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A global attracting set for the Kuramoto-Sivashinsky equation. (English) Zbl 0777.35073

The authors give new bounds for the \(L^ 2\)-norm of the solution of the Kuramoto-Sivashinsky equation. This new bound is obtained by adapting a technique first initiated by R. Temam [Infinite-dimensional dynamical systems in mechanics and physics (Springer, 1988; Zbl 0662.35001)]. This bound is equal to a constant multiplied by \(L^{8/5}\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37C75 Stability theory for smooth dynamical systems

Citations:

Zbl 0662.35001
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References:

[1] [Felen] Frisch, U., She, Z. S., Thual, O.: Viscoelastic behaviour of cellular solutions to the Kuramoto-Sivashinsky model. J. Fluid Mech.168, 221–240 (1986) · Zbl 0597.76006
[2] [Il] Ilyashenko, Yu.S.: Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation. J. Dyn. Differ. Equations, in print
[3] [M] Manneville, P.: Dissipative Structures and Weak Turbulence, San Francisco-London, Academic Press, 1989 · Zbl 0839.76035
[4] [Nelen] Nicolaenko, B., Scheurer, B., Temam, R.: Some global dynamical properties of the Kuramoto-Sivashinsky equations: Nonlinear, stability and attractors. PhysicaD16, 155–183 (1985) · Zbl 0592.35013
[5] [T] Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. Berlin, Heidelberg, New York: Springer, 1988 · Zbl 0662.35001
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