# American Institute of Mathematical Sciences

December  2018, 38(12): 6305-6325. doi: 10.3934/dcds.2018269

## Attractors for model of polymer solutions motion

 1 Peoples Friendship University of Russia, 6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation 2 Voronezh State University, 1 Universitetskaya sq., Voronezh, 394018, Russian Federation

To Professor Rafael de la Llave

Received  July 2017 Revised  January 2018 Published  September 2018

Existence of trajectory, global and pullback attractors for an incompressible non-Newtonian fluid (namely, for the mathematical model which describes a weak aqueous polymer solutions motion) in 2D and 3D bounded domains is studied in this paper. For this aim the approximating topological method is effectively combined with the theory of attractors of trajectory spaces.

Citation: Andrey Zvyagin. Attractors for model of polymer solutions motion. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6305-6325. doi: 10.3934/dcds.2018269
##### References:
 [1] V. B. Amfilokhiev, Ya. I. Voitkunskiy, N. P. Mazaeva and Ya. S. Khodorkovskiy, Flows of polymer solutions under convective accelerations, Trudy Leningradskogo Korablestroitel'nogo Instituta, 96 (1975), 3-9. [2] V. B. Amfilokhiev and V. A. Pavlovsky, Experimental data on laminar-turbulent transition for motion of polymer solutions in pipes, Trudy Leningradskogo Korablestroitel'nogo Instituta, 104 (1976), 3-5. [3] M. Boukrouche, G. Lukaszewicz and J. Real, On pullback attractors for a class of two-dimensional turbulent shear flows, International Journal of Engineering Science, 44 (2006), 830-844.  doi: 10.1016/j.ijengsci.2006.05.012. [4] C. Cao, E. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Communications in Mathematical Sciences, 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8. [5] T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis: Theory, Methods and Applications, 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111. [6] V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, Comptes Rendus de l Acad$\acute{e}$mie des Sciences - Series I - Mathematics, 321 (1995), 1309-1314.  doi: 10.1016/S0021-7824(97)89978-3. [7] V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, Journal de Math$\acute{e}$matiques Pures et Appliqu$\acute{e}$es, 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3. [8] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society. Colloquium Publications, 2002. [9] D. Cioranescu and E. H. Quazar, Nonlinear partial differential equations and their applications: College de france seminar, Vol. VI, H. Brezis and J.L. Lions (eds.), Research Notes in Mathematics, Pitman, Boston, 109 (1984), 178-197. [10] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705. [11] A. V. Fursikov, Optimal Control of Distributed Systems. Theorey and Applications, Providence, 2000. [12] G. P. Galdi, M. Grobbelaar-Van Dalsen and N. Sauer, Existence and uniqueness of classical solutions of the equations of motion for second-grade fluids, Archive for Rational Mechanics and Analysis, 124 (1993), 221-237.  doi: 10.1007/BF00953067. [13] I. Gyarmati, Non-equilibrium Thermodynamics. Field Theory and Variational Principles, Heidelberg, Springer, 1970. [14] P. E. Kloeden and B. Schmalfuv, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numerical Algorithms, 14 (1997), 141-152.  doi: 10.1023/A:1019156812251. [15] O. A. Ladyzhenskaya, On some gaps in two of my papers on the Navier-Stokes equations and the way of closing them, Zapiski Nauchnykh Seminarov POMI, 271 (2000), 151-155.  doi: 10.1023/A:1023321903383. [16] R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of H${\ddot{o}}$lder functions, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 5 (1999), 157-184. [17] R. de la Llave, Variational methods for quasi-periodic solutions of partial differential equations, Hamiltonian and Celestial Mechanics, 6 (2000), 214-228.  doi: 10.1142/9789812792099_0013. [18] G. Lukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211. [19] A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aueous solutions of polymers, Zapiski Nauchnykh Seminarov Leningradskogo Otdelenia Matematicheskogo Instituta Steklova (LOMI), 38 (1973), 98-136. [20] A. P. Oskolkov, On some quasilinear systems occuring in the study of motion of viscous fluids, Zapiski Nauchnykh Seminarov Leningradskogo Otdelenia Matematicheskogo Instituta Steklova (LOMI), 52 (1975), 128-157. [21] V. A. Pavlovsky, On the theoretical description of weak water solutions of polymers, Doklady Akademii Nauk SSSR, (1971), 809-812. [22] R. S. Rivlin and J. L. Ericksen, Stress-deformation relations for isotropic materials, Journal of Rational Mechanics and Analysis, 4 (1955), 323-425. [23] C. Le Roux, Existence and uniqueness of the flow of second-grade fluids with slip boundary conditions, Archive for Rational Mechanics and Analysis, 148 (1999), 309-356.  doi: 10.1007/s002050050164. [24] G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, Journal of Dynamics and Differential Equations, 8 (1996), 1-33.  doi: 10.1007/BF02218613. [25] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, American Mathematical Society, 2002. doi: 10.1007/978-1-4757-5037-9. [26] A. Sequeira and J. Videman, Global existence of classical solutions for the equations of third grade fluids, J. Math. Phys. Sci., 29 (1995), 47-69. [27] J. Simon, Compact sets in the space Lp(0,T;B), Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360. [28] R. Temam, Navier-Stokes Equations. Theory and Nymerical Analysis, Providence, RI, 2001. doi: 10.1090/chel/343. [29] C. Truesdell, A First Course in Rational Continuum Mechanics, The John Hopkins Unirsity, 1972. [30] M. I. Vishik and V. V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Russian Mathematical Surveys, 66 (2011), 637-731.  doi: 10.1070/RM2011v066n04ABEH004753. [31] D. A. Vorotnikov and V. G. Zvyagin, Trajectory and global attractors of the boundary value problem for autonomous motion equations of viscoelastic medium, Journal of Mathematical Fluid Mechanics, 10 (2008), 19-44.  doi: 10.1007/s00021-005-0215-1. [32] D. A. Vorotnikov, Asymptotic behaviour of the non-autonomous 3D Navier-Stokes problem with coercive force, Journal of Differential Equations, 251 (2011), 2209-2225.  doi: 10.1016/j.jde.2011.07.008. [33] C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid, Journal of Differential Equations, 238 (2007), 394-425.  doi: 10.1016/j.jde.2007.04.001. [34] C. Zhao, Y. Li and S. Zhou, Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid, Journal of Differential Equations, 247 (2009), 2331-2363.  doi: 10.1016/j.jde.2009.07.031. [35] C. Zhao, Pullback asymptotic behavior of solutions for a non-autonomous non-Newtonian fluid on two-dimensional unbounded domains, Journal of Mathematical Physics, 53 (2012), 122702, 22pp. doi: 10.1063/1.4769302. [36] C. Zhao, G. Liu and W. Wang, Smooth pullback attractors for a non-autonomous 2D non-Newtonian fluid and their tempered behaviors, Journal of Mathematical Fluid Mechanics, 16 (2014), 243-262.  doi: 10.1007/s00021-013-0153-2. [37] C. Zhao and L. Yang, Pullback attractors and invariant measures for the non-autonomous globally modified Naviertokes equations, Communications in Mathematical Sciences, 15 (2017), 1565-1580.  doi: 10.4310/CMS.2017.v15.n6.a4. [38] Z. Zhu and C. Zhao, Pullback attractor and invariant measures for the three-dimensional regularized MHD equations, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 38 (2018), 1461-1477.  doi: 10.3934/dcds.2018060. [39] A. V. Zvyagin, Solvability for the equations of motion of weak aqueous polymer solution with objective derivative, Nonlinear Analysis: Theory, Methods and Applications, 90 (2013), 70-85.  doi: 10.1016/j.na.2013.05.022. [40] A. V. Zvyagin, An optimal control problem with feedback for a mathematical model of the motion of weakly concentrated water polymer solutions with objective derivative, Siberian Mathematical Journal, 54 (2013), 640-655. [41] A. V. Zvyagin, Attractors for a model of polymer motion with objective derivative in the rheological relation, Doklady Mathematics, 88 (2013), 730-733. [42] A. V. Zvyagin and V. G. Zvyagin, Pullback attractors for a model of weakly concentrated aqueous polymer solution motion with a rheological relation satisfying the objectivity principle, Doklady Mathematics, 95 (2017), 247-249. [43] V. G. Zvyagin and S. K. Kondratiev, Attractors of equation of non-Newtinian fluid dynamics, Russian Mathematical Surveys, 69 (2014), 845-913.  doi: 10.4213/rm9615. [44] V. G. Zvyagin and S. K. Kondratiev, Pullback attractors for the model of motion of dilute aqueous polymer solutions, Izvestiya: Mathematics, 79 (2015), 645-667.  doi: 10.4213/im8163. [45] V. G. Zvyagin and M. V. Turbin, The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, Journal of Mathematical Sciences, 168 (2010), 157-308.  doi: 10.1007/s10958-010-9981-2. [46] V. G. Zvyagin and D. A. Vorotnikov, Approximating-topological methods in some problems of hydrodinamics, Journal of Fixed Point Theory and Applications, 3 (2008), 23-49.  doi: 10.1007/s11784-008-0056-7. [47] V. G. Zvyagin and D. A. Vorotnikov, Topological Approximayion Methods for Evolutionary Problems of Nonlinear Hydrodinamics, De Gruyter Series In Nonlinear Analysis and Applications, 2008. doi: 10.1515/9783110208283.

show all references

##### References:
 [1] V. B. Amfilokhiev, Ya. I. Voitkunskiy, N. P. Mazaeva and Ya. S. Khodorkovskiy, Flows of polymer solutions under convective accelerations, Trudy Leningradskogo Korablestroitel'nogo Instituta, 96 (1975), 3-9. [2] V. B. Amfilokhiev and V. A. Pavlovsky, Experimental data on laminar-turbulent transition for motion of polymer solutions in pipes, Trudy Leningradskogo Korablestroitel'nogo Instituta, 104 (1976), 3-5. [3] M. Boukrouche, G. Lukaszewicz and J. Real, On pullback attractors for a class of two-dimensional turbulent shear flows, International Journal of Engineering Science, 44 (2006), 830-844.  doi: 10.1016/j.ijengsci.2006.05.012. [4] C. Cao, E. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Communications in Mathematical Sciences, 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8. [5] T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis: Theory, Methods and Applications, 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111. [6] V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, Comptes Rendus de l Acad$\acute{e}$mie des Sciences - Series I - Mathematics, 321 (1995), 1309-1314.  doi: 10.1016/S0021-7824(97)89978-3. [7] V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, Journal de Math$\acute{e}$matiques Pures et Appliqu$\acute{e}$es, 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3. [8] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society. Colloquium Publications, 2002. [9] D. Cioranescu and E. H. Quazar, Nonlinear partial differential equations and their applications: College de france seminar, Vol. VI, H. Brezis and J.L. Lions (eds.), Research Notes in Mathematics, Pitman, Boston, 109 (1984), 178-197. [10] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705. [11] A. V. Fursikov, Optimal Control of Distributed Systems. Theorey and Applications, Providence, 2000. [12] G. P. Galdi, M. Grobbelaar-Van Dalsen and N. Sauer, Existence and uniqueness of classical solutions of the equations of motion for second-grade fluids, Archive for Rational Mechanics and Analysis, 124 (1993), 221-237.  doi: 10.1007/BF00953067. [13] I. Gyarmati, Non-equilibrium Thermodynamics. Field Theory and Variational Principles, Heidelberg, Springer, 1970. [14] P. E. Kloeden and B. Schmalfuv, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numerical Algorithms, 14 (1997), 141-152.  doi: 10.1023/A:1019156812251. [15] O. A. Ladyzhenskaya, On some gaps in two of my papers on the Navier-Stokes equations and the way of closing them, Zapiski Nauchnykh Seminarov POMI, 271 (2000), 151-155.  doi: 10.1023/A:1023321903383. [16] R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of H${\ddot{o}}$lder functions, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 5 (1999), 157-184. [17] R. de la Llave, Variational methods for quasi-periodic solutions of partial differential equations, Hamiltonian and Celestial Mechanics, 6 (2000), 214-228.  doi: 10.1142/9789812792099_0013. [18] G. Lukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211. [19] A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aueous solutions of polymers, Zapiski Nauchnykh Seminarov Leningradskogo Otdelenia Matematicheskogo Instituta Steklova (LOMI), 38 (1973), 98-136. [20] A. P. Oskolkov, On some quasilinear systems occuring in the study of motion of viscous fluids, Zapiski Nauchnykh Seminarov Leningradskogo Otdelenia Matematicheskogo Instituta Steklova (LOMI), 52 (1975), 128-157. [21] V. A. Pavlovsky, On the theoretical description of weak water solutions of polymers, Doklady Akademii Nauk SSSR, (1971), 809-812. [22] R. S. Rivlin and J. L. Ericksen, Stress-deformation relations for isotropic materials, Journal of Rational Mechanics and Analysis, 4 (1955), 323-425. [23] C. Le Roux, Existence and uniqueness of the flow of second-grade fluids with slip boundary conditions, Archive for Rational Mechanics and Analysis, 148 (1999), 309-356.  doi: 10.1007/s002050050164. [24] G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, Journal of Dynamics and Differential Equations, 8 (1996), 1-33.  doi: 10.1007/BF02218613. [25] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, American Mathematical Society, 2002. doi: 10.1007/978-1-4757-5037-9. [26] A. Sequeira and J. Videman, Global existence of classical solutions for the equations of third grade fluids, J. Math. Phys. Sci., 29 (1995), 47-69. [27] J. Simon, Compact sets in the space Lp(0,T;B), Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360. [28] R. Temam, Navier-Stokes Equations. Theory and Nymerical Analysis, Providence, RI, 2001. doi: 10.1090/chel/343. [29] C. Truesdell, A First Course in Rational Continuum Mechanics, The John Hopkins Unirsity, 1972. [30] M. I. Vishik and V. V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Russian Mathematical Surveys, 66 (2011), 637-731.  doi: 10.1070/RM2011v066n04ABEH004753. [31] D. A. Vorotnikov and V. G. Zvyagin, Trajectory and global attractors of the boundary value problem for autonomous motion equations of viscoelastic medium, Journal of Mathematical Fluid Mechanics, 10 (2008), 19-44.  doi: 10.1007/s00021-005-0215-1. [32] D. A. Vorotnikov, Asymptotic behaviour of the non-autonomous 3D Navier-Stokes problem with coercive force, Journal of Differential Equations, 251 (2011), 2209-2225.  doi: 10.1016/j.jde.2011.07.008. [33] C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid, Journal of Differential Equations, 238 (2007), 394-425.  doi: 10.1016/j.jde.2007.04.001. [34] C. Zhao, Y. Li and S. Zhou, Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid, Journal of Differential Equations, 247 (2009), 2331-2363.  doi: 10.1016/j.jde.2009.07.031. [35] C. Zhao, Pullback asymptotic behavior of solutions for a non-autonomous non-Newtonian fluid on two-dimensional unbounded domains, Journal of Mathematical Physics, 53 (2012), 122702, 22pp. doi: 10.1063/1.4769302. [36] C. Zhao, G. Liu and W. Wang, Smooth pullback attractors for a non-autonomous 2D non-Newtonian fluid and their tempered behaviors, Journal of Mathematical Fluid Mechanics, 16 (2014), 243-262.  doi: 10.1007/s00021-013-0153-2. [37] C. Zhao and L. Yang, Pullback attractors and invariant measures for the non-autonomous globally modified Naviertokes equations, Communications in Mathematical Sciences, 15 (2017), 1565-1580.  doi: 10.4310/CMS.2017.v15.n6.a4. [38] Z. Zhu and C. Zhao, Pullback attractor and invariant measures for the three-dimensional regularized MHD equations, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 38 (2018), 1461-1477.  doi: 10.3934/dcds.2018060. [39] A. V. Zvyagin, Solvability for the equations of motion of weak aqueous polymer solution with objective derivative, Nonlinear Analysis: Theory, Methods and Applications, 90 (2013), 70-85.  doi: 10.1016/j.na.2013.05.022. [40] A. V. Zvyagin, An optimal control problem with feedback for a mathematical model of the motion of weakly concentrated water polymer solutions with objective derivative, Siberian Mathematical Journal, 54 (2013), 640-655. [41] A. V. Zvyagin, Attractors for a model of polymer motion with objective derivative in the rheological relation, Doklady Mathematics, 88 (2013), 730-733. [42] A. V. Zvyagin and V. G. Zvyagin, Pullback attractors for a model of weakly concentrated aqueous polymer solution motion with a rheological relation satisfying the objectivity principle, Doklady Mathematics, 95 (2017), 247-249. [43] V. G. Zvyagin and S. K. Kondratiev, Attractors of equation of non-Newtinian fluid dynamics, Russian Mathematical Surveys, 69 (2014), 845-913.  doi: 10.4213/rm9615. [44] V. G. Zvyagin and S. K. Kondratiev, Pullback attractors for the model of motion of dilute aqueous polymer solutions, Izvestiya: Mathematics, 79 (2015), 645-667.  doi: 10.4213/im8163. [45] V. G. Zvyagin and M. V. Turbin, The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, Journal of Mathematical Sciences, 168 (2010), 157-308.  doi: 10.1007/s10958-010-9981-2. [46] V. G. Zvyagin and D. A. Vorotnikov, Approximating-topological methods in some problems of hydrodinamics, Journal of Fixed Point Theory and Applications, 3 (2008), 23-49.  doi: 10.1007/s11784-008-0056-7. [47] V. G. Zvyagin and D. A. Vorotnikov, Topological Approximayion Methods for Evolutionary Problems of Nonlinear Hydrodinamics, De Gruyter Series In Nonlinear Analysis and Applications, 2008. doi: 10.1515/9783110208283.
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