Stochastic stability of Burgers equation

The current paper is devoted to stochastic Burgers equation with driving forcing given by white noise type in time and periodic in space. Motivated by the numerical results of Hairer and Voss, we prove that the Burgers equation is stochastic stable in the sense that statistically steady regimes of fluid flows of stochastic Burgers equation converge to that of determinstic Burgers equation as noise tends to zero.     

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.     

Stochastic Burgers PDEs with random coefficients and a generalization of the Cole–Hopf transformation

This paper studies forward and backward versions of the random Burgers equation (RBE) with stochastic coefficients. First, the celebrated Cole–Hopf transformation reduces the forward RBE to a forward random heat equation (RHE) that can be treated pathwise. Next we provide a connection between the backward Burgers equation and a system of FBSDEs. Exploiting this connection, we derive a generalization of the Cole–Hopf transformation which links the backward RBE with the backward RHE and investigate the range of its applicability. Stochastic Feynman–Kac representations for the solutions are provided. Explicit solutions are constructed and applications in stochastic control and mathematical finance are discussed.     

SPDE in Hilbert space with locally monotone coefficients

The aim of this paper is to extend the usual framework of SPDE with monotone coefficients to include a large class of cases with merely locally monotone coefficients. This new framework is conceptually not more involved than the classical one, but includes many more fundamental examples not included previously. Thus our main result can be applied to various types of SPDEs such as stochastic reaction-diffusion equations, stochastic Burgers type equation, stochastic 2-D Navier-Stokes equation, stochastic p-Laplace equation and stochastic porous media equation with non-monotone perturbations.     

Strong convergence for explicit space–time discrete numerical approximation methods for stochastic Burgers equations

In this paper we propose and analyze explicit space–time discrete numerical approximations for additive space–time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers equation with space–time white noise. The main result of this paper proves that the proposed explicit space–time discrete approximation method converges strongly to the solution process of the stochastic Burgers equation with space–time white noise. To the best of our knowledge, the main result of this work is the first result in the literature which establishes strong convergence for a space–time discrete approximation method in the case of the stochastic Burgers equations with space–time white noise.     

Exponential ergodicity for stochastic Burgers and 2D Navier-Stokes equations

It is shown that transition measures of the stochastic Navier-Stokes equation in 2D converge exponentially fast to the corresponding invariant measures in the distance of total variation. As a corollary we obtain the existence of spectral gap for a related semigroup obtained by a sort of ground state transformation. Analogous results are proved for the stochastic Burgers equation.     

Irreducibility and strong Feller property for non-linear SPDEs

Under some non-degeneracy condition, the strong Feller property and irreducibility are studied for non-linear stochastic partial differential equations driven by multiplicative noise within the framework called ‘variational approach’. Our result for irreducibility can be applied to equations with locally monotone coefficients. In some special cases, we discuss the Hölder continuity of the associated Markov semigroups. The main results are applied to several examples such as stochastic Burgers equation, stochastic porous media equation and stochastic fast diffusion equation.     

The Burgers equation with a random force and a general model for directed polymers in random environments

Summary. The study of the Burgers equation with a random force leads via a Hopf-Cole type transformation to a stochastic heat equation having a white noise with spatial parameters type potential. The latter can be studied by means of a general model of directed polymers in random environments with two point random potentials. These models exhibit a Gaussian behavior at large times and have certain stationary distributions which yield the corresponding results for the above stochastic heat and Burgers equations. Received: 18 July 1995 / In revised form: 5 August 1995     

Multiobjective Stochastic Control in Fluid Dynamics via Game Theory Approach: Application to the Periodic Burgers Equation

The purpose of the present work is to implement well-known statistical decision and game theory strategies into multiobjective stochastic control problems of fluid dynamics. Such goal is first justified by the fact that deterministic (either singleobjective or multiobjective) control problems that are obtained without taking into account the uncertainty of the model are usually unreliable. Second, in most real-world problems, several goals must be satisfied simultaneously to obtain an optimal solution and, as a consequence, a multiobjective control approach is more appropriate. Therefore, we develop a multiobjective stochastic control algorithm for general fluid dynamics applications, based on the Bayes decision, adjoint formulation and the Nash equilibrium strategies. The algorithm is exemplified by the multiobjective stochastic control of a periodic Burgers equation.     

Stochastic analysis method for hopf's equation ut+ uux=0

It is proved that the stochastic charactcrtstics of Burgers’ equation u _t+uu_x=μu_xxconverge in probability to the character is tcs of Hopf ’ s cqua lion u_t+uu_x=0 as the viscosity μ →0. It follows naturally that the solution of Burgers’ equation converges to the solution of Hopf 's equation satisfying entropy condition. This is well known result due to E. Hopf in 1950. The method here is new. This paper suggests that this method may be useful for proving the validity of “vanishing viscosity method” in certain cases. Those problems are usually extremely difficult.     

Regularity of Ornstein–Uhlenbeck Processes Driven by a Lévy White Noise

The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by Lévy white noise “obtained by subordination of a Gaussian white noise”. Sufficient conditions for spatial continuity are derived. It is also shown that solutions do not have in general cádlág modifications. General results are applied to equations with fractional Laplacian. Applications to Burgers stochastic equations are considered as well.     

Well-posedness of the stochastic KdV–Burgers equation

We are interested in rigorously proving the invariance of white noise under the flow of a stochastic KdV–Burgers equation. This paper establishes a result in this direction. After smoothing the additive noise (by a fractional spatial derivative), we establish (almost sure) local well-posedness of the stochastic KdV–Burgers equation with white noise as initial data. Next we observe that spatial white noise is invariant under the projection of this system to the first N>0$N>0$ modes of the trigonometric basis. Finally, we prove a global well-posedness result under an additional smoothing of the noise.     

Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.     

The Burgers equation with affine linear noise: Dynamics and stability

We study the dynamics of the Burgers equation on the unit interval driven by affine linear noise. Mild solutions of the Burgers stochastic partial differential equation generate a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. We establish a local stable manifold theorem near a hyperbolic stationary point, as well as the existence of local smooth invariant manifolds with finite codimension and a countable global invariant foliation of the energy space relative to an ergodic stationary point.     

A Note on Stochastic Burgers’ System of Equations

Abstract

We consider a stochastic version of a system of two equations formulated by Burgers [Burgers, J.M. Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Verh. Kon. Nerderl. Akad. Weten-Schappen Amsterdam, Afdeel Natuurkunde, 1939, 17 (2), 1–53] with the aim to describe the laminar and turbulent motions of a fluid in a channel. The existence and uniqueness theorem for a global solution is established. The paper generalizes the result from the paper by Da Prato and Ga¸tarek [Da Prato, G.; Ga¸tarek, D. Stochastic Burgers equation with correlated noise. Stochastics Stochastics Rep. 1995, 52, 29–41] dealing with the equation describing only the turbulent motion.     

Ergodicity for a class of semilinear stochastic partial differential equations

In this paper, we establish the existence and uniqueness of invariant measures for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to SPDEs of various types such as the stochastic Burgers equation and the reaction-diffusion equations perturbed by space-time white noise.     

Strong solution of the stochastic Burgers equation

This work introduces a pathwise notion of solution for the stochastic Burgers equation, in particular, our approach encompasses the Cole–Hopf solution. The developments are based on regularization arguments from the theory of distributions.     

Sliding mode control of the forced generalized Burgers equation

^** Email: smaoui{at}mcs.sci.kuniv.edu.kw This paper deals with the sliding mode control (SMC) of theforced generalized Burgers equation via the Karhunen-Loève(K-L) Galerkin method. The decomposition procedure of the K-Lmethod is presented to illustrate the use of this method inanalysing the numerical simulations data which represent thesolutions of the forced generalized Burgers equation for viscosityranging from 1 to 100. The K-L Galerkin projection is used asa model reduction technique for non-linear systems to derivea system of ordinary differential equations (ODEs) that mimicsthe dynamics of the forced generalized Burgers equation. Thedata coefficients derived from the ODE system are then usedto approximate the solutions of the forced Burgers equation.Finally, static and dynamic SMC schemes with the objective ofenhancing the stability of the forced generalized Burgers equationare proposed. Simulations of the controlled system are givento illustrate the developed theory.     

The Kolmogorov Operator Associated to a Burgers SPDE in Spaces of Continuous Functions

We are concerned with a viscous Burgers equation forced by a perturbation of white noise type. We study the corresponding transition semigroup in a space of continuous functions weighted by a proper potential, and we show that the infinitesimal generator is the closure (with respect to a suitable topology) of the Kolmogorov operator associated to the stochastic equation. In the last part of the paper we use this result to solve the corresponding Fokker-Planck equation.     

Mild Solutions for the Stochastic Generalized Burgers–Huxley Equation

Journal of Theoretical Probability - In this work, we consider the stochastic generalized Burgers–Huxley equation perturbed by space–time white noise and discuss the global solvability...