Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-15T01:24:42.020Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation of attractors for nongradient systems

Published online by Cambridge University Press:  14 November 2011

I. N. Kostin
I. N. Kostin
Affiliation:
Steklov Mathematical Institute, St. Petersburg Branch, 27 Fontanka, St. Petersburg 191011, Russia e-mail: kostin@pdmi.ras.ru
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of approximation of attractors for semidynamical systems (SDSs) in a metric space is studied. Let some (exact) SDS possessing an attractor M be inaccurately defined, i.e. let another (approximate) SDS, which is close in some sense to the exact one, be given. The problem is to construct a set , which is close to M in the Hausdorff metric.

The suggested procedure for constructing is finite, which makes it possible to use it in computations. The results obtained are suitable for numerical approximation of attractors for a rather large class of semidynamical systems, including ones generated by the Lorenz equations and the Navier–Stokes equations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 4 , 1995 , pp. 739 - 758
Copyright
Copyright © Royal Society of Edinburgh 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Babin, A. V. and Vishik, M. I.. Unstable invariant manifolds of semigroups of nonlinear operators and their perturbations. Uspekhi Mat. Nauk 41(4) (1986), 3–34; translated in Russian Math. Surveys 41(4) (1986), 1–41.Google Scholar
2Babin, A. V. and Vishik, M. I.. Attractors of Evolution Equations (Amsterdam: North Holland, 1992).Google Scholar
3Hale, J. K., Lin, X. B. and Raugel, G.. Upper-semicontinuity of attractors for approximations of semigroups and partial differential equations. Math. Comp. 5 (1988), 89–123.CrossRefGoogle Scholar
4Hale, J. K. and Raugel, G.. Lower-semicontinuity of attractors of gradient systems and applications. Ann. Math. Pura Appl. 154 (1989), 281–326.CrossRefGoogle Scholar
5Kapitanskii, L. V. and Kostin, I. N.. Attractors of nonlinear evolution equations and their approximations. Algebra i Analiz 2(1) (1990), 114–40; translated in Leningrad Math. J. 2(1) (1991), 97–117.Google Scholar
6Kostin, I. N.. On a certain method for approximation of attractors. Zap. Nauchn. Semin. LOMI 188 (1991), 87–104 (in Russian).Google Scholar
7Kostin, I. N.. Approximation of attractors for evolution equations with the help of attractors for finite systems. Zap. Nauchn. Semin. LOMI 197 (1992), 71–86 (in Russian).Google Scholar
8Kostin, I. N.. Numerical approximation of the attractor for the Navier-Stokes equations. Zap. Nauchn. Semin. LOMI 200 (1992), 91–7 (in Russian).Google Scholar
9Kuratowski, K.. Topology (New York: Academic Press, 1968).Google Scholar
10Ladyzhenskaya, O. A.. Mathematical Theory of Viscous Incompressible Flow (New York: Gordon and Breach, 1963).Google Scholar
11Ladyzhenskaya, O. A.. On the dynamical system generated by the Navier–Stokes equations. Zap. Nauchn. Semin. LOMI 27 (1972), 91–114 (in Russian).Google Scholar
12Ladyzhenskaya, O. A.. On attractors of nonlinear evolution problems with dissipation. Zap. Nauchn. Semin. LOMI 152 (1986), 72–86 (in Russian).Google Scholar
13Ladyzhenskaya, O. A.. On determination of minimal global attractors for the Navier–Stokes equations and other partial differential equations. Uspekhi Mat. Nauk 42(6) (1987), 25–60; translated in Russian Math. Surveys 42(6) (1987), 27–73.Google Scholar
14Ladyzhenskaya, O. A.. Globally stable difference schemes and their attractors (Preprint, LOMI R-5-91, Leningrad, 1991 (in Russian)).Google Scholar
15Lorenz, E. N.. Deterministic nonperiodic flows. J. Atmos. Sci. 20 (1963), 130–41.2.0.CO;2>CrossRefGoogle Scholar
16Raugel, G.. Continuity of attractors. Math. Mod. Numer. Anal. 23 (1989), 519–33.CrossRefGoogle Scholar
17Saperstone, S. H.. Semidynamical Systems in Infinite Dimensional Space, Applied Mathematical Scienes 37 (Berlin: Springer, 1981).CrossRefGoogle Scholar