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Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attractors. (English) Zbl 0592.35013

The Kuramoto-Sivashinsky equations model pattern formations on unstable flame fronts and thin hydrodynamic films. They are characterized by the coexistence of coherent spatial structures with temporal chaos. We investigate some global dynamical properties, including nonlinear stability. We demonstrate their low modal behavior, in terms of determining modes; and that the fractal dimension of all attractors is bounded by a universal constant times \(\tilde L^{13/8}\), where \(\tilde L\) is a dimensionless pattern cell size (in the one-dimensional case). Such equations are indeed a paradigm of low-dimensional behavior for infinite-dimensional systems.
Reviewer: M.Biroli

MSC:

35B35 Stability in context of PDEs
35K55 Nonlinear parabolic equations
35Q99 Partial differential equations of mathematical physics and other areas of application
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