We revisit the Near Equidiffusional Flames (NEF) model introduced by Matkowsky and Sivashinsky in 1979 and consider a simplified, quasisteady version of it. This simplification allows, near the planar front, an explicit derivation of the front equation. The latter is a pseudodifferential fully nonlinear parabolic equation of the fourth-order. First, we study the (orbital) stability of the null ...

We analyze a Crank–Nicolson–type finite difference scheme for the Kuramoto– Sivashinsky equation in one space dimension with periodic boundary conditions. We discuss linearizations of the scheme and derive second–order error estimates.

We consider the Kuramoto-Sivashinsky (KS) equation in one dimension with periodic boundary conditions. We apply a Lyapunov function argument similar to the one first introduced by Nicolaenko, Scheurer, and Temam [18], and later improved by Collet, Eckmann, Epstein and Stubbe[1], and Goodman [10] to prove that lim sup t→∞ ||u||2 ≤ CL 3 2 . This result is slightly weaker than that recently announ...

We analyse the nonlinear Kuramoto-Sivashinsky equation to develop an accurate finite difference approximation to its dynamics. The analysis is based upon centre manifold theory so we are assured that the finite difference model accurately models the dynamics and may be constructed systematically. The theory is applied after dividing the physical domain into small elements by introducing insulat...

For the 1-dimensional Kuramoto–Sivashinsky equation with random forcing term, existence and uniqueness of solutions is proved. Then, the Markovian semigroup is well defined; its properties are analyzed, in order to provide sufficient conditions for existence and uniqueness of invariant measures for this stochastic equation. Finally, regularity results are obtained by means of Girsanov theorem.

A model consisting of a mixed Kuramoto-Sivashinsky-Kortewegde Vries equation, linearly coupled to an extra linear dissipative equation has been proposed in [1] in order to describe the surface waves on multi-layered liquid films and stability criteria are discussed using wave mode analysis. In this paper, we study the linear stability of solutions to the model from the viewpoint of energy estim...