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Remarks on the Kuramoto-Sivashinsky equation

Description: We report here a joint work in progress on the Kuramoto-Sivashinsky equation. The question we address is the analytical study of a fourth order nonlinear evolution equation. This equation has been obtained by Sivashinsky in the context of combustion and independently by Kuramoto in the context of reaction diffusion-systems. Both were motivated by (nonlinear) stability of travelling waves. Numerical calculations have been done on this equation. All the results seem to indicate a chaotic behavior of the solution. Therefore, the analytical study is of interest in analogy with the Burger's and Navier-Stokes equations. Here we give some existence and uniqueness results for the equation in space dimension one, and we also study a fractional step method of numerical resolution. In a forthcoming joint paper with R. Temam, we will study the asymptotic behavior, as t approaches infinity, of the solution of (0.1) and give an estimate on the number of determining modes.
Date: January 1, 1983
Creator: Nicolaenko, B. & Scheurer, B.
Partner: UNT Libraries Government Documents Department

Traveling-wave solutions to reaction-diffusion systems modeling combustion

Description: We consider the deflagration-wave problem for a compressible reacting gas, with species involved in a single-step chemical reaction. In the limit of small Mach numbers, the one-dimensional traveling-wave problem reduces to a system of reaction-diffusion equations. Thermomechanical coefficients are temperature-dependent. Existence is proved by first considering the problem in a bounded domain, and taking an infinite-domain limit. In the singular limit of high activation energy in the Arrhenius exponential term, we prove strong convergence to a limiting free-boundary problem (discontinuity of the derivatives on the free boundary).
Date: January 1, 1982
Creator: Berestycki, H.; Nicolaenko, B. & Scheurer, B.
Partner: UNT Libraries Government Documents Department