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Localization and approximation of attractors for the Kuramoto-Sivashinsky equations. (English) Zbl 0971.34042

The authors provide a sequence \((M_{m,j})_{j \in \mathbb N}\) of \(m\)-dimensional approximate inertial manifolds \(M_{m,j}\) for the Kuramoto-Sivashinsky equations. Approximate inertial manifolds exponentially attract and absorb all orbits in a thin neighborhood of the manifold. It is shown that the thickness of these neighborhoods, which localize the global attractor, decreases with increasing dimension \(m\) for fixed order \(j\). Thus they aid in the approximate computation of large-time solutions of the Kuramoto-Sivashinsky equations.

MSC:

34G20 Nonlinear differential equations in abstract spaces
34C40 Ordinary differential equations and systems on manifolds
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35B41 Attractors
35B42 Inertial manifolds
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
34D45 Attractors of solutions to ordinary differential equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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