Geodesic
Effective finite parametrization in~phase spaces of parabolic
Izvestiya. Mathematics , Tome 70 (2006) no. 5, pp. 1015-1029.

Voir la notice de l'article provenant de la source Math-Net.Ru

For evolution equations of parabolic type in a Hilbert phase space $E$, consideration is given to the problem of the effective parametrization (with a Lipschitzian estimate) of the sets $\mathcal K\subset E$ by functionals $\varphi_1,\dots,\varphi_m$ in $E^*$ or, in other words, the problem of the linear Lipschitzian embedding of $\mathcal K$ in $\mathbb R^m$. If $\mathcal A$ is the global attractor for the equation, then this kind of parametrization turns out to be equivalent to the finite dimensionality of the dynamics on $\mathcal A$. Some tests are established for the parametrization (in various metrics) of subsets in $E$ and, in particular, of manifolds $\mathcal M\subset E$ by linear functionals of different classes. We outline a range of physically significant parabolic problems with a fundamental domain $\Omega\subset\mathbb R^N$ that admit a parametrization of the elements $u(x)\in\mathcal A$ by their values $u(x_i)$ at a finite system of points $x_i\in\Omega$. 
@article{IM2_2006_70_5_a6,
     author = {A. V. Romanov},
     title = {Effective finite parametrization in~phase spaces of parabolic},
     journal = {Izvestiya. Mathematics },
     pages = {1015--1029},
     publisher = {mathdoc},
     volume = {70},
     number = {5},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2006_70_5_a6/}
}
TY  - JOUR
AU  - A. V. Romanov
TI  - Effective finite parametrization in~phase spaces of parabolic
JO  - Izvestiya. Mathematics 
PY  - 2006
SP  - 1015
EP  - 1029
VL  - 70
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2006_70_5_a6/
LA  - en
ID  - IM2_2006_70_5_a6
ER  - 
%0 Journal Article
%A A. V. Romanov
%T Effective finite parametrization in~phase spaces of parabolic
%J Izvestiya. Mathematics 
%D 2006
%P 1015-1029
%V 70
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2006_70_5_a6/
%G en
%F IM2_2006_70_5_a6
A. V. Romanov. Effective finite parametrization in~phase spaces of parabolic. Izvestiya. Mathematics , Tome 70 (2006) no. 5, pp. 1015-1029. http://geodesic.mathdoc.fr/item/IM2_2006_70_5_a6/

[1] Ladyzhenskaya O. A., “O dinamicheskoi sisteme, porozhdaemoi uravneniyami Nave-Stoksa”, Zap. nauch. semin. LOMI, 27, 1972, 91–115 | MR | Zbl

[2] Mañé R., “On the dimension of the compact invariant sets of certain non-linear maps”, Lecture Notes in Math., 898, Springer-Verlag, Berlin–N. Y., 1981, 230–242 | MR | Zbl

[3] Foias C., Temam R., “Determination of the solutions of the Navier–Stokes equations by a set of nodal values”, Math. Comput., 43:167 (1984), 117–133 | DOI | MR | Zbl

[4] Friz P. K., Robinson J. C., “Parametrising the attractor of the two-dimensional Navier–Stokes equations with a finite number of nodal values”, Phys. D, 148:3–4 (2001), 201–220 | DOI | MR | Zbl

[5] Friz P. K., Kukavica I., Robinson J. C., “Nodal parametrisation of analytic attractors”, Discrete Contin. Dyn. Syst., 7:3 (2001), 643–657 | DOI | MR | Zbl

[6] Foias C., Titi E., “Determining nodes, finite difference schemes and inertial manifolds”, Nonlinearity, 4:1 (1991), 135–153 | DOI | MR | Zbl

[7] Romanov A. V., “Konechnomernaya predelnaya dinamika dissipativnykh parabolicheskikh uravnenii”, Matem. sb., 191:3 (2000), 99–112 | MR | Zbl

[8] Romanov A. V., “Konechnomernost dinamiki na attraktore dlya nelineinykh parabolicheskikh uravnenii”, Izv. RAN. Ser. matem., 65:5 (2001), 129–152 | MR | Zbl

[9] Temam R., Infinite-dimensional dynamical systems in mechanics and physics, 2-nd ed., Appl. Math. Sci., 68, Springer-Verlag, N. Y., 1997 | MR | Zbl

[10] Constantin P., Foias C., Nicolaenko B., Temam R., “Spectral barriers and inertial manifolds for dissipative partial differential equations”, J. Dynam. Differential Equations, 1:1 (1989), 45–73 | DOI | MR | Zbl

[11] Bartucelli M., Constantin P., Doerling C. R., Gibbon J. D., Gisselfalt M., “On the possibility of soft and hard turbulence in the complex Ginzburg–Landau equation”, Phys. D, 44:3 (1990), 421–444 | DOI | MR | Zbl

[12] Nicolaenko B., Scheurer B., Temam R., “Some global dynamical properties of a class of pattern formation equations”, Comm. Partial Differential Equations, 14:2 (1989), 245–297 | DOI | MR | Zbl

[13] Mallet-Paret J., Sell G. R., “Inertial manifolds for reaction diffusion equations in higher space dimensions”, J. Amer. Math. Soc., 1:4 (1988), 805–866 | DOI | MR | Zbl

[14] Kwean H., “An inertial manifold and the principle of spatial averaging”, Int. J. Math. Math. Sci., 28:5 (2001), 293–299 | DOI | MR | Zbl

[15] Khenri D., Geometricheskaya teoriya polulineinykh parabolicheskikh uravnenii, Mir, M., 1985 | MR | Zbl

[16] Ladyzhenskaya O. A., “O nakhozhdenii minimalnykh globalnykh attraktorov dlya uravnenii Nave–Stoksa i drugikh uravnenii s chastnymi proizvodnymi”, UMN, 42:6 (1987), 25–60 | MR | Zbl

[17] Brunovsky P., Terescak I., “Regularity of invariant manifolds”, J. Dynam. Differential Equations, 3:3 (1991), 313–337 | DOI | MR | Zbl

[18] Movahedi-Lankarani H., “On the inverse of Mañé's projection”, Proc. Amer. Math. Soc., 116:2 (1992), 555–560 | DOI | MR | Zbl

[19] Krasnoselskii M. A., Zabreiko P. P., Geometricheskie metody nelineinogo analiza, Nauka, M., 1975 | MR | Zbl

[20] Danford N., Shvarts Dzh. T., Lineinye operatory. Ch. 1. Obschaya teoriya, 2-e izd., URSS, M., 2004