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The wellposedness of stochastic Kawahara equation: fixed point argument and Fourier restriction method
Journal of the Egyptian Mathematical Society volume 27, Article number: 5 (2019)
Abstract
In this paper, we investigate the Cauchy problem for the stochastic Kawahara equation, which is a fifthorder shallow water wave equation. We prove local wellposedness for data in \(H^{s}(\mathbb {R})\), s>−7/4. Moreover, we get global existence for \(L^{2}(\mathbb {R})\) solutions. Due to the nonzero singularity of the phase function, a fixed point argument and Fourier restriction method are proposed.
Introduction
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 twoparameter 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 wellposedness 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)=αξ^{5}−βξ^{3}+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 nonzero singularity. This differs from the phase function of the linear Kortewegde 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 fifthorder 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 nonconstant 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 fifthorder 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 wellposed 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 wellposed [16].
In [17], Huo obtained a local wellposedness result in \(H^{s}(\mathbb {R})(s>11/8)\) for the Kawahara equation. Moreover, Jia and Huo [18] proved the local wellposedness 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 wellposedness result for the KaupKupershmidt equations was presented by Tao and Cui [19]. They proved that their Cauchy problems are locally wellposed 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 wellposedness result for the KaupKupershmidt 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 wellposedness of Eq. (8) was studied by Kenig, Ponce, and Vega [21]. They proved that its Cauchy problem is locally wellposed in \(H^{s}(\mathbb {R})\) for s>−3/4. Also, Ponce [1] discussed the general fifthorder shallow water wave equation:
and gave a global wellposedness result of its Cauchy problem for data in \(H^{4}(\mathbb {R})\). The wellposedness 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 wellposedness 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 wellposed 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 wellposedness: 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 wellposedness result of Eq. (1). In the section “Global wellposedness: proof of Theorem 2”, we extend our technique and show global wellposedness 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
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 u_{0}is\(\mathcal {F}_{0}\)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 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
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 S_{2}, we have:
Now, we limit I_{1}, I_{2}, and I_{3} 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).
Local wellposedness: proof of Theorem 1
According to the stochastic estimation proved in the above section and the Banach fixed point theorem, we deduce a local wellposedness result of Eq. (1). That is, this section is devoted to the proof of Theorem 1. Let v(t)=U(t)u_{0} 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 \(1a>\frac {1}{2}\), 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 wellposedness: proof of Theorem 2
Fix 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
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,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 wellposedness results for their solutions in convenient function spaces. Our attention is focused on the stochastic Kawahara Eq. (1), which is a fifthorder shallow water wave equation considered in random environment. We prove that Eq. (1) is locally wellposed 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 nonzero singularity of the phase function ϕ.
The deterministic Kawahara Eq. (7) was discussed by Jia and Huo in [18]. They proved local wellposedness 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 fifthorder 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, HirotaSatsuma coupled KdV, and SwadaKotera equations.
Abbreviations
 KdV:

Kortewegde Vries
 SPDEs:

Stochastic partial differential equations
References
 1
Ponce, G.: Lax pairs and higher order models for water waves. J. Differ. Equat. 102, 360–381 (1993).
 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(107156), 209–262 (1993).
 3
Bona, J. L., Smith, R. S.: A model for the twoways propagation of water waves in a channel. Math. Proc. Cambridge Philos. Soc. 79, 167–182 (1976).
 4
Kawahara, T.: Oscillatory solitary waves in dispersive media. J. Phys. Soc. Jpn. 33, 260–264 (1972).
 5
Kichenassamy, S., Olver, P. J.: Existence and nonexistence of solitary wave solutions to higherorder model evolution equations. SIAM J. Math. Anal. 23, 1141–1166 (1992).
 6
Akylas, T. R.: On the excitation of long nonlinear water waves by a moving pressure distribution. J. Fluid Mech. 141, 455–466 (1984).
 7
Wu, T. Y.: Generation of upstream advancing solitons by moving disturbances. J. Fluid Mech. 184, 75–99 (1987).
 8
Ghany, H. A., Hyder, A.: White noise functional solutions for the Wicktype twodimensional stochastic ZakharovKuznetsov equations. Int. Rev. Phys. 6, 153–157 (2012).
 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).
 10
Ghany, H. A., Hyder, A.: Exact solutions for the Wicktype stochastic timefractional KdV equations. Kuwait J. Sci. 41, 75–84 (2014).
 11
Ghany, H. A., Hyder, A.: Abundant solutions of Wicktype stochastic fractional 2D KdV equations, Vol. 23 (2014).
 12
Ghany, H. A., Elagan, S. K., Hyder, A.: Exact travelling wave solutions for stochastic fractional HirotaSatsuma coupled KdV equations. Chin. J. Phys. 53, 1–14 (2015).
 13
Ghany, H. A., Hyder, A., Zakarya, M.: NonGaussian white noise functional solutions of χWicktype stochastic KdV equations. Appl. Math. Inf. Sci. 11, 915–924 (2017).
 14
Hyder, A., Zakarya, M.: NonGaussian Wick calculus based on hypercomplex systems. Int. J. Pure Appl. Math. 109, 539–556 (2016).
 15
Ghany, H. A., Zakarya, M.: Generalized solutions of Wicktype stochastic KdVBurgers equations using expfunction method. Int. Rev. Phys. 8, 38–46 (2014).
 16
Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Cambridge University Press, Cambridge (1992).
 17
Huo, Z.: The Cauchy problem for the fifthorder shallow water equation. Acta Math. Appl. Sin. Engl. Ser. 21, 441–454 (2005).
 18
Jia, Y., Huo, Z.: Wellposedness for the fifthorder shallow water equations. J. Diff. Equat. 246, 2448–2467 (2009).
 19
Tao, S. P., Cui, S. B.: Local and global existence of solutions to initial value problems of nonlinear KaupKupershmidt equations. J. Acta Math. Sin. Engl. Ser. 21, 881–892 (2005).
 20
Zhao, X. Q., GU, S. M.: Local solvability of Cauchy problem for KaupKupershmidt equation. J. Math. Res. Exposition. 30, 543–551 (2010).
 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).
 22
de Bouard, A., Debussche, A.: On the stochastic Kortewegde Vries equation. J. Funct. Anal. 154, 215–251 (1998).
 23
de Bouard, A.: White noise driven Kortewegde Vries equation. J. Funct. Anal. 169, 532–558 (1999).
 24
Ghany, H. A., Hyder, A.: Local and global wellposedness of stochastic Zakharov Kuznetsov equation. J. Comput. Anal. Appl. 15, 1332–1343 (2013).
 25
Printems, J.: The stochastic Kortewegde Vries equation in\(L^{2}(\mathbb {R})\). J. Diff. Equat. 153, 338–373 (1999).
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Keywords
 Kawahara equation
 Wellposedness
 Wiener process
 Fixed point theorem
 Fourier restriction method
AMS Subject Classification
 60H15
 49K40
 60H40