Invariant sets of solutions of Navier-Stokes and related evolution equations - A survey

Biler, P.

Annales scientifiques de l'Université de Clermont-Ferrand 2. Série Probabilités et applications, Tome 88 (1986) no. 5, pp. 25-47.

1 A.V. Babin, M.I. Vishik, Attractor of the Navier-Stokes system and the parabolic equations and an estimate of their dimension, Zap. Nauch. Sem. LOMI, 115, 1982, 3-15. | MR | Zbl

2 -,,-, Attractors of the evolution partial differential equations and estimates of their dimension, Usp. Mat. Nauk, 38,4, 1983, 133-187. | MR | Zbl

3 -,,-, Regular attractors of semigroups and evolution equations, J. Math. pures appl. 62, 1983, 441-491. | MR | Zbl

4 -,,-, Estimates from above and below for the dimension of attractors of evolution equations, Sib.Mat. Zh. 24,5, 1983, 15-30.

5 -,,-, Attracteurs maximaux dans les equations aux dérivées partielles, Collège de France, Séminaire 1984, vol. VI, Pitman, Res.Notes in Math. and (in this volume ) A. Haraux, Two remarks on dissipative hyperbolic problems.

6 -,,-, Maximal attractors of semigroups corresponding to evolution differential equations, Mat. Sb. 126,3, 1985, 397-419. | MR | Zbl

7 Z.B. Berkaliev, Attractors of evolution equations of nonlinear viscoelasticity, Vestnik MGU, 5, 1985, 69-71. | Zbl

8 P. Biler, Large time behaviour of periodic solutions to dissipative equations of Korteweg-de Vries-Burgers type, Bull.Pol.Acad.Sc. Math. 32, 1984, 401-405. | MR | Zbl

9 A.N. Chetaev, Neuron networks and Markov chains, Nauka, Moscow, 1985. | MR

10 P. Constantin, C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D-Navier-Stokes equations, Comm. Pure Appl.Math. 38, 1985, 1-27. | MR | Zbl

11 P. Constantin, C. Foias, R. Temam, Attractors representing turbulent flows, Memoirs of AMS, 314, 1985. | MR | Zbl

12 A. Douady, J. Oesterlé, Dimension de Hausdorff des attracteurs, C.R.Acad.Sc. Paris, 290 A-B, 1980, 1135-1138. | MR | Zbl

13 H. Federer, Geometric Measure Theory, Springer 1969. | MR | Zbl

14 C. Foias, O.P. Manley, R. Temam, Y.M. Trève, Asymptotic analysis of the Navier-Stokes equations, Physica 9D, 1983, 157-188. | MR | Zbl

15 C. Foias, B. Nicolaenko, G.R. Sell, R. Temam, variétés inertielles pour l'équation de Kuramoto-Sivashinski, C.R.Acad.Sc. Paris, 301, 1985, 285-288. | MR | Zbl

16 C. Foias, G.R. Sell, R. Temam, Variétés inertielles des equations différentielles dissipatives, C.R. Acad.Sc. Paris, 301, 1985, 139-141. | MR | Zbl

17 C. Foias, R. Temam, Some analytic and geometric properties of the solutions of the Navier-Stokes equations, J.Math. pures appl. 58, 1979, 339-368. | MR | Zbl

18 C. Foias, R. Temam, A specifically nonlinear property of the operator semigroup of the Navier-Stokes equations, Comm. Pure Appl. Math. 35, 1982, 197-207. | MR | Zbl

19 C. Foias, R. Temam, On the Hausdorff dimension of an attractor for the two dimensional Navier-Stokes equations, Physics Letters 93A, 1983, 431-434. | MR

20 A. Fora, A fixed point theorem for product spaces, Pacific. J. Math. 99, 1982, 327-335. | MR | Zbl

21 P. Frederickson, J.L. Kaplan, E.D. Yorke, A. Yorke, The Liapunov dimension of strange attractors, J.Diff.Eq. 49, 1983, 185-207. | MR | Zbl

22 J.-M. Ghidaglia, On the fractal dimension of attractors for viscous incompressible fluid flows, SIAM J.Math. Anal. /to appear/. | MR | Zbl

23 J.-M. Ghidaglia, R. Temam, Propriétés des attracteurs associés à des équations hyperboliques non linéaires amorties, C.R.Acad.Sc. Paris, 300, 1985, 185-188. | MR | Zbl

24 C. Guillopé, Comportement à l'infini des solutions des equations de Navier-Stokes et propriété des ensembles fonctionnels invariants ( ou attracteurs ), Ann. Inst. Fourier, 32, 1982, 1-37. | EuDML | Numdam | MR | Zbl

25 J.K. Hale, Theory of Functional Differential Equations, Springer, 1977. | MR | Zbl

26 J.K. Hale, L.T. Magalhães, W.M. Oliva, An Introduction to Infinite Dimensional Dynamical Systems - Geometric Theory, Springer, 1984. | MR | Zbl

27 W. Hurewicz, H. Wallman, Dimension Theory, Princeton 1948. | JFM | Zbl

28 Yu. S. Il'Yashenko, Weakly contracting systems and attractors of Galerkin approximations of Navier-Stokes equations, Usp. Mat. Nauk, 36, 3, 1981, 243-244. | Zbl

29 -,,-, Weakly contracting systems and their attractors, Usp. Mat. Nauk, 37, 1, 1982, 166.

30 -,,-,On the dimension of attractors of k-contracting systems in an infinite dimensional space, Vestnik MGU, 3, 1983, 52-59. | MR | Zbl

31 -,,-, A.N. Chetaev, On the dimension of attractors for a class of dissipative systems, Prikl. Mat. Mech., 46, 3, 1982, 374-381. limit | MR | Zbl

32 D.A. Kamaev, Hyperbolic limit sets of evolution equations and Galerkin method, Usp. Mat. Nauk 35, 3, 1980. | MR | Zbl

33 -,,-, Hopf hypothesis for a class of chemical reaction equations, Zap. Nauch. Sem. LOMI, 110, 1981,57-73. | MR

34 N. Kopell, D. Ruelle, Bounds on complexity in reaction-diffusion systems, Preprint IHES p/ 42, 1984. | MR | Zbl

35 O.A. Ladyzhenskaya, On a dynamical system generated by Navier-Stokes equations, Zap.Nauch.Sem. LOMI, 27, 1972, 91-115 (J. Soviet Math. 3, 1975, 455-479). | MR | Zbl

36 -,,-, On the finiteness of the dimension of bounded invariant sets for the Navier-Stokes equations and other related dissipative systems, Zap. Nauch. Sem. LOMI, 115, 1982, 137-155. | MR | Zbl

37 A. Lafon, Borne sur la dimension de Hausdorff de l'attracteur pour les équations de Navier-Stokes à deux dimensions, C. R. Acad. Ac. Paris, 298, 1984, 453-456. | MR | Zbl

38 F. Ledrappier, Some relations between dimension and Lyapunov exponents, Comm.Math.Phys. 81, 1981, 229-238. | MR | Zbl

39. E.H. Lieb, On characteristic exponents in turbulence, Comm.Math.Phys. 92, 1984, 473-480. | MR | Zbl

40 J. Mallet-Paret, Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, J.Diff. Eq. 22, 1976, 331-348. | MR | Zbl

41 R. Mañé, On the dimension of the compact invariant sets of certain non-linear maps; Dynamical Systems and Turbulence, Warwick 1980, ed. by D.A. Rand, L.-S. Young, L.N.Math. 898, Springer 1981, 230-242. | MR | Zbl

42 B. Nicolaenko, B. Scheurer, R. Temam, Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attractors, Physica 16D, 155-184, 1985. | MR | Zbl

43 D. Ruelle, Differentiable dynamical systems and the problem of turbulence, Bull. AMS, 5, 1981, 29-42. | MR | Zbl

44 D. Ruelle, Large volume limit of the distribution of characteristic exponents in turbulence, Comm. Math. Phys. 87, 1982, 287-302. | MR | Zbl

45 D. Ruelle, Characteristic exponents for a viscous fluid subjected to time dependent forces, Comm. Math. Phys. 93, 1984, 285-300. | MR | Zbl

46 M. Sermange, R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math. 36, 1983, 635-664. | MR | Zbl

47 V.S. Stepanov, Regular attractors of quasilinear hyperbolic systems with dissipation, Vestnik MGU 5, 1985, 74-77. | MR | Zbl

48 R. Temam, Navier-Stokes equations and nonlinear functional analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983. | MR | Zbl

49 R. Temam, Infinite dimensional dynamical systems in fluid mechanics, Proc. AMS Summer Research Institute "Nonlinear Functional Analysis and Applications", Berkeley 1983, ed. F. Browder. | Zbl

50 R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North Holland, 1984. | MR | Zbl