White-Noise-Driven KdV-Type Boussinesq System

Abstract

The white-noise-driven KdV-type Boussinesq system is a class of stochastic partial differential equations (SPDEs) that describe nonlinear wave propagation under the influence of random noise—specifically white noise—and generalize features from both the Korteweg–de Vries (KdV) and Boussinesq equations. We consider a Cauchy problem for two stochastic systems based on the KdV-type Boussinesq equations. For these systems, we determine sufficient conditions to ensure that this problem is locally and globally well posed for initial data in Sobolev spaces by the linear and bilinear estimates and their modification together with the Banach fixed point.

1. Introduction and Preliminaries

Let $v=v(x,t),x,t\in \mathbb{R}$. The classical Boussinesq equation, derived by Joseph Boussinesq (1872), is a nonlinear dispersive partial differential equation given by

$${\partial }_t^{2}v-c^2{\partial }_x^{2}v+{\partial }_x^2{v}^2-\alpha {\partial }_x^{4}v=0.$$

It is a wave-type equation of second order in time, capturing both nonlinear steepening and dispersive spreading. If it is linearized and scaled under unidirectional wave assumptions, the KdV equation is recovered. The equation

$${\partial }_{t}v+a{\partial }_{x}v+{\partial }_x^{3}v+g\left(v\right){\partial }_{x}v=0$$

includes and models important phenomena in the propagation of nonlinear waves and is applied in energy and industry; this equation is named the Korteweg–de Vries (KdV) equation for $g\left(v\right)=v$ and the modified Korteweg–de Vries (mKdV) equation for $g\left(v\right)=\pm v^2$. This model describes the growth of monodimensional nonlinear waves in media with and without dissipation. Similar issues were previously considered in other articles [1,2]. The dynamics of solutions in the Korteweg–de Vries (KdV) equations and the modified Korteweg–de Vries (mKdV) equations have been studied quite well due to the complete integrability of these equations. The solutions are stable wave formations that retain their shape after interacting with each other and with wave packets. The description of the solution interaction process for mKdV is given in many works [3,4,5], and for KdV, the main results were published in the 1970s [6], although some effects have been newly discovered since then [7].

The KdV-type Boussinesq system models long, small-amplitude surface waves in shallow water. It can be written in many forms; the simplest one is

$$\left\{\begin{array}{c}{\partial }_{t}v+{\partial }_{x}u+{\partial }_x^{3}u+{\partial }_x\left(uv\right)=0\hfill \\ {\partial }_{t}u+{\partial }_{x}v+{\partial }_x^{3}v+u{\partial }_{x}u=0,\hfill \end{array}\right.$$

where $v=v(x,t)$ is the surface elevation and $u=u(x,t)$ is the horizontal velocity. With the addition of white noise, the system becomes stochastically perturbed as follows:

$$\left\{\begin{array}{c}{\partial }_{t}v+{\partial }_{x}u+{\partial }_x^{3}u+{\partial }_x\left(uv\right)={\alpha }_1{B}_1(x,t)\hfill \\ {\partial }_{t}u+{\partial }_{x}v+{\partial }_x^{3}v+u{\partial }_{x}u={\alpha }_2{B}_2(x,t),\hfill \end{array}\right.$$

where $B_1(x,t),B_2(x,t)$ are space–time white noise and ${\alpha }_1,{\alpha }_2$ are noise intensities. Until the mid-1960s, the study of nonlinear partial differential equations was carried out mainly in four directions: existence and uniqueness theorems; construction of solutions of nonlinear equations with weak nonlinearity by perturbation theory methods; study of some classes of self-similar solutions; and numerical modeling. Certain methods are used, such as functional Analysis and Semigroup Theory in Sobolev space estimates and fixed-point arguments; Stochastic Calculus (Itô/SPDE Techniques), where energy is estimated by using Itô’s formula; Galerkin approximations to construct solutions; the Krylov–Bogoliubov method; or Doob’s theorem. At present, along with these methods, the linear and bilinear estimates and their modifications together with the Banach fixed point are also used for a wide class of nonlinear evolution equations. These methods were initiated by [8,9,10,11,12], in which the solution of the Korteweg–de Vries (KdV) equation (coupled system) was described in terms of the quantitative and qualitative properties. In 1974, the works of Novikov [13] and Lax [14] laid the foundation for a new direction in the finite-zone integration method: finding periodic and almost periodic solutions of KdV-type equations. The emergence of this direction, which was designated the method of finite-zone (algebraic–geometric) integration, is associated primarily with the names of Dubrovin, Novikov, Matveev, and Krichever; see [15,16,17]. Both the Korteweg–de Vries equation and the Boussinesq equation for wave propagation in shallow water have algebraic–geometric solutions. [18,19,20,21], and therefore, from our point of view, an interesting question is what has a stronger effect on the shape of the wave process—one of the parameters of the curve used to construct the solutions, or the type of nonlinear wave equation. Recall that the Korteweg–de Vries equation describes wave propagation in one direction, while the Boussinesq equation is the simplest nonlinear wave equation describing wave propagation in both directions; see [22,23,24,25].

In a convenient set of coordinates, the stochastic coupled KdV-type Boussinesq equation is written as

$$\left\{\begin{array}{c}{\partial }_{t}u+{\partial }_{x}u+{\partial }_x^{3}u+3u{\partial }_{x}u-v{\partial }_{x}v+{\partial }_x\left(uv\right)=\Lambda {\partial }_{tx}^{2}B\hfill \\ {\partial }_{t}v-{\partial }_{x}v-{\partial }_x^{3}v-u{\partial }_{x}u+3v{\partial }_{x}v+{\partial }_x\left(uv\right)=\Lambda {\partial }_{tx}^{2}B.\hfill \end{array}\right.$$

(1)

Here, the surface elevation $u=u(x,t)$ and horizontal velocity $v=v(x,t)$ are random processes defined for $x\in \mathbb{R},t\in {\mathbb{R}}^{+}$; the operator $\Lambda$ is linear; and B represents two-parameter Brownian motion on $\mathbb{R}\times {\mathbb{R}}^{+}$, given a zero-mean Gaussian process in which the correlation function is defined as

$$\mathbb{E}\left(B(a,t)B(b,s)\right)=(t\wedge s)(a\wedge b),t,s\in {\mathbb{R}}^{+},a,b\in \mathbb{R}.$$

We begin with function spaces. Let $s\in \mathbb{R},H^s\left(\mathbb{R}\right)$ represent the usual Sobolev space, defined by

$${\parallel f\parallel }_{H^s\left(\mathbb{R}\right)}={\left({\int }_{\mathbb{R}}{(1+|\nu \left|\right)}^{2s}{\left|\widehat{f}\right|}^{2}d\nu \right)}^{1/2}.$$

Here, $\widehat{f}$ denotes the spatial Fourier transform

$$\widehat{f}\left(\nu \right)={\int }_{\mathbb{R}}f\left(x\right)e^{-ix\nu }dx.$$

Similarly, for $s,b\in \mathbb{R},$ the Bourgain spaces $X^{s,b}\left({\mathbb{R}}^2\right)$ and $X_1^{s,b}\left({\mathbb{R}}^2\right)$ are defined by the norms

$$\begin{array}{ccc}\hfill {\parallel u\parallel }_{X^{s,b}\left({\mathbb{R}}^2\right)}& =& {\parallel u\parallel }_{X^{s,b}}\hfill \\ & =& {\left({\int }_{{\mathbb{R}}^2}{(1+|\nu \left|\right)}^{2s}(1+|\tau +\nu -{\nu }^3{\left|\right)}^{2b}{\left|\tilde{u}(\tau ,\nu )\right|}^{2}d\nu d\tau \right)}^{1/2},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\parallel u\parallel }_{X_1^{s,b}\left({\mathbb{R}}^2\right)}& =& {\parallel u\parallel }_{X_1^{s,b}}\hfill \\ & =& {\left({\int }_{{\mathbb{R}}^2}{(1+|\nu \left|\right)}^{2s}(1+|\tau -\nu +{\nu }^3{\left|\right)}^{2b}{\left|\tilde{u}(\tau ,\nu )\right|}^{2}d\nu d\tau \right)}^{1/2},\hfill \end{array}$$

where $\tilde{u}$ denotes the space–time Fourier transform

$$\tilde{u}(\nu ,\tau )={\int }_{{\mathbb{R}}^2}u(x,t)e^{-i(x\nu +t\tau )}dxdt.$$

For $T>0$, we also use the spaces $X_{1,T}^{s,b}$ and $X_T^{s,b}$ of restrictions to the time interval $[0,T]$ of functions in $X_1^{s,b}$ and $X^{s,b}$. They are endowed with the norms

$${\parallel u\parallel }_{X_T^{s,b}}=inf\left\{{\parallel w\parallel }_{X^{s,b}}:u=w,x\in \mathbb{R},t\in [0,T]\right\},$$

$${\parallel v\parallel }_{X_{1,T}^{s,b}}=inf\left\{{\parallel z\parallel }_{X_1^{s,b}}:v=z,x\in \mathbb{R},t\in [0,T]\right\}.$$

Because we are dealing with systems of equations, we will need to consider product function spaces. The product spaces are defined as

$${\mathcal{X}}^{b,s}=X^{b,s}\times X_1^{b,s},$$

$${\mathcal{X}}_T^{b,s}=X_T^{b,s}\times X_{1,T}^{b,s},$$

and

$${\mathcal{H}}^s=H^s\times H^s,$$

where

$${\parallel (u,v)\parallel }_{{\mathcal{X}}^{b,s}}={max\{\parallel u\parallel }_{X^{s,b}}{,\parallel v\parallel }_{X_1^{s,b}}\},$$

$${\parallel (u,v)\parallel }_{{\mathcal{X}}_T^{b,s}}={max\{\parallel u\parallel }_{X_T^{s,b}}{,\parallel v\parallel }_{X_{1,T}^{s,b}}\},$$

and

$$\parallel (u_0,v_0){\parallel }_{{\mathcal{H}}^s}=max\{\parallel u_0{\parallel }_{H^s},\parallel v_0{\parallel }_{H^s}\}.$$

Finally, we denote as

$$L_2^{0,s}=L_2^0(L^2\left(\mathbb{R}\right),H^s\left(\mathbb{R}\right))$$

the space of Hilbert–Schmidt operators from $L^2\left(\mathbb{R}\right)$ to $H^s\left(\mathbb{R}\right)$ with

$${\parallel \Lambda \parallel }_{L_2^{0,s}}^2=\sum _{i=1}^{\infty }{\parallel \Lambda e_i\parallel }_{H^s\left(\mathbb{R}\right)}^2,$$

where ${\left(e_i\right)}_{i\ge 1}$ is an orthonormal basis in $L^2\left(\mathbb{R}\right)$.

With this notation in place, we may finally state the results to be proved. In the subsequent work, let $(\Omega ,\mathcal{F},P)$ be a fixed probability space adapted to a filtration ${\left({\mathcal{F}}_t\right)}_{t\ge 0}$; see [26,27,28]. For Equation (1), we will prove the following local and global results.

To conclude our preliminary discussion, we remark that we will use the notation $A{\lesssim }_{k_1,\dots ,k_n}B$ to denote $A\le cB$ for some constant $c>0$ depending on $k_1,\dots ,k_n$. If c is an absolute constant, we shall write $A\lesssim B$.

2. Linear and Bilinear Estimates

To prove the next main results, we will need to introduce several important estimates. To state these estimates, the Itô form of the system in (1) is given, namely,

$$\left\{\begin{array}{c}du+\left({\partial }_{x}u+{\partial }_x^{3}u+3u{\partial }_{x}u-v{\partial }_{X}v+{\partial }_x\left(uv\right)\right)dt=\Lambda dW\hfill \\ \\ dv+\left(-{\partial }_{x}v-{\partial }_x^{3}v-u{\partial }_{x}u+3v\partial v_x+{\partial }_x\left(uv\right)\right)dt=\Lambda dW.\hfill \end{array}\right.$$

(2)

where $W\left(t\right)={\partial }_{x}B$ is a cylindrical Wiener process on $L^2\left(\mathbb{R}\right)$, which can also be given by

$$W\left(t\right)=\sum _{i=0}^{\infty }{\beta }_i\left(t\right)e_i.$$

Here, ${\left(e_i\right)}_{i\in \mathbb{N}}$ is an orthonormal basis of $L^2\left(\mathbb{R}\right)$, and ${\left({\beta }_i\right)}_{i\in \mathbb{N}}$ is a sequence of mutually independent real Brownian motions in a fixed probability space. System (2) is supplemented with the initial conditions

$$\begin{array}{c}\hfill \begin{array}{cc}u(x,0)=u_0\left(x\right)& \\ \\ v(x,0)=v_0\left(x\right).\end{array}\end{array}$$

(3)

We consider the system

$$\left\{\begin{array}{c}du+{\partial }_{x}u+{\partial }_x^{3}udt=\Lambda dW\hfill \\ \\ dv-{\partial }_{x}u-{\partial }_x^{3}udt=\Lambda dW\hfill \\ \\ u_0\left(x\right)=0,v_0\left(x\right)=0.\hfill \end{array}\right.$$

(4)

System (4) can be written as a stochastic Itô integral

$$\left\{\begin{array}{c}{u}_l\left(t\right)={\int }_0^{t}U(t-\tau )\Lambda dW\left(\tau \right)\hfill \\ \\ v_l\left(t\right)={\int }_0^t{U}_1(t-\tau )\Lambda dW\left(\tau \right),\hfill \end{array}\right.$$

(5)

where

$$U\left(t\right)=e^{-t({\partial }_x-{\partial }_x^3)},$$

$$U_1\left(t\right)=e^{-t({\partial }_x^3-{\partial }_x)}.$$

By the unitarity of $U\left(t\right)$ and $U_1\left(t\right)$, it can be easily shown that $u\left(t\right)$ and $v\left(t\right)$ are in $H^s\left(\mathbb{R}\right)$ only if $\Lambda$ is a Hilbert–Schmidt operator from $L^2\left(\mathbb{R}\right)$ to $H^s\left(\mathbb{R}\right)$.

We will solve (2)–(3) by considering a related mild form.

$$\left\{\begin{array}{cc}{\Delta }_1\left(u\right)\left(t\right)\hfill & =u\left(t\right)\hfill \\ & =U\left(t\right)u_0+{\int }_0^{t}U(t-\tau )(3u{\partial }_{x}u-v{\partial }_{x}v+{\partial }_x\left(uv\right))\left(\tau \right)d\tau \hfill \\ & +{\int }_0^{t}U(t-\tau )\Lambda dW\left(\tau \right)\hfill \\ \\ {\Delta }_2\left(v\right)\left(t\right)\hfill & =v\left(t\right)\hfill \\ & =U_1\left(t\right)v_0+{\int }_0^t{U}_1(t-\tau )(-u{\partial }_{x}u+3v{\partial }_{x}v+{\partial }_x\left(uv\right))\left(\tau \right)d\tau \hfill \\ & +{\int }_0^t{U}_1(t-\tau )\Lambda dW\left(\tau \right),\hfill \end{array}\right.$$

To construct mild solutions, we will need the following estimates. They will be the key to the proofs.

Proposition 1 

(Linear Estimates [29]). For any $s,b\in \mathbb{R}$, we have

$${∥U\left(t\right)u_0∥}_{X_T^{s,b}}\lesssim {\parallel u_0\parallel }_{H^s},$$

$${∥U_1\left(t\right)v_0∥}_{X_{1,T}^{s,b}}\lesssim {\parallel v_0\parallel }_{H^s}.$$

Furthermore, if

$$-{\displaystyle \frac{1}{2}}<b^{\prime }\le 0\le b<1+b^{\prime }$$

and $0\le T\le 1$, then

$${∥{\int }_0^{t}U\left(t-\tau \right)F\left(u\left(\tau \right)\right)\mathrm{d}\tau ∥}_{X_T^{s,b}}\lesssim T^{1-b^{\prime }+b}{\parallel F\left(u\right)\parallel }_{X_T^{s,b^{\prime }}}$$

and

$${∥{\int }_0^t{U}_1\left(t-\tau \right)F\left(u\left(\tau \right)\right)\mathrm{d}\tau ∥}_{X_{1,T}^{s,b}}\lesssim T^{1-b^{\prime }+b}{\parallel F\left(u\right)\parallel }_{X_{1,T}^{s,b^{\prime }}}.$$

Lemma 1 

(Bilinear Estimates [29]). Let $s\ge 0,{\displaystyle \frac{1}{2}}<b<{\displaystyle \frac{3}{4}}$, and $b-1<b^{\prime }<-{\displaystyle \frac{1}{4}}$, such that

$$\begin{array}{c}\hfill \parallel {\partial }_x\left(v^2\right){\parallel }_{X^{s,b^{\prime }}}{\lesssim }_{b,b^{\prime }}{\parallel v\parallel }_{X_1^{s,b}}^2,\end{array}$$

$$\begin{array}{c}\hfill \parallel {\partial }_x\left(u^2\right){\parallel }_{X^{s,b^{\prime }}}{\lesssim }_{b,b^{\prime }}{\parallel u\parallel }_{X^{s,b}}^2,\end{array}$$

$$\begin{array}{c}\hfill \parallel {\partial }_x{\left(uv\right)\parallel }_{X^{s,b^{\prime }}}{\lesssim }_{b,b^{\prime }}{\parallel u\parallel }_{X^{s,b}}{\parallel v\parallel }_{X_1^{s,b}},\end{array}$$

$$\begin{array}{c}\hfill \parallel {\partial }_x\left(u^2\right){\parallel }_{X_1^{s,b^{\prime }}}{\lesssim }_{b,b^{\prime }}{\parallel u\parallel }_{X^{s,b}}^2,\end{array}$$

$$\begin{array}{c}\hfill \parallel {\partial }_x\left(v^2\right){\parallel }_{X_1^{s,b^{\prime }}}{\lesssim }_{b,b^{\prime }}{\parallel v\parallel }_{X_1^{s,b}}^2,\end{array}$$

$$\begin{array}{c}\hfill \parallel {\partial }_x{\left(uv\right)\parallel }_{X_1^{s,b^{\prime }}}{\lesssim }_{b,b^{\prime }}{\parallel u\parallel }_{X^{s,b}}{\parallel v\parallel }_{X_1^{s,b}},\end{array}$$

hold for any $s\ge 0$.

We choose a function $\varpi$ such that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\varpi \left(t\right)=0,\hfill & t<0,\left|t\right|>2,\hfill \\ \\ \varpi \left(t\right)=1,\hfill & t\in [0,1],\hfill \end{array}\right.\end{array}$$

(6)

with $\varpi \in C_0^{\infty }$. Here, we mention that

$$\varpi \in H_t^b=H^b([0,T],\mathbb{R}),\forall b>{\displaystyle \frac{1}{2}},$$

where

$${\parallel \varpi \parallel }_{H_t^b}^2={\parallel \varpi \parallel }_{L^2}^2+{\int }_{{\mathbb{R}}^2}{\displaystyle \frac{|\varpi \left({\eta }_1\right)-\varpi \left({\eta }_2\right){|}^2}{|{\eta }_1-{\eta }_2{|}^{1+2b}}}d{\eta }_{1}d{\eta }_2.$$

(7)

We show the following lemma to deal with stochastic convolution.

Lemma 2. 

Let $b,s\in \mathbb{R}$, where $b<{\displaystyle \frac{1}{2}}$, and suppose that $\Lambda \in L_2^{0,s}$. Then, $u_l\left(t\right),v_l\left(t\right)$, introduced in (5), satisfies

$$\varpi u_l\in L^2\left(\Omega ,X^{b,s}\right),$$

$$\varpi v_l\in L^2\left(\Omega ,X_1^{b,s}\right),$$

and

$$\mathbb{E}\left(\parallel \varpi u_l{\parallel }_{X^{b,s}}^2\right){\lesssim }_{b,\varpi }{\parallel \Lambda \parallel }_{L_2^{0,s}}^2,$$

$$\mathbb{E}\left(\parallel \varpi v_l{\parallel }_{X_1^{b,s}}^2\right){\lesssim }_{b,\varpi }{\parallel \Lambda \parallel }_{L_2^{0,s}}^2.$$

Proof. 

Define the functions

$$f(·,t)=\varpi \left(t\right){\int }_0^{t}U(-\tau )\Lambda dW\left(\tau \right),t\in {\mathbb{R}}^{+}.$$

(8)

$$g(·,t)=\varpi \left(t\right){\int }_0^t{U}_1(-\tau )\Lambda dW\left(\tau \right),t\in {\mathbb{R}}^{+}.$$

(9)

Then,

$$U\left(t\right)f(·,t)=\varpi \left(t\right)u_l,$$

$$U_1\left(t\right)g(·,t)=\varpi \left(t\right)v_l,$$

and

$$\begin{array}{cc}\hfill \mathbb{E}\left({∥\varpi u_l∥}_{X^{s,b}}^2\right)& =\mathbb{E}\left({\int }_{{\mathbb{R}}^2}{\left(\right|\nu |+1)}^{2s}{\left(\right|\tau |+1)}^{2b}{\left|\widehat{f}(\nu ,t)\right|}^{2}d\tau d\nu \right)\hfill \\ & ={\int }_{\mathbb{R}}{(1+|\nu \left|\right)}^{2s}\mathbb{E}\left({∥\widehat{f}(\nu ,·)∥}_{H_t^b}^2\right)d\nu .\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathbb{E}\left({∥\varpi v_l∥}_{X^{s,b}}^2\right)& =\mathbb{E}\left({\int }_{{\mathbb{R}}^2}{\left(\right|\nu |+1)}^{2s}{\left(\right|\tau |+1)}^{2b}{\left|\widehat{g}(\nu ,t)\right|}^{2}d\tau d\nu \right)\hfill \\ & ={\int }_{\mathbb{R}}{(1+|\nu \left|\right)}^{2s}\mathbb{E}\left({∥\widehat{g}(\nu ,·)∥}_{H_t^b}^2\right)d\nu .\hfill \end{array}$$

The proof is similar for f and g, and so we will restrict our argument to f only. According to the expansion

$$W\left(t\right)=\sum _{i=0}^{\infty }{\beta }_i\left(t\right)e_i$$

of the cylindrical Wiener process and (7), we have

$$\mathbb{E}\left({∥\widehat{f}(\nu ,·)∥}_{H_t^b}^2\right)={𝒮}_1+{𝒮}_2,$$

where

$${𝒮}_1=\sum _{i=0}^{\infty }{\left|\widehat{\Lambda e_i}\right|}^2\left[\mathbb{E}\left({∥\varpi \left(t\right){\int }_0^t{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)∥}_{L^2\left(\mathbb{R}\right)}^2\right)\right],$$

$${𝒮}_2=\sum _{i=0}^{\infty }{\left|\widehat{\Lambda e_i}\right|}^2\left[\mathbb{E}\left({\int }_{{\mathbb{R}}^2}{\displaystyle \frac{\left|\begin{array}{c}\varpi \left({\eta }_1\right){\int }_0^{{\eta }_1}{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)-\varpi \left({\eta }_2\right){\int }_0^{{\eta }_2}{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)\end{array}\right|}{{\left|{\eta }_1-{\eta }_2\right|}^{1+2b}}}d{\eta }_{1}d{\eta }_2\right)\right].$$

From the Itô isometry formula, we have

$$\begin{array}{cc}\hfill {𝒮}_1& =\sum _{i=0}^{\infty }{\left|\widehat{\Lambda e_i}\right|}^2{\int }_0^2{\left|\varpi \left(t\right)\right|}^2\mathbb{E}\left({\left|{\int }_0^t{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)\right|}^2\right)dt\hfill \\ & ={∥{\left|t\right|}^{{\displaystyle \frac{1}{2}}}\varpi ∥}_{L_t^2}^2\sum _{i=0}^{\infty }{\left|\widehat{\Lambda e_i}\right|}^2.\hfill \end{array}$$

To estimate ${𝒮}_2$, we obtain

$$\begin{array}{cc}& {𝒮}_2=\sum _{i=0}^{\infty }{\left|\widehat{\Lambda e_i}\right|}^2\left[\mathbb{E}\left({\int }_{{\mathbb{R}}^2}{\displaystyle \frac{\left|\begin{array}{c}\varpi \left({\eta }_1\right){\int }_0^{{\eta }_1}{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)-\varpi \left({\eta }_2\right){\int }_0^{{\eta }_2}{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)\end{array}\right|}{{\left|{\eta }_1-{\eta }_2\right|}^{1+2b}}}d{\eta }_{1}d{\eta }_2\right)\right]\hfill \\ & =2\sum _{i=0}^{\infty }{\left|\widehat{\Lambda e_i}\right|}^2{\int }_{{\eta }_2>0}{\int }_{{\eta }_1<{\eta }_2}{\displaystyle \frac{\mathbb{E}{\left(\left|\varpi \left({\eta }_1\right){\int }_0^{{\eta }_1}{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)-\varpi \left({\eta }_2\right){\int }_0^{{\eta }_2}{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)\right|\right)}^2}{{\left|{\eta }_1-{\eta }_2\right|}^{1+2b}}}d{\eta }_{1}d{\eta }_2\hfill \\ & \le \sum _{i=0}^{\infty }{\left|\widehat{\Lambda e_i}\right|}^2[2{\int }_{{\eta }_2>0}{\int }_{{\eta }_1<0}{\displaystyle \frac{{\left|\varpi \left({\eta }_2\right)\right|}^2\mathbb{E}\left({\left|{\int }_0^{{\eta }_2}{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)\right|}^2\right)}{{\left|{\eta }_1-{\eta }_2\right|}^{1+2b}}}d{\eta }_{1}d{\eta }_2\hfill \\ & +2{\int }_{{\eta }_2>0}{\int }_{0<{\eta }_1<{\eta }_2}\times \hfill \\ & {\displaystyle \frac{\mathbb{E}{\left(\left|\varpi \left({\eta }_1\right){\int }_0^{{\eta }_1}{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)-\varpi \left({\eta }_2\right){\int }_0^{{\eta }_1}{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)+\varpi \left({\eta }_2\right){\int }_{{\eta }_1}^{{\eta }_2}{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)\right|\right)}^2}{{\left|{\eta }_1-{\eta }_2\right|}^{1+2b}}}\times \hfill \\ & d{\eta }_{1}d{\eta }_2]\hfill \\ & \le \sum _{i=0}^{\infty }{\left|\widehat{\Lambda e_i}\right|}^2[2{\int }_{{\eta }_2>0}{\int }_{{\eta }_1<0}{\displaystyle \frac{{\left|\varpi \left({\eta }_2\right)\right|}^2\mathbb{E}\left({\left|{\int }_0^{{\eta }_2}{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)\right|}^2\right)}{{\left|{\eta }_1-{\eta }_2\right|}^{1+2b}}}d{\eta }_{1}d{\eta }_2\hfill \\ & +4{\int }_{{\eta }_2>0}{\int }_{0<{\eta }_1<{\eta }_2}{\displaystyle \frac{{\left|\varpi \left({\eta }_1\right)-\varpi \left({\eta }_2\right)\right|}^2\mathbb{E}\left({\left|{\int }_0^{{\eta }_1}{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)\right|}^2\right)}{{\left|{\eta }_1-{\eta }_2\right|}^{1+2b}}}d{\eta }_{1}d{\eta }_2\hfill \\ & +4{\int }_{{\eta }_2>0}{\int }_{0<{\eta }_1<{\eta }_2}{\displaystyle \frac{{\left|\varpi \left({\eta }_2\right)\right|}^2\mathbb{E}\left({\left|{\int }_{{\eta }_1}^{{\eta }_2}{e}^{i\tau (\nu -{\nu }^3)}d{\beta }_i\left(\tau \right)\right|}^2\right)}{{\left|{\eta }_1-{\eta }_2\right|}^{1+2b}}}d{\eta }_{1}d{\eta }_2]\hfill \\ & =\sum _{i=0}^{\infty }{\left|\widehat{\Lambda e_i}\right|}^2\left[{\mathcal{I}}_1+{\mathcal{I}}_2+{\mathcal{I}}_3\right].\hfill \end{array}$$

We separately estimate ${\mathcal{I}}_1,{\mathcal{I}}_2$, and ${\mathcal{I}}_3$ as follows:

$${\mathcal{I}}_1\le 2{\int }_0^2{\eta }_1{\left|\varpi \left({\eta }_2\right)\right|}^2{\int }_{{\eta }_1<0}{\displaystyle \frac{1}{{\left|{\eta }_1-{\eta }_2\right|}^{1+2b}}}d{\eta }_{1}d{\eta }_2\le {\mathcal{M}}_b{∥{\left|t\right|}^{{\displaystyle \frac{1}{2}}-b}\varpi ∥}_{L_t^2}^2.$$

By (8) and the fact that $0<2b<1$, we have

$$\begin{array}{cc}\hfill {\mathcal{I}}_2& \le 4{\int }_0^{\infty }{\int }_0^{{\eta }_2}{\displaystyle \frac{{\eta }_1{\left|\varpi \left({\eta }_1\right)-\varpi \left({\eta }_2\right)\right|}^2}{{∥{\eta }_1-{\eta }_2∥}^{1+2b}}}d{\eta }_{1}d{\eta }_2\hfill \\ & \le 4{\int }_0^2{\int }_0^{{\eta }_2}{\displaystyle \frac{{\eta }_1{\left|\varpi \left({\eta }_1\right)-\varpi \left({\eta }_2\right)\right|}^2}{{∥{\eta }_1-{\eta }_2∥}^{1+2b}}}d{\eta }_{1}d{\eta }_2\hfill \\ & +4{\int }_2^{\infty }{\int }_0^2{\displaystyle \frac{{\eta }_1{\left|\varpi \left({\eta }_1\right)\right|}^2}{{∥{\eta }_1-{\eta }_2∥}^{1+2b}}}d{\eta }_{1}d{\eta }_2\hfill \\ & \le {8\parallel \varpi \parallel }_{H_t^b}^2+4{∥{\left|t\right|}^{{\displaystyle \frac{1}{2}}}\varpi ∥}_{L_t^{\infty }}^2{\int }_0^{\infty }{\int }_0^2{\displaystyle \frac{1}{{\left|{\eta }_1-{\eta }_2\right|}^{1+2b}}}d{\eta }_{1}d{\eta }_2\hfill \\ & \le {8\parallel \varpi \parallel }_{H_t^b}^2+{\mathcal{M}}_b{∥{\left|t\right|}^{{\displaystyle \frac{1}{2}}}\varpi ∥}_{L_t^{\infty }}^2.\hfill \end{array}$$

Similarly,

$$\begin{array}{ccc}\hfill {\mathcal{I}}_3& \le & 4{\int }_0^2{\int }_0^{{\eta }_2}{\displaystyle \frac{{\left|\varpi \left({\eta }_2\right)\right|}^2}{{\left|{\eta }_1-{\eta }_2\right|}^{2b}}}d{\eta }_{1}d{\eta }_2\hfill \\ & \le & {\mathcal{M}}_b{∥{\left|t\right|}^{{\displaystyle \frac{1}{2}}-b}\varpi ∥}_{L_t^2}^2.\hfill \end{array}$$

Thus, we have

$$\mathbb{E}\left({∥\widehat{f}(\nu ,·)∥}_{H_t^b}^2\right)\le \mathcal{K}(b,\varpi )\sum _{i=0}^{\infty }{\left|\widehat{\Lambda e_i}\right|}^2,$$

where

$$\mathcal{K}(b,\varpi )={\mathcal{M}}_b\left({\parallel \varpi \parallel }_{H_t^b}+{∥{\left|t\right|}^{{\displaystyle \frac{1}{2}}}\varpi ∥}_{L_t^2}+{∥{\left|t\right|}^{{\displaystyle \frac{1}{2}}}\varpi ∥}_{L_t^{\infty }}\right).$$

□

3. Local Well-Posedness

We need the following theorem to prove the local existence of a solution.

Theorem 1 

([30]). Assume that $\mathcal{A}$ generates a contraction semi-group and

$$\Lambda \in {\mathcal{N}}_{\mathcal{W}}^2\left([0,T],L_2^{0,s}\right).$$

Then, the process ${\mathcal{W}}_{\mathcal{A}}^{\Lambda }(·)$ has a continuous modification, and there exists a constant $C>0$ such that

$$\mathbb{E}\left(\underset{\tau \in [0,t]}{sup}{\left|{\mathcal{W}}_{\mathcal{A}}^{\Lambda }\left(\tau \right)\right|}^2\right)\le C\mathbb{E}\left({\int }_0^t{\parallel \Lambda \left(\tau \right)\parallel }_{L_2^0}^{2}d\tau \right),0\le t\le T.$$

Our first main result is as follows.

Theorem 2. 

Assume that $s\ge 0$, $\Lambda \in L_2^{0,s}$, $b>{\displaystyle \frac{1}{2}}$, and b is close enough to ${\displaystyle \frac{1}{2}}$. If

$$(u_0,v_0)\in H^s\left(\mathbb{R}\right)\times H^s\left(\mathbb{R}\right)$$

for $\varrho \in \Omega$ and $u_0,v_0$ are ${\mathcal{F}}_0$-measurable, then, for $\varrho \in \Omega$, there exist a stopping time $T_{\varrho }>0$ and a unique solution $(u,v)$ of (1) on $\left[0,T_{\varrho }\right]\times \left[0,T_{\varrho }\right]$ such that

$$(u,v)\in C\left(\left[0,T_{\varrho }\right],H^s\left(\mathbb{R}\right)\right)\times C\left(\left[0,T_{\varrho }\right],H^s\left(\mathbb{R}\right)\right)\cap {\mathcal{X}}_{T_{\varrho }}^{s,b}.$$

Proof. 

To prove the local well-posedness of (1) given in Theorem 2, we define

$${\Pi }_1\left(t\right)=U\left(t\right)u_0,$$

$${\Pi }_2\left(t\right)=U_1\left(t\right)v_0,$$

$$u_l\left(t\right)={\int }_0^{t}U(t-\tau )\Lambda dW\left(\tau \right),$$

$$v_l\left(t\right)={\int }_0^t{U}_1(t-\tau )\Lambda dW\left(\tau \right),$$

$${\varphi }_1\left(t\right)={\int }_0^{t}U(t-\tau )(3u{\partial }_{x}u-v{\partial }_{x}v+{\partial }_x\left(uv\right))\left(\tau \right)d\tau ,$$

and

$${\varphi }_2\left(t\right)={\int }_0^t{U}_1(t-\tau )(-u{\partial }_{x}u+3v{\partial }_{x}v+{\partial }_x\left(uv\right))\left(\tau \right)d\tau .$$

Now, assume that

$$\left\{\begin{array}{c}{\varphi }_1\left(t\right)=u\left(t\right)-{\Pi }_1\left(t\right)-u_l\left(t\right)\hfill \\ \\ {\varphi }_2\left(t\right)=v\left(t\right)-{\Pi }_2\left(t\right)-v_l\left(t\right).\hfill \end{array}\right.$$

(10)

Therefore,

$$\left\{\begin{array}{c}{\displaystyle {\varphi }_1\left(t\right)=\frac{1}{2}{\int }_0^{t}U(t-\tau )(3u{\partial }_{x}u-v{\partial }_{x}v+{\partial }_x\left(uv\right))\left(\tau \right)d\tau }\hfill \\ \\ {\displaystyle {\varphi }_2\left(t\right)={\int }_0^t{U}_1(t-\tau )(-u{\partial }_{x}u+3v{\partial }_{x}v+{\partial }_x\left(uv\right))\left(\tau \right)d\tau .}\hfill \end{array}\right.$$

(11)

Then, (11) is equivalent to

$$\left\{\begin{array}{cc}\hfill & {\Gamma }_1\left({\varphi }_1\left(t\right)\right)={\varphi }_1\left(t\right)\hfill \\ \\ & {\displaystyle =\frac{1}{2}{\int }_0^{t}U(t-\tau )[\frac{3}{2}{\partial }_x\left({\varphi }_1^2+{\Pi }_1^2+u_l^2+2{\varphi }_1{\Pi }_2+2{\varphi }_1{u}_l+2{\Pi }_2{u}_l\right)}\hfill \\ \\ & -{\displaystyle \frac{1}{2}}{\partial }_x\left({\varphi }_2^2+{\Pi }_2^2+v_l^2+2{\varphi }_2{\Pi }_2+2{\varphi }_2{v}_l+2{\Pi }_2{v}_l\right)\hfill \\ \\ & +{\partial }_x\left({\varphi }_1{\varphi }_2+{\Pi }_1{\Pi }_2+u_l{v}_l+{\varphi }_1({\Pi }_2+v_l)+{\varphi }_2({\Pi }_1+u_l)+{\Pi }_1{v}_l+{\Pi }_2{u}_l\right)]\left(\tau \right)d\tau \hfill \\ \\ \hfill & {\Gamma }_2\left({\varphi }_2\left(t\right)\right)={\varphi }_2\left(t\right)\hfill \\ \\ & {\displaystyle ={\int }_0^t{U}_1(t-\tau )[-\frac{1}{2}{\partial }_x\left({\varphi }_1^2+{\Pi }_1^2+u_l^2+2{\varphi }_1{\Pi }_2+2{\varphi }_1{u}_l+2{\Pi }_2{u}_l\right)}\hfill \\ \\ & +{\displaystyle \frac{3}{2}}{\partial }_x\left({\varphi }_2^2+{\Pi }_2^2+v_l^2+2{\varphi }_2{\Pi }_2+2{\varphi }_2{v}_l+2{\Pi }_2{v}_l\right)\hfill \\ \\ & +{\partial }_x\left({\varphi }_1{\varphi }_2+{\Pi }_1{\Pi }_2+u_l{v}_l+{\varphi }_1({\Pi }_2+v_l)+{\varphi }_2({\Pi }_1+u_l)+{\Pi }_1{v}_l+{\Pi }_2{u}_l\right)]\left(\tau \right)d\tau .\hfill \end{array}\right.$$

(12)

Next, we define the ball ${\mathcal{B}}_{\mathcal{R},T}$ as

$${\mathcal{B}}_{\mathcal{R},T}=\{({\varphi }_1,{\varphi }_2):\parallel ({\varphi }_1,{\varphi }_2){\parallel }_{{\mathcal{X}}_T^{b,s}}\le \mathcal{R},\mathcal{R}>0\}.$$

At this stage, our aim is to prove that $({\varphi }_1\left(t\right),{\varphi }_2\left(t\right))$ is a contraction mapping in ${\mathcal{B}}_{\mathcal{R},T}$. According to Proposition 1, Lemma 1 and Lemma 2, we obtain

$$\begin{array}{ccc}\hfill \parallel {\Gamma }_1\left({\varphi }_1\left(t\right)\right){\parallel }_{X_T^{b,s}}& \lesssim & T^{1-b^{\prime }+b}\left({\mathcal{R}}^2+\parallel (u_l,v_l){\parallel }_{{\mathcal{X}}_T^{b,s}}+{\parallel (u_0,v_0)\parallel }_{{\mathcal{H}}^s}\right),\hfill \\ \\ \hfill \parallel {\Gamma }_2\left({\varphi }_2\left(t\right)\right){\parallel }_{X_{1,T}^{b,s}}& \lesssim & T^{1-b^{\prime }+b}\left({\mathcal{R}}^2+\parallel (u_l,v_l){\parallel }_{{\mathcal{X}}_T^{b,s}}+{\parallel (u_0,v_0)\parallel }_{{\mathcal{H}}^s}\right);\hfill \end{array}$$

thus,

$$\begin{array}{ccc}\hfill \parallel ({\Gamma }_1\left({\varphi }_1\left(t\right)\right),{\Gamma }_2\left({\varphi }_2\left(t\right)\right)){\parallel }_{{\mathcal{X}}_T^{b,s}}& \lesssim & T^{1-b^{\prime }+b}\left({\mathcal{R}}^2+\parallel (u_l,v_l){\parallel }_{{\mathcal{X}}_T^{b,s}}+{\parallel (u_0,v_0)\parallel }_{{\mathcal{H}}^s}\right).\hfill \end{array}$$

Therefore, for

$$({\varphi }_{1.1},{\varphi }_{2.1}),({\varphi }_{1.2},{\varphi }_{2.2})\in {\mathcal{B}}_{\mathcal{R},T},$$

we obtain

$$\begin{array}{ccc}\hfill \parallel {\Gamma }_1({\varphi }_{1.1}-{\varphi }_{1.2}){\parallel }_{X_T^{b,s}}& \lesssim & T^{1-b^{\prime }+b}\left(\mathcal{R}+\parallel (u_l,v_l){\parallel }_{{\mathcal{X}}_T^{b,s}}+{\parallel (u_0,v_0)\parallel }_{{\mathcal{H}}^s}\right)\hfill \\ \\ & & \times \parallel ({\varphi }_{1.1}-{\varphi }_{1.2},{\varphi }_{2.1}-{\varphi }_{2.2}){\parallel }_{{\mathcal{X}}_T^{b,s}},\hfill \\ \\ \hfill \parallel {\Gamma }_2({\varphi }_{2.1}-{\varphi }_{2.2}){\parallel }_{X_{1,T}^{b,s}}& \lesssim & T^{1-b^{\prime }+b}\left(\mathcal{R}+\parallel (u_l,v_l){\parallel }_{{\mathcal{X}}_T^{b,s}}+{\parallel (u_0,v_0)\parallel }_{{\mathcal{H}}^s}\right)\hfill \\ \\ & & \times \parallel ({\varphi }_{1.1}-{\varphi }_{1.2},{\varphi }_{2.1}-{\varphi }_{2.2}){\parallel }_{{\mathcal{X}}_T^{b,s}},\hfill \end{array}$$

and thus,

$$\begin{array}{ccc}\hfill \parallel ({\Gamma }_1({\varphi }_{1.1}-{\varphi }_{1.2}),{\Gamma }_2({\varphi }_{2.1}-{\varphi }_{2.2})){\parallel }_{X_T^{b,s}}& \lesssim & T^{1-b^{\prime }+b}\left(\mathcal{R}+\parallel (u_l,v_l){\parallel }_{{\mathcal{X}}_T^{b,s}}+{\parallel (u_0,v_0)\parallel }_{{\mathcal{H}}^s}\right)\hfill \\ \\ & & \times \parallel ({\varphi }_{1.1}-{\varphi }_{1.2},{\varphi }_{2.1}-{\varphi }_{2.2}){\parallel }_{{\mathcal{X}}_T^{b,s}}.\hfill \end{array}$$

Let us choose $T_{\varrho }$ such that

$$4C{T}^{1-b^{\prime }+b}\left({\mathcal{R}}_{\varrho }+\parallel (u_l,v_l){\parallel }_{{\mathcal{X}}_T^{b,s}}+{\parallel (u_0,v_0)\parallel }_{{\mathcal{H}}^s}\right)\le 1,$$

where

$${\mathcal{R}}_{\varrho }=\parallel (u_l,v_l){\parallel }_{{\mathcal{X}}_T^{b,s}}+{\parallel (u_0,v_0)\parallel }_{{\mathcal{H}}^s}.$$

It is easily checked that $({\Gamma }_1,{\Gamma }_2)$ maps ${\mathcal{B}}_{\mathcal{R},T}$ to itself and is a strict contraction in ${\mathcal{B}}_{\mathcal{R},T}$ for the norm $\parallel ({\varphi }_1,{\varphi }_2){\parallel }_{{\mathcal{X}}_T^{b,s}}$:

$$\begin{array}{ccc}\hfill \parallel ({\Gamma }_1({\varphi }_{1.1}-{\varphi }_{1.2}),{\Gamma }_2({\varphi }_{2.1}-{\varphi }_{2.2})){\parallel }_{X_T^{b,s}}& \le & {\displaystyle \frac{1}{4}}{\parallel ({\varphi }_{1.1}-{\varphi }_{1.2},{\varphi }_{2.1}-{\varphi }_{2.2})\parallel }_{{\mathcal{X}}_T^{b,s}}.\hfill \end{array}$$

Hence, $({\Gamma }_1,{\Gamma }_2)$ has a unique fixed point in ${\mathcal{X}}_T^{b,s}$ that is a solution of (12) on $\left[0,T_{\varrho }\right]\times \left[0,T_{\varrho }\right]$.

Now, observe that

$$\left\{\begin{array}{c}u\left(t\right)={\Pi }_1\left(t\right)+{\varphi }_1\left(t\right)+u_l\left(t\right)\in X_{T_{\varrho }}^{b^{\prime },s}+X_{T_{\varrho }}^{b,s}\hfill \\ \\ v\left(t\right)={\Pi }_2\left(t\right)+{\varphi }_2\left(t\right)+v_l\left(t\right)\in X_{1,T_{\varrho }}^{b^{\prime },s}+X_{1,T_{\varrho }}^{b,s}.\hfill \end{array}\right.$$

(13)

We complete the proof by showing that

$$(u,v)\in C([0,T_{\varrho }],H^s\left(\mathbb{R}\right))\times C([0,T_{\varrho }],H^s\left(\mathbb{R}\right)).$$

We take into account that $b^{\prime }<{\displaystyle \frac{1}{2}}$, $b>{\displaystyle \frac{1}{2}}$.

By the Sobolev embedding theorem, we have

$$({\Pi }_1,{\Pi }_2)\in C([0,T_{\varrho }],H^s\left(\mathbb{R}\right))\times C([0,T_{\varrho }],H^s\left(\mathbb{R}\right)).$$

As in Theorem 1, we operate under the condition that $\Lambda \in L_2^{0,s}$ and the fact that $U\left(t\right)$ and $U_1\left(t\right)$ are a unitary group in $H^s\left(\mathbb{R}\right)$; the use of Theorem 1 implies that

$$(u_l,v_l)\in C([0,T_{\varrho }],H^s\left(\mathbb{R}\right))\times C([0,T_{\varrho }],H^s\left(\mathbb{R}\right)).$$

By Lemma 1, we have

$$\tilde{u}{\partial }_x\tilde{u},\tilde{v}{\partial }_x\tilde{v},{\partial }_x\left(\tilde{u}\tilde{v}\right)\in X^{s,b^{\prime }},$$

and

$$\tilde{u}{\partial }_x\tilde{u},\tilde{v}{\partial }_x\tilde{v},{\partial }_x\left(\tilde{u}\tilde{v}\right)\in X_1^{s,b^{\prime }},$$

for any prolongation $\tilde{u}$ of u in

$$X^{s,b^{\prime }}+X^{s,b}$$

and $\tilde{v}$ of v in

$$X_1^{s,b^{\prime }}+X_1^{s,b},$$

where

$$-{\displaystyle \frac{1}{2}}<b^{\prime }\le 0\le b<1+b.$$

Therefore,

$${∥{\int }_0^{t}U(t-\tau )(3\tilde{u}{\partial }_x\tilde{u}-\tilde{v}{\partial }_x\tilde{v}+{\partial }_x\left(\tilde{u}\tilde{v}\right))\left(\tau \right)d\tau ∥}_{X^{s,b}}\le C{∥3\tilde{u}{\partial }_x\tilde{u}-\tilde{v}{\partial }_x\tilde{v}+{\partial }_x\left(\tilde{u}\tilde{v}\right)∥}_{X^{s,b^{\prime }}},$$

$${∥{\int }_0^{t}U(t-\tau )(-\tilde{u}{\partial }_x\tilde{u}+3\tilde{v}{\partial }_x\tilde{v}+{\partial }_x\left(\tilde{u}\tilde{v}\right))\left(\tau \right)d\tau ∥}_{X_1^{s,b}}\le C{∥-\tilde{u}{\partial }_x\tilde{u}+3\tilde{v}{\partial }_x\tilde{v}+{\partial }_x\left(\tilde{u}\tilde{v}\right)∥}_{X_1^{s,b^{\prime }}}.$$

Since $1+b^{\prime }>{\displaystyle \frac{1}{2}}$, it follows that

$$(\tilde{u},\tilde{v})\in {\mathcal{X}}^{1+b^{\prime },s}\subset C([0,T_{\varrho }],H^s\left(\mathbb{R}\right))\times C([0,T_{\varrho }],H^s\left(\mathbb{R}\right)).$$

□

4. Global Well-Posedness of KdV-Type Boussinesq System in ${\mathit{L}}^{\mathbf{2}}\left(\mathbb{R}\right)\times {\mathit{L}}^{\mathbf{2}}\left(\mathbb{R}\right)$

Our second main result reads as follows.

Theorem 3. 

Let

$$(u_0,v_0)\in L^2\left(\Omega ,L^2\left(\mathbb{R}\right)\right)\times L^2\left(\Omega ,L^2\left(\mathbb{R}\right)\right)$$

be ${\mathcal{F}}_0$-measurable initial data, and let $\Lambda \in L_2^{0,0}$. Then, the solution u given by Theorem 2 is global and satisfies

$$(u,v)\in L^2\left(\Omega ,C\left([0,T_0],H^s\left(\mathbb{R}\right)\right)\right)\times L^2\left(\Omega ,C\left([0,T_0],H^s\left(\mathbb{R}\right)\right)\right)forany{T}_0>0.$$

Proof. 

We assume here that

$$(u_0,v_0)\in L^2\left(\Omega ,L^2\left(\mathbb{R}\right)\right)\times L^2\left(\Omega ,L^2\left(\mathbb{R}\right)\right)$$

and that the operator

$$\Lambda \in L_2^0\left(L^2\left(\mathbb{R}\right),L^2\left(\mathbb{R}\right)\right).$$

As in [31], we follow the same argument to prove that the solution $(u,v)$ can be continued on $[0,1]$. To this end, we take a sequence

$${\left({\Lambda }_n\right)}_{n\in \mathbb{N}}\in L_2^0\left(L^2\left(\mathbb{R}\right),H^4\left(\mathbb{R}\right)\right)$$

such that

$${\Phi }_{1,n}\to {\Phi }_1,\mathrm{in}{L}_2^0\left(L^2\left(\mathbb{R}\right),L^2\left(\mathbb{R}\right)\right),$$

$${\Phi }_{2,n}\to {\Phi }_2,\mathrm{in}{L}_2^0\left(L^2\left(\mathbb{R}\right),L^2\left(\mathbb{R}\right)\right),$$

and another sequence

$${\left(u_{0,n},v_{0,n}\right)}_{n\in \mathbb{N}}\in L^2\left(\Omega ,H^3\left(\mathbb{R}\right)\right)\times L^2\left(\Omega ,H^3\left(\mathbb{R}\right)\right)$$

such that

$$(u_{0,n},v_{0,n})\to (u_0,v_0)\mathrm{in}{L}^2\left(\Omega ,L^2\left(\mathbb{R}\right)\right)\times L^2\left(\Omega ,L^2\left(\mathbb{R}\right)\right).$$

It is well known from Lemma 3.2 in [31] that there exists a unique solution

$$(u_n,v_n)\in C\left([0,1],H^3\left(\mathbb{R}\right)\right)\times C\left([0,1],H^3\left(\mathbb{R}\right)\right).$$

$$\left\{\begin{array}{cc}{u}_n\left(t\right)\hfill & {\displaystyle =U\left(t\right)u_{0,n}-\frac{1}{2}{\int }_0^{t}U(t-\tau )(3u_n\left(\tau \right){\partial }_x{u}_n\left(\tau \right)-v_n\left(\tau \right){\partial }_x{v}_n\left(\tau \right)+{\partial }_x\left(u_n\left(\tau \right)v_n\left(\tau \right)\right))d\tau }\hfill \\ \\ \hfill & +{\int }_0^{t}U(t-\tau ){\Lambda }_{n}dW\left(\tau \right),\hfill \\ \\ v_n\left(t\right)\hfill & {\displaystyle =U_1\left(t\right)v_{0,n}-\frac{1}{2}{\int }_0^t{U}_1(t-\tau )(-u_n\left(\tau \right){\partial }_x{u}_n\left(\tau \right)+3v_n\left(\tau \right){\partial }_x{v}_n\left(\tau \right)}\hfill \\ \\ & +{\partial }_x\left(u_n\left(\tau \right)v_n\left(\tau \right)\right))d\tau +{\int }_0^t{U}_1(t-\tau ){\Lambda }_{n}dW\left(\tau \right).\hfill \end{array}\right.$$

We then use Itô’s formula on $\parallel u_n{\parallel }_{L^2\left(\mathbb{R}\right)}^2$, $\parallel v_n{\parallel }_{L^2\left(\mathbb{R}\right)}^2$ and a martingale inequality that can be seen in Theorem 3.14 of [30]; as a result, we have

$$\begin{array}{cc}& \mathbb{E}\left(\underset{t\in [0,1]}{sup}{\int }_0^t\left(u_n\left(\tau \right),{\Lambda }_{n}dW\left(\tau \right)\right)\right)\le {\displaystyle \frac{1}{2}}\mathbb{E}\left(\underset{t\in [0,1]}{sup}{\left|u_n\left(t\right)\right|}_{L_x^2}^2\right)+C{∥{\Lambda }_n∥}_{L_2^{0,0}}^2,\hfill \\ \\ & \mathbb{E}\left(\underset{t\in [0,1]}{sup}{\int }_0^t\left(v_n\left(\tau \right),{\Lambda }_{n}dW\left(\tau \right)\right)\right)\le {\displaystyle \frac{1}{2}}\mathbb{E}\left(\underset{t\in [0,1]}{sup}{\left|v_n\left(t\right)\right|}_{L_x^2}^2\right)+C{∥{\Lambda }_n∥}_{L_2^{0,0}}^2.\hfill \end{array}$$

We deduce that

$$\mathbb{E}\left(\underset{t\in [0,1]}{sup}{\parallel u_n\left(t\right)\parallel }_{L_x^2}^2\right)\le \mathbb{E}\left(\parallel u_{0,n}{\parallel }_{L_x^2}^2\right)+C{∥{\Lambda }_n∥}_{L_2^{0,0}}^2,$$

$$\mathbb{E}\left(\underset{t\in [0,1]}{sup}{\parallel v_n\left(t\right)\parallel }_{L_x^2}^2\right)\le \mathbb{E}\left(\parallel v_{0,n}{\parallel }_{L_x^2}^2\right)+C{∥{\Lambda }_n∥}_{L_2^{0,0}}^2.$$

Then

$${\left(u_n\right)}_{n\in \mathbb{N}},{\left(v_n\right)}_{n\in \mathbb{N}}isboundedin{L}^2\left(\Omega ,L^{\infty }\left([0,1],L^2\left(\mathbb{R}\right)\right)\right),$$

where it is weakly star-convergent to a function $\tilde{u},\tilde{v}$, which satisfies

$$\mathbb{E}\left(\underset{t\in [0,1]}{sup}{\parallel \tilde{u}\left(t\right)\parallel }_{L_x^2}^2\right)\le \mathbb{E}\left(\parallel u_0{\parallel }_{L_x^2}^2\right)+C{\parallel \Lambda \parallel }_{L_2^{0,0}}^2,$$

$$\mathbb{E}\left(\underset{t\in [0,1]}{sup}{\parallel \tilde{v}\left(t\right)\parallel }_{L_x^2}^2\right)\le \mathbb{E}\left(\parallel v_0{\parallel }_{L_x^2}^2\right)+C{\parallel \Lambda \parallel }_{L_2^{0,0}}^2.$$

We define the mapping $({\Gamma }_{1,n},{\Gamma }_{2,n})$ in the same way as $({\Gamma }_1,{\Gamma }_2)$; it is not hard to see that $({\Gamma }_{1,n},{\Gamma }_{2,n})$ is a strict contraction uniformly on ${\mathcal{B}}_{{\mathcal{R}}_1,T_{\varrho 1}}$ where

$${\mathcal{R}}_1⩾2C\left(\underset{n\in \mathbb{N}}{\overset{2}{sup}}\left({∥(\varpi {\varphi }_{1,n},\varpi {\varphi }_{2,n})∥}_{{\mathcal{X}}_{b,0}}\right)+C_1^2{\parallel (\tilde{u},\tilde{v})\parallel }_{L^{\infty }{\left([0,1],L^2\left(\mathbb{R}\right)\right)}^2}^2\right)$$

and

$$4C{T}_{\varrho 1}^{(1-b^{\prime }+b)}\left({\mathcal{R}}_1+\underset{n\in \mathbb{N}}{sup}\left({∥(\varpi {\varphi }_{1,n},\varpi {\varphi }_{2,n})∥}_{{\mathcal{X}}_{b,0}}\right)+C_1{\parallel (\tilde{u},\tilde{u})\parallel }_{L^{\infty }{\left([0,1],L^2\left(\mathbb{R}\right)\right)}^2}\right)\le 1.$$

Owing to the fixed point theorem, we have a unique coupled function

$$(u,v)\in {\mathcal{X}}_{T_{\varrho }}^{b,s},$$

where

$$(u,v)=(\tilde{u},\tilde{v})\mathrm{on}[0,T_{\varrho 1}]\times [0,T_{\varrho 1}]$$

and

$$\parallel u\left(T_{\varrho 1}\right),v\left(T_{\varrho 1}\right){)\parallel }_{L^2\left(\mathbb{R}\right)}\le {\parallel (\tilde{u},\tilde{v})\parallel }_{L^{\infty }{\left([0,1],L^2\left(\mathbb{R}\right)\right)}^2}.$$

□