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Numerical approximations for a smectic-A liquid crystal flow model: First-order, linear, decoupled and energy stable schemes
Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China |
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 [
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. |
[3] |
H. Bellout and F. Bloom,
J. Nečas, Existence, uniqueness and stability of solutions to the initial boundary value problem for bipolar viscous fluids, Differential Integral Equations, 8 (1995), 453-464.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amercian Mathematical Society, Providence, RI, 2002. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
A. Eden, C. Foias, B. Nicolaenko and R. Teman, Exponential Attractors for Dissipative Evilution Equations, John Wiley-Sons, Ltd, Chichester, 1994. |
[16] |
M. Efendiev, A. 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. |
[17] |
M. Efendiev, Attractors for Degenerate Parabolic type equations, American Mathematical Society, Providence, RI, Madrid, 2013.
doi: 10.1090/surv/192. |
[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. |
[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. |
[20] |
J. García-Luengo, P. 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. |
[21] |
J. García-Luengo, P. 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. |
[22] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. |
[23] |
P. E. Kloden, J. 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. |
[24] |
J. L. Lion, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969. |
[25] |
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd, Gordon and Breach, New York, 1969. |
[26] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolutions, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
[27] |
J. A. Langa, A. Miranville and J. Real,
Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357.
doi: 10.3934/dcds.2010.26.1329. |
[28] |
G. Liu, C. 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. |
[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. |
[30] |
J. Málek, J. Nečas, M. Rokyta and M. R${\rm{\dot u}}$žička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman-Hall, London, 1996.
doi: 10.1007/978-1-4899-6824-1. |
[31] |
A. Miranville,
Exponential attractors for nonautonomous evolution equations, Appl. Math. Lett., 11 (1998), 19-22.
doi: 10.1016/S0893-9659(98)00004-4. |
[32] |
M. Pokorný,
Cauchy problem for the non-Newtonian viscous incompressible fluid, Appl. Math., 41 (1996), 169-201.
|
[33] |
J. C. Robinson, Infinite-dimensional Dynamical System, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
[34] |
G. Sell and Y. You,
Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[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. |
[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. |
[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. |
[38] |
C. Zhao, Y. 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. |
[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. |
[40] |
C. Zhao, G. 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. |
[41] |
C. Zhao, G. 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. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. |
[3] |
H. Bellout and F. Bloom,
J. Nečas, Existence, uniqueness and stability of solutions to the initial boundary value problem for bipolar viscous fluids, Differential Integral Equations, 8 (1995), 453-464.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amercian Mathematical Society, Providence, RI, 2002. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
A. Eden, C. Foias, B. Nicolaenko and R. Teman, Exponential Attractors for Dissipative Evilution Equations, John Wiley-Sons, Ltd, Chichester, 1994. |
[16] |
M. Efendiev, A. 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. |
[17] |
M. Efendiev, Attractors for Degenerate Parabolic type equations, American Mathematical Society, Providence, RI, Madrid, 2013.
doi: 10.1090/surv/192. |
[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. |
[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. |
[20] |
J. García-Luengo, P. 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. |
[21] |
J. García-Luengo, P. 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. |
[22] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. |
[23] |
P. E. Kloden, J. 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. |
[24] |
J. L. Lion, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969. |
[25] |
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd, Gordon and Breach, New York, 1969. |
[26] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolutions, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
[27] |
J. A. Langa, A. Miranville and J. Real,
Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357.
doi: 10.3934/dcds.2010.26.1329. |
[28] |
G. Liu, C. 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. |
[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. |
[30] |
J. Málek, J. Nečas, M. Rokyta and M. R${\rm{\dot u}}$žička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman-Hall, London, 1996.
doi: 10.1007/978-1-4899-6824-1. |
[31] |
A. Miranville,
Exponential attractors for nonautonomous evolution equations, Appl. Math. Lett., 11 (1998), 19-22.
doi: 10.1016/S0893-9659(98)00004-4. |
[32] |
M. Pokorný,
Cauchy problem for the non-Newtonian viscous incompressible fluid, Appl. Math., 41 (1996), 169-201.
|
[33] |
J. C. Robinson, Infinite-dimensional Dynamical System, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
[34] |
G. Sell and Y. You,
Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[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. |
[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. |
[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. |
[38] |
C. Zhao, Y. 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. |
[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. |
[40] |
C. Zhao, G. 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. |
[41] |
C. Zhao, G. 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. |
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