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May  2018, 23(3): 1155-1176. doi: 10.3934/dcdsb.2018146

Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows

Oleksiy V. Kapustyan 1, , Pavlo O. Kasyanov 2, , José Valero 3,, and  Michael Z. Zgurovsky 2,

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

* Corresponding author

Received  March 2017 Revised  June 2017 Published  May 2018 Early access  February 2018

Fund Project: The first two authors were partially supported by the State Fund for Fundamental Research of Ukraine under grant GP/F66/14921 and by the Grant of the National Academy of Sciences of Ukraine 2290/2018. The third author was partially supported by Spanish Ministry of Economy and Competitiveness and FEDER, projects MTM2015-63723-P and MTM2016-74921-P, and by Junta de Andalucía (Spain), project P12-FQM-1492.

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.

Keywords: Non-Newtonian fluids, parabolic equations, global attractors, infinite-dimensional dynamical systems.
Mathematics Subject Classification: 35B40, 35B41, 35K55, 37B25.
Citation: Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero, Michael Z. Zgurovsky. Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1155-1176. doi: 10.3934/dcdsb.2018146
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]

K. Yosida, Functional Analysis, Springer, Berlin, 1980.

[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]

K. Yosida, Functional Analysis, Springer, Berlin, 1980.

[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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