Stability and attractors for the quasi-steady equation of cellular flames
Summary: We continue to study a simple integro-differential equation: the quasi-steady equation \[ \Phi_t+ \tfrac12 \Phi_+x^2= (I-\alpha A)\Phi_{xx}, \qquad A:= (I-\partial_x^2)^{-1}, \] of flame front dynamics. This equation is dynamically similar to the Kuramoto-Sivashinsky (KS) equation. In [\textit{M. L. Frankel, P. V. Gordon} and \textit{G. I. Sivashinsky} [On disintegration of near-limit cellular flames, Phys. Lett. A 310, 389--392 (2003)], where it was introduced, its well-posedness and proximity for finite time intervals to the KS equation in Sobolev spaces of periodic functions were established. Here we demonstrate that QS possesses a universal absorbing set, and a compact attractor. Furthermore we show that the attractor is of a finite Hausdorff dimension, and give an estimate on it. We discuss relationship with the Kuramoto-Sivashinsky and Burgers-Sivashinsky equations.
scientific article; zbMATH DE number 5074423