Backward Compact Random Attractors for Stochastic Kuramoto-Sivashinsky Lattice Equation

QIAO Shanshan; LI Yangrong

doi:10.13718/j.cnki.xsxb.2022.08.007

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It is mainly used to study the non-autonomous random Kuramoto-Sivashinsky lattice equation in the case that the external force is backward slowly increasing. First, by estimating the solution, it is proved that the Kuramoto-Sivashinsky lattice equation has random absorption set on the space $\ell^{2} $, then deduce the existence of the backward uniform absorption set. Secondly, it is proved that the lattice equation is asymptotically backward compact on the absorption set. Finally, by means of the existence theorem of the attractor, it is proved that the non-autonomous random Kuramoto-Sivashinsky lattice equation has a backward compact random attractor on the space $\ell^{2} $.

Corresponding author: LI Yangrong

Received Date: 10/12/2021

Available Online: 20/08/2022

Key words:

Kuramoto-Sivashinsky lattice equations /
multiplicative noise /
backward compact random attractors

Abstract: It is mainly used to study the non-autonomous random Kuramoto-Sivashinsky lattice equation in the case that the external force is backward slowly increasing. First, by estimating the solution, it is proved that the Kuramoto-Sivashinsky lattice equation has random absorption set on the space $\ell^{2} $, then deduce the existence of the backward uniform absorption set. Secondly, it is proved that the lattice equation is asymptotically backward compact on the absorption set. Finally, by means of the existence theorem of the attractor, it is proved that the non-autonomous random Kuramoto-Sivashinsky lattice equation has a backward compact random attractor on the space $\ell^{2} $.

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文献[1-4]对Kuramoto-Sivashinsky方程的吸引子进行了研究，并建立了相对完善的理论体系. 文献[5-9]对非自治动力系统的拉回吸引子的存在性与后向紧性做了研究. 文献[10-11]研究了非自治格系统吸引子的后向紧性. 本文将研究具有乘性噪音的Kuramoto-Sivashinsky方程

的空间离散化. 其中$ \mathbb{Z}$代表整数集，υ>0，λ>8，β>2，W是完备概率空间(Ω，$\widetilde{\mathscr{F}} $，P)上的双边实值Wiener过程，Ω={ω∈C($\mathbb{R} $，$\mathbb{R} $)：ω(0)=0}，$\widetilde{\mathscr{F}} $是由Ω的紧开拓朴生成的σ-代数，P是在(Ω，$\widetilde{\mathscr{F}} $)上相应的Wiener测度. 对于非自治项g=(g_i)_{i∈$ \mathbb{Z}$}，有如下假设条件：

(G)g∈L_loc²($\mathbb{R} $，$\ell^{2} $)，且满足

本文主要研究随机Kuramoto-Sivashinsky格方程的后向紧随机吸引子.

1. 非自治随机动力系统

为了方便，定义$\ell^{2} $上的有界算子

其中涉及的三线性形式为

微分方程(1)可整理为

下面证明方程(4)能生成随机动力系统.

做变量替换v(t)=e^-z(θ_tω)u(t). 其中u(t)是方程(4)的解，$z\left(\theta_{t} \omega\right)=-\int_{-\infty}^{0} \mathrm{e}^{r} \theta_{t} \omega(r) \mathrm{d} r $是方程

的稳态解. 由文献[10, 12]可知，对任意ω∈Ω，z(θ_tω)关于t连续，且满足

因此方程(4)可转化为关于v的随机微分方程

由文献[1]和Galevkin逼近法，容易证明对任意T>0，v₀∈ $\ell^{2} $，ω∈Ω，方程(6)存在唯一的解v(∙，τ，ω，v₀)∈C([τ，+∞)，$\ell^{2} $)，且依赖初值v₀连续. 因此方程(6)在(Ω，$\widetilde{\mathscr{F}} $，P，{θ_t}_{t∈ $\mathbb{R} $})上能生成一个连续的随机动力系统{Φ(t)}_t≥0，即对v₀∈$\ell^{2} $，t≥0，τ∈$\mathbb{R} $和ω∈Ω，有

2. 解的估计

在下文中，设$\mathscr{D}_{0} $是X中所有缓增集构成的集合，$\mathscr{D} $是X中所有后向缓增集构成的集合. 若集合D满足

则称集合D为后向缓增集.

引理1  若条件(G)成立，则对任意后向缓增集D∈ $\mathscr{D} $，τ∈$\mathbb{R} $，ω∈Ω，存在T=T(D，τ，ω)≥1，使得t≥T且v_s-t∈D(s-t，θ_-tω)，有

成立，其中

证  对任意固定的τ∈ $\mathbb{R} $，ω∈Ω，v_s-t∈D(s-t，θ_-tω)，令

其中s≤τ. v(r)与方程(6)做内积，可得

其中

利用Young不等式，有

由文献[11]知‖Bv‖≤2‖v‖，故2‖Bv‖²≤8‖v‖². 又由(10)，(11)，(12)式以及Young不等式，整理可得

对(13)式利用Gronwall不等式，计算可得

对(14)式关于s∈(-∞，τ]取上确界，由于v_s-t∈D(s-t，θ_-tω)(s≤τ)，结合(5)，(7)式可知，存在T=T(s，ω，D)≥1，使得当t≥T时，有

因此可以得到

即(8)式得证.

命题1  若条件(G)成立，则对$\forall $ε>0，(τ，ω，D)∈($\mathbb{R} $ ×Ω× $\mathscr{D} $)，v_s-t∈D(s-t，θ_-tω)，存在T(ε，τ，ω，D)>0，K(ε，τ，ω，D)≥1，使得

证  构造光滑函数ρ，满足0≤ρ≤1，且当|s|≤1时ρ=0，当|s|≥2时ρ=1. 假设存在常数c₀，使得对任意s∈ $ \mathbb{R} $，有|ρ′(s)|≤c₀. 令K是一个固定的整数，设

ψ与方程(6)做内积，可得

其中

由于|ρ′(s)|≤c₀，因此

由Young不等式可知

由(17)-(21)式可得

对(22)式运用Gronwall引理，计算整理可得

根据(5)式可知，对$\forall $ε>0，存在C=C(ε，ω)>0，使得

由于v_s-t∈D(s-t，θ_-tω)(s≤τ)，因此，在(24)式中令$ \varepsilon <\frac{\lambda-8}{4}$，结合(5)，(7)式可得

由引理1与条件(G)可知，存在T>0，当t>T时有

因此，结合(26)-(28)式可得，对$ \forall$ε>0，(τ，ω，D)∈($ \mathbb{R}$ ×Ω× $\mathscr{D}$)，v_s-t∈D(s-t，θ_-tω)，存在T(ε，τ，ω，D)>0，K(ε，τ，ω，D)≥1，使得

3. 后向紧随机吸引子

定理1 若条件(G)成立，则方程(1)生成的动力系统存在后向紧随机吸引子.

证 {Φ(t)}_t≥0满足文献[13]中定理3.9的拉回吸引子的两个存在性条件：

(i) 非自治动力系统{Φ(t)}_t≥0存在$\mathscr{D}_0$ -拉回随机吸收集$\mathscr{K}_0$∈ $\mathscr{D}_0$，其中

(ii) 非自治动力系统{Φ(t)}_t≥0存在$\mathscr{D}$-拉回后向一致吸收集$\mathscr{K}$ ∈ $\mathscr{D}$，其中

由文献[6]可得非自治动力系统{Φ(t)}_t≥0在吸收集$\mathscr{K}$ ∈ $\mathscr{D}$上是后向紧的.

又因为随机吸引子的后向并是预紧的，则称该吸引子为后向紧随机吸引子. 因此方程(6)生成的非自治随机动力系统Φ(t)存在唯一的后向紧$\mathscr{D}$ -拉回吸引子$\mathscr{A}$ ∈ $\mathscr{D}$和唯一的可测$\mathscr{D}_0$-拉回吸引子$\mathscr{A}_0$∈ $\mathscr{D}_0$. 再由文献[12]的定理6.1知$\mathscr{A}$ =$\mathscr{A}_0$，故吸引子$\mathscr{A}$也是随机的，即Φ(t)存在唯一的后向紧$\mathscr{D}$ -拉回随机吸引子$\mathscr{A}$ ∈ $\mathscr{D}$. 再由文献[14-15]知方程(1)与(6)生成的随机动力系统共轭，从而可知方程(1)存在后向紧随机吸引子.

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Backward Compact Random Attractors for Stochastic Kuramoto-Sivashinsky Lattice Equation

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