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Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms
Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows
1. | Taras Shevchenko National University of Kyiv, Kyiv, Ukraine |
2. | National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine |
3. | Universidad Miguel Hernandez de Elche, Centro de Investigación Operativa, Avda. Universidad s/n, 03202-Elche (Alicante), Spain |
In this paper we prove the existence of global attractors in the strong topology of the phase space for semiflows generated by vanishing viscosity approximations of some class of complex fluids. We also show that the attractors tend to the set of all complete bounded trajectories of the original problem when the parameter of the approximations goes to zero.
References:
[1] |
J. M. Amigó, I. Catto, A. Giménez and J. Valero,
Attractors for a non-linear parabolic equation modelling suspension flows, Discrete Contin. Dyn. Sist., Series B, 11 (2009), 205-231.
|
[2] |
J. M. Amigó, A. Giménez, F. Morillas and J. Valero,
Attractors for a lattice dynamical system generated by non-Newtonian fluids modeling suspensions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2681-2700.
doi: 10.1142/S0218127410027295. |
[3] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989. |
[4] |
E. Cancès, I. Catto and Yo. Gati,
Mathematical analysis of a nonlinear parabolic equation arising in the modelling of non-Newtonian flows, SIAM J. Math. Anal., 37 (2005), 60-82.
doi: 10.1137/S0036141003430044. |
[5] |
E. Cancès and C. Le Bris,
Convergence to equilibrium of a multiscale model for suspensions, Discrete Contin. Dyn. Sist., Series B, 6 (2006), 449-470.
doi: 10.3934/dcdsb.2006.6.449. |
[6] |
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. |
[7] |
E. A. Feinberg, P. O. Kasyanov and M. Z. Zgurovsky,
Uniform Fatou's lemma, J. Math. Anal. Appl., 444 (2016), 550-567.
doi: 10.1016/j.jmaa.2016.06.044. |
[8] |
H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1975.
doi: 10.1002/mana.19750672207. |
[9] |
A. Giménez, F. Morillas, J. Valero and J. M. Amigó, Stability and numerical analysis of the Hebraud-Lequeux model for suspensions, Discrete Dyn. Nat. Soc. , 2011 (2011), Art. ID 415921, 24 pp. |
[10] |
N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov,
Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Carathéodory's nonlinearity, Nonlinear Anal., 98 (2014), 13-26.
doi: 10.1016/j.na.2013.12.004. |
[11] |
P. Hébraud and F. Lequeux,
Mode-coupling theory for the pasty rheology of soft glassy materials, Phys. Rev. Lett., 81 (1998), 2934-2937.
|
[12] |
O. V. Kapustyan, J. Valero, P. O. Kasyanov, A. Giménez and J. M. Amigó, Convergence of numerical approximations for a non-Newtonian model of suspensions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1540022, 24 pp. |
[13] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. |
[14] |
R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1979. |
[15] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. |
[16] |
J. Valero, A. Giménez, O. V. Kapustyan, P. Kasyanov and J. M. Amigó,
Convergence of equilibria for numerical approximations of a suspension model, Comput. Math. Appl., 72 (2016), 856-878.
doi: 10.1016/j.camwa.2016.05.034. |
[17] | |
[18] |
M. Z. Zgurovsky, P. O. Kasyanov and J. Valero,
Noncoercive evolution inclusions for Sk type operators, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2823-2834.
doi: 10.1142/S0218127410027386. |
[19] |
M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and N. V. Zadoianchuk, Evolution Inclusions and variation inequalities for earth data processing Ⅲ, Springer-Verlag, Berlin, 2012. |
[20] |
M. Z. Zgurovsky and P. O. Kasyanov, Multivalued dynamics of solutions for autonomous operator differential equations in strongest topologies, in Continuous and Distributed Systems: Theory and Applications, Solid Mechanics and Its Applications, Springer, Cham, 211 (2014), 149-162. |
show all references
References:
[1] |
J. M. Amigó, I. Catto, A. Giménez and J. Valero,
Attractors for a non-linear parabolic equation modelling suspension flows, Discrete Contin. Dyn. Sist., Series B, 11 (2009), 205-231.
|
[2] |
J. M. Amigó, A. Giménez, F. Morillas and J. Valero,
Attractors for a lattice dynamical system generated by non-Newtonian fluids modeling suspensions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2681-2700.
doi: 10.1142/S0218127410027295. |
[3] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989. |
[4] |
E. Cancès, I. Catto and Yo. Gati,
Mathematical analysis of a nonlinear parabolic equation arising in the modelling of non-Newtonian flows, SIAM J. Math. Anal., 37 (2005), 60-82.
doi: 10.1137/S0036141003430044. |
[5] |
E. Cancès and C. Le Bris,
Convergence to equilibrium of a multiscale model for suspensions, Discrete Contin. Dyn. Sist., Series B, 6 (2006), 449-470.
doi: 10.3934/dcdsb.2006.6.449. |
[6] |
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. |
[7] |
E. A. Feinberg, P. O. Kasyanov and M. Z. Zgurovsky,
Uniform Fatou's lemma, J. Math. Anal. Appl., 444 (2016), 550-567.
doi: 10.1016/j.jmaa.2016.06.044. |
[8] |
H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1975.
doi: 10.1002/mana.19750672207. |
[9] |
A. Giménez, F. Morillas, J. Valero and J. M. Amigó, Stability and numerical analysis of the Hebraud-Lequeux model for suspensions, Discrete Dyn. Nat. Soc. , 2011 (2011), Art. ID 415921, 24 pp. |
[10] |
N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov,
Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Carathéodory's nonlinearity, Nonlinear Anal., 98 (2014), 13-26.
doi: 10.1016/j.na.2013.12.004. |
[11] |
P. Hébraud and F. Lequeux,
Mode-coupling theory for the pasty rheology of soft glassy materials, Phys. Rev. Lett., 81 (1998), 2934-2937.
|
[12] |
O. V. Kapustyan, J. Valero, P. O. Kasyanov, A. Giménez and J. M. Amigó, Convergence of numerical approximations for a non-Newtonian model of suspensions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1540022, 24 pp. |
[13] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. |
[14] |
R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1979. |
[15] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. |
[16] |
J. Valero, A. Giménez, O. V. Kapustyan, P. Kasyanov and J. M. Amigó,
Convergence of equilibria for numerical approximations of a suspension model, Comput. Math. Appl., 72 (2016), 856-878.
doi: 10.1016/j.camwa.2016.05.034. |
[17] | |
[18] |
M. Z. Zgurovsky, P. O. Kasyanov and J. Valero,
Noncoercive evolution inclusions for Sk type operators, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2823-2834.
doi: 10.1142/S0218127410027386. |
[19] |
M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and N. V. Zadoianchuk, Evolution Inclusions and variation inequalities for earth data processing Ⅲ, Springer-Verlag, Berlin, 2012. |
[20] |
M. Z. Zgurovsky and P. O. Kasyanov, Multivalued dynamics of solutions for autonomous operator differential equations in strongest topologies, in Continuous and Distributed Systems: Theory and Applications, Solid Mechanics and Its Applications, Springer, Cham, 211 (2014), 149-162. |
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