We study the effects of multiplicative noise on a spatio-temporal pattern forming nonlinear Partial Differential Equation (PDE) model for premixed flame instability, known as the Kuramoto-Sivashinsky equation, in a circular domain. Modifications of a previously developed numerical integration scheme allow for longer time integration in the presence of noise. In order to gain additional insight,...

We present a heuristic model for the energy spectrum of the one-dimensional phase turbulence in the steady state of the Kuramoto-Sivashinsky equation. Our model contains an energy transfer mechanism from low- to high-wave-vector modes. The energy transfer is written as the sum of local and nonlocal interactions. Our analytical results show good agreement with numerical simulations, particularly...

Abstract: We study a dynamic system described by the Kuramoto-Sivashinsky equation with an external excitation f posed on a finite domain. Firstly it shows that under the given boundary feedback conditions it admits a unique solution and the solution is stable. Secondly it proves that if the external excitation f is a time periodic function, then the system under the boundary conditions admits ...

In this article, we study the boundary controllability of the linear Kuramoto-Sivashinsky equation on a bounded interval. The control acts on the first spatial derivative at the left endpoint. First, we prove that this control system is null controllable. It is done using a spectral analysis and the method of moments. Then, we introduce a boundary feedback law stabilizing to zero the solution o...

The Kuramoto-Sivashinsky equation with Ehrilch-Schwoebel effects models the evolution of surface morphology during Molecular Beam Epitaxy growth, provoked by an interplay between deposition atoms onto and relaxation profile through diffusion. It consists a nonlinear fourth order partial differential equation. Using energy methods we prove well-posedness (i.e., existence, uniqueness stability re...

We analyze and implement fully discrete schemes for periodic initial value problems for a general class of dispersively modified Kuramoto–Sivashinsky equations. Time discretizations are constructed using linearly implicit schemes and spectral methods are used for the spatial discretization. The general case analyzed covers several physical applications arising in multi-phase hydrodynamics and t...

Abstract It is shown that an oblique projection-based feedback control able to stabilize the state of Kuramoto–Sivashinsky equation, evolving in rectangular domains, a given time-dependent trajectory. The actuators consist finite number indicator functions supported small subdomains. Simulations are presented, one-dimensional case under periodic boundary conditions and two-dimensional Neumann c...