1 Introduction
The Kuramoto–Sivashinsky (KS) equation is an important model to describe some physical phenomena. It properly describes the turbulence of chemical reactions, the movement of peaks in flames and the changes of unstable interfaces [Reference Kuramoto and Tsuzuki14]. The relevant properties of deterministic KS equation solutions have been extensively studied, such as the well-fitting problems and dynamical properties of one-dimensional KS solutions with periodic boundary conditions. We focus on its nonlocal variant. Here, a nonlocal term refers to an operator in the equation that depends on the global behaviour of the solution. The nonlocal operator arising is assumed by [Reference Duan and Ervin6, Reference Yang26]
One important application of the nonlocal operator arises in modelling the dynamics of a viscous liquid film flowing down inclined planes under electric fields. Specifically, Duan and Ervin [Reference Duan and Ervin6] used the semigroup method to prove the long-term dynamics of the nonlocal deterministic KS equation. However, the deterministic KS equation cannot fully describe chaos and turbulent flow. The main purpose of this paper is to consider the long time dynamics of the stochastic nonlocal KS (SNKS) equation with multiplicative white noise. We study the random attractor of the SNKS equation under the impact of nonlocal operator and multiplicative noise, which is given by
$$ \begin{align} du+(D^{4}u+D^{2}u+u\,Du+\alpha O(D^{3}u))\,dt=\sigma u\circ dW(t) \end{align} $$
with initial condition
$$ \begin{align*} u(x,t_{0})=u_{0}(x),\quad x\in G, \end{align*} $$
where
$D:={\partial }/{\partial x}$
,
$\circ $
is the Stratonovich integral,
$\alpha $
is a positive constant,
$x\in G=(-l,l)$
, u is periodic on G,
$\sigma $
is a constant,
$W(t)$
is a standard Brownian motion,
$t\in [0,T]$
. Generally, a term in an equation that contains a nonlocal operator is referred to as a nonlocal term. The Hilbert transform applied to the third spatial derivative of the unknown solution is to destabilize the liquid films falling down an inclined plane, subjected to a uniform electric field at infinity. This destabilization will lead to an enhancement of heat transfer. Random attractors are a compact set in the phase space, attracting the image of particular sets of initial states under the evolution of the stochastic system. We prove the existence of random attractors for the SNKS equation (1.1) in this paper. This work extends the results of Duan and Ervin [Reference Duan and Ervin6] who considered the maximal attractor due to the nonlocal term in the deterministic KS equation. They also proved the global well posedness of the one-dimensional stochastic KS equation. Wu et al. [Reference Wu, Cui and Duan22] proved that the stochastic generalized KS equation has globally well-posed solutions, but without the nonlocal term. The global existence of the solution was proved by Liu et al. [Reference Liu16–Reference Liu and Röckner19] for a large class of stochastic partial differential equations by using the variational approach. The variational framework has been used intensively for studying the existence of the solution for stochastic partial differential equations with nonlinear noise, where the coefficients satisfy some monotonicity and coercivity conditions. One method to prove the existence of random attractors is based on the global existence and regularity results of the solution for the stochastic partial differential equations.
The existence of random attractors for a large class of stochastic partial differential equations has been receiving much attention recently. For example, the existence of random attractors has been investigated by applying the theory of random dynamical systems [Reference Crauel and Flandoli5, Reference Gess, Liu and Röckner9, Reference Gess, Liu and Schenke10, Reference Yang26]. The theoretical framework of random dynamical systems in the sense of classical orbits can be seen in Arnold’s book [Reference Arnold1]. Crauel and Flandoli [Reference Crauel and Flandoli5] studied the dynamical properties of a series of stochastic partial differential equations driven by additive noise or linear multiplicative noise under this framework. For the existence of random attractors for the stochastic KS equation with additive noise, a common idea is to transform the stochastic KS equation into a random dynamic system by an auxiliary process called the Ornstein–Uhlenbeck process. For example, Liu and Röckner [Reference Liu and Röckner18] used the framework of random dynamical systems to consider the random attractors of stochastic KS equations driven by additive Lévy processes. Yang [Reference Yang26] proved that there exist finite-dimensional random attractors of the SNKS, where the noise is additive noise. However, research on random attractors for systems with multiplicative noise remains limited, with only specific classes of equations having been investigated. For example, Xu [Reference Xu and Caraballo23] considered the existence of random attractors for nonlocal stochastic reaction–diffusion equations with multiplicative and additive noise. Li [Reference Li and Xia15] obtained the (pullback) random attractors for the stochastic nonautonomous KS equations with multiplicative white noise and coloured coefficients recently, but they do not consider the impact of a nonlocal term on the equations.
However, the random attractors are only suitable to deal with the stochastic partial differential equations with additive or linear multiplicative noise. A weak version of a random attractor is used to deal with the general nonlinear term. Regarding research on random attractors, Kloeden and Lorenz [Reference Kloeden and Lorenz13] pioneered the concept of mean-square random attractors, which first resolved the existence problem of random attractors in infinite-dimensional systems. Wang extended the concept of mean-square random attractor to weak pullback mean random attractor in Bochner space and applied it to the stochastic Navier–Stokes equation [Reference Wang20]. There are more papers on the weak version of random attractors [Reference Gu11, Reference Kinra and Mohan12, Reference Wang21, Reference Xu and Caraballo23, Reference Xu and Caraballo24] to deal with the stochastic partial differential equations driven by the nonlinear noise. We also notice that Bo et al. [Reference Bo, Shi and Wang2] proved the existence and uniqueness of the weak solution and invariant measure of the SNKS equation by compensated Poisson random measures. In particular, Fan and Chen [Reference Fan and Chen8] considered the existence of weak pullback mean random attractor for the stochastic KS equation with additive noise without the nonlocal term. In this paper, we prove the existence of random attractor since the multiplicative noise is linear. Our method is based on the random dynamical system via a transformation into the associated conjugated random differential equation.
The content of this paper is as follows. We study the existence of random attractor for the SNKS equation with linear multiplicative noise. In Section 2, we introduce the basic space and the solution of the SNKS equation. In Section 3, we recall the theory of random dynamical systems and random attractors. It is shown that the unique solution of the SNKS equation generates a random dynamical system. In Section 4, we prove the existence and uniqueness of the random attractor for the SNKS equation and in the final section, we summarize the results.
2 Preliminaries
In this section, we introduce some basic concepts and symbols in the SNKS equation.
Denote the set of periodic functions with zero mean as
$$ \begin{align*} H=\bigg\{u\in L^{2}(G)\mid u(x+l)=u(x-l),\int _{G}u(x)\,dx=0\bigg\}, \end{align*} $$
which is endowed with the usual Sobolev space
$L^{2}(G)$
in the inner product
$(\cdot ,\cdot )$
and corresponding norm
$\|\cdot \|$
. Let
$V:= H^{2}\cap H$
, where
$H^{2}$
is the usual Sobolev space of periodic functions on G equipped with the norm
$\|\cdot \|$
,
$V^{*}$
is the dual space of V,
${}_{V^*}\langle z,v \rangle _{V} :=z(v) $
for
$z\in V^{*}, v\in V$
.
We denote the differential operator D by
$D:={\partial }/{\partial x}$
,
$A=-D^{2}$
,
$$ \begin{align*} A^{2}u = u_{xxxx}. \end{align*} $$
The eigenvalues and eigenvectors of self-adjoint operator A are
$\{\lambda _i\}_{i \in {Z}^+}$
and
$\{e_{i,1} , e_{i,2}\}_{i \in {Z}^+}$
, where
$$ \begin{gather*} \lambda_i = \bigg(\frac{\pi i}{l}\bigg)^2 ,\quad e_{i,1} = \sqrt{\frac{1}{l}}\sin\bigg(\frac{\pi ix}{l}\bigg),\quad e_{i,2}= \sqrt{\frac{1}{l}}\cos\bigg(\frac{\pi ix}{l}\bigg). \end{gather*} $$
Furthermore, we define the fractional index of operator A as
$$ \begin{align*} A^m u = \sum_{i=1}^{\infty} \lambda_i^m(u,e_i)e_i \quad \text{and} \quad ||A^m u||^2 = \sum_{i=1}^{\infty} \lambda_{i}^{2m}(u,e_i) < \infty, \end{align*} $$
$m \in {\mathbb {R}}$
. For all integer n, the dual space of inner product in Domain
$(A^{n})$
on
${H}$
is defined by (Domain
$(A^{n}))^*=$
Domain
$(A^{-n})$
. Let
$H^{n}(G)$
denote the Sobolev space of order n with periodic boundary conditions. Specifically, for natural numbers
$n \in \mathbb {N}$
, the space Domain
$(A^{n/2})$
is equivalent to
${H}^n(G)$
. The domain of
$A^{n/2}$
is
$$ \begin{align*} {V}^n = \text{Domain}(A^{n/2}) \quad \text{and} \quad |x|_{V^n} = (D^n x,D^n x)^{{1}/{2}} = ||D^n x||. \end{align*} $$
We will use the following Poincaré inequality:
$$ \begin{align} |x|_{V^n}\leq \lambda_1 |x|_{V^m}, \quad n\leq m,\quad x\in V^m. \end{align} $$
In particular, when
$m> 0$
, self-operator
$A^m$
generates a
$C_0$
linear semigroup in
${H}$
, which is defined by
$$ \begin{align*} S(t)u = e^{-A^m t}u = \sum_{i=1}^{\infty}e^{-{\lambda_i^{m}}t}u_i e_i \end{align*} $$
for all
$t>0$
, and
$$ \begin{align*} u = \sum_{i=1}^{\infty}u_i e_i \in H. \end{align*} $$
We define the operator,
$$ \begin{align} K(u):=-(D^{4}u+D^{2}u+u\,Du+\alpha H(D^{3}u)). \end{align} $$
The bilinear operator B is defined by
$$ \begin{align} (B(u,v),z) = b(u,v,z): = \int_{-l}^{l} u(x)(Dv(x))z(x)\, dx. \end{align} $$
By
$H^1(G) \subset L^\infty (G)$
, for all
$u_1,u_3 \in {H}$
and
$u_2 \in {V}$
, there exists a constant c such that
$$ \begin{align*} |b(u_1,u_2,u_3)| \le ||u_1|||Du_2|_{L^\infty}||u_3|| \le c||u_1||||D^{2}u_2||||u_3||. \end{align*} $$
Through integration by parts of (2.3), we have
$$ \begin{align*} \begin{cases} b(u,u,u) = 0,\\ b(u_1,u_2,u_2) = b(u_2,u_2,u_1) = -\frac{1}{2}b(u_2,u_1,u_2),\\ b(u_1,u_2,u_3) = - b(u_2,u_1,u_3) -b(u_1,u_3,u_2). \end{cases} \end{align*} $$
Nonlocal operator O is a linear operator which has periodic Hilbert transform and which possesses the following properties [Reference Duan and Ervin6, Reference Yang26]:
$$ \begin{align*} \begin{aligned} &DO=OD;\\ &O^{-1}=-O;\\ &\int uO(v)\, dx=\int vO(u)\, dx;\\ &\int O(u)O(v)\, dx=\int uv\, dx;\\ &\int uO(u)\, dx=0;\\ &\| O(u) \| =\|u\|. \end{aligned} \end{align*} $$
Let
$(\Omega , \mathscr {F}, \{\mathscr {F}_t\}_{t \in {\mathbb {R}}}, \mathbb {P})$
be a complete filtered probability space satisfying the usual condition, that is,
$ \{\mathscr {F}_t\}_{t \in {\mathbb {R}}}$
is an increasing and right continuous family of sub-
$\sigma $
-algebras of
$\mathscr {F}$
that contains all
$\mathbb {P}$
-null sets.
Definition 2.1. A stochastic process
$\{X_t\}_{t \in \mathbb {R}}$
is said to be
$\mathscr {F}_t$
-measurable if for each
$t \in \mathbb {R}$
, the random variable
$X_t$
is measurable with respect to the
$\sigma $
-algebra
$\mathscr {F}_t$
.
This implies
$X_t$
is completely determined by the information available up to time t in the filtration
$\{\mathscr {F}_s\}_{s \leq t}$
. We are now in a position to state the following definition.
Definition 2.2. [Reference Yang25, Reference Yang26] A stochastic process
$(u(t))_{t \ge 0}$
is called a weak solution of the SNKS (1.1) if it satisfies the following:
-
(1)
$u(t)$
is
$\mathscr {F}_t$
-adapted, which is
$\mathscr {F}_t$
-measurable for every t; -
(2)
$u(t)\in L^{\infty }(0,T;H)\cap L^{2}(0,T;V)$
, almost surely for any
$T> 0$
; -
(3) the identity
holds
$$ \begin{align*} u(t)=u(0)+\int_{0}^{t}K(u)\,ds+\int_{0}^{t}\sigma u\circ dW_{s} \end{align*} $$
$\mathbb {P}$
-almost surely in
$V^{2,*}$
, the dual space of
$V^2$
,
$K(u)$
is defined by (2.2).
3 Random dynamical systems
Let
$(X,\|\cdot \|_{X})$
be a Banach space with Borel
$\sigma $
-algebra
$\mathscr {B}(X)$
, and
$({\Omega },\mathscr {F}, \mathbb {P})$
be the corresponding probability space.
Definition 3.1.
$(\Omega ,\mathscr {F}, \mathbb {P},(\theta )_{t\in \mathbb {R}})$
is called a metric dynamical system if
${\theta :\mathbb {R}{\kern-1pt}\times{\kern-1pt} \Omega {\kern-1pt}\to{\kern-1pt} \Omega }$
is
$ ( \mathscr {B}(\mathbb {R}) \times \mathscr {F},\mathscr {F} )$
-measurable,
$\theta _{0}$
is the identity on
$\Omega $
,
$\theta _{s+t}=\theta _{s}\circ \theta _{t}$
for all
$s,t\in \mathbb {R}$
and
$\theta _{t} \mathbb {P}= \mathbb {P}$
for all
$t\in \mathbb {R}$
.
Definition 3.2. A random dynamical system (RDS) on Banach space X over a metric dynamical system
$(\Omega ,\mathscr {F}, \mathbb {P},(\theta )_{t\in \mathbb {R}})$
is a mapping
$$ \begin{align*} \varphi : \mathbb{R}^{+}\times \Omega \times X \to X , \quad (t,\omega,x) \mapsto \varphi(t,\omega,x), \end{align*} $$
which is
$(\mathscr {B}(\mathbb {R}^{+})\times \mathscr {F}\times \mathscr {B}(X),\mathscr {B}(X))$
-measurable and satisfies, for almost all
$\omega \in \Omega $
:
-
(1)
$\varphi (0,\omega ,\cdot )$
is the identity on X; -
(2)
$\varphi (t+s,\omega ,\cdot )=\varphi (t,\theta _{t}\omega ,\cdot )\circ \varphi (s,\omega ,\cdot )$
for all
$t,s\in \mathbb {R}^{+}$
.
When a stochastic differential equation is driven by a two-sided Wiener process
$W_{t}$
, the probability space can be identified with the canonical space of continuous mappings
$\Omega = C_0(\mathbb {R}; \mathbb {R})$
, that is, an element
$\omega $
of the space
$\Omega $
is a continuous function
$\omega : \mathbb {R}\to \mathbb {R}$
such that
$\omega (0)=0$
. Define the time shift by
$$ \begin{align*} \theta_{t}\omega(\cdot)=\omega(t+\cdot)-\omega(t), \quad t\in \mathbb{R}. \end{align*} $$
Moreover, we can identify
$W_{t}=\omega (t)$
for every
$\omega \in \Omega $
.
It is well known that stochastic differential equations driven by additive noise generate random dynamical systems. A random dynamical system for a particular noise is obtained by the following lemma [Reference Caraballo, Kloeden and Schmalfuss4].
Lemma 3.3. Let
$\psi $
be an RDS. Suppose that the mapping
$T:\Omega \times X\to X$
possesses the following properties: for every fixed
$\omega \in \Omega $
, the mapping
$T(\omega ,\cdot )$
is a homeomorphism on X, and for fixed
$x\in X$
, mappings
$T(\cdot ,x),T^{-1}(\cdot ,x)$
are measurable. Then, the mapping
$$ \begin{align*} (t,\omega,x)\to \varphi(t,\omega,x):=T^{-1}(\theta_{t}\omega,\psi(t,\omega,T(\omega,x))) \end{align*} $$
is an (conjugated) RDS.
We will transform our stochastic KS equation driven by multiplicative white noise into a random equation without noise but random coefficients.
We recall the definition of random sets and random attractors [Reference Crauel and Flandoli5].
Definition 3.4. Let
$(\Omega ,\mathscr {F}, \mathbb {P})$
be a probability space. A random set C on X is a measurable subset of
$X\times \Omega $
with respect to the product
$\sigma $
-algebra of the Borel
$\sigma $
-algebra of X and
$\mathscr {F}$
.
A random set
$\mathscr {C}$
also can be rewritten as
$$ \begin{align*} \mathscr{C}(\omega)=\{x\in X\mid(x,\omega)\in \mathscr{C}\}, \quad \omega\in\Omega. \end{align*} $$
When a random set
$ \mathscr {C}(\omega )$
is closed, it is said to be a closed random set if and only if for every
$x\in X$
, for all
$\omega \in \Omega $
, the mapping
$$ \begin{align*} \omega \mapsto d(x,\mathscr{C}(\omega))\in [0,+\infty) \end{align*} $$
is measurable and
$ d(x,\mathscr {C}(\omega ))=\inf _{y\in \mathscr {C}(\omega )}||x-y||_{X}.$
Similarly, when the fibres
$\mathscr {C}(\omega )$
are compact,
$\mathscr {C}$
is said to be a compact random set.
Definition 3.5. A random bounded set
$\mathscr {B}(\omega )\subset X$
is called tempered with respect to
$(\theta _{t})_{t\in R}$
if for almost every
$\omega \in \Omega $
,
$$ \begin{align*} \lim_{t\to \infty}e^{-\beta t}d(0,\mathscr{B}(\theta_{-t}\omega))=0\quad \mbox{for all } \, \beta>0. \end{align*} $$
Definition 3.6. Let
$\mathscr {D}$
be a universe, that is, a collection of random sets in X. A random set
${\mathscr {K}(\omega )}$
is said to be an absorbing set for a random dynamical system
$\varphi $
with respect to
$\mathscr {D}$
if for all
$\omega \in \Omega $
almost surely, there exists
$t_{\mathscr {B}}(\omega )>0$
such that
$$ \begin{align*} \varphi(t,\theta_{-t}\omega,\mathscr{B}(\theta_{-t}\omega))\subset \mathscr{K}(\omega) \quad \mbox{for all } t \geq t_{\mathscr{B}}(\omega). \end{align*} $$
Definition 3.7. Let
$\mathscr {D}$
be a universe in X. A random set
${\mathscr {A}}(\omega )$
of X is called a global random
$\mathscr {D}$
-attractor (pullback
$\mathscr {D}$
-attractor) for random dynamical system
$\varphi $
if the following conditions are satisfied for almost every
$\omega \in \Omega $
:
-
(1)
${\mathscr {A}(\omega )}$
is a random compact set; -
(2)
${\mathscr {A}(\omega )}$
is strictly invariant, that is,
$$ \begin{align*} \varphi(t,\omega,{\mathscr{A}}(\omega))={\mathscr{A}}(\theta_{t}\omega) \quad \mbox{for all } t\geq 0; \end{align*} $$
-
(3)
${\mathscr {A}(\omega )}$
attracts all sets in
$\mathscr {D}$
, that is, for all
$\mathscr {B}={\mathscr {B}(\omega )}\in \mathscr {D}$
, we have where
$$ \begin{align*} \lim_{t\to +\infty}d(\varphi(t,\theta_{-t}\omega,\mathscr{B}(\theta_{-t}\omega)),{\mathscr{A}}(\omega))=0, \end{align*} $$
is the Hausdorff semi-metric for
$$ \begin{align*} d(A,B)=\sup_{x\in A}\inf_{y\in B}||x-y||_{X} \end{align*} $$
$ A,B\subset X$
.
The existence result of random attractors can be stated as follows [Reference Crauel and Flandoli5].
Theorem 3.8. Suppose there exists a random compact set that absorbs every bounded deterministic set
$B\subset X$
. Then, the set
$$ \begin{align*} {\mathscr{A}}(\omega)=\overline{\bigcup_{B\subset X}\Lambda_{B}(\omega)} \end{align*} $$
is a random attractor for
$\varphi $
, where the union is taken over all bounded B in X, and
$\Lambda _{B}(\omega )$
is the omega-limit set of B given by
$$ \begin{align*} \Lambda_{B}(\omega)=\bigcap_{T\geq 0}\overline{\bigcup_{t\geq T}\varphi(t,\theta_{-t}\omega,B)}. \end{align*} $$
Next, we prove the existence and uniqueness of the weak solution of the SNKS equation with multiplicative white noise.
Suppose that the process
$W(t)$
is a two-sided standard Brownian motion which is constant in space. Consider the following Langevin equation:
$$ \begin{align} \begin{aligned} &dz=-z\,dt+dW_{t},\quad t\geq t_{0},\\ &z(t_{0})=z_{0}. \end{aligned} \end{align} $$
The solution of (3.1) has the form
$$ \begin{align*} z(t;t_{0},z_{0})=e^{-(t-t_{0})}z_{0}+W_{t}-e^{-(t-t_{0})}W_{t_{0}}-\int_{t_{0}}^{t}e^{-(t-s)}W_{s}\,ds. \end{align*} $$
Let
$t_{0}\to -\infty $
; then the solution of (3.1) has the form
$$ \begin{align*} z^{*}(\theta_{t}\omega):=\lim_{t_{0}\to -\infty}z(t;t_{0},z_{0})=W_{t}-\int_{-\infty}^{t}e^{-(t-s)}W_{s}\,ds. \end{align*} $$
Therefore,
$z^{*}(\omega )=-\int _{-\infty }^{0}e^{s}W_{s}(\omega )\,ds$
. Let
$z(t,\omega )=z^{*}(\theta _{t}\omega )$
; note that
$z(t,\omega )$
is a stationary solution of the Langevin equation (3.1). In addition,
$\overline {\Omega }$
is a subset of
$\Omega $
satisfying
$\mathbb {P}(\overline {\Omega })=1$
for
$\omega \in \overline {\Omega } $
[Reference Xu and Caraballo23],
$$ \begin{align} \begin{aligned} &\lim_{t\to \pm\infty}\frac{|z^{*}(\theta_{t}\omega)|}{|t|}=0,\\ &\lim_{t\to \pm\infty}\frac{1}{t}\int_{0}^{t}z^{*}(\theta_{\tau}\omega)\,d\tau=0,\\ &\lim_{t\to \pm\infty}\frac{1}{t}\int_{0}^{t}|z^{*}(\theta_{\tau}\omega)|\,d\tau=E|z^{*}|< \infty. \end{aligned} \end{align} $$
Remark 3.9. The process
$z(\cdot ,\omega )$
is a stationary process which is called an Ornstein–Uhlenbeck process [Reference Duan, Lu and Schmalfuss7]. Here,
$E|z^{*}|$
is the expectation of realizations
$\omega $
. Since
$\mathbb {P}(\bar {\Omega }) = 1$
, we may restrict our analysis to
$\bar {\Omega }$
without loss of generality. Therefore, we will continue to use the same probability space notation when referring to the restricted space
$\bar {\Omega }$
, inheriting all its established properties.
Next, we move into the change of variable
$v(t)=u(t)e^{-\sigma z^{*}(\omega )}$
, we get the process
$v(\cdot ):=v(\cdot ;t_{0},\omega ,v_{t_{0}})$
with initial value
$v(t_{0})=v_{0}=u_{0}e^{-\sigma z^{*}(\theta _{t_{0}}\omega )}$
; by integration by parts, we have
$$ \begin{align} \begin{aligned} &dv+(D^{4}v+D^{2}v+e^{-\sigma z^{*}(\theta_{t}\omega)}v\,Dv+\alpha O(D^{3}v))\,dt=z^{*}(\theta_{t}\omega)\sigma v\,dt,\\ &v(x,t_{0})=v_{0}=u_{0}e^{-\sigma z^{*}(\theta_{t_{0}}\omega)},\quad x\in G. \end{aligned} \end{align} $$
Let
$\{e_k\}_{k=1}^\infty $
be an orthonormal basis of H. For fixed
$n \in \mathbb {N}$
, define
${H_n := \operatorname {span}\{ e_1, \dots , e_n \}}$
. Here,
$P_n$
is the orthogonal projection operator from
$H$
onto
$H_n$
.
The Galerkin approximation to (3.3) is
$$ \begin{align} \begin{aligned} &\frac{dv_{n}}{dt} + P_{n} ( D^{4}v_n + D^{2}v_n + e^{-\sigma z^{*}(\theta_{t}\omega)}v_n Dv_n + \alpha \mathcal{O}(D^{3}v_n) ) = \sigma z^{*}(\theta_{t}\omega)v_{n}(t), \\ &v_{n}(0) = P_n ( u_{0} e^{-\sigma z^{*}(\omega)} ). \end{aligned} \end{align} $$
An a priori estimate is needed to guarantee the global existence of a solution
$v_n$
[Reference Yang26].
Lemma 3.10. (Energy estimate) Let
$v_{n}$
be a solution of the stochastic ordinary differential equation (ODE) (3.4), then there exists a constant
$C>0$
such that for all
$n\in \mathbb {N}$
, we have
$$ \begin{align} E(\sup_{t\in[0,T]}\|v_{n}(t)\|^{2})+E\int_{0}^{T}\|D^{2}v_{n}(t)\|^{2}\,dt\leq C. \end{align} $$
Proof. Since
$B(u,u,u)=0$
, we obtain the energy equality,
$$ \begin{align} \frac 1 2 \frac {d\|v_{n}\|^{2}}{dt}+\|D^{2}v_{n}\|^{2}-\|Dv_{n}\|^{2}+\alpha \int v_{n}\cdot O(D^{3}v_{n})=\sigma z^{*}(\theta_{t}\omega)\|v_{n}\|^2. \end{align} $$
Let
$F(\omega , v)=\sigma z^{*}(\omega )\|v_{n}\|^2$
, then there exist
$a(\omega )$
and
$b(\omega )$
such that [Reference Wu, Cui and Duan22]
$$ \begin{align*} \begin{aligned} &F(\omega,t)\leq a(\omega)(1+|t|),\\ &F(\omega,t)-F(\omega,s)(t-s) \leq b(\omega) |t-s|^2. \end{aligned} \end{align*} $$
Moreover, for the nonlocal term, we get the estimation
$$ \begin{align*} \int v_{n}\cdot O(D^{3}v_{n})= \int Dv_{n}\cdot O(D^{2}v_{n})\leq \|Dv_{n}\|\|D^2v_{n}\|\leq \frac 1 4\|D^2v_{n}\|^2+\|Dv_{n}\|^2. \end{align*} $$
We apply the similar result in [Reference Caraballo, Herrera-Cobos and Marín-Rubio3] to obtain the energy inequality (3.5).
The next step to take the limit of n to infinity on both sides of (3.6) is standard, so the stochastic nonlocal equation (3.3) has a weak unique solution. Finally, we obtain the following theorem by the transformation
$v(t)=u(t)e^{-\sigma z^{*}(\omega )}$
.
Theorem 3.11. Suppose
$u_0\in H$
. There exists a unique weak solution to (1.1) in the sense of Definition 2.2. In addition, this solution behaves continuously in H with respect to the time t.
Now, we prove that the solution of the nonlocal KS equation (1.1) generates a random dynamical system. We transform (1.1) with multiplicative noise into the random equation. Define a homeomorphic mapping
$T(\omega ):L^{2}(G) \to L^{2}(G)$
by
$T(\omega )u=e^{\sigma z^{*}(\omega )}u$
. By the following change of variable,
$$ \begin{align*} v(t)=u(t)e^{\sigma z^{*}(\theta_{t}\omega)}=T^{-1}(\theta_{t}\omega)u(t), \end{align*} $$
where
$u(t)$
and
$v(t)$
are the solutions of (1.1) and (3.3), respectively.
Since
$T(\omega )$
is a homeomorphism, if (3.3) generates a random dynamical system, then (1.1) induced by
$T(\omega )$
also constitutes a random dynamical system by Lemma 3.3.
Theorem 3.12. To obtain a solution
$v(t)$
for (3.3), a continuous random dynamical system
$\psi (t)$
over
$(\Omega , \mathscr {F},\mathbb {P}, {(\theta _t)}_{t \in {\mathbb {R}}} )$
is generated by (3.3), where
$$ \begin{align*} \psi(t,\omega,v_{0})=v(t;0,\omega,v_{0})\quad \mbox{for all } v_{0}\in L^{2}(G), \ \mbox{and}\ t\geq 0. \end{align*} $$
Define
$$ \begin{align*} \varphi(t,\omega,v_{0})=T(\theta_{t}\omega)\psi(t,\omega,T^{-1}(\omega)v_{0}), \end{align*} $$
then
$\varphi $
is also a random dynamical system generated by (1.1).
Proof. By the initial condition
$\psi (0,\omega ,v_{0})=v(0;0,\omega ,v_{0})=0$
,
$ \psi (0,\omega ,\cdot )$
is the identity on
$L^{2}(G)$
. Then, we verify the cocycle property
$$ \begin{align*} \psi(t,\theta_{r}\omega,\psi(r,\omega,\cdot))= \psi(t+r,\omega,\cdot),\quad t,r\geq 0. \end{align*} $$
Since
$$ \begin{align*}\psi(t,\theta_{r}\omega,\psi(r,\omega,v_{0}))= v(t;0,\theta_{r}\omega,v(r;0,\omega,v_{0})), \end{align*} $$
$$ \begin{align*} \psi(t+r,\omega,v_{0})= v(t+r;0,\omega,v_{0}), \end{align*} $$
both the solutions
$v(t;0,\theta _{r}\omega ,v(r;0,\omega ,v_{0}))$
and
$v(t+r;0,\omega ,v_{0})$
have the same initial value
$v(0)=v_{0}$
. Thus, we obtain the cocycle property by the uniqueness of the solution in Theorem 3.11. Since
$T(\omega )$
is a homeomorphism,
$\varphi $
is a conjugated random dynamical system by Lemma 3.3. This completes the proof.
4 Existence of random attractors
We now apply Theorem 3.8 to prove the existence of a random attractor for the KS equation (1.1).
Theorem 4.1. There exists a random attractor
${\mathscr {A}}(\omega )$
for a random dynamical system associated to solutions of the SNKS equation (1.1).
Proof. We will prove Theorem 4.1 by defining a random compact absorbing set
$K(\omega )$
. We need to derive the boundedness of
$v(t)$
in
$L^{2}(G)$
.
Suppose
$ u$
is a solution of (1.1) in
$L^{2}(G)$
with initial value
$u(s)$
. Let
$v(t)=v(t;s,\omega , v(s))$
with initial value
$v(s)=u(s)e^{-\sigma z^{*}(\theta _{t}\omega )}$
, then
$$ \begin{align*} \begin{aligned} dv+(D^{4}v+D^{2}v+e^{-\sigma z^{*}(\theta_{t}\omega)}v\,Dv+\alpha O(D^{3}v))=z^{*}(\theta_{t}\omega)v\,dt. \end{aligned} \end{align*} $$
Through periodicity and inner product, we have the energy equality of (3.3),
$$ \begin{align*} \frac 1 2 \frac {d\|v\|^{2}}{dt}+\|D^{2}v\|^{2}-\|Dv\|^{2}+(\alpha O(D^{3}v),v)=z^{*}(\theta_{t}(\omega))\|v\|^{2}. \end{align*} $$
We obtain the following inequality:
$$ \begin{align*} \begin{aligned} 2|(\alpha O(D^{3}v,v))|&\leq 2\alpha \|O(D^{2}v)\|\|Dv\| \\ &=2\alpha \|D^{2}v\|\|Dv\| \\ &\leq \|D^{2}v\|^{2}+\alpha^{2}\|Dv\|^{2}. \end{aligned} \end{align*} $$
By the above inequality, we get
$$ \begin{align*} \begin{aligned} \frac{d \|v(t)\|^{2}}{dt}+ 2\|D^{2}v\|^{2} &\leq (2+\alpha^{2})\|Dv(t)\|^{2}+\|D^{2}v\|^{2}+2z^{*}(\theta_{\tau}(\omega))\|v\|^{2},\\ \frac{d \|v(t)\|^{2}}{dt} &\leq (2+\alpha^{2})\|Dv(t)\|^{2}+2z^{*}(\theta_{\tau}(\omega))\|v\|^{2}-\|D^{2}v\|^{2}\\ &\leq [\lambda_{1}(2+\alpha^{2}-\lambda_{1})+2z^{*}(\theta_{t}(\omega))]\|v(t)\|^{2}, \end{aligned} \end{align*} $$
where
$\lambda _1$
is selected large enough from the Poincaré inequality (2.1). We obtain
$$ \begin{align*} \begin{aligned} \|v(t)\|^{2}\leq \exp\bigg\{\kern-2pt\int_{s}^{t}[\lambda_{1}(2+\alpha^{2}-\lambda_{1})+2z^{*}(\theta_{\tau}(\omega))]\,d\tau \bigg\}\|v(s)\|^{2}. \end{aligned} \end{align*} $$
Then, we conclude that
$$ \begin{align*} \|u(t)\|^{2}\leq \exp\{z^{*}(\theta_{t}\omega)-z^{*}(\theta_{s}\omega)\} \exp\bigg\{\kern-2pt\int_{s}^{t}[\lambda_{1}(2+\alpha^{2}-\lambda_{1})+2z^{*}(\theta_{\tau}(\omega))]\,d\tau \bigg\}\|u(s)\|^{2}. \end{align*} $$
By (3.2), let
$t=-1$
for a given deterministic bounded set
$D\subset B(0,\rho )$
. Then, there exists a small enough
${T}(\omega , \rho )<-1$
such that for all
$s< {T}$
,
$u_0\in D$
,
$$ \begin{align*} \begin{aligned} \exp \{z^{*}(\theta_{-1}\omega)-z^{*}(\theta_{s}\omega)\} \exp\bigg\{\kern-2pt\int_{s}^{-1}[\lambda_{1}(2+\alpha^{2}-\lambda_{1})+2z^{*}(\theta_{\tau}(\omega))]\, d\tau\bigg\}\|u_0\|^{2}\leq 1, \end{aligned} \end{align*} $$
which implies the following result:
$$ \begin{align*} \begin{aligned} \|u(t)\|^{2}\leq 1. \end{aligned} \end{align*} $$
When
$t\in [-1,1]$
, for a random variable
$$ \begin{align*} r_{t}(\omega)=\exp\bigg\{\kern-2pt\int_{-\infty}^{t}z^{*}(\theta_{\tau}(\omega))\,d\tau\bigg\}, \end{align*} $$
we have
$$ \begin{align*} \|u(t)\|^{2}\leq r_{t}(\omega). \end{align*} $$
Next, we prove the existence and uniqueness of the random attractor. Let
${r_{0}(\omega )=:r(\omega )}$
,
$$ \begin{align*} \mathscr D=\{u(t)\in L^{\infty}(0,T;H)\cap L^{2}(0,T;V)\mid \|u(t)\|^{2}\leq r(\omega)\}. \end{align*} $$
Since
$\mathscr D $
is a bounded closed convex subset of
$L^{2}(0,T;V)$
, it is compact in
$L^{2}(0,T;H)$
. In addition, it is shown that for all
$t\in [0,+\infty )$
, there exists
${T}>0$
such that for all
$s< {T}$
,
$u(s)\in B(0,r_{0})\subset \mathscr {D}$
and
$$ \begin{align*} \varphi(s,\theta_{-s}\omega, B(0,r_{0}))\subset \mathscr D. \end{align*} $$
Then, the random dynamical system
$\varphi $
for (1.1) has a compact absorbing set
$\mathscr {D}$
. Through Theorem 3.8, the random dynamical system
$\varphi $
for (1.1) has a unique random attractor
$\mathcal {A}(\omega )$
in H over
$(\Omega ,\mathscr {F}, \mathbb {P},(\theta )_{t\in \mathbb {R}})$
. This completes the proof.
5 Conclusion
This paper investigates the long-time dynamics of the SNKS equation driven by multiplicative white noise. By transforming the stochastic equation into a conjugated random differential equation, we establish the existence and uniqueness of solutions within the framework of RDS and prove the existence of random attractors. These results characterize the system’s asymptotic behaviour under multiplicative noise. These theoretical advances provide a foundation for modelling interfacial phenomena in viscous liquid films under electric fields. Future research should investigate systems with nonlinear multiplicative noise or perturbations driven by Lévy processes, explore complex nonlocal effects and noise interaction mechanisms in higher spaces, and develop efficient numerical methods for simulation.
Acknowledgements
This work was supported by the Guangdong Province Ordinary Universities Characteristic Innovation Project (2022KTSCX037) and the Guangdong–Dongguan Joint Research Grant (2023A1515140016).
