Averaging Principle for Two Time-Scales Stochastic Partial Differential Equations with Reflection

Abstract

In this work, we consider a system of fast and slow time-scale stochastic partial differential equations with reflection, where the slow component is the one-dimensional stochastic Burgers equation, the fast component is the stochastic reaction-diffusion equation, and both the fast and slow components have two reflecting walls. The well-posedness of this system is established. Our approach is based on the penalized method by giving the delicate estimation of the penalized terms, which do not resort to splitting the reflected system into stochastic system without reflection and deterministic system with reflection. Then by means of penalized method and combining the classical Khasminskii’s time discretization, we prove the averaging principle for a class of reflected stochastic partial differential equations. In particular, due to the existence and uniqueness of invariant measure for fast component with frozen slow component, the ergodicity for frozen equations are given for different initial function spaces, which plays an important role.

Appendix

The following several properties of $b(\cdot ,\cdot ,\cdot )$ and $B(\cdot ,\cdot )$ are well-known (for example see [49]) and will be used later on.

Lemma A.1

For any $x, y \in H^1_0(0, 1)$,

$$\begin{aligned} b(x,x,y)=-\frac{1}{2}b(x,y,x),\quad b(x,x,x)=0. \end{aligned}$$

Lemma A.2

Suppose $\alpha _{i}\ge 0~(i=1,2,3)$ satisfies one of the following conditions

$(1) ~\alpha _{i}\ne \frac{1}{2}(i=1,2,3), \alpha _{1}+\alpha _{2}+\alpha _{3}\ge \frac{1}{2}$,

$(2) ~\alpha _{i}=\frac{1}{2}$ for some i, $\alpha _{1}+\alpha _{2}+\alpha _{3}>\frac{1}{2}$,

then b is continuous from $H^{\alpha _{1}}(0,1)\times H^{\alpha _{2}+1}(0,1)\times H^{\alpha _{3}}(0,1)$ to ${\mathbb {R}}$, i.e.

$$\begin{aligned} \big |b(x,y,z)\big |\le C|x|_{\alpha _{1}}|y|_{\alpha _{2}+1}|z|_{\alpha _{3}}. \end{aligned}$$

The following inequalities can be derived by the above lemma.

Lemma A.3

For any $x\in H_{0}^{1}(0,1)$, we have

$(1) |B(x)|\le C||x||^{2}$.

$(2) |B(x)|_{-1}\le C|x|\cdot ||x||.$

Lemma A.4

For any $x,y\in H_{0}^{1}(0,1)$, we have

$(1) |B(x)-B(y)|\le C|x-y|_{1}(||x||+||y||)$.

$(2) |B(x)-B(y)|_{-1}\le C|x-y|\left( ||x||+||y||\right) .$

$(3) \langle B(x)-B(y), x-y\rangle \le C|x-y|(||x-|y||)||x||.$