A Stochastic Weakly Damped, Forced KdV-BO Equation

and Applied Analysis 3 and then u satisfies (5) if and only if V is a solution of V t + αHV xx + βV xxx + λV + VV x + uu x + Vu x + uV x = f. (14) This is a weakly damped, forced KdV-BO equation with random parameter with the following initial condition: V (0, x) = u (0, x) + u (0, x) = u (0, x) . (15) The estimation about the random parameter u and the bilinear term VV x in (14) can be obtained by using themethod in [6]. Then by the fixed point argument the global existence results to the random parameter Cauchy problem (14)-(15) can be obtained. Theorem 3. Assume that Φ ∈ L 2 (or Φ ∈ L 0,2 2 ); let V 0 ∈ L 2 (R) (or V 0 ∈ H 1 (R)). Then, the solution V(t) of problem (14) and (15) is global and belongs to C([0, T]; L(R)) (or C([0, T];H 1 (R)) for any T > 0. We summarize the above existence results for P-a.s. ω ∈ Ω of (14) with initial condition V(s, x) = u(s, x) = V s , s ∈ R, as follows. (i) Under the assumption of Theorem 3, for s < T, and any T ∈ R and V s ∈ L 2 (R), there exists a unique solution V ∈ C([s, T]; L(R)). (ii) Under the assumption of Theorem 3, for s < T, and any T ∈ R, and V s ∈ H 1 (R), there exists a unique solution V ∈ C([s, T];H(R)). (iii) Denoting such a solution V(t, ω; s, V s ), the mapping V s 󳨃→ V(t, ω; s, V s ) is continuous for all s ⩽ T. Nowwe construct anRDSmodeling the stochasticweakly damped, forced KdV-BO equation. For example, consider a set of continuous functions with value 0 at t = 0 Ω = {ω ∈ C (R,R) : ω (0) = 0} . (16) Let F be the Borel sigma-algebra induced by the compact open topology ofΩ, and letP be aWienermeasure on (Ω,F). We write (β 1 (t, ω), . . . , β k (t, ω), . . .) = ω(t). The time shift is simply defined by θ s ω (t) = ω (t + s) − ω (s) , t, s ∈ R, (17) and then (Ω,F, P, θ t ) is an ergodic metric dynamical system which models white noise. Having the mapping V s 󳨃→ V(t, ω; s, V s ), we define u (t, ω; s, u s ) = S (t, s; ω) u s = V (t, ω; s; V s ) + u (t, ω) , (18) where V(t, ω; s; V s ) is a solution to (14) with V(s) = V s and u(t) satisfies du + {αH∂ 2 x u + β∂ 3 x u + λu} dt = ΦdW, u (s) = 0. (19) Obviously, for s ⩽ r ⩽ t, we have S (t, s; ω) = S (t, r; ω) S (r, s; ω) . (20) Thanks to (17), for any s, t ∈ R, u s ∈ H 1 (R), we have P-a.s. S (t + s, 0; ω) u 0 = S (t, 0; θ s ω) S (s, 0; ω) u 0 . (21) Therefore, the process Ψ : R × Ω × V → V defined by Ψ (t, ω) u 0 = S (t, 0 : ω) u 0 (22) is cocycle. It is continuous RDS on H(R) over (Ω,F, P, (θ t ) t∈R) andmodels the dynamical system associated with the stochastic equation (5) with initial value u(s) = u s . 4. Compact Random Absorbing Set In the following computations ω ∈ Ω is fixed. We usually denote L(R) (1 ⩽ p ⩽ ∞) by L x ; L([0, T]; L(R)) (1 ⩽ p ⩽ ∞, 1 ⩽ q ⩽ ∞) by Lp t (L q x ); and L(R; L([0, T]))(1 ⩽ p ⩽ ∞, 1 ⩽ q ⩽ ∞) by L x (L q t ) for any T > 0 in this part. First note that, for the Hilbert transform, we have for any f, g ∈ L 2 (R) H 2 f = −f, H (fg) =H (HfHg) + fHg + gHf, (f,Hg) = − (Hf, g) , (Hf, f) = 0, (Hf,Hg) = (f, g) , 󵄩󵄩󵄩󵄩Hf 󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩 , ∀f ∈ H 1 (R) , Hf x = (Hf) x . (23) Before we prove the existence of a compact random absorbing set, we first give some estimates about the solution u of problem (P). Let us introduce the following space to study the solution of problem (P): X σ (T) = {u ∈ C (0, T;H σ (R)) ∩ L 2 (R; L ∞ ([0, T])) , D σ ∂ x u ∈ L ∞ (R; L 2 ([0, T])) , ∂ x u ∈ L 4 ([0, T] ; L ∞ (R))} , (24) for some σ < 1. Lemma 4 (see [11]). Assume that Φ ∈ L 2 for some σ > 3/4; then u is almost surely in X σ (T) for any T > 0 and any σ such that 3/4 < σ < σ. More precisely, E( sup t∈[0,T] ‖u‖ 2 H σ x ) ⩽ C‖Φ‖ 2 L 0,σ 2 , for any 3 4 < σ ⩽ σ,


Introduction
The deterministic Korteweg-de Vries-Benjamin-Ono (KdV-BO) equation describes a large class of internal waves in the ocean and stratified fluid.The well-posedness for this equation was studied in [1,2].When the surface of the fluid is submitted to a nonconstant pressure, or when the bottom of the layer is not flat, a forcing term has to be added to the equation [3,4].The long time behavior of the forced generalized KdV-BO was studied in [5].In this paper we are interested in the case when the forcing term is random.The well-posedness for the stochastic KdV-BO driven by the additive noise was studied in [6].We have found no studies on the long time behavior of the stochastic KdV-BO equation.
In this paper, we consider the long time behavior of the following stochastic damped, forced KdV-BO equation: where , , and  are real constants and  ̸ = 0. H denotes the Hilbert transform The forcing term  is time independent, and  is a random process defined on (, ) ∈ R + × R. Φ is a linear operator.Also,  is a two-parameter Brownian motion on R + × R, that is, a zero mean Gaussian process whose correlation function is given by E ( (, )  (, )) = ( ∧ ) ( ∧ ) , for ,  ⩾ 0, ,  ∈ R.
Alternatively we consider a cylindrical Wiener process  by setting where {  } ∈N is an orthonormal basis of  2 (R) and {  } ∈N is a sequence of mutually independent real Brownian motions in a fixed probability space (Ω, F, ) adapted to a filtration {F} ⩾0 .
The purpose of this paper is to study the long time behavior of the problem (5) with initial data.Before describing our works, we recall some facts related to this paper.The Cauchy problem for the deterministic KdV-BO equation, that is,  =  = Φ = 0 in (5), was considered in [1].There the authors obtained well-posedness results by using Fourier restriction norm method in Bourgain's type spaces  , with  > 1/2.

Abstract and Applied Analysis
Based on the global existence results given in [1], the attractor of the damped forced KdV-BO equation was obtained in [5].The stochastic KdV-BO equation (i.e.,  =  = 0 in ( 5)) was studied in space  , with  < 1/2 in [6].By introducing some useful inequalities to deal with the irregularity caused by stochastic term, well-posedness results were obtained in [6].Following the work in [6], we will construct the attractor to the Cauchy problem for the stochastic damped, forced KdV-BO equation.
Before stating our main result precisely, we introduce some notations.
Denote by (⋅, ⋅) and | ⋅ | the inner product and the norm in  2 (R), respectively.And ‖ ⋅ ‖  is the norm of the Banach space .
Given two separable Hilbert spaces  and H, we denote by  0 2 (; H) the space of Hilbert-Schmidt operators from  into H.Its norm is given by where (  ) ∈N is any orthonormal basis of .When  =  2 (R), H =   ,  0 2 ( 2 (R);   ) is simply denoted by  0, 2 .The proof of the global well-posedness of the solution ( 5)-( 6) is similar to [6].Here, we only give the following global existence results.
We now give our main result about the long time behavior of the KdV-BO equation based on its global existence results.Theorem 1.Under the assumption that  > 0,  ∈  1 (R), Φ ∈  0,2 2 , and   ∈  2 (Ω;  1 (R)) is F  -measurable, then the random dynamical system associated with the stochastic equation (5) with initial value () =   possesses a universal weak random attractor A in  1 (R).
The paper is organized as follows.Section 2 contains some concepts about the random dynamical system, and Lemma 2 which gives the existence conditions and the structure of the attractor.Then we show that the unique solution of problem ( 5)-( 6) generates a random dynamical system in Section 3. In Section 4, we prove that there exists a compact random absorbing set, which leads to the existence of a random attractor, that is, Theorem 1.

Preliminaries on Random Dynamical System
We now recall some concepts and results from [7][8][9].Let (Ω, F, ) be a probability space and {  : Ω → Ω,  ∈ R} a family of measure preserving transformations such that (, )  →    is measurable,  0 = , and  + =   ∘   for all ,  ∈ R. The flow   together with the corresponding probability space (Ω, F, ) is called a metric dynamical system.
A set-valued map  : Ω → 2  , the set of all subsets of , is called a random compact set, if () is a compact -a.s. and if   → (, ()) is measurable with respect to F for each  ∈ , where (, ) = inf ∈ (, ).
Let A() be a random set and  ⊂ ; one says A() A random set A() is said to be a random attractor for the RDS Ψ if -a.s.
Similar to the deterministic theory, the existence result of random attractors can be stated as follows (see [8,9]).

Solve the Equations and Generate an RDS
We consider the following linear problem to ( 5)-( 6): whose solution is given by the stochastic integral (see [10]) From now on we turn our attention to study the wellposedness of a weakly damped, forced KdV-BO equation with random parameter by change of variable.
Let us study (5) by means of the change of variable and then  satisfies (5) if and only if V is a solution of This is a weakly damped, forced KdV-BO equation with random parameter with the following initial condition: The estimation about the random parameter  and the bilinear term VV  in ( 14) can be obtained by using the method in [6].Then by the fixed point argument the global existence results to the random parameter Cauchy problem ( 14)-( 15) can be obtained.
Now we construct an RDS modeling the stochastic weakly damped, forced KdV-BO equation.For example, consider a set of continuous functions with value 0 at t = 0 Let F be the Borel sigma-algebra induced by the compact open topology of Ω, and let  be a Wiener measure on (Ω, F).
Having the mapping V   → V(, ; , V  ), we define

Compact Random Absorbing Set
In the following computations  ∈ Ω is fixed.We usually denote for any  > 0 in this part.First note that, for the Hilbert transform, we have for any ,  ∈  2 (R) Before we prove the existence of a compact random absorbing set, we first give some estimates about the solution  of problem ().
Let us introduce the following space to study the solution of problem (): for some  < 1.
Remark 5. We have to impose a stronger condition on Φ ∈  0,2 2 in the present paper than that on Φ ∈  0,1 2 in [11].Thus, the solution  of the linear problem is more regular, which will be used in proving the boundedness of  in  1 (R).More precisely, we can get (28) Let  < −1 and   ∈  2 (R) be given, and let V be the solution of (14) with initial condition V(, ) = V  .Multiplying (14) in  2 (R) by V, we obtain It follows that Using Gronwall Lemma for  ⩽ −1 and noticing Lemma 4, we get Noticing we get that ).Then we have the following proposition.Proposition 6.There exists a random radius  1 () > 0, such that for all  > 0 there exists () ⩽ −1 such that the following holds -a.s.For all  ⩽ () and all   ∈  2 (R) with ‖  ‖ ⩽ , the solution  of problem (5) with initial condition (, ) =   satisfies the inequality Proof.Given  > 0, there exists () such that for all  ⩽ ().Put Then the proof of the proposition is completed.
We can also get an auxiliary estimate from (30) by the Gronwall Lemma with  ⩽  ⩽ 0.
Consider the following: This inequality will be useful in the following proof.

4.2.
Absorption in  1 (R) at Time =0.To obtain the  1 estimate, we multiply ( 14) by V  and integrate by part to get Moreover Combining ( 37), (38), and (39) (2(37) + (38) + (39)), we have Denote And noticing that we deduce that Now we stop to estimate the right hand side of (43) term by term: Applying the Gronwall Lemma for  ⩽ 0, we find We can read the boundedness of the right hand side of (46) from the following estimates.Using (36), we have where (50) Then we get the following proposition.