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Wellposedness of stochastic modified Kawahara equation
Advances in Difference Equations volume 2020, Article number: 18 (2020)
Abstract
In this paper we consider the Cauchy problem for the stochastic modified Kawahara equation, which is a fifthorder shallow water wave equation. We prove local wellposedness for data in \(H^{s}(\mathbb{R})\), \(s\geq 1/4\). Moreover, we get the global existence for \(L^{2}( \mathbb{R})\) solutions. Due to the nonzero singularity of the phase function, a fixed point argument and the Fourier restriction method are proposed.
1 Introduction
In this paper, we investigate the Cauchy problem for the stochastic modified Kawahara equation:
where \(\alpha \neq 0\), 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 \((\varOmega ,\mathcal{F},P)\) adapted to a filtration \((\mathcal{F}_{t})_{t\geq 0}\). As usual, we can rewrite the righthand 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 \((\varOmega , \mathcal{F},P)\). 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):
Here, \(U(t)=\mathfrak{F}_{x}^{1}\exp (it\phi (\xi ) ) \mathfrak{F}_{x}\) is the unitary group of operators related to the linearized equation:
where \(\phi (\xi )=\alpha \xi ^{5}\beta \xi ^{3}+\gamma \xi \) is the phase function and \(\mathfrak{F}_{x}\) (or “ \(\hat{\cdot}\) ”) 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 Korteweg–de Vries (KdV) equation (see [1]) and causes some difficulties in solving 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 \(\varPhi \equiv 0\) (effect of the noise does not exist), Eq. (1) is reduced to the deterministic modified 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 equation (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 \(u\in C ([0,T];H ^{s}(\mathbb{R}) )\). 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 \(s\geq 1/4\), respectively. The first wellposedness result for the Kaup–Kupershmidt 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 Kaup–Kupershmidt equations was established in [18] for data in \(H^{s}(\mathbb{R})\) with \(s>0\) and \(s>1/4\), respectively.
If \(\alpha = \gamma =0\), the model (7) is reduced to the famous modified 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 \(s\geq 3/4\). 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 martingales 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 [15, 23–25].
The goal of this paper is to investigate the Cauchy problem of the stochastic modified Kawahara equation (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 \(H^{s}(\mathbb{R}), s \geq 1/4\). Also, we give global existence for \(L^{2}(\mathbb{R})\) solutions. An outline of this paper is as follows. Section 2 contains precise statement of our new results and some important function spaces. In Sect. 3, we give an estimation of the stochastic convolution term via a Fourier restriction method and some basic inequalities. In Sect. 4, we use the stochastic estimation proved in Sect. 3 and the Banach fixed point theorem to obtain a local wellposedness result of Eq. (1). In Sect. 5, we extend our technique and show global wellposedness result of Eq. (1). Section 6 is devoted to a summary and a discussion.
2 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 \(\mathcal{S} (\mathbb{R}^{2} )\) with respect to the norm
where \(\langle \cdot \rangle =1+\cdot \).
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
Definition 3
Let H be a Hilbert space, \(L_{2}^{0}:=L_{2} ^{0} (L^{2}(\mathbb{R}); H )\) is the space of Hilbert–Schmidt operators from \(L^{2}(\mathbb{R})\) into H with the norm
where \((e_{i})_{i\geq 1}\) is an orthonormal basis in \(L^{2}( \mathbb{R})\). When \(H=H^{s}(\mathbb{R})\), we write \(L_{2}^{0}=L_{2} ^{0,s}\).
Theorem 1
Assume that \(s\geq \frac{1}{4}\), \(\varPhi \in L_{2}^{0,s}\) \(b>\frac{1}{2}\)andbis close enough to \(\frac{1}{2}\). If \(u_{0}\in H^{s}(\mathbb{R})\)for almost surely \(\omega \in \varOmega \)and \(u_{0}\)is \(\mathcal{F}_{0}\) measurable. Then, for almost surely \(\omega \in \varOmega \), there exist a constant \(T_{\omega }>0\)and a unique solutionuof the Cauchy problem (4) on \([0,T_{\omega }]\)which satisfies
In fact the \(L^{2}\)norm is preserved for a solution of the modified Kawahara equation [4]. Therefore, in the case of \(s=0\), we can obtain a global existence result for Eq. (1). Precisely, we have the following.
Theorem 2
Let \(u_{0}\in L^{2} (\varOmega ,L^{2}( \mathbb{R}) )\)be an \(\mathcal{F}_{0}\)measurable initial data, and let \(\varPhi \in L_{2}^{0,0}\). Then the solutionugiven by Theorem 1is global and satisfies
3 The stochastic convolution estimate
In this section, using 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 \(\chi \in C_{0}^{\infty } (\mathbb{R}_{+} )\) such that
Hence, \(\chi \in H^{b}(\mathbb{R})\) for any \(b>\frac{1}{2}\). Let \(H_{t}^{b}:=H^{b} ([0,T];\mathbb{R} )\) be the Sobolev space in the time variable t with the norm:
Now, we state and prove the estimation of the stochastic convolution (13) as follows.
Lemma 1
Assume that \(s,b\in \mathbb{R}\)with \(b>\frac{1}{2}\), and let \(\varPhi \in L_{2}^{0,s}\). Then \(u_{l}\)defined by (13) satisfies
and
where \(N(b,\chi )\)is a constant that depends onb, \(\\chi \_{H^{b}_{t}}\), \(\t^{\frac{1}{2}}\chi \_{L^{2}_{t}}\)and \(\t^{\frac{1}{2}}\chi \_{L^{\infty }_{t}}\).
Proof
Let us introduce the function
this implies that \(U(t)w(t,\cdot)=\chi (t)u_{l}(t)\). Thus, by Eq. (10), we have
According to the expansion (3) of the cylindrical Wiener process and Eq. (14), 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. (16), we have
Similarly,
Combining (21)–(25) with (18), we get
where \(N(b,\chi )=M_{b} (\\chi \_{H^{b}_{t}}+\t^{ \frac{1}{2}}\chi \_{L^{2}_{t}}+\t^{\frac{1}{2}}\chi \_{L^{\infty }_{t}} )\). Hence, the estimate (15) comes from (17) and (26). □
4 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 \(a>0\), \(b>\frac{1}{2}\)andbis 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 \(0< a<1\), \(\frac{1}{2}< b<\frac{7}{10}\)andbis close enough to \(\frac{1}{2}\). For \(b'>\frac{1}{2}\), \(s\geq \frac{1}{4}\)and \(u_{1},u_{2},u_{3} \in \mathcal{S}(\mathbb{R}^{2})\), we have
provided that the righthand 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 \(T_{\omega }\) by
where \(R_{\omega }^{T}= \Vert u_{l} \Vert _{\mathfrak{X}^{T} _{s,b}}+ \Vert u_{0} \Vert _{H^{s}}\). 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 \(u=v+\bar{u}+u_{l}\in \mathfrak{X}^{T_{\omega }}_{s,b'}+ \mathfrak{X}^{T_{\omega }}_{s,b}\).
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 \(b,b'>\frac{1}{2}\). By virtue of the Sobolev imbedding theorem, we have \(v\in C ([0,T_{\omega }],H^{s}(\mathbb{R}) )\). Under the condition that \(\varPhi \in L_{2}^{0,s}\) 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 \(u_{l}\in C ([0,T_{\omega }];H^{s}( \mathbb{R}) )\).
Now choose a cutoff function \(\chi _{T}\in C_{0}^{\infty }(\mathbb{R})\) such that \(\chi _{T}(t)=1\) on \([0,2]\), \(\operatorname{supp} \chi _{T}\subset [1,2]\) and \(\chi _{T}(t)=0\) on \((\infty ,1]\cup [2,\infty )\). Denote \(\chi _{q}(\cdot)=\chi (q ^{1}(\cdot) )\) for some \(q\in \mathbb{R}\). By Lemma 3, we have \(\tilde{u}^{2}\tilde{u}_{x}\in \mathfrak{X}_{s,a}\) for any prolongation ũ of u in \(\mathfrak{X}_{s,c}+\mathfrak{X}_{s,b}\). Therefor
Since \(1a>\frac{1}{2}\), then \(\tilde{u}\in \mathfrak{X}_{s,b}\subset C ([0,T_{\omega }];H^{s}(\mathbb{R}) )\). This completes the proof of Theorem 1.
5 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 \((\varPhi _{n} )_{n\in \mathbb{N}}\) be a sequence in \(L_{0}^{0,4}\) such that
and let \((u_{0,n} )_{n\in \mathbb{N}}\) be another sequence in \(L^{2} (\varOmega ,H^{s}(\mathbb{R}) )\) such that
By using a reasoning similar to that in[23], we can find a unique solution \(u_{n}\) in \(C ([0,T_{0}], H^{3}(\mathbb{R}) )\) 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 \(u^{\ast }\in L^{2} (\varOmega ;L ^{\infty } ([0,T_{0}];L^{2}(\mathbb{R}) ) )\), 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 \([T_{\omega },2T_{\omega } ]\). Hence, the solution u can be extended to \([0,T_{0} ]\) almost surely by reiteration. This completes the proof of Theorem 2.
6 Summary and discussion
This paper is devoted to employing 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 to the stochastic modified Kawahara equation (1), which is a fifthorder shallow water wave equation considered in a random environment. We prove that Eq. (1) is locally wellposed for data in \(H^{s}(\mathbb{R})\), \(s\geq 1/4\) 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 modified Kawahara equation (7) was discussed by Jia and Huo in [18]. They proved local wellposedness result for data in \(H^{s}(\mathbb{R})\), \(s\geq 1/4\). In this paper, we extend their result and handle the stochastic version of the modified Kawahara equation by choosing new appropriate stochastic function spaces (such as the space \(\mathfrak{X}_{s,b}^{T}\)) and estimating the stochastic convolution (13) 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 KdV, the generalized KdV, the Hirota–Satsuma coupled KdV and the Swada–Kotera equations.
Abbreviations
 KdV:

Korteweg–de Vries
 SPDEs:

Stochastic Partial Differential Equations
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Acknowledgements
The authors would like to extend their appreciations to the Deanship of Scientific Research at King Khalid University for funding their work through Research Groups Program under grant number (R. G. P. 1/160/40).
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Agarwal, P., Hyder, AA. & Zakarya, M. Wellposedness of stochastic modified Kawahara equation. Adv Differ Equ 2020, 18 (2020). https://doi.org/10.1186/s1366201924856
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DOI: https://doi.org/10.1186/s1366201924856