Averaging principle for one dimensional stochastic Burgers equation

Abstract

In this paper, we consider the averaging principle for one dimensional stochastic Burgers equation with slow and fast time-scales. Under some suitable conditions, we show that the slow component strongly converges to the solution of the corresponding averaged equation. Meanwhile, when there is no noise in the slow component equation, we also prove that the slow component weakly converges to the solution of the corresponding averaged equation with the order of convergence 1 − r, for any 0 < r < 1.

Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients

“…The advantage of using the variational approach is that it can cover some nonlinear SPDEs for slow component, such as stochastic power law fluids, and some quasilinear SPDEs for slow component, such as the stochastic porous medium equation and the stochastic p-Laplace equation, which can not be handled by the mild solution approach and thus have not been studied yet. Furthermore, our result also generalizes some known results of the cases that the slow component is a semilinear stochastic partial differential equation, such as the stochastic Burgers equation (see [9]) and stochastic two dimensional Navier-Stokes equation (see [19]). Besides some known results, our result can also be applied to many other unstudied hydrodynamical models in [8], such as the stochastic magneto-hydrodynamic equations, the stochastic Boussinesq model for the Bénard convection, the stochastic 2D magnetic Bénard problem, the stochastic 3D Leray-α model and some stochastic shell models of turbulence.…”

“…For some further results on this topic, we refer to [4,12,22,23] and the references therein.However, the references we mentioned above always assume that the coefficients satisfy Lipschitz conditions, and there are few results on the average principle for SPDEs with nonlinear terms. For example, stochastic reaction-diffusion equations with polynomial coefficients [5], stochastic Burgers equation [9], stochastic two dimensional Navier-Stokes equations [19], stochastic Kuramoto-Sivashinsky equation [14], stochastic Schrödinger equation [15] and stochastic Klein-Gordon equation [16]. But all these papers consider semilinear SPDEs (i.e., for operators A = A 1 + A 2 with A 1 a linear operator and A 2 a nonlinear perturbation), and use the mild solution approach to SPDEs exploiting the smoothing properties of the C 0 -semigroup e A 1 t generated by the linear operator A 1 in an essential way.…”

Averaging principle for multiscale stochastic reaction‐diffusion‐advection equations

“…In order to deal with the above estimate, we will use the skill of stopping times; this is inspired from Dong et al 9 We have…”

“…The averaging principle in the stochastic ordinary differential equations setup was first considered by Khasminskii, which proved that an averaging principle holds in weak sense and has been an active research field on which there is a great deal of literature. There are some results on the averaging principle for infinite dimensional systems . The main difficulty for us to establish the averaging principles for is the non‐Lipschitz terms: the cubic nonlinear term u 3 and advection term u u x .…”

Averaging principle for stochastic real Ginzburg-Landau equation driven by $ \alpha $-stable process

“…It is well known that it is non-trival to deal with nonlinear terms. Recently, averaging principle for stochastic reaction-diffusion systems with nonlinear term has become an active research area which attracted much attention, see, for instance, stochastic reaction-diffusion equations with polynomial coefficients [6], stochastic Burgers equation [7], stochastic two dimensional Navier-Stokes equations [21], stochastic Kuramoto-Sivashinsky equation [13], stochastic Schrödinger equation [14] and stochastic Klein-Gordon equation [15]. The noise considered in above references are all Wiener noise, and it is natural to study the stochastic systems driven by jump noise.…”