Journal ArticleDOI
Attractors of partial differential evolution equations and estimates of their dimension
A V Babin,M I Vishik +1 more
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In this paper, an upper bound for the Hausdorff dimension of an attractor of a two-dimensional Navier-Stokes system has been derived, and a lower bound has been established for the same system.Abstract:
CONTENTSIntroduction § 1. Maximal attractors of semigroups generated by evolution equations § 2. Examples of parabolic equations and systems having a maximal attractor § 3. The Hausdorff dimension of invariant sets § 4. Estimate of the change in volume under the action of shift operators generated by linear evolution equations § 5. An upper bound for the Hausdorff dimension of attractors of semigroups corresponding to evolution equations § 6. A lower bound for the dimension of an attractor § 7. Differentiability of shift operators § 8. Estimates of the Hausdorff dimension of an attractor of a two-dimensional Navier-Stokes system § 9. Upper and lower bounds for the Hausdorff dimension of attractors of parabolic equations and parabolic systems § 10. Attractors of semigroups having a global Lyapunov function § 11. Regular attractors of semigroups having a Lyapunov functionReferencesread more
Citations
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Mathematical problems for the next century
TL;DR: Arnabels invitation is inspired in part by Hilbert's list of 1900 (see e.g. [Browder, 1976]) and I have used that list to help design this essay.
Journal ArticleDOI
Some global dynamical properties of a class of pattern formation equations
TL;DR: On etudie deux equations d'evolution non lineaires as discussed by the authors : l'equation de Kuramoto-Sevashinsky and theequation of Cahn-Hilliard.
Journal ArticleDOI
On the determination of minimal global attractors for the Navier-Stokes and other partial differential equations
TL;DR: In this article, problems with dissipation of parabolic type (semigroups of class 1) and problems with disentanglement of class 2 are discussed, as well as problems with hyperbolic types.
Journal ArticleDOI
Dimension of the attractors associated to the Ginzburg-Landau partial differential equation
Jean-Michel Ghidaglia,B. Herón +1 more
TL;DR: In this paper, the authors studied the long-time behavior of solutions to the Ginzburg-Landau partial differential equation and showed that a finite-dimensional attractor captures all the solutions.
Journal ArticleDOI
Attractors for reaction-diffusion equations: existence and estimate of their dimension
TL;DR: In this paper, the existence of a maximal attractor which describes the long-time behavior of the solutions was proved for two types of reaction-diffusion equations: an equation with a polynomial growth nonlinearity and systems admitting a positively invariant region.
References
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Book
Geometric Theory of Semilinear Parabolic Equations
TL;DR: The neighborhood of an invariant manifold near an equilibrium point is a neighborhood of nonlinear parabolic equations in physical, biological and engineering problems as mentioned in this paper, where the neighborhood of a periodic solution is defined by the invariance of the manifold.
Book
Navier-Stokes Equations: Theory and Numerical Analysis
TL;DR: This paper presents thediscretization of the Navier-Stokes Equations: General Stability and Convergence Theorems, and describes the development of the Curl Operator and its application to the Steady-State Naviers' Equations.
Book ChapterDOI
Differentiable dynamical systems
TL;DR: A survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold M is presented in this paper.