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Comparison of Inertial Manifolds and Application to Modulated Systems
Author(s) -
Mielke A.,
Schneider G.,
Ziegra A.
Publication year - 2000
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200006)214:1<53::aid-mana53>3.0.co;2-4
Subject(s) - dissipative system , mathematics , inertial frame of reference , manifold (fluid mechanics) , mathematical analysis , boundary (topology) , interval (graph theory) , periodic boundary conditions , dimension (graph theory) , boundary value problem , pure mathematics , classical mechanics , physics , mechanical engineering , quantum mechanics , combinatorics , engineering
We consider two dissipative systems having inertial manifolds and give estimates which allow us to compare the flows on the two inertial manifolds. As an example of a modulated system we treat the Swift–Hohenberg equation $\partial _\tau w = -(1+\partial ^2 _y )^2 w+\varepsilon ^2 w-w^3$ , $w(\tau, y)$ ∈ ℝ, with periodic boundary conditions on the interval $(0, l/\varepsilon )$ . Recent results in the theory of modulation equation show that the solutions of this equation can be described over long time scales by those of the associated Ginzburg–Landau equation $\partial _t v = 4\partial ^2 _x v+v-3 \, |v|^2v, \, v(t,x)$ ∈ ℂ, with suitably generalized periodic boundary conditions on $(0, l)$ . We prove that both systems have an inertial manifold of the same dimension and that the flows on these finite dimensional manifolds converge against each other for $\varepsilon \to 0$ .