Related papers: Polynomial mixing for white-forced Kuramoto-Sivash…
Our goal in this paper is to investigate ergodicity of the white-forced wave equation on the whole line. Under the assumption that sufficiently many directions of the phase space are stochastically forced, we prove the uniqueness of…
In this paper, we study the large-time behaviors of the Kuramoto-Sivashinsky equation (KSE) on the 1D torus while being subjected to random perturbation via additive Gaussian noise. It is well-known that under suitable assumptions on the…
This paper is devoted to proving the polynomial mixing for a weakly damped stochastic nonlinear Schr\"{o}dinger equation with additive noise on a 1D bounded domain. The noise is white in time and smooth in space. We consider both focusing…
This article introduces and analyzes a new explicit, easily implementable, and full discrete accelerated exponential Euler-type approximation scheme for additive space-time white noise driven stochastic partial differential equations…
We investigate a fully discrete finite element approximation for the stochastic Kuramoto-Sivashinsky equation, combining the standard finite element methods in spatial discretization with the implicit Euler-Maruyama scheme in time. Rigorous…
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…
We study the long-time mixing behavior of the stochastic nonlinear Schr\"odinger equation in $\mathbb{R}^d$, $d\le 3$. It is well known that, under a sufficiently strong damping force, the system admits unique ergodicity, although the rate…
The Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and with parameter $\nu=0.1212$ is considered. We give a computer-assisted proof the existence of symbolic dynamics and countable infinity of periodic orbits…
We propose an approximate model for the 2D Kuramoto-Sivashinsky equations (KSE) of flame fronts and crystal growth. We prove that this new ``calmed'' version of the KSE is globally well-posed, and moreover, its solutions converge to…
This paper presents numerical results for the two-dimensional isotropic Kuramoto-Sivashinsky equation (KSE) with an additional nonlinear term and a single independent parameter. Surfaces generated by this equation exhibit a certain…
We study numerically the Kuramoto-Sivashinsky (KS) equation forced by external white noise in two space dimensions, that is a generic model for e.g. surface kinetic roughening in the presence of morphological instabilities. Large scale…
We consider parabolic stochastic partial differential equations driven by white noise in time. We prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions. These conditions…
Finite-dimensional observer-based controller design for PDEs is a challenging problem. Recently, such controllers were introduced for the 1D heat equation, under the assumption that one of the observation or control operators is bounded.…
In this paper we study the problem of ergodicity for the complex Ginzburg--Landau equation perturbed by an unbounded random kick-force. Randomness is introduced both through the kicks and through the times between the kicks. We show that…
We establish a general criterion which ensures exponential mixing of parabolic Stochastic Partial Differential Equations (SPDE) driven by a non additive noise which is white in time and smooth in space. We apply this criterion on two…
We prove strengthenings of the Birkhoff Ergodic Theorem for weakly mixing and strongly mixing measure preserving systems. We show that our pointwise theorem for weakly mixing systems is strictly stronger than the Wiener-Wintner Theorem. We…
The generalized Kuramoto-Sivashinsky equation in the case of the power nonlinearity with arbitrary degree is considered. New exact solutions of this equation are presented.
A noisy stabilized Kuramoto-Sivashinsky equation is analyzed by stochastic decomposition. For values of control parameter for which periodic stationary patterns exist, the dynamics can be decomposed into diffusive and transverse parts which…
We study a general stochastic probing problem defined on a universe V, where each element e in V is "active" independently with probability p_e. Elements have weights {w_e} and the goal is to maximize the weight of a chosen subset S of…
This paper presents a finite difference method combined with the Crank--Nicolson scheme of the Kuramoto--Sivashinsky equation defined on an expanding circle (\cite{KUY}), and the existence, uniqueness, and second-order error estimate of the…