-5042. doi: 10.1016/j.jde.2015.11.038.

[32]

V. G. Zvyagin and S. K. Kondratyev, Approximating topological approach to the existence of attractors in fluid mechanics, J. Fixed Point Theory Appl., 13 (2013), 359-395. doi: 10.1007/s11784-013-0122-7.

[33]

V. G. Zvyagin and D. A. Vorotnikov, Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics, Walter de Gruyter, Berlin, New York, 2008. doi: 10.1515/9783110208283.

show all references

References:
[1]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractor for Infinite-dimensional Nonautonomous Dynamical Systems, Appl. Math. Sciences, 182, Springer, 2013.

[2]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964. doi: 10.1016/S0021-7824(97)89978-3.

[3]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, C. R. Acad. Sci. Paris, 321 (1995), 1309-1314. doi: 10.1016/S0021-7824(97)89978-3.

[4]

I. Chueshov and A. Shcherbina, Semi-weak well-posedness and attractors for 2D Schrodinger-Boussinesq equations, Evolution Equ. Control Theory, 1 (2012), 57-80. doi: 10.3934/eect.2012.1.57.

[5]

H. CuiJ. A. Langa and Y. Li, Regularity and structure of pullback attractors for reactiondiffusion type systems without uniqueness, Nonlinear Anal., 140 (2016), 208-235. doi: 10.1016/j.na.2016.03.012.

[6]

H. CuiJ. A. LangaY. Li and J. Valero, Attractors for multi-valued non-autonomous dynamical systems: relationship, characterization and robustness, Set-Valued Var. Anal., 26 (2018), 493-530. doi: 10.1007/s11228-016-0395-2.

[7]

H. CuiY. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 304-324. doi: 10.1016/j.na.2015.08.009.

[8]

T. A. Cung and M. T. Vu, Local exact controllability to trajectories of the magneto-micropolar fluid equations, Evolution Equ. Control Theory, 6 (2017), 357-379. doi: 10.3934/eect.2017019.

[9]

D. S. JorgeA. Marcio and V. Narciso, Long-time Dynamics for a class of extensible beams with nonlocal nonlinear damping, Evolution Equ. Control Theory, 6 (2017), 437-470. doi: 10.3934/eect.2017023.

[10]

P. E. Kloeden and P. Marin-Rubio, Negatively invariant sets and entire trajectories of set-valued dynamical systems, Set-Valued Anal., 19 (2011), 43-57. doi: 10.1007/s11228-009-0123-2.

[11]

P. E. KloedenP. Marin-Rubio and J. Valero, The envelope attractor of non-strict multivalued dynamical systems with application to the 3D Navier-Stokes and reaction-diffusion equations, Set-Valued Var. Anal., 21 (2013), 517-540. doi: 10.1007/s11228-012-0228-x.

[12]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, 176, Providence, 2011. doi: 10.1090/surv/176.

[13]

P. E. KloedenJ. Real and C. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differ. Equ., 246 (2009), 4702-4730. doi: 10.1016/j.jde.2008.11.017.

[14]

Y. LiA. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021.

[15]

Y. LiH. Cui and J. Li, Upper semi-continuity and regularity of random attractors on p-times integrable spaces and applications, Nonlinear Anal., 109 (2014), 33-44. doi: 10.1016/j.na.2014.06.013.

[16]

Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differ. Equ., 245 (2008), 1775-1800. doi: 10.1016/j.jde.2008.06.031.

[17]

Y. LiR. Wang and J. Yin, Backward compact attractors for non-autonomous BenjaminBona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. B, 22 (2017), 2569-2586. doi: 10.3934/dcdsb.2017092.

[18]

M. PaicuG. Raugel and A. Rekalo, Regularity of the global attractor and finite-dimensional behavior for the second grade fluid equations, J. Differ. Equ., 252 (2012), 3695-3751. doi: 10.1016/j.jde.2011.10.015.

[19]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd edition, Society for Industrial and Applied Mathematics, 1995. doi: 10.1137/1.9781611970050.

[20]

M. I. Vishik and V. V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Russian Math. Surveys, 66 (2011), 637-731. doi: 10.1070/RM2011v066n04ABEH004753.

[21]

M. I. Vishik and V. V. Chepyzhov, Trajectory and global attractors of the three-dimensional Navier-Stokes system, Mat. Zametki, 71 (2002), 194-213. doi: 10.1023/A:1014190629738.

[22]

D. A. Vorotnikov, Asymptotic behaviour of the non-autonomous 3D Navier-Stokes problem with coercive force, J. Differ. Equ., 251 (2011), 2209-2225. doi: 10.1016/j.jde.2011.07.008.

[23]

D. A. Vorotnikov and V. G. Zvyagin, Trajectory and global attractors of the boundary value problem for autonomous motion equations of viscoelastic medium, J. Math. Fluid Mech., 10 (2008), 19-44. doi: 10.1007/s00021-005-0215-1.

[24]

D. A. Vorotnikov and V. G. Zvyagin, Uniform attractors for non-autonomous motion equations of viscoelastic medium, J. Math. Anal. Appl., 325 (2007), 438-458. doi: 10.1016/j.jmaa.2006.01.078.

[25]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.

[26]

Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differ. Equ., 259 (2015), 728-776. doi: 10.1016/j.jde.2015.02.026.

[27]

J. YinA. Gu and Y. Li, Backwards compact attractors for non-autonomous damped 3D Navier-Stokes equations, Dynamics of PDE, 14 (2017), 201-218. doi: 10.4310/DPDE.2017.v14.n2.a4.

[28]

J. YinY. Li and H. Cui, Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations on an unbounded domain, J. Math. Anal. Appl., 450 (2017), 1180-1207. doi: 10.1016/j.jmaa.2017.01.064.

[29]

J. YinY. Li and H. Zhao, Random attractors for stochastic semi-linear degenerate parabolic equations with additive noise in Lq, Appl. Math. Comput., 225 (2013), 526-540. doi: 10.1016/j.amc.2013.09.051.

[30]

M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM J. Math. Anal., 47 (2015), 1530-1561. doi: 10.1137/140978995.

[31]

V. Zvyagin and S. Kondratyev, Pullback attractor of Jeffreys-Oldroyd equations, J. Differ. Equ., 260 (2016), 5020-5042. doi: 10.1016/j.jde.2015.11.038.

[32]

V. G. Zvyagin and S. K. Kondratyev, Approximating topological approach to the existence of attractors in fluid mechanics, J. Fixed Point Theory Appl., 13 (2013), 359-395. doi: 10.1007/s11784-013-0122-7.

[33]

V. G. Zvyagin and D. A. Vorotnikov, Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics, Walter de Gruyter, Berlin, New York, 2008. doi: 10.1515/9783110208283.

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