V. I. Arnold, on behalf of the International Mathematical Union has written to a number of mathematicians with a suggestion that they describe some great problems for the next century. This report is my response. Arnold's 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. I have listed 18 problems, chosen with these criteria:

Mathematical problems for the next century

On etudie deux equations d'evolution non lineaires: l'equation de Kuramoto-Sevashinsky et l'equation de Cahn-Hilliard Analyse non lineaire de la stabilite Existence et dimension des attracteurs Variete inertielle pour l'equation de Cahn-Hilliard

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Some global dynamical properties of a class of pattern formation equations

CONTENTS ??1. Introduction ??2. Problems with dissipation of parabolic type (semigroups of class 1) ??3. Semigroups of class 2. Equations of hyperbolic type References

http://iopscience.iop.org/article/10.1070/RM1987v042n06ABEH001503/pdf

On the determination of minimal global attractors for the Navier-Stokes and other partial differential equations

Dimension of the attractors associated to the Ginzburg-Landau partial differential equation

In this paper, we study some questions related to attractors for two types of reaction-diffusion equations : an equation with a polynomial growth nonlinearity and systems admitting a positively invariant region. For these problems, we prove the existence of a maximal attractor which describes the long-time behaviour of the solutions and we derive estimates of its Hausdorff and fractal dimensions in terms of the data. Our results are applied to several classical equations.

Attractors for reaction-diffusion equations: existence and estimate of their dimension

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 functionReferences

https://iopscience.iop.org/article/10.1070/RM1983v038n04ABEH004209/pdf

A V Babin

Papers

Attractors of partial differential evolution equations and estimates of their dimension