Lyapunov exponents and the dimension of the attractor for 2D shear-thinning incompressible flow

Discrete and Continuous Dynamical Systems

Volumn 20, Issue 4, 2008, Pages 961-974

  Kaplický, Petr ^a   Pražák, Dalibor ^a  

^a CHARLES UNIVERSITY   (Czech Republic)

Author keywords

Fractal dimension; Global attractor; Lieb Thirring inequality; Lyapunov exponents; Power law fluids; Shear thinning fluids

Indexed keywords

EID: 44849107745     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2008.20.961     Document Type: Article

References (20)

1. M. Bulíček, J. Málek and D. Pražák, On the dimension of the attractor for a class of fluids with pressure dependent viscosities, Commun. Pure Appl. Anal., 4 (2005), 805-822.
2. P. Constantin and C. Foias, "Navier-Stokes Equations," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
3. A. Eden, C. Foias and B. Nicolaenko, Exponential attractors of optimal Lyapunov dimension for Navier-Stokes equations, J. Dynam. Differential Equations, 6 (1994), 301-323.
4. A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations," Masson, Paris, 1994.
5. P. Kaplický, J. Málek and J. Stará, Global-in-time Holder continuity of the velocity gradients for fluids with shear-dependent viscosities, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 175-195.
6. P. Kaplický and D. Pražák, Differentiability of the solution operator and the dimension of the attractor for certain power-law fluids, J. Math. Anal. Appl., 326 (2007), 75-87.
7. O. A. Ladyzhenskaya, New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them, Trudy Mat. Inst. Steklov., 102 (1967), 85-104.
8. O. A. Ladyzhenskaya, Attractors for the modifications of the three-dimensional Navier-Stokes equations, Philos. Trans. Roy. Soc. London Ser. A, 346 (1994), 173-190.
9. E. Lieb and W. Thirring, Inequalities for the moments of the eigenvalues of the schrödinger equations and their relation to sobolev inequalities, in " Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann," Princeton University Press, (1976), 269-303.
10. P. L. Lions, "Mathematical Topics in Fluid Mechanics, Vol. 1: Incompressible Models," The Clarendon Press, Oxford University Press, New York, 1996.
11. J. Málek, J. Nečas, M. Rokyta and M. Růž ička, "Weak and Measure-Valued Solutions to Evolutionary PDEs," Chapman & Hall, London, 1996.
12. J. Málek, K. R. Rajagopal and M. Růžička, Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity, Math. Models Methods Appl. Sci., 5 (1995), 789-812.
13. J. Málek and J. Nečas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids, J. Differential Equations, 127 (1996), 498-518.
14. J. Málek and D. Pražák, Large time behavior via the method of l-trajectories, J. Differential Equations, 181 (2002), 243-279.
15. J. Málek and D. Pražák, On the dimension of the global attractor for the modified Navier-Stokes equations, in "Nonlinear Problems in Mathematical Physics and Related Topics, II," Kluwer/Plenum, New York (2002), 267-283.
16. D. Pražák, Exponential attractor for a planar shear-thinning flow, Math. Meth. Appl. Sci., 30 (2007), 2197-2214.
17. D. Pražák, Exponential attractor for the delayed logistic equation with a nonlinear diffusion, in "Dynamical Systems and Differential Equations (Wilmington, NC, 2002)," Discrete Contin. Dyn. Syst. 2003, suppl., 717-726.
18. D. Pražák, On the dimension of the attractor for the wave equation with nonlinear damping, Commun. Pure Appl. Anal., 4 (2005), 165-174.
19. H.-J. Schmeisser and H. Triebel, "Topics in Fourier Analysis and Function Spaces," Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987.
20. R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics And Physics," second edition, Springer-Verlag, New York, 1997.