August  2018, 23(6): 2193-2216. doi: 10.3934/dcdsb.2018231

Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Guowei Liu

Received  November 2016 Revised  April 2018 Published  July 2018

Fund Project: The first author is partially supported by the National NSFC grant No.11771284 and Weng Hongwu Academic Innovation Research Fund of Peking University and Original Research Fund of Peking University

This paper studies the pullback asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid on 2D bounded domains. We show existence of the pullback exponential attractor introduced by Langa, Miranville and Real [27], moreover, give existence of the global pullback attractor with finite fractal dimension and reveal the relationship between the global pullback attractor and the pullback exponential attractor. These results improve our previous associated results in papers [29,40] for the non-Newtonian fluid.

Citation: Guowei Liu, Rui Xue. Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2193-2216. doi: 10.3934/dcdsb.2018231
References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. Google Scholar

[3]

H. Bellout and F. Bloom, J. Nečas, Existence, uniqueness and stability of solutions to the initial boundary value problem for bipolar viscous fluids, Differential Integral Equations, 8 (1995), 453-464. Google Scholar

[4]

F. Bloom and W. Hao, Regularization of a non-Newtonian system in unbounded channel: Existence and uniqueness of solutions, Nonlinear Anal., 44 (2001), 281-309. doi: 10.1016/S0362-546X(99)00264-3. Google Scholar

[5]

F. Bloom and W. Hao, Regularization of a non-Newtonian system in an unbounded channel: Existence of a maximal compact attractor, Nonlinear Anal., 43 (2001), 743-766. doi: 10.1016/S0362-546X(99)00232-1. Google Scholar

[6]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0. Google Scholar

[7]

H. Bellout and F. Bloom, Incompressible Bipolar and Non-Newtonian Viscous Fluid Flow, Springer, Cham, 2014. doi: 10.1007/978-3-319-00891-2. Google Scholar

[8]

S. Bosia and S. Gatti, Pullback exponential attractor for a Cahn-Hillard-Navier-Stokes system in 2D, Dyn. Partial Differ Equ., 11 (2014), 1-38. doi: 10.4310/DPDE.2014.v11.n1.a1. Google Scholar

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amercian Mathematical Society, Providence, RI, 2002. Google Scholar

[10]

R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations Part Ⅰ: Semilinear parabolic problems, J. Math. Anal. Appl., 381 (2011), 748-765. doi: 10.1016/j.jmaa.2011.03.053. Google Scholar

[11]

A. N. Carvalho and S. Sonner, Pullback exponential attractor for evolution process in Banach spaces: theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071. doi: 10.3934/cpaa.2013.12.3047. Google Scholar

[12]

A. N. Carvalho and S. Sonner, Pullback exponential attractor for evolution process in Banach spaces: Properties and applications, Commun. Pure Appl. Anal., 12 (2014), 1141-1165. doi: 10.3934/cpaa.2014.13.1141. Google Scholar

[13]

B. Dong and Y. Li, Large time behavior to the system of incompressible non-Newtonian fluds in $\mathbb{R}^2$, J. Math. Anal. Appl., 298 (2004), 667-676. doi: 10.1016/j.jmaa.2004.05.032. Google Scholar

[14]

B. Dong and Z. Chen, Time decay rates of non-Newtonian flows in $\mathbb{R}^n_+$, J. Math. Anal. Appl., 324 (2006), 820-833. doi: 10.1016/j.jmaa.2005.12.070. Google Scholar

[15]

A. Eden, C. Foias, B. Nicolaenko and R. Teman, Exponential Attractors for Dissipative Evilution Equations, John Wiley-Sons, Ltd, Chichester, 1994. Google Scholar

[16]

M. EfendievA. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy Soc. Edinburgh Sect. A., 135 (2005), 703-730. doi: 10.1017/S030821050000408X. Google Scholar

[17]

M. Efendiev, Attractors for Degenerate Parabolic type equations, American Mathematical Society, Providence, RI, Madrid, 2013. doi: 10.1090/surv/192. Google Scholar

[18]

P. Fabrie and A. Miranville, Exponential attractors for nonautonomous first-order evolution equation, Discrete Contin. Dyn. Syst., 4 (1998), 225-240. doi: 10.3934/dcds.1998.4.225. Google Scholar

[19]

Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S. Google Scholar

[20]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors in Ⅴ for non-autonomous 2D-Navier-Stokes equations and their tempered behavior, J. Differential Equations, 252 (2012), 4333-4356. doi: 10.1016/j.jde.2012.01.010. Google Scholar

[21]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for the non-autonomous 2D Navier-Stokes equations for minimally regular forcing, Discrete Contin. Dyn. Syst., 34 (2014), 203-227. doi: 10.3934/dcds.2014.34.203. Google Scholar

[22]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. Google Scholar

[23]

P. E. KlodenJ. A. Langa and J. Real, Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937. Google Scholar

[24]

J. L. Lion, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969. Google Scholar

[25]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd, Gordon and Breach, New York, 1969. Google Scholar

[26]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolutions, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418. Google Scholar

[27]

J. A. LangaA. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357. doi: 10.3934/dcds.2010.26.1329. Google Scholar

[28]

G. LiuC. Zhao and J. Cao, $H^4$-boundedness of pullback attractor for a 2D non-Newtonian fluid flow, Front. Math. China, 8 (2013), 1377-1390. doi: 10.1007/s11464-013-0250-9. Google Scholar

[29]

G. Liu, Pullback asymptotic behavior of solutions for a 2D non-autonomous non-Newtonian fluid, J. Math. Fluid Mech., 19 (2017), 623-643. doi: 10.1007/s00021-016-0299-9. Google Scholar

[30]

J. Málek, J. Nečas, M. Rokyta and M. R${\rm{\dot u}}$žička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman-Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1. Google Scholar

[31]

A. Miranville, Exponential attractors for nonautonomous evolution equations, Appl. Math. Lett., 11 (1998), 19-22. doi: 10.1016/S0893-9659(98)00004-4. Google Scholar

[32]

M. Pokorný, Cauchy problem for the non-Newtonian viscous incompressible fluid, Appl. Math., 41 (1996), 169-201. Google Scholar

[33]

J. C. Robinson, Infinite-dimensional Dynamical System, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0. Google Scholar

[34]

G. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. Google Scholar

[35]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar

[36]

C. Zhao and Y. Li, $H^2$-compact attractor for a non-Newtonian system in two-dimensional unbound domains, Nonlinear Anal., 56 (2004), 1091-1103. doi: 10.1016/j.na.2003.11.006. Google Scholar

[37]

C. Zhao and S. Zhou, Pullback attractors for nonautonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425. doi: 10.1016/j.jde.2007.04.001. Google Scholar

[38]

C. ZhaoY. Li and S. Zhou, Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid, J. Differential Equations, 247 (2009), 2331-2363. doi: 10.1016/j.jde.2009.07.031. Google Scholar

[39]

C. Zhao, Pullback asymptotic behavior of solutions for a non-autonomous non-Newtonian fluid on two-dimensional unbounded domains, J. Math. Phys., 53 (2012), 122702, 22pp. doi: 10.1063/1.4769302. Google Scholar

[40]

C. ZhaoG. Liu and W. Wang, Smooth pullback attractors for a non-autonomous 2D non-Newtonian fluid and their tempered behaviors, J. Math. Fluid Mech., 16 (2014), 243-262. doi: 10.1007/s00021-013-0153-2. Google Scholar

[41]

C. ZhaoG. Liu and R. An, Global well-posedness and pullback attractors for an incompressible non-Newtonian fluid with infinite delays, Differ. Equ. Dyn. Syst., 25 (2017), 39-64. doi: 10.1007/s12591-014-0231-9. Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. Google Scholar

[3]

H. Bellout and F. Bloom, J. Nečas, Existence, uniqueness and stability of solutions to the initial boundary value problem for bipolar viscous fluids, Differential Integral Equations, 8 (1995), 453-464. Google Scholar

[4]

F. Bloom and W. Hao, Regularization of a non-Newtonian system in unbounded channel: Existence and uniqueness of solutions, Nonlinear Anal., 44 (2001), 281-309. doi: 10.1016/S0362-546X(99)00264-3. Google Scholar

[5]

F. Bloom and W. Hao, Regularization of a non-Newtonian system in an unbounded channel: Existence of a maximal compact attractor, Nonlinear Anal., 43 (2001), 743-766. doi: 10.1016/S0362-546X(99)00232-1. Google Scholar

[6]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0. Google Scholar

[7]

H. Bellout and F. Bloom, Incompressible Bipolar and Non-Newtonian Viscous Fluid Flow, Springer, Cham, 2014. doi: 10.1007/978-3-319-00891-2. Google Scholar

[8]

S. Bosia and S. Gatti, Pullback exponential attractor for a Cahn-Hillard-Navier-Stokes system in 2D, Dyn. Partial Differ Equ., 11 (2014), 1-38. doi: 10.4310/DPDE.2014.v11.n1.a1. Google Scholar

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amercian Mathematical Society, Providence, RI, 2002. Google Scholar

[10]

R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations Part Ⅰ: Semilinear parabolic problems, J. Math. Anal. Appl., 381 (2011), 748-765. doi: 10.1016/j.jmaa.2011.03.053. Google Scholar

[11]

A. N. Carvalho and S. Sonner, Pullback exponential attractor for evolution process in Banach spaces: theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071. doi: 10.3934/cpaa.2013.12.3047. Google Scholar

[12]

A. N. Carvalho and S. Sonner, Pullback exponential attractor for evolution process in Banach spaces: Properties and applications, Commun. Pure Appl. Anal., 12 (2014), 1141-1165. doi: 10.3934/cpaa.2014.13.1141. Google Scholar

[13]

B. Dong and Y. Li, Large time behavior to the system of incompressible non-Newtonian fluds in $\mathbb{R}^2$, J. Math. Anal. Appl., 298 (2004), 667-676. doi: 10.1016/j.jmaa.2004.05.032. Google Scholar

[14]

B. Dong and Z. Chen, Time decay rates of non-Newtonian flows in $\mathbb{R}^n_+$, J. Math. Anal. Appl., 324 (2006), 820-833. doi: 10.1016/j.jmaa.2005.12.070. Google Scholar

[15]

A. Eden, C. Foias, B. Nicolaenko and R. Teman, Exponential Attractors for Dissipative Evilution Equations, John Wiley-Sons, Ltd, Chichester, 1994. Google Scholar

[16]

M. EfendievA. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy Soc. Edinburgh Sect. A., 135 (2005), 703-730. doi: 10.1017/S030821050000408X. Google Scholar

[17]

M. Efendiev, Attractors for Degenerate Parabolic type equations, American Mathematical Society, Providence, RI, Madrid, 2013. doi: 10.1090/surv/192. Google Scholar

[18]

P. Fabrie and A. Miranville, Exponential attractors for nonautonomous first-order evolution equation, Discrete Contin. Dyn. Syst., 4 (1998), 225-240. doi: 10.3934/dcds.1998.4.225. Google Scholar

[19]

Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S. Google Scholar

[20]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors in Ⅴ for non-autonomous 2D-Navier-Stokes equations and their tempered behavior, J. Differential Equations, 252 (2012), 4333-4356. doi: 10.1016/j.jde.2012.01.010. Google Scholar

[21]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for the non-autonomous 2D Navier-Stokes equations for minimally regular forcing, Discrete Contin. Dyn. Syst., 34 (2014), 203-227. doi: 10.3934/dcds.2014.34.203. Google Scholar

[22]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. Google Scholar

[23]

P. E. KlodenJ. A. Langa and J. Real, Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937. Google Scholar

[24]

J. L. Lion, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969. Google Scholar

[25]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd, Gordon and Breach, New York, 1969. Google Scholar

[26]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolutions, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418. Google Scholar

[27]

J. A. LangaA. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357. doi: 10.3934/dcds.2010.26.1329. Google Scholar

[28]

G. LiuC. Zhao and J. Cao, $H^4$-boundedness of pullback attractor for a 2D non-Newtonian fluid flow, Front. Math. China, 8 (2013), 1377-1390. doi: 10.1007/s11464-013-0250-9. Google Scholar

[29]

G. Liu, Pullback asymptotic behavior of solutions for a 2D non-autonomous non-Newtonian fluid, J. Math. Fluid Mech., 19 (2017), 623-643. doi: 10.1007/s00021-016-0299-9. Google Scholar

[30]

J. Málek, J. Nečas, M. Rokyta and M. R${\rm{\dot u}}$žička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman-Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1. Google Scholar

[31]

A. Miranville, Exponential attractors for nonautonomous evolution equations, Appl. Math. Lett., 11 (1998), 19-22. doi: 10.1016/S0893-9659(98)00004-4. Google Scholar

[32]

M. Pokorný, Cauchy problem for the non-Newtonian viscous incompressible fluid, Appl. Math., 41 (1996), 169-201. Google Scholar

[33]

J. C. Robinson, Infinite-dimensional Dynamical System, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0. Google Scholar

[34]

G. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. Google Scholar

[35]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar

[36]

C. Zhao and Y. Li, $H^2$-compact attractor for a non-Newtonian system in two-dimensional unbound domains, Nonlinear Anal., 56 (2004), 1091-1103. doi: 10.1016/j.na.2003.11.006. Google Scholar

[37]

C. Zhao and S. Zhou, Pullback attractors for nonautonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425. doi: 10.1016/j.jde.2007.04.001. Google Scholar

[38]

C. ZhaoY. Li and S. Zhou, Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid, J. Differential Equations, 247 (2009), 2331-2363. doi: 10.1016/j.jde.2009.07.031. Google Scholar

[39]

C. Zhao, Pullback asymptotic behavior of solutions for a non-autonomous non-Newtonian fluid on two-dimensional unbounded domains, J. Math. Phys., 53 (2012), 122702, 22pp. doi: 10.1063/1.4769302. Google Scholar

[40]

C. ZhaoG. Liu and W. Wang, Smooth pullback attractors for a non-autonomous 2D non-Newtonian fluid and their tempered behaviors, J. Math. Fluid Mech., 16 (2014), 243-262. doi: 10.1007/s00021-013-0153-2. Google Scholar

[41]

C. ZhaoG. Liu and R. An, Global well-posedness and pullback attractors for an incompressible non-Newtonian fluid with infinite delays, Differ. Equ. Dyn. Syst., 25 (2017), 39-64. doi: 10.1007/s12591-014-0231-9. Google Scholar

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