The Kuramoto-Sivashinsky equation and related models of cellular flame instabilities: Dissipation and pattern formation
Speaker: Victor Roytburd, NSF and RPI
Abstract: Attempts to capture salient features of cellular flame instabilities have led to a variety of mathematical models of a very geometrical nature, including the Kuramotoâ€”Sivashinsky (KS) equation; its lesser relative is the Burgersâ€”Sivashinsky (BS) equation (which is just a linearly forced Burgers equation). The linear dispersion relations for both equations admit exponential mode growth for a range of long waves. Nonetheless, the equations are dissipative due to the nonlinear mixing. Here dissipativity is understood as the property that the eventual time evolution of solutions is confined to a bounded (actually compact) absorbing set. I'll discuss a recently introduced model of quasi-steady evolution of cellular flames, the QS equation, which is intermediate between BS and KS. Its dispersion relation coincides with that for BS for short waves, and is virtually identical to that of KS for long waves. The equation is a perturbation of the BS by a 0-th order pseudo-differential operator. In contrast with the BS, whose dynamics are more or less trivial, QS demonstrate a very rich dynamical behavior (similarly to KS). The proof of dissipativity and generalizations to elliptic pseudo-differential operators will be discussed.
Room Reservation Information
Room Number: 106 McAllister
Time: 4:00pm - 4:55pm