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

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## Abstract

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.

## Keywords

Kawahara equation Well-posedness Wiener process Fixed point theorem Fourier restriction method## Abbreviations

- KdV
Korteweg-de Vries

- SPDEs
Stochastic partial differential equations

## AMS Subject Classification

60H15 49K40 60H40## Introduction

*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:

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

*δ*is the Dirac function and

*ϕ*(

*ξ*)=

*α*

*ξ*

^{5}−

*β*

*ξ*

^{3}+

*γ*

*ξ*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]:

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

As aforesaid in [3, 4, 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, 9, 10, 11, 12, 13, 14, 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.

*α*=

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

*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, 24, 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.

## Main results

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

### **Definition 1**

where 〈·〉=1+|·|.

### **Definition 2**

*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*∈

*Ω*

*and*

*u*

_{0}

*is*\(\mathcal {F}_{0}-\)

*measurable. Then for almost surely*

*ω*∈

*Ω*,

*there exists a constant*

*T*

_{ω}>0

*and a unique solution u of the Cauchy problem*(4)

*on*[0,

*T*

_{ω}]

*which satisfies:*

In fact the *L*^{2}−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:*

## The stochastic convolution estimate

*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**

*u*

_{l}defined by (12) satisfies:

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*

*U*(

*t*)

*w*(

*t*,.)=

*χ*(

*t*)

*u*

_{l}(

*t*). Thus, by Eq. (10), we have:

□

*S*

_{2}, we have:

*I*

_{1},

*I*

_{2}, and

*I*

_{3}separately,

*b*∈(0,1), we have

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).

## Local well-posedness: proof of Theorem 1

*v*(

*t*)=

*U*(

*t*)

*u*

_{0}and \(\bar {u}=u(t)-v(t)-u_{l}(t)\), then Eq. (5) is equivalent to

*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**

*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:

### **Lemma 3**

*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.

*T*

_{ω}by:

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 \(u_{l}\in C\left ([0,T_{\omega }];H^{s}(\mathbb {R})\right)\).

*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.

## Global well-posedness: proof of Theorem 2

*T*

_{0}>0 and assume that

*u*

_{0}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,

*T*

_{0}]. Let \(\left (\Phi _{n}\right)_{n\in \mathbb {N}}\) be a sequence in \(L_{0}^{0,4}\) such that

*u*

_{n}in \(C\left ([0,T_{0}],H^{3}(\mathbb {R})\right)\) for

*u*

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

Thus, we can emerge a solution on [*T*_{ω},2*T*_{ω}]. Hence, the solution *u* can be extended to [0,*T*_{0}] almost surely by reiteration. This completes the proof of Theorem 2.

## Summary and discussion

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,*T*_{0}]. 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.

## Notes

### Acknowledgements

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.

### Authors’ contributions

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

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## References

- 1.Ponce, G.: Lax pairs and higher order models for water waves. J. Differ. Equat. 102, 360–381 (1993).MathSciNetCrossRefGoogle Scholar
- 2.Bourgain, J.: Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equation, part II: the KdV equation. Geom. Funct. Anal. 2(107-156), 209–262 (1993).CrossRefGoogle Scholar
- 3.Bona, J. L., Smith, R. S.: A model for the two-ways propagation of water waves in a channel. Math. Proc. Cambridge Philos. Soc. 79, 167–182 (1976).MathSciNetCrossRefGoogle Scholar
- 4.Kawahara, T.: Oscillatory solitary waves in dispersive media. J. Phys. Soc. Jpn. 33, 260–264 (1972).CrossRefGoogle Scholar
- 5.Kichenassamy, S., Olver, P. J.: Existence and nonexistence of solitary wave solutions to higher-order model evolution equations. SIAM J. Math. Anal. 23, 1141–1166 (1992).MathSciNetCrossRefGoogle Scholar
- 6.Akylas, T. R.: On the excitation of long nonlinear water waves by a moving pressure distribution. J. Fluid Mech. 141, 455–466 (1984).MathSciNetCrossRefGoogle Scholar
- 7.Wu, T. Y.: Generation of upstream advancing solitons by moving disturbances. J. Fluid Mech. 184, 75–99 (1987).MathSciNetCrossRefGoogle Scholar
- 8.Ghany, H. A., Hyder, A.: White noise functional solutions for the Wick-type two-dimensional stochastic Zakharov-Kuznetsov equations. Int. Rev. Phys. 6, 153–157 (2012).Google Scholar
- 9.Ghany, H. A., Okb El Bab, A. S., Zabal, A. M., Hyder, A.: The fractional coupled KdV equations: exact solutions and white noise functional approach, Vol. 22 (2013).Google Scholar
- 10.Ghany, H. A., Hyder, A.: Exact solutions for the Wick-type stochastic time-fractional KdV equations. Kuwait J. Sci. 41, 75–84 (2014).MathSciNetGoogle Scholar
- 11.Ghany, H. A., Hyder, A.: Abundant solutions of Wick-type stochastic fractional 2D KdV equations, Vol. 23 (2014).Google Scholar
- 12.Ghany, H. A., Elagan, S. K., Hyder, A.: Exact travelling wave solutions for stochastic fractional Hirota-Satsuma coupled KdV equations. Chin. J. Phys. 53, 1–14 (2015).MathSciNetGoogle Scholar
- 13.Ghany, H. A., Hyder, A., Zakarya, M.: Non-Gaussian white noise functional solutions of
*χ*-Wick-type stochastic KdV equations. Appl. Math. Inf. Sci. 11, 915–924 (2017).MathSciNetCrossRefGoogle Scholar - 14.Hyder, A., Zakarya, M.: Non-Gaussian Wick calculus based on hypercomplex systems. Int. J. Pure Appl. Math. 109, 539–556 (2016).CrossRefGoogle Scholar
- 15.Ghany, H. A., Zakarya, M.: Generalized solutions of Wick-type stochastic KdV-Burgers equations using exp-function method. Int. Rev. Phys. 8, 38–46 (2014).Google Scholar
- 16.Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Cambridge University Press, Cambridge (1992).CrossRefGoogle Scholar
- 17.Huo, Z.: The Cauchy problem for the fifth-order shallow water equation. Acta Math. Appl. Sin. Engl. Ser. 21, 441–454 (2005).MathSciNetCrossRefGoogle Scholar
- 18.Jia, Y., Huo, Z.: Well-posedness for the fifth-order shallow water equations. J. Diff. Equat. 246, 2448–2467 (2009).MathSciNetCrossRefGoogle Scholar
- 19.Tao, S. P., Cui, S. B.: Local and global existence of solutions to initial value problems of nonlinear Kaup-Kupershmidt equations. J. Acta Math. Sin. Engl. Ser. 21, 881–892 (2005).MathSciNetCrossRefGoogle Scholar
- 20.Zhao, X. Q., GU, S. M.: Local solvability of Cauchy problem for Kaup-Kupershmidt equation. J. Math. Res. Exposition. 30, 543–551 (2010).MathSciNetzbMATHGoogle Scholar
- 21.Kenig, C. E., Ponce, G., Vega, L.: A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc. 9, 573–603 (1996).MathSciNetCrossRefGoogle Scholar
- 22.de Bouard, A., Debussche, A.: On the stochastic Korteweg-de Vries equation. J. Funct. Anal. 154, 215–251 (1998).MathSciNetCrossRefGoogle Scholar
- 23.de Bouard, A.: White noise driven Korteweg-de Vries equation. J. Funct. Anal. 169, 532–558 (1999).MathSciNetCrossRefGoogle Scholar
- 24.Ghany, H. A., Hyder, A.: Local and global well-posedness of stochastic Zakharov- Kuznetsov equation. J. Comput. Anal. Appl. 15, 1332–1343 (2013).MathSciNetzbMATHGoogle Scholar
- 25.Printems, J.: The stochastic Korteweg-de Vries equation in\(L^{2}(\mathbb {R})\). J. Diff. Equat. 153, 338–373 (1999).MathSciNetCrossRefGoogle Scholar

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