The well-posedness of stochastic Kawahara equation: fixed point argument and Fourier restriction method

In this paper, we investigate the Cauchy problem for the stochastic Kawahara equation, which is a fifth-order shallow water wave equation. We prove local well-posedness for data in \(H^{s}(\mathbb {R})\), s>−7/4. Moreover, we get global existence for \(L^{2}(\mathbb {R})\) solutions. Due to the non-zero singularity of the phase function, a fixed point argument and Fourier restriction method are proposed.

In this paper, we consider the Cauchy problem for the stochastic Kawahara equation:

where α≠0, β, and γ are real numbers; μ is a complex number; u is a stochastic process defined on \((x,t)\in \mathbb {R}\times \mathbb {R_{+}}\); Φ is a linear operator; and B is a two-parameter Brownian motion on \(\mathbb {R}\times \mathbb {R_{+}}\), that is, a zero mean Gaussian process whose correlation function is given by:

In general, the covariance operator Φ can be described by a kernel \(\mathcal {K}(x,y).\) The correlation function of the noise is then given by

where \(t,s\geq 0,\ x,y\in \mathbb {R}\), δ is the Dirac function and

Consider a fixed probability space \((\Omega,\mathcal {F},P)\) adapted to a filtration \((\mathcal {F}_{t})_{t\geq 0}\). As usual, we can rewrite the right hand side of Eq. (1) as the time derivative of a cylindrical Wiener process on \(L^{2}(\mathbb {R})\) by setting:

where \((e_{i})_{i\in \mathbb {N}}\) is an orthonormal basis of \(L^{2}(\mathbb {R})\) and \((\beta _{i})_{i\in \mathbb {N}}\) is a sequence of mutually independent real Brownian motions in \((\Omega,\mathcal {F},P)\). Let us rewrite Eq. (1) in its Itô form as follows:

In order to obtain local well-posedness of Eq. (1), we mainly work on the general mild formulation of Cauchy problem (4) as below:

Here, \(U(t)=\mathfrak {F}_{x}^{-1}\text {exp}\left (-it\phi (\xi)\right)\mathfrak {F}_{x}\) is the unitary group of operators related to the linearized equation:

where ϕ(ξ)=αξ⁵−βξ³+γξ is the phase function and \(\mathfrak {F}_{x}\) (or “. ̂”) is the usual Fourier transform in the x variable. We note that the phase function ϕ has non-zero singularity. This differs from the phase function of the linear Korteweg-de Vries (KdV) equation (see [1]) and causes some difficulties in the problem. To avoid these difficulties, we eliminate the singularity of the phase function ϕ by using the Fourier restriction operators [2]:

In the case of Φ≡0 (effect of the noise does not exist), Eq. (1) is reduced to the deterministic Kawahara equation:

As aforesaid in [3–5], Eq. (7) is a fifth-order shallow water wave equation. It arises in study of the water waves with surface tension, in which the Bond number takes on the critical value, where the Bond number represents a dimensionless magnitude of surface tension in the shallow water regime. If we consider a realistic situation, in which a non-constant pressure affects on the surface of the fluid or the bottom of the layer is not flat, it is meaningful to add a forcing term to Eq. (7). This term can be given by the gradient of the exterior pressure or of the function whose graph defines the bottom [6, 7]. This paper focuses on the case when the forcing term is of additive white noise type. This leads us to study the stochastic fifth-order shallow water wave Eq. (1). By means of white noise functional analysis, the analytical white noise functional solutions for the nonlinear stochastic partial differential equations (SPDEs) can be investigated. This subject is attracting more and more attention [8–15].

It is well known that the Cauchy problem (4) is locally well-posed for data in \(H^{s}(\mathbb {R}),\ s\in \mathbb {R}\), if for any finite time T, there exists a locally continuous mapping that transfers \(u_{0}\in H^{s}(\mathbb {R})\) to a unique solution \(u\in C\left ([0,T];H^{s}(\mathbb {R})\right)\). If the solution mapping exists for all time, we say that the Cauchy problem (4) is globally well-posed [16].

In [17], Huo obtained a local well-posedness result in \(H^{s}(\mathbb {R})(s>-11/8)\) for the Kawahara equation. Moreover, Jia and Huo [18] proved the local well-posedness of the Kawahara and modified Kawahara equations for data in \(H^{s}(\mathbb {R})\) with s>−7/4 and s≥−1/4 respectively. The first well-posedness result for the Kaup-Kupershmidt equations was presented by Tao and Cui [19]. They proved that their Cauchy problems are locally well-posed in \(H^{s}(\mathbb {R})\) for s>5/4 and s>301/108, respectively. Thereafter, Zhao and Gu [20] lowered the regularity of the initial data space to s>9/8 and improved the preceding result in [19]. Also, using a Fourier restriction method, a local well-posedness result for the Kaup-Kupershmidt equations was established in [18] for data in \(H^{s}(\mathbb {R})\) with s>0 and s>−1/4, respectively.

If α=γ=0, the model (7) is minified to the famous KdV equation:

The well-posedness of Eq. (8) was studied by Kenig, Ponce, and Vega [21]. They proved that its Cauchy problem is locally well-posed in \(H^{s}(\mathbb {R})\) for s>−3/4. Also, Ponce [1] discussed the general fifth-order shallow water wave equation:

and gave a global well-posedness result of its Cauchy problem for data in \(H^{4}(\mathbb {R})\). The well-posedness of the SPDEs has been the subject of a large amount of work. de Bouard and Debussche [22] considered the stochastic KdV equation forced by a random term of white noise type. They proved existence and uniqueness of solutions in \(H^{1}(\mathbb {R})\) and existence of martingale solutions in \(L^{2}(\mathbb {R})\) in the case of additive and multiplicative noise, respectively. Since that time, many researchers paid more attention to investigate the Cauchy problems for some SPDEs and have obtained a number of local and global well-posedness results [23–25].

The goal of this paper is to investigate the Cauchy problem of the stochastic Kawahara Eq. (1), where the random force is of additive white noise type. By employing a Fourier restriction method, a Banach fixed point theorem, and some basic inequalities, we show that Eq. (1) is locally well-posed for data in \(H^{s}(\mathbb {R}),\ s>-7/4\). Also, we give global existence for \(L^{2}(\mathbb {R})\) solutions. An outline of this paper is as follows. The “Main results” section contains precise statement of our new results and some important function spaces. In the section “The stochastic convolution estimate”, we give an estimation of the stochastic convolution term via a Fourier restriction method and some basic inequalities. In the section “Local well-posedness: proof of Theorem 1”, we use the stochastic estimation proved in the section “The stochastic convolution estimate” and the Banach fixed point theorem to obtain a local well-posedness result of Eq. (1). In the section “Global well-posedness: proof of Theorem 2”, we extend our technique and show global well-posedness result of Eq. (1). The “Summary and discussion” section is devoted to the summary and discussion.

Before giving the precise statement of our main results, we introduce some notations and assumptions.

Definition 1

For \(s,b\in \mathbb {R}\) the space \(\mathfrak {X}_{s,b}\) is defined to be the completion of the Schwartz function space \(\mathcal {S}\left (\mathbb {R}^{2}\right)\) with respect to the norm:

where 〈·〉=1+|·|.

Definition 2

For T>0, \(\mathfrak {X}_{s,b}^{T}\) is the space of restrictions to [0,T] of functions in \(\mathfrak {X}_{s,b}\) endowed with the norm:

Theorem 1

Assume that\(s>-\frac {7}{4}\), \(\Phi \in L_{2}^{0,s}\), \(b\in \left (0,\frac {1}{2}\right)\)and b is close enough to\(\frac {1}{2}\). If\(u_{0}\in H^{s}(\mathbb {R})\)for almost surely ω∈Ωand u₀is\(\mathcal {F}_{0}-\)measurable. Then for almost surely ω∈Ω, there exists a constant T_ω>0and a unique solution u of the Cauchy problem (4) on [0,T_ω]which satisfies:

In fact the L²−norm is preserved for a solution of the Kawahara equation [4]. Therefore, in the case of s=0, we can obtain a global existence result for Eq. (1). Precisely, we have:

Theorem 2

Let\(u_{0}\in L^{2}\left (\Omega,L^{2}(\mathbb {R})\right)\)be an\(\mathcal {F}_{0}-\)measurable initial data, and let\(\Phi \in L_{2}^{0,0}\). Then, the solution u given by Theorem 1 is global and satisfies:

In this section, using the Fourier restriction method, the properties of Itô stochastic integral and some basic inequalities, we give an estimation for the last term in Eq. (5), which is the stochastic convolution:

Choose \(\chi \in C_{0}^{\infty }\left (\mathbb {R}_{+}\right)\) such that χ(t)=0 for t>0,χ(t)=1 for 0<t<1 and χ(t)=0 for t≥2. Hence, \(\chi \in H^{b}(\mathbb {R})\) for any \(b<\frac {1}{2}\). Let \(H_{t}^{b}:=H^{b}\left ([0,T];\mathbb {R}\right)\) be the Sobolev space in the time variable t with the norm:

Now, we state and prove the estimation of the stochastic convolution (12) as follows:

Lemma 1

Assume that \(s,b\in \mathbb {R}\) with \(b\in \left (0,\frac {1}{2}\right)\), and let \(\Phi \in L_{2}^{0,s}\) Then, u_l defined by (12) satisfies:

and

where N(b,χ)is a constant that depends on b, \(\|\chi \|_{H^{b}_{t}}\), \(\||t|^{\frac {1}{2}}\chi \|_{L^{2}_{t}}\) and \(\||t|^{\frac {1}{2}}\chi \|_{L^{\infty }_{t}}\),

Proof

Let us introduce the function

This implies that U(t)w(t,.)=χ(t)u_l(t). Thus, by Eq. (10), we have:

According to the expansion (3) of the cylindrical Wiener process and Eq. (13), we have:

where,

□

From the Itô isometry formula, we get:

To estimate S₂, we have:

Now, we limit I₁, I₂, and I₃ separately,

Using Eq. (15) and the assumption that 2b∈(0,1), we have

Similarly,

Combining (20)–(24) with (17), we get

where \(N(b,\chi)=M_{b}\left (\|\chi \|_{H^{b}_{t}}+\||t|^{\frac {1}{2}}\chi \|_{L^{2}_{t}}+\||t|^{\frac {1}{2}}\chi \|_{L^{\infty }_{t}}\right)\). Hence, the estimate (14) comes from (16) and (25).

According to the stochastic estimation proved in the above section and the Banach fixed point theorem, we deduce a local well-posedness result of Eq. (1). That is, this section is devoted to the proof of Theorem 1. Let v(t)=U(t)u₀ and \(\bar {u}=u(t)-v(t)-u_{l}(t)\), then Eq. (5) is equivalent to

Therefore, the goal of this section becomes to prove that \(\mathcal {A}\) is a contraction mapping in

where R and T are sufficiently large and small, respectively. Before doing this, we recall some previous results on the linear and bilinear estimates.

Lemma 2

[23] Assume that a>0, \(b<\frac {1}{2}\) and b is close enough to \(\frac {1}{2}\). For \(s\in \mathbb {R}\), \(u_{0}\in H^{s}(\mathbb {R})\), and \(f\in \mathfrak {X}_{s,-a}^{T}\), we have:

and

Lemma 3

[18] Assume that a>0, \(b<\frac {1}{2}\), and b is close enough to \(\frac {1}{2}\). For \(b'>\frac {1}{2}\), \(s>-\frac {7}{4}\), and \(u_{1},u_{2}\in \mathcal {S}(\mathbb {R}^{2})\), we have:

provided that the right hand side is finite.

According to Lemmas 1, 2, and 3, we obtain

Therefore, for \(\bar {u}_{1},\bar {u}_{2}\in \mathfrak {Y}_{R}^{T}\), we get

Now, define the stopping time T_ω by:

where \(R_{\omega }^{T}=\left \|u_{l}\right \|_{\mathfrak {X}^{T}_{s,b}}+\left \|u_{0}\right \|_{H^{s}}\). Then, \(\mathcal {A}\) maps the ball with center zero and radius \(R_{\omega }^{T}\) in \(\mathfrak {X}_{s,b}^{T_{\omega }}\) into itself, and

From the fixed point theory, \(\mathcal {A}\) has a unique fixed point, which is the solution of (5) in \(\mathfrak {X}_{s,b}^{T_{\omega }}\). Observe that \(u=v+\bar {u}+u_{l}\in \mathfrak {X}^{T_{\omega }}_{s,b^{\prime }}+\mathfrak {X}^{T_{\omega }}_{s,b}\).

In the remaining part of this section, we complete the proof by showing that \(u\in C([0,T_{\omega }],H^{s}(\mathbb {R}))\). Taking in attention that \(b<\frac {1}{2}, b^{\prime } >\frac {1}{2}\). By virtue of the Sobolev imbedding theorem, we have \(v\in C\left ([0,T_{\omega }],H^{s}(\mathbb {R})\right)\). Under the condition that \(\Phi \in L_{2}^{0,s}\) and the fact that U(t) is a unitary group in \(H^{s}(\mathbb {R})\), an application of Theorem 6.10 in [16] implies that \(u_{l}\in C\left ([0,T_{\omega }];H^{s}(\mathbb {R})\right)\).

Now, choose a cutoff function \(\chi _{T}\in C_{0}^{\infty }(\mathbb {R})\) such that χ_T(t)=1 on [0,2], supp χ_T⊂[−1,2], and χ_T(t)=0 on (−∞,−1]∪[2,∞). Denote χ_q(.)=χ(q⁻¹(.)) for some \(q\in \mathbb {R}\). By Lemma 3, we have \(\tilde {u}\tilde {u}_{x}\in \mathfrak {X}_{s,-a}\) for any prolongation \(\tilde {u}\) of u in \(\mathfrak {X}_{s,c}+\mathfrak {X}_{s,b}\). Therefore,

Since \(1-a>\frac {1}{2}\), then \(\tilde {u}\in \mathfrak {X}_{s,b}\subset C\left ([0,T_{\omega }];H^{s}(\mathbb {R})\right)\). This completes the proof of Theorem 1.

Fix T₀>0 and assume that u₀ satisfies the conditions of Theorem 1. In this section, we present a proof of Theorem 2, that is, we show that the solution u can be extended to the whole interval [0,T₀]. Let \(\left (\Phi _{n}\right)_{n\in \mathbb {N}}\) be a sequence in \(L_{0}^{0,4}\) such that

and let \(\left (u_{0,n}\right)_{n\in \mathbb {N}}\) be another sequence in \(L^{2}\left (\Omega,H^{s}(\mathbb {R})\right)\) such that

By using reasoning similar to that in [23], we can find a unique solution u_n in \(C\left ([0,T_{0}],H^{3}(\mathbb {R})\right)\) for

By using the Itô formula on \(\|u_{n}\|^{2}_{L^{2}(\mathbb {R})}\) and martingale inequality (see [16]), we have

Therefore, the sequence \((u_{n})_{n\in \mathbb {N}}\) is bounded and weakly star convergent to a function \(u^{\ast }\in L^{2}\left (\Omega ;L^{\infty }\left (\left [0,T_{0}\right ];L^{2}(\mathbb {R})\right)\right)\), which satisfies

In the same way as \(\mathcal {A}\), define the mapping \(\mathcal {A}_{n}\). It is easy to show that \(\mathcal {A}_{n}\) is uniformly strict contraction on \(\mathfrak {Y}_{r(\omega)}^{t(\omega)}\) in \(\mathfrak {X}_{s,b}^{T_{\omega }}\). According to the fixed point theorem, there exists a unique function \(u\in \mathfrak {X}_{s,b}^{T_{\omega }}\) such that

where u_n is the unique fixed point of \(\mathcal {A}_{n}\). Also, we have

Thus, we can emerge a solution on [T_ω,2T_ω]. Hence, the solution u can be extended to [0,T₀] almost surely by reiteration. This completes the proof of Theorem 2.

This paper is devoted to employ the Fourier restriction method, the Banach contraction principle, and some basic inequalities for investigating nonlinear SPDEs and for proving local and global well-posedness results for their solutions in convenient function spaces. Our attention is focused on the stochastic Kawahara Eq. (1), which is a fifth-order shallow water wave equation considered in random environment. We prove that Eq. (1) is locally well-posed for data in \(H^{s}(\mathbb {R})\), s>−7/4 and its solution can be extended to a global one on [0,T₀]. The Fourier restriction method is proposed due to the non-zero singularity of the phase function ϕ.

The deterministic Kawahara Eq. (7) was discussed by Jia and Huo in [18]. They proved local well-posedness result for data in \(H^{s}(\mathbb {R})\), s>−7/4. In this paper, we extend their result and handle the stochastic version of the Kawahara equation by choosing new appropriate stochastic function spaces (such as the space \(\mathfrak {X}_{s,b}^{T})\) and estimating the stochastic convolution (12) in these spaces. That is, we consider a realistic situation of the fifth-order shallow water wave equations. We believe that the ideas which we have suggested in this paper can be also applied to a wide class of stochastic nonlinear evolution equations in the field of mathematical physics, for instance, the stochastic modified Kawahara, generalized KdV, Hirota-Satsuma coupled KdV, and Swada-Kotera equations.

KdV:

Korteweg-de Vries

SPDEs:

Stochastic partial differential equations

The authors is very thankful to the editor and referees for their valuable comments and suggestions.

Funding

Not applicable.

Availability of data and materials

Not applicable.

Affiliations

1. Department of Mathematics, College of Science, King Khalid University, Abha, P.O. Box 9004, Saudi Arabia
   • Abd-Allah Hyder
   •  & M. Zakarya
2. Department of Engineering Mathematics and Physics, Faculty of Engineering, Al-Azhar University, Cairo, 11371, Egypt
   • Abd-Allah Hyder
3. Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, 71524, Egypt
   • M. Zakarya

Contributions

All authors jointly worked on the results and they read and approved the final manuscript.

Corresponding author

Correspondence to Abd-Allah Hyder.

Competing interests

The authors declare that they have no competing interests.

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