Stochastic Kuramoto-Sivashinsky Equation Research Articles

A backward problem for stochastic Kuramoto-Sivashinsky equation: Conditional stability and numerical solution

In this paper, we consider a backward problem in time for a linear stochastic Kuramoto-Sivashinsky equation. Firstly, we present two Carleman estimates incorporating weight functions independent of the variable x for the stochastic Kuramoto-Sivashinsky equation. Subsequently, we employ these two Carleman estimates to establish conditional stability for the backward problem in two distinct scenarios: when 0<t0<T and when t0=0. Lastly, we transform the backward problem in time into the minimization of a regularized Tikhonov functional. This functional is solved by the conjugate gradient algorithm based on the gradient formula tailored for the regularized functional. Numerical examples related to the recovery of continuous and discontinuous initial values illustrate the effectiveness of the conjugate gradient algorithm.

Continuity in expectation of odd random attractors for stochastic Kuramoto-Sivashinsky equations

Continuity in expectation of odd random attractors for stochastic Kuramoto-Sivashinsky equations

A numerical method for a backward problem of a linear stochastic Kuramoto-Sivashinsky equation

This paper concerns a backward problem for a linear stochastic Kuramoto–Sivashinsky equation, which aims to reconstruct the initial value from the mean measurement at the terminal time. By transforming the backward problem into a regularized optimization one, a regularization method together with its numerical implementation in a finite dimensional space is proposed for solving the backward problem. Finally, we show effectiveness of the proposed reconstruction method by several numerical examples.

Moderate deviations for stochastic Kuramoto–Sivashinsky equation

This paper aims to establish the central limit theorem and moderate deviation principle for the stochastic Kuramoto–Sivashinsky equation driven by multiplicative noise on a bounded domain. The moderate deviation principle is investigated using the weak convergence approach based on a variational representation for expected values of positive functionals of the Brownian motion. The approach relies on proving basic qualitative properties of controlled versions of the original stochastic partial differential equation which is under consideration.

The Analytical Solutions of the Stochastic Fractional Kuramoto–Sivashinsky Equation by Using the Riccati Equation Method

In this work, we consider the stochastic fractional-space Kuramoto–Sivashinsky equation using conformable derivative. The Riccati equation method is used to get the analytical solutions to the space-fractional stochastic Kuramoto–Sivashinsky equation. Because this equation has never been examined with space-fractional and multiplicative noise at the same time, we generalize some previous results. Moreover, we display how the multiplicative noise influences on the stability of obtained solutions of the space-fractional stochastic Kuramoto–Sivashinsky equation.

The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation

Abstract In this article, we take into account the stochastic Kuramoto-Sivashinsky equation forced by multiplicative noise in the Itô sense. To obtain the exact stochastic solutions of the stochastic Kuramoto-Sivashinsky equation, we apply the G ′ G \frac{{G}^{^{\prime} }}{G} -expansion method. Furthermore, we extend some previous results where this equation has not been previously studied in the presence of multiplicative noise. Also, we show the influence of multiplicative noise on the analytical solutions of the stochastic Kuramoto-Sivashinsky equation.

Large deviations for stochastic Kuramoto–Sivashinsky equation with multiplicative noise

The Kuramoto–Sivashinsky equation is a nonlinear parabolic partial differential equation, which describes the instability and turbulence of waves in chemical reactions and laminar flames. The aim of this work is to prove the large deviation principle for the stochastic Kuramoto–Sivashinsky equation driven by multiplicative noise. To establish the large deviation principle, the weak convergence approach is used, which relies on proving basic qualitative properties of controlled versions of the original stochastic partial differential equation.

Well-posedness and large deviation principle for a class of SPDEs

In this paper, we study a class of stochastic partial differential equations with general locally monotone coefficients under thevariational framework. By proving the existence of martingale solutions and pathwise uniqueness, we first get the existence and uniqueness of strong solutions. Then we prove the large deviation principle by using the stochastic control and weakconvergence approach. The mainresults are applicable to various concrete SPDE models, such as stochastic Cahn-Hilliard equation, stochastic Kuramoto-Sivashinsky equation and stochastic 3D tamedNavier-Stokes equation.

Random attractors for locally monotone stochastic partial differential equations

We prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive Lévy noise. The main result is applicable to various types of SPDE such as stochastic Burgers type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-α model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard type equations, stochastic Kuramoto-Sivashinsky type equations, stochastic generalized porous media equations and stochastic generalized p-Laplace equations.

Null controllability with constraints on the state for the linear stochastic Kuramoto–Sivashinsky equation

This paper is addressed to study the null controllability with constraints on the state for the linear stochastic Kuramoto–Sivashinsky equation. We transform the problem into an equivalent null controllability problem with constraint on the control. Using a Carleman inequality adapted to the constraints, we prove the null controllability with constraints on the state for the linear stochastic Kuramoto–Sivashinsky equation holds.

Exponential moments for numerical approximations of stochastic partial differential equations

Stochastic partial differential equations (SPDEs) have become a crucial ingredient in a number of models from economics and the natural sciences. Many SPDEs that appear in such applications include non-globally monotone nonlinearities. Solutions of SPDEs with non-globally monotone nonlinearities are in nearly all cases not known explicitly. Such SPDEs can thus only be solved approximatively and it is an important research problem to construct and analyze discrete numerical approximation schemes which converge with positive strong convergence rates to the solutions of such infinite dimensional SPDEs. In the case of finite dimensional stochastic ordinary differential equations (SODEs) with non-globally monotone nonlinearities it has recently been revealed that exponential integrability properties of the discrete numerical approximation scheme are a key instrument to establish positive strong convergence rates for the considered approximation scheme. To the best of our knowledge, there exists no result in the scientific literature which proves exponential integrability properties for a time discrete approximation scheme in the case of an infinite dimensional SPDE. In this paper we propose a new class of tamed space-time-noise discrete exponential Euler approximation schemes that admit exponential integrability properties in the case of infinite dimensional SPDEs. In particular, we establish exponential moment bounds for the proposed approximation schemes in the case of stochastic Burgers equations, stochastic Kuramoto-Sivashinsky equations, and two-dimensional stochastic Navier-Stokes equations.

Global Carleman estimates for the linear stochastic Kuramoto–Sivashinsky equations and their applications

In this paper, we establish two new global Carleman estimates for the linear stochastic Kuramoto–Sivashinsky equations. The first one is for the backward linear stochastic Kuramoto–Sivashinsky equation. Based on this estimate and the duality argument, we obtain the null controllability of the forward linear stochastic Kuramoto–Sivashinsky equation by three controls, one is an internal control in the diffusion term and the others are boundary controls. The second one is for the forward linear stochastic Kuramoto–Sivashinsky equation. According to this estimate, we obtain a unique continuation property for the forward linear stochastic Kuramoto–Sivashinsky equation.

Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation

This work concerns the problem associated with averaging principle for a stochastic Kuramoto-Sivashinsky equation with slow and fast time-scales. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the stochastic Kuramoto-Sivashinsky equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single stochastic Kuramoto-Sivashinsky equation with a modified coefficient.

Numerical Simulation of Stochastic Kuramoto-Sivashinsky Equation

In this paper, the random Kuramoto-Sivashinsky equation with additive noise is studied numerically, using the finite difference method to simulate the effect of different amplitude of noise on the solitary wave. And numerical experiments show that the white noise does not affect the propagation of the solitary wave, but can increase the amplitude of the solitary wave.

Long time stability of nonlocal stochastic Kuramoto–Sivashinsky equations with jump noises

In this paper, we consider a class of nonlocal stochastic Kuramoto–Sivashinsky equations driven by compensated Poisson random measures. Under appropriate conditions, we prove almost sure exponential stability and second moment stability of the solution.

Controlling roughening processes in the stochastic Kuramoto–Sivashinsky equation

We present a novel control methodology to control the roughening processes of semilinear parabolic stochastic partial differential equations in one dimension, which we exemplify with the stochastic Kuramoto-Sivashinsky equation. The original equation is split into a linear stochastic and a nonlinear deterministic equation so that we can apply linear feedback control methods. Our control strategy is then based on two steps: first, stabilize the zero solution of the deterministic part and, second, control the roughness of the stochastic linear equation. We consider both periodic controls and point actuated ones, observing in all cases that the second moment of the solution evolves in time according to a power-law until it saturates at the desired controlled value.

Long time behavior for nonlocal stochastic Kuramoto–Sivashinsky equations

Abstract In this paper, we consider a class of nonlocal stochastic Kuramoto–Sivashinsky equations driven by additive noises. Under some appropriate conditions, we investigate long time behavior, i.e., stability and growth bounds of the solutions to the equation. Finally, several examples are given to illustrate our results.

Kolmogorov equation associated to a stochastic Kuramoto–Sivashinsky equation

This paper investigates the relation between the Kolmogorov operator associated to a stochastic Kuramoto–Sivashinsky equation and the infinitesimal generator for the corresponding transition semigroup. We prove that the infinitesimal generator is the closure of Kolmogorov operator in the space of continuous functions with kth-polynomial growth with respect to π-convergence topology and the space L2(H,ν) respectively. The proof depends on various estimates on the solution, invariant measure and transition semigroup. As a product, we also obtains smoothing properties of the transition semigroup.

A random map implementation of implicit filters

Implicit particle filters for data assimilation generate high-probability samples by representing each particle location as a separate function of a common reference variable. This representation requires that a certain underdetermined equation be solved for each particle and at each time an observation becomes available. We present a new implementation of implicit filters in which we find the solution of the equation via a random map. As examples, we assimilate data for a stochastically driven Lorenz system with sparse observations and for a stochastic Kuramoto–Sivashinsky equation with observations that are sparse in both space and time.

Absolute continuity of laws for semilinear stochastic equations with additive noise

We present the Girsanov theorem for a non linear Ito equation in an infinite dimensional Hilbert space with a non linearity of polynomial growth and an infinite dimensional additive noise. We assume a condition weaker than Novikov one, as done by Mikulevicius and Rozovskii in the study of more general stochastic PDE's. The equivalence of the laws of the linear equation and of the non linear equation implies results on weak solutions and on invariant measures for the given non linear equation. Two examples are presented: a stochastic Kuramoto-Sivashinsky equation and a stochastic hyperviscosity-regularized Navier-Stokes equation.