Well-Posedness for a System of Generalized KdV-Type Equations Driven by White Noise

Abstract

In this paper, we investigate the Cauchy problem for the coupled generalized Korteweg–de Vries system driven by a cylindrical Wiener process. We prove local well-posedness for data in $H^s\times H^s,$ with $s>{\displaystyle \frac{1}{2}}$. The key methods that we used in this paper are multilinear estimates in Bourgain spaces, the Itô formula, and a fixed-point argument.

1. Introduction

Dispersive nonlinear systems appear in many physical applications. They can be used, for example, to model wave propagation on the surface of water or to describe the interaction of nonlinear internal waves. In this study, we focus on Hamiltonian systems.

$$\left\{\begin{array}{c}{\partial }_t\varphi +{\partial }_x^3\varphi +\mu {\partial }_x\left(N_{\varphi }(\varphi ,\phi )\right)=0\hfill \\ \\ {\partial }_t\phi +\alpha {\partial }_x^3\phi +\mu {\partial }_x\left(N_{\phi }(\varphi ,\phi )\right)=0,0<\alpha <1,(x,t)\in {\mathbb{R}}^2,\hfill \end{array}\right.$$

(1)

where $\varphi =\varphi (x,t)$ and $\phi =\phi (x,t)$ are real valued functions, N is a smooth function, $N_{\varphi }$ and $N_{\phi }$ denote the derivative of N with respect to $\varphi$ and $\phi$, respectively, and $\mu$ is real constant which we normalize to $\pm 1$.

This system generalizes the classical Korteweg–de Vries equation [1], originally introduced to model shallow water waves.

The question of the correct local and global definition of the initial value problem associated with (1) has been a major topic in the theory of dispersive equations in the last few years. We shall briefly recall some results of interest to us that are available in the current literature.

For $N_{\varphi }(\varphi ,\phi )={\phi }^2,N_{\phi }(\varphi ,\phi )=\varphi \phi$, the system (1) is a particular case of the Majda–Biello system [2], which models the nonlinear interaction of long-wavelength equatorial Rossby waves and barotropic Rossby waves. The well-posedness problem associated with initial value problem (1) in this case has been studied by many authors. Tadahiro Oh [3] proved that the initial value problem is locally and globally well-posed in both cases:

• In the periodic case, it is locally well-posed for $H^s\left({\mathbb{T}}_{\lambda }\right)\times H^s\left({\mathbb{T}}_{\lambda }\right),s\ge s*$, where $s*\in ({\displaystyle \frac{1}{2}},1])$, and globally well-posed in $H^1\left({\mathbb{T}}_{\lambda }\right)\times H^1\left({\mathbb{T}}_{\lambda }\right)$ due to the conservation of the Hamiltonian.
• In the nonperiodic case, it is locally well-posed for $H^s\left(\mathbb{R}\right)\times H^s\left(\mathbb{R}\right)$, where $s\ge 0$, and globally well-posed in $L^2\left(\mathbb{R}\right)\times L^2\left(\mathbb{R}\right)$ due to the $L^2$ conservation law.

For $N_{\varphi }(\varphi ,\phi )=\varphi {\phi }^2,N_{\phi }(\varphi ,\phi )={\varphi }^2\phi$, system (1) reduces to a special case of a broad class of nonlinear evolution equations considered by Ablowiz et al. [4] in the context of inverse scattering. In this case, questions regarding the proper positioning as well as the existence and stability of solitary waves for this system have been extensively studied in the literature. Carvajal and Panthee [5] proved that the initial value problem (1) is locally well-posed for given data $H^s\left(\mathbb{R}\right)\times H^s\left(\mathbb{R}\right)$, where $s>-{\displaystyle \frac{1}{2}}$.

For $N_{\varphi }(\varphi ,\phi )$ and $N_{\phi }(\varphi ,\phi )$ given by

$$N_{\varphi }(\varphi ,\phi )=A{\varphi }^{2k+1}+B{\varphi }^k{\phi }^{k+1}+{\displaystyle \frac{k+2}{k}}C{\varphi }^{k+1}{\phi }^k+D{\varphi }^{k-1}{\phi }^{k+2}$$

$$N_{\phi }(\varphi ,\phi )=A{\phi }^{2k+1}+B{\phi }^k{\varphi }^{k+1}+{\displaystyle \frac{k+2}{k}}D{\phi }^{k+1}{\varphi }^k+C{\phi }^{k-1}{\varphi }^{k+2},$$

Gomes and Pastor [6] proved that the system is locally well-posed for given data $H^s\left(\mathbb{R}\right)\times H^s\left(\mathbb{R}\right)$, where $s\ge 1$, and globally well-posed in $H^1\left(\mathbb{R}\right)\times H^1\left(\mathbb{R}\right)$.

In [7], Grujić and Kalisch establish local well-posedness for the initial-value problem

$${\partial }_t\varphi +{\partial }_x^3\varphi +{\varphi }^k{\partial }_x\varphi =0,$$

for initial data in $G^{\sigma ,s}$, where $s\ge 0$ for $p=1$, and $s>{\displaystyle \frac{3}{2}}$ for $p\ge 2$. The solution persists in $C\left([0,T],G^{\sigma ,s}\right)$ for some $T>0$, remaining analytic on the same fixed strip of width $2\sigma$ throughout the time interval $[0,T]$.

In [8], de Bouard and Debussche studied the stochastic KdV equation with white noise forcing. They proved existence and uniqueness of solutions in $H^1\left(\mathbb{R}\right)$ for additive noise, and existence of martingale solutions in $L^2\left(\mathbb{R}\right)$ for multiplicative noise (e.g., [9,10,11,12]).

Boukarou et al. [13] proved that the white-noise-driven KdV-type Boussinesq system

$$\left\{\begin{array}{c}{\partial }_t\phi +{\partial }_x\varphi +{\partial }_x^3\varphi +{\partial }_x\left(\varphi \phi \right)={\alpha }_1{B}_1(x,t)\hfill \\ \\ {\partial }_t\varphi +{\partial }_x\phi +{\partial }_x^3\phi +\varphi {\partial }_x\varphi ={\alpha }_2{B}_2(x,t),\hfill \end{array}\right.$$

is locally well-posed for given data $H^s\left(\mathbb{R}\right)\times H^s\left(\mathbb{R}\right)$, where $s\ge 0$, and globally well-posed in $L^2\left(\Omega ,L^2\left(\mathbb{R}\right)\right)\times L^2\left(\Omega ,L^2\left(\mathbb{R}\right)\right)$ (e.g., [14,15,16]).

We are interested in stochastic system with the Hamiltonian structure, which is a coupled system of generalized Korteweg–de Vries equations given by (2) and (3).

$$\left\{\begin{array}{c}{\displaystyle \frac{d\varphi \left(t\right)}{dt}}+{\partial }_x^3\varphi +{\partial }_x\left(N_{\varphi }(\varphi ,\phi )\right)={\displaystyle \frac{dW\left(t\right)}{dt}}\hfill \\ \\ {\displaystyle \frac{d\phi \left(t\right)}{dt}}+\alpha {\partial }_x^3\phi +{\partial }_x\left(N_{\phi }(\varphi ,\phi )\right)={\displaystyle \frac{dW\left(t\right)}{dt}},0<\alpha <1\hfill \\ \\ (\varphi (x,0),\phi (x,0))=\left({\varphi }_0\left(x\right),{\phi }_0\left(x\right)\right)\in H^s\left(\mathbb{R}\right)\times H^s\left(\mathbb{R}\right),\hfill \end{array}\right.$$

(2)

with

$$\left\{\begin{array}{c}{N}_{\varphi }(\varphi ,\phi )=A{\varphi }^{2k+1}+B{\varphi }^k{\phi }^{k+1}+{\displaystyle \frac{k+2}{k}}C{\varphi }^{k+1}{\phi }^k+D{\varphi }^{k-1}{\phi }^{k+2}\hfill \\ \\ N_{\phi }(\varphi ,\phi )=A{\phi }^{2k+1}+B{\phi }^k{\varphi }^{k+1}+{\displaystyle \frac{k+2}{k}}D{\phi }^{k+1}{\varphi }^k+C{\phi }^{k-1}{\varphi }^{k+2},\hfill \end{array}\right.$$

(3)

where $k\ge 1$ is a natural number, and $A,B,C$ and D non-negative real constants. Here, $\varphi =\varphi (x,t)$ and $\phi =\phi (x,t)$ are random processes defined for $(x,t)\in \mathbb{R}\times {\mathbb{R}}^{+}$, $\Xi$ is a linear operator, and $W\left(t\right)={\displaystyle \frac{\partial \mathbb{B}}{\partial x}}$ is a cylindrical Wiener process on $L^2\left(\mathbb{R}\right)$, which can also be given by $W\left(t\right)={\sum }_{i=0}^{\infty }{\beta }_i\left(t\right)e_i$, where ${\left(e_i\right)}_{i\in \mathbb{N}}$ 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.

In this work, we improve the local well-posedness result of Gomes and Pastor [6] by extending their deterministic analysis to the stochastic setting and lowering the required Sobolev regularity. Building on the methods introduced in [7,8,11], we prove that the stochastic system of (2) and (3) is locally well-posed for initial data in $H^s\left(\mathbb{R}\right)\times H^s\left(\mathbb{R}\right)$ with $s>{\displaystyle \frac{1}{2}}$.

The regularity threshold $s>{\displaystyle \frac{1}{2}}$ is critical for KdV-type systems, as it reflects the intrinsic balance between dispersive effects and nonlinear interactions within the Bourgain space framework. Improving the previously known requirement of $s\ge 1$ to $s>{\displaystyle \frac{1}{2}}$ constitutes a meaningful advancement in the low-regularity theory for coupled stochastic dispersive equations.

To state our results precisely, let us first introduce the notation to be used. We begin with function spaces. For $s\in \mathbb{R},H^s\left(\mathbb{R}\right)$ denotes the usual Sobolev space of order s, defined by the norm

$${\parallel \omega \parallel }_{H^s\left(\mathbb{R}\right)}={\left({\int }_{\mathbb{R}}{(1+|\zeta \left|\right)}^{2s}{\left|\widehat{\omega }\right|}^{2}d\zeta \right)}^{{\displaystyle \frac{1}{2}}},$$

where $\widehat{\omega }$ is the spatial Fourier transform

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

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

$${\parallel w\parallel }_{X^{s,b}\left({\mathbb{R}}^2\right)}={\parallel w\parallel }_{X^{s,b}}={\left({\int }_{{\mathbb{R}}^2}{(1+|\zeta \left|\right)}^{2s}(1+|\gamma -{\zeta }^3{\left|\right)}^{2b}{\left|\tilde{w}(\gamma ,\zeta )\right|}^{2}d\zeta d\gamma \right)}^{{\displaystyle \frac{1}{2}}},$$

$${\parallel w\parallel }_{X_{\alpha }^{s,b}\left({\mathbb{R}}^2\right)}={\parallel w\parallel }_{X^{s,b}}={\left({\int }_{{\mathbb{R}}^2}{(1+|\zeta \left|\right)}^{2s}(1+|\gamma -\alpha {\zeta }^3{\left|\right)}^{2b}{\left|\tilde{w}(\gamma ,\zeta )\right|}^{2}d\zeta d\gamma \right)}^{{\displaystyle \frac{1}{2}}},$$

where $\tilde{w}$ denotes the spacetime Fourier transform

$$\tilde{w}(\zeta ,\gamma )={\int }_{{\mathbb{R}}^2}w(x,t)e^{-i(x\zeta +t\gamma )}dxdt.$$

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

$${\parallel w\parallel }_{X_T^{s,b}}=inf\left\{{\parallel z\parallel }_{X^{s,b}}:w(x,t)=z(x,t)\mathrm{on}\mathbb{R}\times [0,T]\right\},$$

$${\parallel w\parallel }_{X_{\alpha ,T}^{s,b}}=inf\left\{{\parallel z\parallel }_{X_{\alpha }^{s,b}}:w(x,t)=z(x,t)\mathrm{on}\mathbb{R}\times [0,T]\right\}.$$

Since we are dealing with system of equations, we will need to consider product function spaces. For this, we define the product spaces

$${\mathcal{X}}_{\alpha }^{b,s}=X^{b,s}\times X_{\alpha }^{b,s},{\mathcal{X}}_{\alpha ,T}^{b,s}=X_T^{b,s}\times X_{\alpha ,T}^{b,s}\mathrm{and}{\mathcal{H}}^s=H^s\times H^s,$$

with norms

$${\parallel (\varphi ,\phi )\parallel }_{{\mathcal{X}}_{\alpha }^{b,s}}={max\{\parallel \varphi \parallel }_{X^{s,b}}{,\parallel \phi \parallel }_{X_{\alpha }^{s,b}}\},$$

$${\parallel (\varphi ,\phi )\parallel }_{{\mathcal{X}}_{\alpha ,T}^{b,s}}={max\{\parallel \varphi \parallel }_{X_T^{s,b}}{,\parallel \phi \parallel }_{X_{\alpha ,T}^{s,b}}\},$$

and

$$\parallel ({\varphi }_0,{\phi }_0){\parallel }_{{\mathcal{H}}^s}=max\{\parallel {\varphi }_0{\parallel }_{H^s},\parallel {\phi }_0{\parallel }_{H^s}\}.$$

The mixed $L^p-L^q$ -norm is defined by

$${\parallel u\parallel }_{L^p{L}^q}={\left({\int }_{-\infty }^{+\infty }{\left|{\int }_{-\infty }^{+\infty }{\left|u(x,t)\right|}^{q}dt\right|}^{{\displaystyle \frac{p}{q}}}dx\right)}^{{\displaystyle \frac{1}{p}}}$$

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 $L^2\left(\mathbb{R}\right)$ into $H^s\left(\mathbb{R}\right)$ with the norm

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

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 proven. In the subsequent work, let $(\Omega ,\mathcal{F},P)$ be a fixed probability space adapted to a filtration ${\left({\mathcal{F}}_t\right)}_{t\ge 0}$. For system (2), we will prove the following local results.

Theorem 1. 

Assume that $k\ge 1$ and $s>{\displaystyle \frac{1}{2}},\Xi \in L_2^{0,s},b\in \left(0,{\displaystyle \frac{1}{2}}\right)$, and b is close enough to ${\displaystyle \frac{1}{2}}$. If $({\varphi }_0,{\phi }_0)\in H^s(\mathbb{R}\times H^s\left(\mathbb{R}\right)$ for almost surely $\varrho \in \Omega$ and ${\varphi }_0,{\phi }_0$ are ${\mathcal{F}}_0-$ measurable, then for almost surely $\varrho \in \Omega$, there exists a constant $T_{\varrho }>0$ and a unique solution u of the Cauchy problem (2) on $\left[0,T_{\varrho }\right]$ which satisfies

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

Remark 1. 

The solution satisfies $(\varphi ,\phi )\in C\left(\left[0,T_{\varrho }\right],H^s\left(\mathbb{R}\right)\right)\times C\left(\left[0,T_{\varrho }\right],H^s\left(\mathbb{R}\right)\right)$, which follows from the embedding $X^{s,b}\hookrightarrow C_t{H}_x^s$ for $b>{\displaystyle \frac{1}{2}}$.

Notation: Throughout this paper, we use the following standard asymptotic notation:

• $a{\lesssim }_{k_1,\dots ,k_n}b$ denotes $a\le cb$ with a constant $c>0$ depending on $k_1,\dots ,k_n$. If c is an absolute constant, we shall write $a\lesssim b$.
• $a\sim b$ means that a and b are asymptotically equivalent.
• $a\approx b$ means that a and b are comparable in size, typically with implicit constants independent of the parameters.
• $a\ll b$ means that a is much smaller than b, typically in the sense that the ratio ${\displaystyle \frac{a}{b}}$ is bounded by a small constant.

This paper is organized as follows: Section 2 introduces the notation and linear estimates. Section 3 establishes multilinear estimates in Bourgain spaces. Section 4 provides stochastic convolution estimates. Section 5 proves local well-posedness via a fixed-point argument. Section 6 concludes the study and presents remarks and future directions.

2. Linear Estimates

To prove our main results, we will need to introduce several important estimates. To state these estimates, let us write the Itô form of the system in Equation (2), namely

$$\left\{\begin{array}{c}d\varphi +\left({\partial }_x^3\varphi +{\partial }_x\left(N_{\varphi }(\varphi ,\phi )\right)\right)dt=\Xi dW\hfill \\ \\ d\phi +\left(\alpha {\partial }_x^3\phi +{\partial }_x\left(N_{\phi }(\varphi ,\phi )\right)\right)dt=\Xi dW.\hfill \end{array}\right.$$

(4)

System (4) is supplemented with the initial conditions

$$\varphi (x,0)={\varphi }_0\left(x\right),\phi (x,0)={\phi }_0\left(x\right).$$

(5)

To understand the assumptions required for $\Xi$, it is useful to first consider the linear equation

$$\left\{\begin{array}{c}d\varphi +{\partial }_x^3\varphi dt=\Xi dW\hfill \\ \\ d\phi +\alpha {\partial }_x^3\phi dt=\Xi dW\hfill \\ \\ {\varphi }_0\left(x\right)=0,{\phi }_0\left(x\right)=0.\hfill \end{array}\right.$$

which is given by the stochastic Itô integral

$$\left\{\begin{array}{c}{\varphi }_l\left(t\right)={\int }_0^t\mathfrak{U}(t-\gamma )\Xi dW\left(\gamma \right)\hfill \\ \\ {\phi }_l\left(t\right)={\int }_0^t{\mathfrak{U}}_{\alpha }(t-\gamma )\Xi dW\left(\gamma \right).\hfill \end{array}\right.$$

(6)

where $\mathfrak{U}\left(t\right)=e^{-t{\partial }_x^3},{\mathfrak{U}}_{\alpha }\left(t\right)=e^{-t\alpha {\partial }_x^3}$ comprise the Airy group. Using the unitarity of $\mathfrak{U}\left(t\right)$ and ${\mathfrak{U}}_{\alpha }\left(t\right)$, one can easily show that $\varphi \left(t\right)$ and $\phi \left(t\right)$ belong to $H^s\left(\mathbb{R}\right)$ only if $\Xi$ is a Hilbert–Schmidt operator from $L^2\left(\mathbb{R}\right)$ into $H^s\left(\mathbb{R}\right)$.

We will solve (4) supplemented with the initial condition (5) by considering its mild form

$$\left\{\begin{array}{c}\varphi \left(t\right)=\mathfrak{U}\left(t\right){\varphi }_0+{\int }_0^t\mathfrak{U}(t-\gamma ){\partial }_x{N}_{\varphi }(\varphi ,\phi )\left(\gamma \right)d\gamma +{\int }_0^t\mathfrak{U}(t-\gamma )\Xi dW\left(\gamma \right)\hfill \\ \\ \phi \left(t\right)={\mathfrak{U}}_{\alpha }\left(t\right){\phi }_0+{\int }_0^t{\mathfrak{U}}_{\alpha }(t-\gamma ){\partial }_x{N}_{\phi }(\varphi ,\phi )\left(\gamma \right)d\gamma +{\int }_0^t{\mathfrak{U}}_{\alpha }(t-\gamma )\Xi dW\left(\gamma \right).\hfill \end{array}\right.$$

(7)

To construct mild solutions, we will need the following estimates:

Proposition 1 

(Linear Estimates [5]). For any $s,b\in \mathbb{R}$, we have

$${∥\mathfrak{U}\left(t\right){\varphi }_0∥}_{X_T^{s,b}}\lesssim {\parallel {\varphi }_0\parallel }_{H^s},$$

$${∥{\mathfrak{U}}_{\alpha }\left(t\right){\phi }_0∥}_{X_{\alpha ,T}^{s,b}}\lesssim {\parallel {\phi }_0\parallel }_{H^s}.$$

Further, if $-{\displaystyle \frac{1}{2}}<b^{\prime }\le 0\le b<b^{\prime }+1$ and $0\le T\le 1$, then

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

and

$${∥{\int }_0^t{\mathfrak{U}}_{\alpha }\left(t-\gamma \right)N_{\phi }(\varphi \left(\gamma \right),\phi \left(\gamma \right))\mathrm{d}\gamma ∥}_{X_{\alpha ,T}^{s,b}}\lesssim T^{1-b+b^{\prime }}{\parallel N_{\phi }(\varphi \left(\gamma \right),\phi \left(\gamma \right))\parallel }_{X_{\alpha ,T}^{s,b^{\prime }}}.$$

3. Multilinear Estimates

The proof of multilinear estimates will be established using a number of auxiliary results. To state these results, we first need to introduce some more notation. We note that the operators $A,\Lambda$$F_{\kappa }$ and $F_{\kappa }^{\alpha }$ are defined as

$$\widehat{Aw}(\zeta ,\gamma )=\left(1+\left|\zeta \right|\right)\widehat{w}(\zeta ,\gamma ),$$

$$\widehat{\Lambda w}(\zeta ,\gamma )=\left(1+\left|\gamma \right|\right)\widehat{w}(\zeta ,\gamma ),$$

$$\widehat{F_{\kappa }}(\zeta ,\gamma )={\displaystyle \frac{f(\zeta ,\gamma )}{{\left(1+|\gamma -{\zeta }^3|\right)}^{\kappa }}},$$

$$\widehat{F_{\kappa }^{\alpha }}(\zeta ,\gamma )={\displaystyle \frac{f(\zeta ,\gamma )}{{\left(1+|\gamma -\alpha {\zeta }^3|\right)}^{\kappa }}}.$$

Lemma 1. 

([7]). Let s and κ be given. There is a constant c depending on s and κ such that

$$If\kappa >{\displaystyle \frac{1}{4}},then\parallel A^{{\displaystyle \frac{1}{2}}}{F}_{\kappa }{\parallel }_{L_x^4{L}_t^2}\le C{\parallel f\parallel }_{L_{\zeta }^2{L}_{\gamma }^2},$$

(8)

$$If\kappa >{\displaystyle \frac{1}{4}},then\parallel A^{{\displaystyle \frac{1}{2}}}{F}_{\kappa }^{\alpha }{\parallel }_{L_x^4{L}_t^2}\le C{\parallel f\parallel }_{L_{\zeta }^2{L}_{\gamma }^2},$$

(9)

$$If\kappa >{\displaystyle \frac{1}{2}},ands>{\displaystyle \frac{1}{2}},\mathit{then}\parallel A^{-s}{F}_{\kappa }{\parallel }_{L_x^{\infty }{L}_t^{\infty }}\le C{\parallel f\parallel }_{L_{\zeta }^2{L}_{\gamma }^2},$$

(10)

$$If\kappa >{\displaystyle \frac{1}{2}},ands>{\displaystyle \frac{1}{2}},\mathit{then}\parallel A^{-s}{F}_{\kappa }^{\alpha }{\parallel }_{L_x^{\infty }{L}_t^{\infty }}\le C{\parallel f\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}.$$

(11)

Lemma 2. 

For $\kappa >{\displaystyle \frac{1}{2}}$, $s>{\displaystyle \frac{1}{2}}$, and ${\displaystyle \frac{1}{2}}<b<\kappa$, there exists a constant C depending only on κ and s, such that

$$\parallel A^{-s}{F}_{\kappa }{\parallel }_{L_x^2{L}_t^{\infty }}\le C{\parallel f\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}.$$

(12)

Proof. 

We begin by applying the Sobolev embedding theorem in the time variable. For $b>{\displaystyle \frac{1}{2}}$, we have the continuous embedding $H^b\left({\mathbb{R}}_t\right)\hookrightarrow L^{\infty }\left({\mathbb{R}}_t\right)$, which gives

$$\parallel A^{-s}{F}_{\kappa }{\parallel }_{L_x^2{L}_t^{\infty }}\le c{∥\parallel {\Lambda }^b\left(A^{-s}{F}_{\kappa }\right)(x,·){\parallel }_{L_t^2}∥}_{L_x^2},$$

where ${\Lambda }^b$ is the Fourier multiplier operator in time defined by

$$\widehat{{\Lambda }^{b}w}\left(\gamma \right)={(1+|\gamma \left|\right)}^b\widehat{w}\left(\gamma \right).$$

Now observe that

$${∥\parallel {\Lambda }^b\left(A^{-s}{F}_{\kappa }\right)(x,·){\parallel }_{L_t^2}∥}_{L_x^2}={\parallel {\Lambda }^b{A}^{-s}{F}_{\kappa }\parallel }_{L_{x,t}^2}.$$

According to Plancherel’s theorem in $x,t$, we compute

$$\begin{array}{cc}\hfill \parallel {\Lambda }^b{A}^{-s}{F}_{\kappa }{\parallel }_{L_{x,t}^2}^2& =\int \int {\left|{\mathcal{F}}_{x,t}\left[{\Lambda }^b{A}^{-s}{F}_{\kappa }\right](\zeta ,\gamma )\right|}^{2}d\zeta d\gamma \hfill \\ \hfill & =\int \int {\displaystyle \frac{{(1+|\gamma \left|\right)}^{2b}}{{(1+|\zeta \left|\right)}^{2s}(1+|\gamma -{\zeta }^3{\left|\right)}^{2\kappa }}}{\left|f(\zeta ,\gamma )\right|}^{2}d\zeta d\gamma .\hfill \end{array}$$

Thus, it suffices to show that the multiplier

$$M(\zeta ,\gamma )={\displaystyle \frac{{(1+|\gamma \left|\right)}^b}{{(1+|\zeta \left|\right)}^s(1+|\gamma -{\zeta }^3{\left|\right)}^{\kappa }}}$$

is bounded on ${\mathbb{R}}_{\zeta }\times {\mathbb{R}}_{\gamma }$. We consider two cases:

• Case 1: $\left|\gamma \right|\lesssim {\left|\zeta \right|}^3$.

• In this case, we have $(1+|\gamma \left|\right)\lesssim {(1+|\zeta \left|\right)}^3$, and also

  $$1+|\gamma -{\zeta }^3{|\le 1+|\gamma |+|\zeta |}^3\lesssim {(1+|\zeta \left|\right)}^3.$$

• Therefore,

$$M(\zeta ,\gamma )\lesssim {\displaystyle \frac{{(1+|\zeta \left|\right)}^{3b}}{{(1+|\zeta \left|\right)}^s{(1+|\zeta \left|\right)}^{3\kappa }}}={(1+|\zeta \left|\right)}^{3b-s-3\kappa }.$$

Since $s>{\displaystyle \frac{1}{2}}$ and $\kappa >{\displaystyle \frac{1}{2}}$, we can choose b such that ${\displaystyle \frac{1}{2}}<b<\kappa$ and $3b-s-3\kappa \le 0$. For example, set b sufficiently close to ${\displaystyle \frac{1}{2}}$ that $3b<s+3\kappa$, which is possible because $s+3\kappa >2$. Then $M(\zeta ,\gamma )$ is bounded in this region.

Case 2: $\left|\gamma \right|\gg {\left|\zeta \right|}^3$.

In this case, we have $|\gamma -{\zeta }^3{|\ge |\gamma |-|\zeta |}^3\gtrsim \left|\gamma \right|$ for large $\left|\gamma \right|$, so that

$$1+|\gamma -{\zeta }^3|\gtrsim 1+|\gamma |.$$

Hence,

$$M(\zeta ,\gamma )\lesssim {\displaystyle \frac{{(1+|\gamma \left|\right)}^b}{{(1+|\zeta \left|\right)}^s{(1+|\gamma \left|\right)}^{\kappa }}}={\displaystyle \frac{1}{{(1+|\zeta \left|\right)}^s{(1+|\gamma \left|\right)}^{\kappa -b}}}.$$

Since $\kappa >b$ and $s>0$, this is bounded as $\left|\gamma \right|\to \infty$ and $\left|\zeta \right|\to \infty$. Therefore, $M(\zeta ,\gamma )\in L^{\infty }({\mathbb{R}}_{\zeta }\times {\mathbb{R}}_{\gamma })$, and we conclude that

$$\parallel {\Lambda }^b{A}^{-s}{F}_{\kappa }{\parallel }_{L_{x,t}^2}\le {\parallel M\parallel }_{L^{\infty }}{\parallel f\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}.$$

□

Lemma 3. 

For $\kappa >{\displaystyle \frac{1}{2}}$, $s>{\displaystyle \frac{1}{2}}$, ${\displaystyle \frac{1}{2}}<b<min(\kappa ,{\displaystyle \frac{s}{3}}+ϵ)$, and $0<\alpha <1$, there exists a constant C depending only on κ, s, and α, such that

$$\parallel A^{-s}{F}_{\kappa }^{\alpha }{\parallel }_{L_x^2{L}_t^{\infty }}\le C{\parallel f\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}.$$

(13)

Proof. 

For $b>{\displaystyle \frac{1}{2}}$, we have $H^b\left({\mathbb{R}}_t\right)\hookrightarrow L^{\infty }\left({\mathbb{R}}_t\right)$, which gives

$$\parallel A^{-s}{F}_{\kappa }^{\alpha }{\parallel }_{L_x^2{L}_t^{\infty }}\le c{∥\parallel {\Lambda }^b\left(A^{-s}{F}_{\kappa }^{\alpha }\right)(x,·){\parallel }_{L_t^2}∥}_{L_x^2},$$

where ${\Lambda }^b$ is the Fourier multiplier operator in time. Then

$${∥\parallel {\Lambda }^b\left(A^{-s}{F}_{\kappa }^{\alpha }\right)(x,·){\parallel }_{L_t^2}∥}_{L_x^2}={\parallel {\Lambda }^b{A}^{-s}{F}_{\kappa }^{\alpha }\parallel }_{L_{x,t}^2}.$$

According to Plancherel’s theorem in $x,t$, we compute

$$\begin{array}{cc}\hfill \parallel {\Lambda }^b{A}^{-s}{F}_{\kappa }^{\alpha }{\parallel }_{L_{x,t}^2}^2& =\int \int {\left|{\mathcal{F}}_{x,t}\left[{\Lambda }^b{A}^{-s}{F}_{\kappa }^{\alpha }\right](\zeta ,\gamma )\right|}^{2}d\zeta d\gamma \hfill \\ \hfill & =\int \int {\displaystyle \frac{{(1+|\gamma \left|\right)}^{2b}}{{(1+|\zeta \left|\right)}^{2s}(1+|\gamma -\alpha {\zeta }^3{\left|\right)}^{2\kappa }}}{\left|f(\zeta ,\gamma )\right|}^{2}d\zeta d\gamma .\hfill \end{array}$$

Thus, it suffices to show that the multiplier

$$M(\zeta ,\gamma )={\displaystyle \frac{{(1+|\gamma \left|\right)}^b}{{(1+|\zeta \left|\right)}^s(1+|\gamma -\alpha {\zeta }^3{\left|\right)}^{\kappa }}}$$

is bounded on ${\mathbb{R}}_{\zeta }\times {\mathbb{R}}_{\gamma }$. We consider two cases:

Case 1: $\left|\gamma \right|\lesssim {\left|\zeta \right|}^3$.

In this case, we have $(1+|\gamma \left|\right)\lesssim {(1+|\zeta \left|\right)}^3$, and also

$$1+|\gamma -\alpha {\zeta }^3{|\le 1+|\gamma |+\alpha |\zeta |}^3\lesssim {(1+|\zeta \left|\right)}^3.$$

Therefore,

$$M(\zeta ,\gamma )\lesssim {\displaystyle \frac{{(1+|\zeta \left|\right)}^{3b}}{{(1+|\zeta \left|\right)}^s{(1+|\zeta \left|\right)}^{3\kappa }}}={(1+|\zeta \left|\right)}^{3b-s-3\kappa }.$$

Since $s>{\displaystyle \frac{1}{2}}$ and $\kappa >{\displaystyle \frac{1}{2}}$, we can choose b such that ${\displaystyle \frac{1}{2}}<b<\kappa$ and $3b-s-3\kappa \le 0$. Then $M(\zeta ,\gamma )$ is bounded in this region.

Case 2: $\left|\gamma \right|\gg {\left|\zeta \right|}^3$.

In this case, we have $|\gamma -\alpha {\zeta }^3{|\ge |\gamma |-\alpha |\zeta |}^3\gtrsim \left|\gamma \right|$ for large $\left|\gamma \right|$, so that

$$1+|\gamma -\alpha {\zeta }^3|\gtrsim 1+|\gamma |.$$

Hence,

$$M(\zeta ,\gamma )\lesssim {\displaystyle \frac{{(1+|\gamma \left|\right)}^b}{{(1+|\zeta \left|\right)}^s{(1+|\gamma \left|\right)}^{\kappa }}}={\displaystyle \frac{1}{{(1+|\zeta \left|\right)}^s{(1+|\gamma \left|\right)}^{\kappa -b}}}.$$

Since $\kappa >b$ and $s>0$, this is bounded as $\left|\gamma \right|\to \infty$ and $\left|\zeta \right|\to \infty$. Therefore, $M(\zeta ,\gamma )\in L^{\infty }({\mathbb{R}}_{\zeta }\times {\mathbb{R}}_{\gamma })$, and we conclude that

$$\parallel {\Lambda }^b{A}^{-s}{F}_{\kappa }^{\alpha }{\parallel }_{L_{x,t}^2}\le {\parallel M\parallel }_{L^{\infty }}{\parallel f\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}.$$

□

Lemma 4 

(Multilinear estimate). Let $k\ge 1$, $0<\alpha <1$; then there exist $b>{\displaystyle \frac{1}{2}}$ and $-{\displaystyle \frac{1}{2}}<b^{\prime }<-{\displaystyle \frac{1}{4}}$ such that for all $\varphi \in X^{s,b}$ and $\phi \in X_{\alpha }^{s,b}$, we have

$$\begin{array}{c}\hfill \parallel {\partial }_x{\varphi }^{2k+1}{\parallel }_{X^{s,b^{\prime }}}\lesssim {\parallel \varphi \parallel }_{X^{s,b}}^{2k+1},\end{array}$$

(14)

$$\begin{array}{c}\hfill \parallel {\partial }_x{\phi }^{2k+1}{\parallel }_{X_{\alpha }^{s,b^{\prime }}}\lesssim {\parallel \phi \parallel }_{X_{\alpha }^{s,b}}^{2k+1},\end{array}$$

(15)

$$\begin{array}{c}\hfill \parallel {\partial }_x\left({\varphi }^k{\phi }^{k+1}\right){\parallel }_{X^{s,b^{\prime }}}\lesssim {\parallel \varphi \parallel }_{X^{s,b}}^k{\parallel \phi \parallel }_{X_{\alpha }^{s,b}}^{k+1},\end{array}$$

(16)

$$\begin{array}{c}\hfill \parallel {\partial }_x\left({\varphi }^{k+1}{\phi }^k\right){\parallel }_{X_{\alpha }^{s,b^{\prime }}}\lesssim {\parallel \varphi \parallel }_{X^{s,b}}^{k+1}{\parallel \phi \parallel }_{X_{\alpha }^{s,b}}^k,\end{array}$$

(17)

$$\begin{array}{c}\hfill \parallel {\partial }_x\left({\varphi }^{k+1}{\phi }^k\right){\parallel }_{X^{s,b^{\prime }}}\lesssim {\parallel \varphi \parallel }_{X^{s,b}}^{k+1}{\parallel \phi \parallel }_{X_{\alpha }^{s,b}}^k,\end{array}$$

(18)

$$\begin{array}{c}\hfill \parallel {\partial }_x\left({\varphi }^k{\phi }^{k+1}\right){\parallel }_{X_{\alpha }^{s,b^{\prime }}}\lesssim {\parallel \varphi \parallel }_{X^{s,b}}^k{\parallel \phi \parallel }_{X_{\alpha }^{s,b}}^{k+1},\end{array}$$

(19)

$$\begin{array}{c}\hfill \parallel {\partial }_x\left({\varphi }^{k-1}{\phi }^{k+2}\right){\parallel }_{X^{s,b^{\prime }}}\lesssim {\parallel \varphi \parallel }_{X^{s,b}}^{k-1}{\parallel \phi \parallel }_{X_{\alpha }^{s,b}}^{k+2},\end{array}$$

(20)

$$\begin{array}{c}\hfill \parallel {\partial }_x\left({\varphi }^{k+2}{\phi }^{k-1}\right){\parallel }_{X_{\alpha }^{s,b^{\prime }}}\lesssim {\parallel \varphi \parallel }_{X^{s,b}}^{k+2}{\parallel \phi \parallel }_{X_{\alpha }^{s,b}}^{k-1},\end{array}$$

(21)

hold for any $s>{\displaystyle \frac{1}{2}}$.

Proof. 

First of all, for $i=1,2,\dots ,2k+1$ and $j=1,2,\dots ,2k+1$, we define

$$\begin{array}{c}\hfill f_i(\zeta ,\gamma )={(1+|\zeta \left|\right)}^s(1+|\gamma -{\zeta }^3{\left|\right)}^b\left|\widehat{{\varphi }_i}(\zeta ,\gamma )\right|,\\ \hfill g_j(\zeta ,\gamma )={(1+|\zeta \left|\right)}^s(1+|\gamma -\alpha {\zeta }^3{\left|\right)}^b\left|\widehat{{\phi }_j}(\zeta ,\gamma )\right|.\end{array}$$

We begin by proving (14) and (15); the proofs of the remaining inequalities are similar.

We first prove the case $k=1$, as the proof for general $2k+1$ will then be more transparent, which means we prove

$$\parallel {\partial }_x\left({\varphi }_1{\varphi }_2{\varphi }_3\right){\parallel }_{X^{s,b^{\prime }}}\le C\parallel {\varphi }_1{\parallel }_{X^{s,b}}\parallel {\varphi }_2{\parallel }_{X^{s,b}}{\parallel {\varphi }_3\parallel }_{X^{s,b}},$$

$$\parallel {\partial }_x\left({\phi }_1{\phi }_2{\phi }_3\right){\parallel }_{X_{\alpha }^{s,b^{\prime }}}\le C\parallel {\phi }_1{\parallel }_{X_{\alpha }^{s,b}}\parallel {\phi }_2{\parallel }_{X_{\alpha }^{s,b}}{\parallel {\phi }_3\parallel }_{X_{\alpha }^{s,b}}.$$

Inequality (14): We have

$$\begin{array}{cc}\hfill \parallel {\partial }_x\left({\varphi }_1{\varphi }_2{\varphi }_3\right){\parallel }_{X^{s,b^{\prime }}}& ={∥{(1+|\zeta \left|\right)}^s(1+|\gamma -{\zeta }^3{\left|\right)}^{b^{\prime }}\left|\widehat{{\partial }_x\left({\varphi }_1{\varphi }_2{\varphi }_3\right)}(\zeta ,\gamma )\right|∥}_{L_{\zeta }^2{L}_{\gamma }^2}\hfill \\ \\ & ={∥{(1+|\zeta \left|\right)}^s(1+|\gamma -{\zeta }^3{\left|\right)}^{b^{\prime }}\left|\zeta \right||\widehat{\left({\varphi }_1{\varphi }_2{\varphi }_3\right)}(\zeta ,\gamma )|∥}_{L_{\zeta }^2{L}_{\gamma }^2}\hfill \\ \\ & ={∥{(1+|\zeta \left|\right)}^s(1+|\gamma -{\zeta }^3{\left|\right)}^{b^{\prime }}\left|\zeta \right||\widehat{{\varphi }_1}*\widehat{{\varphi }_2}*\widehat{{\varphi }_3}(\zeta ,\gamma )|∥}_{L_{\zeta }^2{L}_{\gamma }^2}\hfill \\ \\ & =\parallel (1+{\left|\zeta \right|)}^s(1+|\gamma -{\zeta }^3{\left|\right)}^{b^{\prime }}\left|\zeta \right|{\int }_{{\mathbb{R}}^4}\widehat{{\varphi }_1}({\zeta }_1,{\gamma }_1)\widehat{{\varphi }_2}(\zeta -{\zeta }_2,\gamma -{\gamma }_2)\hfill \\ \\ & \times \widehat{{\varphi }_3}({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1){|d{\zeta }_{1}d{\gamma }_{1}d{\zeta }_{2}d{\gamma }_2\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}\hfill \\ \\ & =\parallel {(1+|\zeta \left|\right)}^s(1+|\gamma -{\zeta }^3{\left|\right)}^{b^{\prime }}\left|\zeta \right|{\int }_{{\mathbb{R}}^4}\left({\displaystyle \frac{(1+|{\zeta }_1{\left|\right)}^{-s}\widehat{f_1}({\zeta }_1,{\gamma }_1)}{(1+|\gamma -{\zeta }^3{\left|\right)}^b}}\right)\hfill \\ \\ & \times \left({\displaystyle \frac{(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}\widehat{f_2}(\zeta -{\zeta }_2,\gamma -{\gamma }_2)}{(1+|(\gamma -{\gamma }_2)-{(\zeta -{\zeta }_2)}^3{\left|\right)}^b}}\right)\hfill \\ \\ & \times \left({\displaystyle \frac{(1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}\widehat{f_3}({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)}{(1+|{\gamma }_2-{\gamma }_1-{({\zeta }_2-{\zeta }_1)}^3{\left|\right)}^b}}\right)d\mu {\parallel }_{L_{\zeta }^2{L}_{\gamma }^2},\hfill \end{array}$$

where $d\mu =d{\zeta }_{1}d{\gamma }_{1}d{\zeta }_{2}d{\gamma }_{2}d\zeta d\gamma$.

Using duality, we prove this estimate, where $m(\zeta ,\gamma )$ is a positive function in $L^2\left({\mathbb{R}}^2\right)$ with norm ${\parallel m\parallel }_{L^2\left({\mathbb{R}}^2\right)}=1$; then

$$\begin{array}{cc}\hfill \parallel {\partial }_x\left({\varphi }_1{\varphi }_2{\varphi }_3\right){\parallel }_{X^{s,b^{\prime }}}& ⩽{\int }_{{\mathbb{R}}^6}{\displaystyle \frac{{(1+|\zeta \left|\right)}^s\left|\zeta \right|m(\zeta ,\gamma )}{(1+|\gamma -{\zeta }^3{\left|\right)}^{-b^{\prime }}}}{\displaystyle \frac{(1+|{\zeta }_1{\left|\right)}^{-s}{f}_1({\zeta }_1,{\gamma }_1)}{(1+|{\gamma }_1-{\zeta }_1^3{\left|\right)}^b}}\hfill \\ \\ & {\displaystyle \frac{(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}{f}_2(\zeta -{\zeta }_2,\gamma -{\gamma }_2)}{(1+|\gamma -{\gamma }_2-{(\zeta -{\zeta }_2)}^3{\left|\right)}^b}}\hfill \\ \\ & {\displaystyle \frac{(1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}{f}_3({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)}{(1+|{\gamma }_2-{\gamma }_1-{({\zeta }_2-{\zeta }_1)}^3{\left|\right)}^b}}d\mu .\hfill \end{array}$$

Now, we split the Fourier space into six regions as follows:

$$\mathrm{region}1:|\zeta -{\zeta }_2|\le |{\zeta }_2-{\zeta }_1|\le |{\zeta }_1|,\mathrm{region}2:|\zeta -{\zeta }_2|\le |{\zeta }_1|\le |{\zeta }_2-{\zeta }_1|,$$

$$\mathrm{region}3:|{\zeta }_1|\le |{\zeta }_2-{\zeta }_1|\le |\zeta -{\zeta }_2|,\mathrm{region}4:|{\zeta }_1|\le |\zeta -{\zeta }_2|\le |{\zeta }_2-{\zeta }_1|,$$

$$\mathrm{region}5:|{\zeta }_2-{\zeta }_1|\le |\zeta -{\zeta }_2\le |{\zeta }_1|,\mathrm{region}6:|{\zeta }_2-{\zeta }_1|\le |{\zeta }_1|\le |\zeta -{\zeta }_2|.$$

We begin with region 1:

$$|\zeta -{\zeta }_2|\le |{\zeta }_2-{\zeta }_1|\le |{\zeta }_1|,$$

so

$$(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}\ge (1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}\ge (1+|{\zeta }_1{\left|\right)}^{-s}.$$

(22)

We assume that $\left|\zeta \right|\le 1$ or $\left|\zeta \right|\ge 1$.

Firstly, for case $\left|\zeta \right|\ge 1$,

$${(1+|\zeta \left|\right)}^s\le \left(\right|\zeta |+{\left|\zeta \right|)}^s=2^s{\left|\zeta \right|}^s=C{\left|\zeta \right|}^s.$$

According to the last inequality and $\left(22\right)$, we obtain

$$\begin{array}{cc}\hfill \parallel {\partial }_x\left({\varphi }_1{\varphi }_2{\varphi }_3\right){\parallel }_{X^{s,b^{\prime }}}& ⩽{\int }_{{\mathbb{R}}^6}{\displaystyle \frac{{(1+|\zeta \left|\right)}^s\left|\zeta \right|m(\zeta ,\gamma )}{(1+|\gamma -{\zeta }^3{\left|\right)}^{-b^{\prime }}}}{\displaystyle \frac{(1+|{\zeta }_1{\left|\right)}^{-s}{f}_1({\zeta }_1,{\gamma }_1)}{(1+|{\gamma }_1-{\zeta }_1^3{\left|\right)}^b}}\hfill \\ \\ & {\displaystyle \frac{(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}{f}_2(\zeta -{\zeta }_2,\gamma -{\gamma }_2)}{(1+|\gamma -{\gamma }_2-{(\zeta -{\zeta }_2)}^3{\left|\right)}^b}}{\displaystyle \frac{(1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}{f}_3({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)}{(1+|{\gamma }_2-{\gamma }_1-{({\zeta }_2-{\zeta }_1)}^3{\left|\right)}^b}}d\mu \hfill \\ \\ & \le C{\int }_{{\mathbb{R}}^6}{\displaystyle \frac{{\left|\zeta \right|}^s\left|\zeta \right|m(\zeta ,\gamma )}{(1+|\gamma -{\zeta }^3{\left|\right)}^{-b^{\prime }}}}{\displaystyle \frac{(1+|{\zeta }_1{\left|\right)}^{-s}{f}_1({\zeta }_1,{\gamma }_1)}{(1+|{\gamma }_1-{\zeta }_1^3{\left|\right)}^b}}\hfill \\ \\ & {\displaystyle \frac{(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}{f}_2(\zeta -{\zeta }_2,\gamma -{\gamma }_2)}{(1+|\gamma -{\gamma }_2-{(\zeta -{\zeta }_2)}^3{\left|\right)}^b}}{\displaystyle \frac{(1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}{f}_3({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)}{(1+|{\gamma }_2-{\gamma }_1-{({\zeta }_2-{\zeta }_1)}^3{\left|\right)}^b}}d\mu \hfill \\ \\ & \le C{\int }_{{\mathbb{R}}^6}{\displaystyle \frac{{\left|\zeta \right|}^{s+1}m(\zeta ,\gamma )}{(1+|\gamma -{\zeta }^3{\left|\right)}^{-b^{\prime }}}}{\displaystyle \frac{(1+|{\zeta }_1{\left|\right)}^{-s}{f}_1({\zeta }_1,{\gamma }_1)}{(1+|{\gamma }_1-{\zeta }_1^3{\left|\right)}^b}}\hfill \\ \\ & {\displaystyle \frac{(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}{f}_2(\zeta -{\zeta }_2,\gamma -{\gamma }_2)}{(1+|\gamma -{\gamma }_2-{(\zeta -{\zeta }_2)}^3{\left|\right)}^b}}{\displaystyle \frac{(1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}{f}_3({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)}{(1+|{\gamma }_2-{\gamma }_1-{({\zeta }_2-{\zeta }_1)}^3{\left|\right)}^b}}d\mu .\hfill \end{array}$$

By calculating

$${\left|\zeta \right|}^{s+1}(1+|{\zeta }_1{\left|\right)}^{-s}\le {\left|\zeta \right|}^{s+1}|{\zeta }_1{|}^{-s}\le {C\left|\zeta \right|}^{{\displaystyle \frac{1}{2}}}{\left|{\zeta }_1\right|}^{{\displaystyle \frac{1}{2}}},$$

we get

$$\begin{array}{cc}\hfill \parallel {\partial }_x\left({\varphi }_1{\varphi }_2{\varphi }_3\right){\parallel }_{X^{s,b^{\prime }}}& \le C{\int }_{{\mathbb{R}}^6}{\displaystyle \frac{{\left|\zeta \right|}^{\frac{1}{2}}m(\zeta ,\gamma )}{(1+|\gamma -{\zeta }^3{\left|\right)}^{-b^{\prime }}}}{\displaystyle \frac{(1+|{\zeta }_1{\left|\right)}^{\frac{1}{2}}{f}_1({\zeta }_1,{\gamma }_1)}{(1+|{\gamma }_1-{\zeta }_1^3{\left|\right)}^b}}\hfill \\ \\ & {\displaystyle \frac{(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}{f}_2(\zeta -{\zeta }_2,\gamma -{\gamma }_2)}{(1+|\gamma -{\gamma }_2-{(\zeta -{\zeta }_2)}^3{\left|\right)}^b}}{\displaystyle \frac{(1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}{f}_3({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)}{(1+|{\gamma }_2-{\gamma }_1-{({\zeta }_2-{\zeta }_1)}^3{\left|\right)}^b}}d\mu .\hfill \end{array}$$

We suppose that

$$\begin{array}{cc}\hfill \widehat{A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}}(\zeta ,\gamma )& ={\displaystyle \frac{{\left|\zeta \right|}^{\frac{1}{2}}m(\zeta ,\gamma )}{(1+|\gamma -{\zeta }^3{\left|\right)}^{-b^{\prime }}}}\hfill \\ \\ \hfill \widehat{A^{{\displaystyle \frac{1}{2}}}{F}_b^1}({\zeta }_1,{\gamma }_1)& ={\displaystyle \frac{(1+|{\zeta }_1{\left|\right)}^{\frac{1}{2}}{f}_1({\zeta }_1,{\gamma }_1)}{(1+|{\gamma }_1-{\zeta }_1^3{\left|\right)}^b}}\hfill \\ \\ \hfill \widehat{A^{-s}{F}_b^2}(\zeta -{\zeta }_2,\gamma -{\gamma }_2)& ={\displaystyle \frac{(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}{f}_2(\zeta -{\zeta }_2,\gamma -{\gamma }_2)}{(1+|\gamma -{\gamma }_2-{(\zeta -{\zeta }_2)}^3{\left|\right)}^b}}\hfill \\ \\ \hfill \widehat{A^{-s}{F}_b^3}({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)& ={\displaystyle \frac{(1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}{f}_3({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)}{(1+|{\gamma }_2-{\gamma }_1-{({\zeta }_2-{\zeta }_1)}^3{\left|\right)}^b}}.\hfill \end{array}$$

Then

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^6}{\displaystyle \frac{{\left(\right|\zeta \left|\right)}^{\frac{1}{2}}m(\zeta ,\gamma )}{(1+|\gamma -{\zeta }^3{\left|\right)}^{-b^{\prime }}}}{\displaystyle \frac{(1+|{\zeta }_1{\left|\right)}^{\frac{1}{2}}{f}_1({\zeta }_1,{\gamma }_1)}{(1+|{\gamma }_1-{\zeta }_1^3{\left|\right)}^b}}{\displaystyle \frac{(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}{f}_2(\zeta -{\zeta }_2,\gamma -{\gamma }_2)}{(1+|\gamma -{\gamma }_2-{(\zeta -{\zeta }_2)}^3{\left|\right)}^b}}\hfill \\ \\ & {\displaystyle \frac{(1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}{f}_3({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)}{(1+|{\gamma }_2-{\gamma }_1-{({\zeta }_2-{\zeta }_1)}^3{\left|\right)}^b}}d\mu \hfill \\ \\ & ={\int }_{{\mathbb{R}}^6}\widehat{A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}}(\zeta ,\gamma )\widehat{A^{{\displaystyle \frac{1}{2}}}{F}_b}({\zeta }_1,{\gamma }_1)\widehat{A^{-s}{G}_b^1}(\zeta -{\zeta }_2,\gamma -{\gamma }_2)\widehat{A^{-s}{G}_b^2}({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)d\mu \hfill \\ \\ & ={\int }_{{\mathbb{R}}^2}\left(\widehat{A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}}(\zeta ,\gamma )\right)\times \hfill \\ \\ & \left({\int }_{{\mathbb{R}}^4}\widehat{A^{{\displaystyle \frac{1}{2}}}{F}_b}({\zeta }_1,{\gamma }_1)\widehat{A^{-s}{G}_b^1}(\zeta -{\zeta }_2,\gamma -{\gamma }_2)\widehat{A^{-s}{G}_b^2}({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)d{\zeta }_{1}d{\gamma }_{1}d{\zeta }_{2}d{\gamma }_2\right)d\zeta d\gamma \hfill \\ \\ & ={\int }_{{\mathbb{R}}^2}\left(\widehat{A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}}(\zeta ,\gamma )\right)\left(\left(\widehat{A^{{\displaystyle \frac{1}{2}}}{F}_b^1}★\widehat{A^{-s}{F}_b^2}★\widehat{A^{-s}{F}_b^3}\right)(\zeta ,\gamma )\right)d\zeta d\gamma \hfill \\ \\ & ={\int }_{{\mathbb{R}}^2}\left(\widehat{A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}}(\zeta ,\gamma )\right)\left(\left(\widehat{A^{{\displaystyle \frac{1}{2}}}{F}_b^1·A^{-s}{F}_b^2·A^{-s}{F}_b^3}\right)(\zeta ,\gamma )\right)d\zeta d\gamma \hfill \\ \\ & ={\int }_{{\mathbb{R}}^2}{A}^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}\left(A^{{\displaystyle \frac{1}{2}}}{F}_b^1·A^{-s}{F}_b^2·A^{-s}{F}_b^3\right)dxdt.\hfill \end{array}$$

By using Cauchy–Schwarz’s inequality for the variables x and t, we get

$$\begin{array}{cc}\hfill & \parallel {\partial }_x\left({\varphi }_1{\varphi }_2{\varphi }_3\right){\parallel }_{X^{s,b^{\prime }}}\le C\parallel A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}{\parallel }_{L_x^4{L}_t^2}\parallel A^{{\displaystyle \frac{1}{2}}}{F}_b^1{\parallel }_{L_x^4{L}_t^2}\parallel A^{-s}{F}_b^2{\parallel }_{L_x^2{L}_t^{\infty }}{\parallel A^{{\displaystyle \frac{1}{2}}}{F}_b^3\parallel }_{L_x^{\infty }{L}_t^{\infty }}.\hfill \end{array}$$

Hence, according to Lemmas 1 and 2,

$$\begin{array}{cc}\hfill \parallel {\partial }_x\left({\varphi }_1{\varphi }_2{\varphi }_3\right){\parallel }_{X^{s,b^{\prime }}}& \le {C\parallel m\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}{\parallel f\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}\parallel f_2{\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}{\parallel f_3\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}\hfill \\ \\ & \le C\parallel {\varphi }_1{\parallel }_{X^{s,b}}\parallel {\varphi }_2{\parallel }_{X^{s,b}}{\parallel {\varphi }_3\parallel }_{X^{s,b}}.\hfill \end{array}$$

Secondly, for the case $\left|\zeta \right|\le 1$, we have

$$\begin{array}{c}\hfill {(1+|\zeta \left|\right)}^s\left|\zeta \right|(1+|{\zeta }_1{\left|\right)}^{-s}\le C{(1+|\zeta \left|\right)}^{{\displaystyle \frac{1}{2}}}{(1+|{\zeta }_1\left|\right)}^{{\displaystyle \frac{1}{2}}}.\end{array}$$

Then

$$\begin{array}{cc}\hfill \parallel {\partial }_x\left({\varphi }_1{\varphi }_2{\varphi }_3\right){\parallel }_{X^{s,b^{\prime }}}& \le {\int }_{{\mathbb{R}}^6}{\displaystyle \frac{{(1+|\zeta \left|\right)}^s\left|\zeta \right|m(\zeta ,\gamma )}{(1+|\gamma -{\zeta }^3{\left|\right)}^{-b^{\prime }}}}{\displaystyle \frac{(1+|{\zeta }_1{\left|\right)}^{-s}{f}_1({\zeta }_1,{\gamma }_1)}{(1+|{\gamma }_1-{\zeta }_1^3{\left|\right)}^b}}\hfill \\ \\ & \times {\displaystyle \frac{(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}{f}_2(\zeta -{\zeta }_2,\gamma -{\gamma }_2)}{(1+|\gamma -{\gamma }_2-{(\zeta -{\zeta }_2)}^3{\left|\right)}^b}}{\displaystyle \frac{(1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}{f}_3({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)}{(1+|{\gamma }_2-{\gamma }_1-{({\zeta }_2-{\zeta }_1)}^3{\left|\right)}^b}}d\mu \hfill \\ \\ & \le C{\int }_{{\mathbb{R}}^6}{\displaystyle \frac{{(1+|\zeta \left|\right)}^{\frac{1}{2}}m(\zeta ,\gamma )}{(1+|\gamma -{\zeta }^3{\left|\right)}^{-b^{\prime }}}}{\displaystyle \frac{(1+|{\zeta }_1{\left|\right)}^{\frac{1}{2}}{f}_1({\zeta }_1,{\gamma }_1)}{(1+|{\gamma }_1-{\zeta }_1^3{\left|\right)}^b}}\hfill \\ \\ & \times {\displaystyle \frac{(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}{f}_2(\zeta -{\zeta }_2,\gamma -{\gamma }_2)}{(1+|\gamma -{\gamma }_2-{(\zeta -{\zeta }_2)}^3{\left|\right)}^b}}{\displaystyle \frac{(1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}{f}_3({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)}{(1+|{\gamma }_2-{\gamma }_1-{({\zeta }_2-{\zeta }_1)}^3{\left|\right)}^b}}d\mu .\hfill \end{array}$$

Then, by calculating the inner product, we have

$$\begin{array}{cc}\hfill \parallel {\partial }_x\left({\varphi }_1{\varphi }_2{\varphi }_3\right){\parallel }_{X^{s,b^{\prime }}}& \le C\langle A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }};A^{{\displaystyle \frac{1}{2}}}{F}_b^1·A^{-s}{F}_b^2·A^{-s}{F}_b^3\rangle \hfill \\ & \le C\parallel A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}{\parallel }_{L_x^4{L}_t^2}\parallel A^{{\displaystyle \frac{1}{2}}}{F}_b^1{\parallel }_{L_x^4{L}_t^2}\parallel A^{-s}{F}_b^2{\parallel }_{L_x^2{L}_t^{\infty }}{\parallel A^{-s}{F}_b^3\parallel }_{L_x^{\infty }{L}_t^{\infty }}.\hfill \end{array}$$

Hence, according to Lemmas 1 and 2,

$$\begin{array}{cc}\hfill \parallel {\partial }_x\left({\varphi }_1{\varphi }_2{\varphi }_3\right){\parallel }_{X^{s,b^{\prime }}}& \le {C\parallel m\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}\parallel f_1{\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}\parallel f_2{\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}{\parallel f_3\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}\hfill \\ \\ & \le C\parallel {\varphi }_1{\parallel }_{X^{s,b}}\parallel {\varphi }_2{\parallel }_{X^{s,b}}{\parallel {\varphi }_3\parallel }_{X^{s,b}}.\hfill \end{array}$$

The inequalities in the other five regions are proven in a similar manner.

The case $k\ge 2$ is handled virtually identically. The only difference is that we need to split the Fourier space in $\left(\right(2k+1)+1)!$. We want to prove that

$$\begin{array}{c}\hfill \parallel {\partial }_x\prod _{i=1}^{2k+1}{\varphi }_i{\parallel }_{X^{s,b^{\prime }}}\le C\prod _{i=1}^{2k+1}{\parallel {\varphi }_i\parallel }_{X^{s,b}}.\end{array}$$

We have

$$\begin{array}{cc}\hfill \parallel {\partial }_x\prod _{i=1}^{2k+1}{\varphi }_i{\parallel }_{X^{s,b^{\prime }}}& =\parallel {(1+|\zeta \left|\right)}^s(1+|\gamma -{\zeta }^3{\left|\right)}^{b^{\prime }}\widehat{{\partial }_x\prod _{i=1}^{2k+1}{\varphi }_i}(\zeta ,\gamma ){\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}\hfill \\ & =\parallel {(1+|\zeta \left|\right)}^s(1+|\gamma -{\zeta }^3{\left|\right)}^{b^{\prime }}\left|\zeta \right|\widehat{\prod _{i=1}^{2k+1}{\varphi }_i}(\zeta ,\gamma ){\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}.\hfill \end{array}$$

In the same way, by calculating the inner product, we have

$$\begin{array}{cc}\hfill \parallel {\partial }_x\prod _{i=1}^{2k+1}{\varphi }_i{\parallel }_{X^{s,b^{\prime }}}& \le C\langle \widehat{A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}};\widehat{A^{{\displaystyle \frac{1}{2}}}{F}_b^1}★\widehat{A^{-s}{F}_b^2}★\prod _{i=3}^{2k+1}\widehat{A^{-s}{F}_b^i}\rangle \hfill \\ & \le C\langle \widehat{A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}};\widehat{A^{{\displaystyle \frac{1}{2}}}{F}_b^1·A^{-s}{F}_b^2\prod _{i=3}^{2k+1}{A}^{-s}{F}_b^i}\rangle \hfill \\ & \le C\parallel A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}{\parallel }_{L_x^4{L}_t^2}\parallel A^{{\displaystyle \frac{1}{2}}}{F}_b^1{\parallel }_{L_x^4{L}_t^2}{\parallel A^{-s}{F}_b^2\parallel }_{L_x^2{L}_t^{\infty }}\parallel \prod _{i=3}^{2k+1}{A}^{-s}{F}_b^i{\parallel }_{L_x^{\infty }{L}_t^{\infty }}\hfill \\ & \le C\prod _{i=1}^{2k+1}{\parallel {\varphi }_i\parallel }_{X^{s,b}}.\hfill \end{array}$$

Inequality (15): We have

$$\begin{array}{cc}\hfill \parallel {\partial }_x\left({\phi }_1{\phi }_2{\phi }_3\right){\parallel }_{X_{\alpha }^{s,b^{\prime }}}& \le {\int }_{{\mathbb{R}}^6}{\displaystyle \frac{{(1+|\zeta \left|\right)}^s\left|\zeta \right|m(\zeta ,\gamma )}{(1+|\gamma -\alpha {\zeta }^3{\left|\right)}^{-b^{\prime }}}}{\displaystyle \frac{(1+|{\zeta }_1{\left|\right)}^{-s}{g}_1({\zeta }_1,{\gamma }_1)}{(1+|{\gamma }_1-\alpha {\zeta }_1^3{\left|\right)}^b}}\hfill \\ \\ & {\displaystyle \frac{(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}{g}_2(\zeta -{\zeta }_2,\gamma -{\gamma }_2)}{(1+|\gamma -{\gamma }_2-\alpha {(\zeta -{\zeta }_2)}^3{\left|\right)}^b}}{\displaystyle \frac{(1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}{g}_3({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)}{(1+|{\gamma }_2-{\gamma }_1-\alpha {({\zeta }_2-{\zeta }_1)}^3{\left|\right)}^b}}d\mu .\hfill \end{array}$$

We begin with region 1: We assume that $\left|\zeta \right|\le 1$ or $\left|\zeta \right|\ge 1$.

Firstly, for case $\left|\zeta \right|\ge 1$,

$$\begin{array}{cc}\hfill \parallel {\partial }_x\left({\phi }_1{\phi }_2{\phi }_3\right){\parallel }_{X_{\alpha }^{s,b^{\prime }}}& \le C{\int }_{{\mathbb{R}}^6}{\displaystyle \frac{{\left|\zeta \right|}^{\frac{1}{2}}m(\zeta ,\gamma )}{(1+|\gamma -\alpha {\zeta }^3{\left|\right)}^{-b^{\prime }}}}{\displaystyle \frac{(1+|{\zeta }_1{\left|\right)}^{\frac{1}{2}}{g}_1({\zeta }_1,{\gamma }_1)}{(1+|{\gamma }_1-\alpha {\zeta }_1^3{\left|\right)}^b}}\hfill \\ & \times {\displaystyle \frac{(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}{g}_2(\zeta -{\zeta }_2,\gamma -{\gamma }_2)}{(1+|\gamma -{\gamma }_2-\alpha {(\zeta -{\zeta }_2)}^3{\left|\right)}^b}}{\displaystyle \frac{(1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}{g}_3({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)}{(1+|{\gamma }_2-{\gamma }_1-\alpha {({\zeta }_2-{\zeta }_1)}^3{\left|\right)}^b}}d\mu .\hfill \end{array}$$

We suppose that

$$\begin{array}{cc}\hfill \widehat{A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}^{\alpha }}(\zeta ,\gamma )& ={\displaystyle \frac{{\left|\zeta \right|}^{\frac{1}{2}}m(\zeta ,\gamma )}{(1+|\gamma -\alpha {\zeta }^3{\left|\right)}^{-b^{\prime }}}}\hfill \\ \\ \hfill \widehat{A^{{\displaystyle \frac{1}{2}}}{G}_{\alpha ,b}^1}({\zeta }_1,{\gamma }_1)& ={\displaystyle \frac{(1+|{\zeta }_1{\left|\right)}^{\frac{1}{2}}{g}_1({\zeta }_1,{\gamma }_1)}{(1+|{\gamma }_1-\alpha {\zeta }_1^3{\left|\right)}^b}}\hfill \\ \\ \hfill \widehat{A^{-s}{G}_{\alpha ,b}^2}(\zeta -{\zeta }_2,\gamma -{\gamma }_2)& ={\displaystyle \frac{(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}{g}_2(\zeta -{\zeta }_2,\gamma -{\gamma }_2)}{(1+|\gamma -{\gamma }_2-\alpha {(\zeta -{\zeta }_2)}^3{\left|\right)}^b}}\hfill \\ \\ \hfill \widehat{A^{-s}{G}_{\alpha ,b}^3}({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)& ={\displaystyle \frac{(1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}{g}_3({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)}{(1+|{\gamma }_2-{\gamma }_1-\alpha {({\zeta }_2-{\zeta }_1)}^3{\left|\right)}^b}}.\hfill \end{array}$$

Then

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^6}{\displaystyle \frac{{\left|\zeta \right|}^{\frac{1}{2}}m(\zeta ,\gamma )}{(1+|\gamma -\alpha {\zeta }^3{\left|\right)}^{-b^{\prime }}}}{\displaystyle \frac{(1+|{\zeta }_1{\left|\right)}^{\frac{1}{2}}{g}_1({\zeta }_1,{\gamma }_1)}{(1+|{\gamma }_1-\alpha {\zeta }_1^3{\left|\right)}^b}}{\displaystyle \frac{(1+|\zeta -{\zeta }_2{\left|\right)}^{-s}{g}_2(\zeta -{\zeta }_2,\gamma -{\gamma }_2)}{(1+|\gamma -{\gamma }_2-\alpha {(\zeta -{\zeta }_2)}^3{\left|\right)}^b}}\hfill \\ \\ & \times {\displaystyle \frac{(1+|{\zeta }_2-{\zeta }_1{\left|\right)}^{-s}{g}_3({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)}{(1+|{\gamma }_2-{\gamma }_1-\alpha {({\zeta }_2-{\zeta }_1)}^3{\left|\right)}^b}}d\mu \hfill \\ \\ & ={\int }_{{\mathbb{R}}^6}\widehat{A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}^{\alpha }}(\zeta ,\gamma )\widehat{A^{{\displaystyle \frac{1}{2}}}{G}_{\alpha ,b}^1}({\zeta }_1,{\gamma }_1)\widehat{A^{-s}{G}_{\alpha ,b}^2}(\zeta -{\zeta }_2,\gamma -{\gamma }_2)\widehat{A^{-s}{G}_{\alpha ,b}^3}({\zeta }_2-{\zeta }_1,{\gamma }_2-{\gamma }_1)d\mu \hfill \\ \\ & ={\int }_{{\mathbb{R}}^2}{A}^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}^{\alpha }\left(A^{{\displaystyle \frac{1}{2}}}{G}_{\alpha ,b}^1·A^{-s}{G}_{\alpha ,b}^2·A^{-s}{G}_{\alpha ,b}^3\right)dxdt.\hfill \end{array}$$

By using Cauchy–Schwarz’s inequality for the variables x and t, we get

$$\begin{array}{cc}\hfill & \parallel {\partial }_x\left({\phi }_1{\phi }_2{\phi }_3\right){\parallel }_{X_{\alpha }^{s,b^{\prime }}}\le C\parallel A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}^{\alpha }{\parallel }_{L_x^4{L}_t^2}\parallel A^{{\displaystyle \frac{1}{2}}}{G}_{\alpha ,b}^1{\parallel }_{L_x^4{L}_t^2}\parallel A^{-s}{G}_{\alpha ,b}^2{\parallel }_{L_x^2{L}_t^{\infty }}{\parallel A^{{\displaystyle \frac{1}{2}}}{G}_{\alpha ,b}^3\parallel }_{L_x^{\infty }{L}_t^{\infty }}.\hfill \end{array}$$

Hence, according to Lemmas 1 and 3,

$$\begin{array}{cc}\hfill \parallel {\partial }_x\left({\phi }_1{\phi }_2{\phi }_3\right){\parallel }_{X_{\alpha }^{s,b^{\prime }}}& \le {C\parallel m\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}\parallel g_1{\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}\parallel g_2{\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}{\parallel g_3\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}\hfill \\ \\ & \le C\parallel {\phi }_1{\parallel }_{X_{\alpha }^{s,b}}\parallel {\phi }_2{\parallel }_{X_{\alpha }^{s,b}}{\parallel {\phi }_3\parallel }_{X_{\alpha }^{s,b}}.\hfill \end{array}$$

Secondly, for the case $\left|\zeta \right|\le 1$, similarly, here we use

$$\begin{array}{c}\hfill {(1+|\zeta \left|\right)}^s\left|\zeta \right|(1+|{\zeta }_1{\left|\right)}^{-s}\le C{(1+|\zeta \left|\right)}^{{\displaystyle \frac{1}{2}}}(1+|{\zeta }_1{\left|\right)}^{{\displaystyle \frac{1}{2}}}.\end{array}$$

For the case $k\ge 2$, we have

$$\begin{array}{cc}\hfill \parallel {\partial }_x\prod _{i=1}^{2k+1}{\phi }_i{\parallel }_{X_{\alpha }^{s,b^{\prime }}}& =\parallel {(1+|\zeta \left|\right)}^s(1+|\gamma -\alpha {\zeta }^3{\left|\right)}^{b^{\prime }}\widehat{{\partial }_x\prod _{i=1}^{2k+1}{\phi }_i}(\zeta ,\gamma ){\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}\hfill \\ & =\parallel {(1+|\zeta \left|\right)}^s(1+|\gamma -\alpha {\zeta }^3{\left|\right)}^{b^{\prime }}\left|\zeta \right|\widehat{\prod _{i=1}^{2k+1}{\phi }_i}(\zeta ,\gamma ){\parallel }_{L_{\zeta }^2{L}_{\gamma }^2}.\hfill \end{array}$$

In the same way, by calculating the inner product, we have

$$\begin{array}{cc}\hfill \parallel {\partial }_x\prod _{i=1}^{2k+1}{\phi }_i{\parallel }_{X_{\alpha }^{s,b^{\prime }}}& \le C\langle \widehat{A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}^{\alpha }};\widehat{A^{{\displaystyle \frac{1}{2}}}{G}_{\alpha ,b}^1}★\widehat{A^{-s}{G}_{\alpha ,b}^2}★\prod _{i=3}^{2k+1}\widehat{A^{-s}{G}_{\alpha ,b}^i}\rangle \hfill \\ & \le C\langle \widehat{A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}^{\alpha }};\widehat{A^{{\displaystyle \frac{1}{2}}}{G}_{\alpha ,b}^1·A^{-s}{G}_{\alpha ,b}^2\prod _{i=3}^{2k+1}{A}^{-s}{G}_{\alpha ,b}^i}\rangle \hfill \\ & \le C\parallel A^{{\displaystyle \frac{1}{2}}}{M}_{-b^{\prime }}^{\alpha }{\parallel }_{L_x^4{L}_t^2}\parallel A^{{\displaystyle \frac{1}{2}}}{G}_{\alpha ,b}^1{\parallel }_{L_x^4{L}_t^2}{\parallel A^{-s}{G}_{\alpha ,b}^2\parallel }_{L_x^2{L}_t^{\infty }}\parallel \prod _{i=3}^{2k+1}{A}^{-s}{G}_{\alpha ,b}^i{\parallel }_{L_x^{\infty }{L}_t^{\infty }}\hfill \\ & \le C\prod _{i=1}^{2k+1}{\parallel {\phi }_i\parallel }_{X_{\alpha }^{s,b}}.\hfill \end{array}$$

The case $k\ge 2$ is calculated similarly though induction; see, e.g., [17] for analogous multilinear estimates in deterministic generalized Korteweg–de Vries systems. □

4. Stochastic Estimates

We introduce a smooth temporal cutoff function $\varpi \in C_0^{\infty }\left(\mathbb{R}\right)$ satisfying

$$\varpi \left(t\right)=0\mathrm{for}t<0\mathrm{and}\left|t\right|>2,\varpi \left(t\right)=1\mathrm{for}t\in [0,1].$$

This function localizes the stochastic convolution to the time interval $[0,T]$, which allows us to work within Bourgain spaces on $\mathbb{R}\times [0,T]$. Note that for any $b>{\displaystyle \frac{1}{2}}$, such a $\varpi$ belongs to the fractional Sobolev space $H_t^b=H^b([0,T],\mathbb{R})$, whose norm is given by

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

We show the following lemma.

Lemma 5 

(Stochastic convolution). Let $s,b\in \mathbb{R}$, with $b>{\displaystyle \frac{1}{2}}$, and assume that $\Xi \in L_2^{0,s}$; then ${\varphi }_l\left(t\right),{\phi }_l\left(t\right)$, defined by (6), satisfies

$$\varpi {\varphi }_l\in L^2\left(\Omega ,X^{b,s}\right),\varpi {\phi }_l\in L^2\left(\Omega ,X_{\alpha }^{b,s}\right)$$

and

$$\mathbb{E}\left(\parallel \varpi {\varphi }_l{\parallel }_{X^{b,s}}^2\right){\lesssim }_{b,\varpi }{\parallel \Xi \parallel }_{L_2^{0,s}}^2,\mathbb{E}\left(\parallel \varpi {\phi }_l{\parallel }_{X_{\alpha }^{b,s}}^2\right){\lesssim }_{b,\varpi }{\parallel \Xi \parallel }_{L_2^{0,s}}^2.$$

Proof. 

Let us introduce the function

$$f(·,t)=\varpi \left(t\right){\int }_0^t{\mathfrak{U}}_{\alpha }(-\gamma )\Xi dW\left(\gamma \right),t\in {\mathbb{R}}^{+}.$$

(23)

This implies that ${\mathfrak{U}}_{\alpha }\left(t\right)f(·,t)=\varpi \left(t\right)\Psi \left(t\right)$. Thus, we have

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

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 (6)₂, we have

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

where

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

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

From the Itô isometry formula, we have

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

To estimate ${\mathfrak{S}}_2$, we decompose the double integral over $({\eta }_1,{\eta }_2)$ into three distinct regions, corresponding to the cases where ${\eta }_1$ and ${\eta }_2$ lie in different intervals relative to the support of the cutoff function $\omega$.

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

We now bound ${\mathfrak{I}}_1,{\mathfrak{I}}_2$, and ${\mathfrak{I}}_3$ separately:

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

Using Equation (23) and the assumption that $2b\in (0,1)$, we have

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

Similarly,

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

So, we have

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

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

5. Local Well-Posedness

Using the stochastic estimates from the previous section and the Banach fixed-point theorem, we now establish a local well-posedness result for system (2). This section is therefore dedicated to the proof of Theorem 1. Let

$${\Pi }_1\left(t\right)=\mathfrak{U}\left(t\right)u_0,{\Pi }_2\left(t\right)={\mathfrak{U}}_{\alpha }\left(t\right){\phi }_0,$$

$${\mathrm{{\rm Y}}}_1\left(t\right)={\int }_0^t\mathfrak{U}(t-\gamma )\Xi dW\left(\gamma \right),{\mathrm{{\rm Y}}}_2\left(t\right)={\int }_0^t{\mathfrak{U}}_{\alpha }(t-\gamma )\Xi dW\left(\gamma \right),$$

and

$${\Psi }_1\left(t\right)={\int }_0^t\mathfrak{U}(t-\gamma ){\partial }_x\left(N_{\varphi }(\varphi ,\phi )\right)\left(\gamma \right)d\gamma ,{\Psi }_2\left(t\right)={\int }_0^t{\mathfrak{U}}_{\alpha }(t-\gamma ){\partial }_x\left(N_{\phi }(\varphi ,\phi )\right)\left(\gamma \right)d\gamma .$$

Then, we may rewrite (7) in terms of

$${\Psi }_1=\varphi -{\mathrm{{\rm Y}}}_1-{\Pi }_1,$$

$${\Psi }_2=\phi -{\mathrm{{\rm Y}}}_2-{\Pi }_2,$$

as

$$\left\{\begin{array}{c}{\Psi }_1\left(t\right)={\displaystyle \frac{1}{2}}{\int }_0^t\mathfrak{U}(t-\gamma ){\partial }_x\left(N_{\varphi }(\varphi ,\phi )\right)\left(\gamma \right)d\gamma \hfill \\ \\ {\Psi }_2\left(t\right)={\int }_0^t{\mathfrak{U}}_{\alpha }(t-\gamma ){\partial }_x\left(N_{\phi }(\varphi ,\phi )\right)\left(\gamma \right)d\gamma ,\hfill \end{array}\right.$$

(24)

with

$$\left\{\begin{array}{c}{N}_{\varphi }(\varphi ,\phi )=A{\varphi }^{2k+1}+B{\varphi }^k{\phi }^{k+1}+{\displaystyle \frac{k+2}{k}}C{\varphi }^{k+1}{\phi }^k+D{\varphi }^{k-1}{\phi }^{k+2}\hfill \\ \\ N_{\phi }(\varphi ,\phi )=A{\phi }^{2k+1}+B{\phi }^k{\varphi }^{k+1}+{\displaystyle \frac{k+2}{k}}D{\phi }^{k+1}{\varphi }^k+C{\phi }^{k-1}{\varphi }^{k+2}.\hfill \end{array}\right.$$

(25)

Then (24) and (25) are equivalent to

$$\left\{\begin{array}{cc}{\Psi }_1\left(t\right)\hfill & ={\displaystyle \frac{1}{2}}{\int }_0^t\mathfrak{U}(t-\gamma ){\partial }_x(A{({\Psi }_1+{\mathrm{{\rm Y}}}_1+{\Pi }_1)}^{2k+1}+B{({\Psi }_1+{\mathrm{{\rm Y}}}_1+{\Pi }_1)}^k{({\Psi }_2+{\mathrm{{\rm Y}}}_2+{\Pi }_2)}^{k+1}\hfill \\ \\ \hfill & +{\displaystyle \frac{k+2}{k}}C{({\Psi }_1+{\mathrm{{\rm Y}}}_1+{\Pi }_1)}^{k+1}{({\Psi }_2+{\mathrm{{\rm Y}}}_2+{\Pi }_2)}^k+D{({\Psi }_1+{\mathrm{{\rm Y}}}_1+{\Pi }_1)}^{k-1}{({\Psi }_2+{\mathrm{{\rm Y}}}_2+{\Pi }_2)}^{k+2})\left(\gamma \right)d\gamma \hfill \\ \\ {\Psi }_2\left(t\right)\hfill & ={\int }_0^t{\mathfrak{U}}_{\alpha }(t-\gamma ){\partial }_x(A{({\Psi }_2+{\mathrm{{\rm Y}}}_2+{\Pi }_2)}^{2k+1}+B{({\Psi }_2+{\mathrm{{\rm Y}}}_2+{\Pi }_2)}^k{({\Psi }_1+{\mathrm{{\rm Y}}}_1+{\Pi }_1)}^{k+1}\hfill \\ \\ \hfill & +{\displaystyle \frac{k+2}{k}}D{({\Psi }_2+{\mathrm{{\rm Y}}}_2+{\Pi }_2)}^{k+1}{({\Psi }_1+{\mathrm{{\rm Y}}}_1+{\Pi }_1)}^k+C{({\Psi }_2+{\mathrm{{\rm Y}}}_2+{\Pi }_2)}^{k-1}{({\Psi }_1+{\mathrm{{\rm Y}}}_1+{\Pi }_1)}^{k+2})\left(\gamma \right)d\gamma ,\hfill \end{array}\right.$$

(26)

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

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

The goal of this section is thus to prove that the mapping $({\Psi }_1\left(t\right),{\Psi }_2\left(t\right))$ is a contraction on ${\mathcal{B}}_{\mathcal{R},T}$. According to Lemmas 1, 4, and 5, we obtain

$$\begin{array}{ccc}\hfill \parallel {\Gamma }_1\left({\Psi }_1\left(t\right)\right){\parallel }_{X_T^{b,s}}& \lesssim & T^{1-b+b^{\prime }}\left({\mathcal{R}}^{2k+1}+\parallel ({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2){\parallel }_{{\mathcal{X}}_{\alpha ,T}^{b,s}}^{2k+1}+{\parallel ({\varphi }_0,{\phi }_0)\parallel }_{{\mathcal{H}}^s}^2\right),\hfill \\ \\ \hfill \parallel {\Gamma }_2\left({\Psi }_2\left(t\right)\right){\parallel }_{X_{\alpha ,T}^{b,s}}& \lesssim & T^{1-b+b^{\prime }}\left({\mathcal{R}}^{2k+1}+\parallel ({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2){\parallel }_{{\mathcal{X}}_{\alpha ,T}^{b,s}}^{2k+1}+{\parallel ({\varphi }_0,{\phi }_0)\parallel }_{{\mathcal{H}}^s}^2\right),\hfill \end{array}$$

so

$$\begin{array}{ccc}\hfill \parallel ({\Gamma }_1\left({\Psi }_1\left(t\right)\right),{\Gamma }_2\left({\Psi }_2\left(t\right)\right)){\parallel }_{{\mathcal{X}}_{\alpha ,T}^{b,s}}& \lesssim & T^{1-b+b^{\prime }}\left({\mathcal{R}}^{2k+1}+\parallel ({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2){\parallel }_{{\mathcal{X}}_{\alpha ,T}^{b,s}}^{2k+1}+{\parallel ({\varphi }_0,{\phi }_0)\parallel }_{{\mathcal{H}}^s}^2\right),\hfill \end{array}$$

Therefore, for $({\Psi }_{1.1},{\Psi }_{2.1}),({\Psi }_{1.2},{\Psi }_{2.2})\in {\mathcal{B}}_{\mathcal{R},T}$, we get

$$\begin{array}{ccc}\hfill \parallel {\Gamma }_1({\Psi }_{1.1}-{\Psi }_{1.2}){\parallel }_{X_T^{b,s}}& \lesssim & T^{1-b+b^{\prime }}\left({\mathcal{R}}^{2k}+\parallel ({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2){\parallel }_{{\mathcal{X}}_{\alpha ,T}^{b,s}}^{2k}+{\parallel ({\varphi }_0,{\phi }_0)\parallel }_{{\mathcal{H}}^s}\right)\hfill \\ \\ & & \times \parallel ({\Psi }_{1.1}-{\Psi }_{1.2},{\Psi }_{2.1}-{\Psi }_{2.2}){\parallel }_{{\mathcal{X}}_{\alpha ,T}^{b,s}},\hfill \\ \\ \hfill \parallel {\Gamma }_2({\Psi }_{2.1}-{\Psi }_{2.2}){\parallel }_{X_{\alpha ,T}^{b,s}}& \lesssim & T^{1-b+b^{\prime }}\left({\mathcal{R}}^{2k}+\parallel ({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2){\parallel }_{{\mathcal{X}}_{\alpha ,T}^{b,s}}^{2k}+{\parallel ({\varphi }_0,{\phi }_0)\parallel }_{{\mathcal{H}}^s}\right)\hfill \\ \\ & & \times \parallel ({\Psi }_{1.1}-{\Psi }_{1.2},{\Psi }_{2.1}-{\Psi }_{2.2}){\parallel }_{{\mathcal{X}}_{\alpha ,T}^{b,s}},\hfill \end{array}$$

so

$$\begin{array}{ccc}\hfill \parallel ({\Gamma }_1({\Psi }_{1.1}-{\Psi }_{1.2}),{\Gamma }_2({\Psi }_{2.1}-{\Psi }_{2.2})){\parallel }_{X_{\alpha ,T}^{b,s}}& \lesssim & T^{1-b+b^{\prime }}\left({\mathcal{R}}^{2k}+\parallel ({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2){\parallel }_{{\mathcal{X}}_{\alpha ,T}^{b,s}}^{2k}+{\parallel ({\varphi }_0,{\phi }_0)\parallel }_{{\mathcal{H}}^s}\right)\hfill \\ \\ & & \times \parallel ({\Psi }_{1.1}-{\Psi }_{1.2},{\Psi }_{2.1}-{\Psi }_{2.2}){\parallel }_{{\mathcal{X}}_{\alpha ,T}^{b,s}}.\hfill \end{array}$$

First, we define

$${\mathcal{R}}_{\varrho }=2C\left(\parallel ({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2){\parallel }_{{\mathcal{X}}_{\alpha ,T}^{b,s}}^{2k+1}+{\parallel ({\varphi }_0,{\phi }_0)\parallel }_{{\mathcal{H}}^s}^2\right).$$

Then, we choose a small enough $T_{\varrho }$ such that

$$4C{T}^{1-b+b^{\prime }}\left({\mathcal{R}}_{\varrho }^{2k}+\parallel ({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2){\parallel }_{{\mathcal{X}}_{\alpha ,T}^{b,s}}^{2k}+{\parallel ({\varphi }_0,{\phi }_0)\parallel }_{{\mathcal{H}}^s}\right)\le 1.$$

One can easily verify that $({\Gamma }_1,{\Gamma }_2)$ maps ${\mathcal{B}}_{\mathcal{R},T}$ into itself and is a strict contraction on ${\mathcal{B}}_{\mathcal{R},T}$ with respect to the norm $\parallel ({\Psi }_1,{\Psi }_2){\parallel }_{{\mathcal{X}}_{\alpha ,T}^{b,s}}$,

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

Hence, $({\Gamma }_1,{\Gamma }_2)$ has a unique fixed point in ${\mathcal{X}}_{\alpha ,T}^{b,s}$, which is a solution to (26) on $[0,T_{\varrho }]$.

Now observe that

$$\left\{\begin{array}{c}\mathfrak{U}\left(t\right)={\Pi }_1\left(t\right)+{\Psi }_1\left(t\right)+{\mathrm{{\rm Y}}}_1\left(t\right)\in X_{T_{\varrho }}^{b^{\prime },s}+X_{T_{\varrho }}^{b,s}\hfill \\ \\ {\mathfrak{U}}_{\alpha }\left(t\right)={\Pi }_2\left(t\right)+{\Psi }_2\left(t\right)+{\mathrm{{\rm Y}}}_2\left(t\right)\in X_{\alpha ,T_{\varrho }}^{b^{\prime },s}+X_{\alpha ,T_{\varrho }}^{b,s}.\hfill \end{array}\right.$$

(27)

We complete the proof by showing that $(\varphi ,\phi )\in C([0,T_{\varrho }],H^s\left(\mathbb{R}\right))\times C([0,T_{\varrho }],H^s\left(\mathbb{R}\right))$. Recall that $b^{\prime }<{\displaystyle \frac{1}{2}}$ and $b>{\displaystyle \frac{1}{2}}$. According to the Sobolev embedding theorem, we have $({\Pi }_1,{\Pi }_2)\in C([0,T_{\varrho }],H^s\left(\mathbb{R}\right))\times C([0,T_{\varrho }],H^s\left(\mathbb{R}\right))$.

We need the following theorem:

Theorem 2 

([9]). Assume that $\mathcal{A}$ generates a contraction semigroup and $\Xi \in {\mathcal{N}}_{\mathcal{W}}^2\left([0,T],L_2^{0,s}\right)$. Then the process ${\mathcal{W}}_{\mathcal{A}}^{\Xi }(·)$ has a continuous modification and there exists a constant C such that

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

Under the condition that $\Xi \in L_2^{0,s}$, and the fact that $\mathfrak{U}\left(t\right)$ and ${\mathfrak{U}}_{\alpha }\left(t\right)$ are a unitary group in $H^s\left(\mathbb{R}\right)$, an application of Theorem 2 implies that $({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)\in C([0,T_{\varrho }],H^s\left(\mathbb{R}\right))\times C([0,T_{\varrho }],H^s\left(\mathbb{R}\right))$.

According to Lemma 4, we have ${\partial }_x\left(N_{\varphi }(\tilde{\varphi },\tilde{\phi })\right)\in X^{s,b^{\prime }}$ and ${\partial }_x\left(G(\tilde{\varphi },\tilde{\phi })\right)\in X_{\alpha }^{s,b^{\prime }}$ for any extension $\tilde{\varphi }$ of $\varphi$ in $X^{s,b^{\prime }}+X^{s,b}$, and $\tilde{\phi }$ of $\phi$ in $X_{\alpha }^{s,b^{\prime }}+X_{\alpha }^{s,b}$, where $-{\displaystyle \frac{1}{2}}<b^{\prime }\le 0\le b<b^{\prime }+1$, and applying Lemma 3.2 in [18], we get

$${∥{\int }_0^t\mathfrak{U}(t-\gamma ){\partial }_x\left(N_{\varphi }(\tilde{\varphi },\tilde{\phi })\right)d\gamma ∥}_{X^{s,b}}\le C{∥{\partial }_x\left(N_{\varphi }(\tilde{\varphi },\tilde{\phi })\right)∥}_{X^{s,b^{\prime }}},$$

$${∥{\int }_0^t{\mathfrak{U}}_{\alpha }(t-\gamma ){\partial }_x\left(N_{\phi }(\tilde{\varphi },\tilde{\phi })\right)d\gamma ∥}_{X_{\alpha }^{s,b}}\le C{∥{\partial }_x\left(N_{\phi }(\tilde{\varphi },\tilde{\phi })\right)∥}_{X_{\alpha }^{s,b^{\prime }}}.$$

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

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

6. Conclusions

In this work, we have established local well-posedness for a system of coupled generalized KdV equations driven by cylindrical noise in the Sobolev space $H^s\left(\mathbb{R}\right)\times H^s\left(\mathbb{R}\right)$ for $s>{\displaystyle \frac{1}{2}}$. The proof relies on a combination of Bourgain space techniques, multilinear estimates, and stochastic analysis, thereby extending and improving previous deterministic well-posedness results by both lowering the required regularity threshold and incorporating stochastic forcing.

A natural extension of this work is the study of the corresponding initial-boundary value problem on the half-line, for which the Fokas unified transform method provides a powerful analytical framework [19]. In this setting, the boundary behavior of solutions to stochastic dispersive systems remains an open and challenging problem, with recent progress in the linear case offering a useful starting point [20]. Further research directions include investigating global well-posedness, blow-up phenomena, and the effect of more singular noise or higher-dimensional settings.