Dissipativity for a semi-linearized system modeling cellular flames

Michael Frankel; Victor Roytburd; Gregory I. Sivashinsky

doi:10.3934/dcdss.2011.4.83

Article Contents

2011, Volume 4, Issue 1: 83-99. Doi: 10.3934/dcdss.2011.4.83

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Dissipativity for a semi-linearized system modeling cellular flames

1.
Department of Mathematical Sciences, IUPUI, Indianapolis, IN 46202-3216
2.
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180-3590
3.
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978

Received: November 2008

Revised: September 2009

Published: February 2011

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Abstract

We study a Semi-Linearized System (SLS) of second order PDEs modeling flame front dynamics. SLS is a simplified version of the weak $\kappa\theta$ model of cellular flames which is dynamically similar to the Kuramoto-Sivashinsky (KS) equation [7, 4]. We prove existence of the solutions at large, and their proximity, for finite time, to the solutions of KS. We demonstrate that SLS possesses a universal absorbing set and a compact attractor. Furthermore, we show that the attractor is of finite Hausdorff dimension.

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Mathematics Subject Classification: Primary: 35K55, 35B25; Secondary: 80A25.

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References

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