October  2016, 21(8): 2687-2702. doi: 10.3934/dcdsb.2016068

Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays

1. 

Department of Mathematics, Busan National University, Busan 609-735, South Korea

2. 

Dong-Eui University, Department of Mathematics, 47340 Pusan, South Korea

Received  October 2015 Revised  April 2016 Published  September 2016

In this paper, we prove the existence of a pullback attractor in higher regularity space for the multi-valued process associated with the $2D$-non-autonomous incompressible non-Newtonian fluid with delays and without the uniqueness of solutions.
Citation: Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068
References:
[1]

R. A. Adams, Sobolev Spaces,, Acasemic Press, (1975).

[2]

H. Bellout, F. Bloom and J. Nečas, Weak and measure-valued solutions for non-Newtonian fluids,, C. R. Acad. Sci. Paris, 317 (1993), 795.

[3]

H. Bellout, F. Bloom and J. Nečas, Young measure-valued solutions for non-Newtonian incompressible viscous fluids,, Comm. Partial Differential Equations, 19 (1994), 1763. doi: 10.1080/03605309408821073.

[4]

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

[5]

M. Boukrouche, G. Lukaszewicz and J. Real, On pullback attractors for a class of two-dimensional turbulent shear flows,, Internat. J. Engrg. Sci., 44 (2006), 830. doi: 10.1016/j.ijengsci.2006.05.012.

[6]

B. T. Caraballo and J. A. Langa, Attractors for differential equations with variable delay,, J. Math. Anal. Appl., 260 (2001), 421. doi: 10.1006/jmaa.2000.7464.

[7]

B. T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484. doi: 10.1016/j.na.2005.03.111.

[8]

T. Caraballo and J. Real, Attractors for $2D$-Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271. doi: 10.1016/j.jde.2004.04.012.

[9]

P. E. Kloedean and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization,, Numer. Algorithms, 14 (1997), 141. doi: 10.1023/A:1019156812251.

[10]

J. A. Langa, G. Lukaszewicz and J. Real, Finite fractal dimension of pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains,, Nonlinear Anal., 66 (2007), 735. doi: 10.1016/j.na.2005.12.017.

[11]

X. Liu and Y. Wang, Pullback attractors for nonautonomous $2D$-Navier Stokes models with variable delays,, Abstr. Appl. Anal., 2003 (2003).

[12]

P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators,, Discrete Contin. Dyn. Syst., 26 (2010), 989.

[13]

M. Pokorny, Cauchy problem for the non-Newtonian viscous incompressible fluids,, Appl. Math., 41 (1996), 169.

[14]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1988). doi: 10.1007/978-1-4684-0313-8.

[15]

Y. J. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain,, Discrete Contin. Dyn. Syst., 34 (2014), 4343. doi: 10.3934/dcds.2014.34.4343.

[16]

Y. Wang and S. Zhou, Kernel sections on multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains,, Quart. Appl. Math., 67 (2009), 343. doi: 10.1090/S0033-569X-09-01150-0.

[17]

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

[18]

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

show all references

References:
[1]

R. A. Adams, Sobolev Spaces,, Acasemic Press, (1975).

[2]

H. Bellout, F. Bloom and J. Nečas, Weak and measure-valued solutions for non-Newtonian fluids,, C. R. Acad. Sci. Paris, 317 (1993), 795.

[3]

H. Bellout, F. Bloom and J. Nečas, Young measure-valued solutions for non-Newtonian incompressible viscous fluids,, Comm. Partial Differential Equations, 19 (1994), 1763. doi: 10.1080/03605309408821073.

[4]

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

[5]

M. Boukrouche, G. Lukaszewicz and J. Real, On pullback attractors for a class of two-dimensional turbulent shear flows,, Internat. J. Engrg. Sci., 44 (2006), 830. doi: 10.1016/j.ijengsci.2006.05.012.

[6]

B. T. Caraballo and J. A. Langa, Attractors for differential equations with variable delay,, J. Math. Anal. Appl., 260 (2001), 421. doi: 10.1006/jmaa.2000.7464.

[7]

B. T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484. doi: 10.1016/j.na.2005.03.111.

[8]

T. Caraballo and J. Real, Attractors for $2D$-Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271. doi: 10.1016/j.jde.2004.04.012.

[9]

P. E. Kloedean and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization,, Numer. Algorithms, 14 (1997), 141. doi: 10.1023/A:1019156812251.

[10]

J. A. Langa, G. Lukaszewicz and J. Real, Finite fractal dimension of pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains,, Nonlinear Anal., 66 (2007), 735. doi: 10.1016/j.na.2005.12.017.

[11]

X. Liu and Y. Wang, Pullback attractors for nonautonomous $2D$-Navier Stokes models with variable delays,, Abstr. Appl. Anal., 2003 (2003).

[12]

P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators,, Discrete Contin. Dyn. Syst., 26 (2010), 989.

[13]

M. Pokorny, Cauchy problem for the non-Newtonian viscous incompressible fluids,, Appl. Math., 41 (1996), 169.

[14]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1988). doi: 10.1007/978-1-4684-0313-8.

[15]

Y. J. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain,, Discrete Contin. Dyn. Syst., 34 (2014), 4343. doi: 10.3934/dcds.2014.34.4343.

[16]

Y. Wang and S. Zhou, Kernel sections on multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains,, Quart. Appl. Math., 67 (2009), 343. doi: 10.1090/S0033-569X-09-01150-0.

[17]

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

[18]

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

[1]

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

[2]

M. Bulíček, F. Ettwein, P. Kaplický, Dalibor Pražák. The dimension of the attractor for the 3D flow of a non-Newtonian fluid. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1503-1520. doi: 10.3934/cpaa.2009.8.1503

[3]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809

[4]

Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212

[5]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195

[6]

T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037

[7]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281

[8]

Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of non-autonomous thermoelastic plate systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3553-3571. doi: 10.3934/dcdsb.2017214

[9]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194

[10]

Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138

[11]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703

[12]

Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543

[13]

Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-14. doi: 10.3934/dcdsb.2019036

[14]

Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483

[15]

Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417

[16]

Li Fang, Zhenhua Guo. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure & Applied Analysis, 2017, 16 (1) : 209-242. doi: 10.3934/cpaa.2017010

[17]

Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743

[18]

Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229

[19]

Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639

[20]

Zhenhua Guo, Wenchao Dong, Jinjing Liu. Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2133-2161. doi: 10.3934/cpaa.2019096

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]