Infinite-Dimensional Dynamical Systems in Mechanics and Physics

Abstract

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Parametrising the attractor of the two-dimensional Navier–Stokes equations with a finite number of nodal values

“…[12]) The same argument applies to the 3D equations, provided that one assumes regularity (in the sense of Constantin et al [7]), although in this case the estimate of the dimension of the attractor and hence of the number of nodes is much larger (see [7,58] or [26]). …”

“…We will compare this to the classical, heuristic, length scale estimates due to [37] (see also [58]; Doering and Gibbon [12] give a much more detailed discussion of such length scales, and Eden et al [13] give a very good summary of the various bounds). Kraichnan's theory constructs a length scale from the viscous enstrophy dissipation and the forcing.…”

Approximate inertial manifolds for the pattern formation Cahn-Hilliard equation

“…The function u^ + l^ satisfies an équation of the form Then, using the induction assumption (A.2) y , we can apply the uniform Gronwall Lemma (see [14] for instance) to (A.4) and this gives successively …”

Analysis of integrodifferential control system with pulse-width modulated sampler on Banach spaces

“…Nevertheless, it can supply a technical-minded mathematician with a number of new and interesting problems of mathematical nature. There are some results such as steady-state control, stability analysis, robust control of pulse-width sampler systems [1][2][3][4][5][6][7][8][9], integral control by variable sampling based on steady-state data and adaptive sampled-data integral control [10][11][12][13][14][15].…”