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Initial boundary value problem for generalized Zakharov equations with nonlinear function terms
Boundary Value Problems volume 2020, Article number: 85 (2020)
Abstract
In this paper, we consider the initial boundary value problem for generalized Zakharov equations. Firstly, we prove the existence and uniqueness of the global smooth solution to the problem by a priori integral estimates, the Galerkin method, and compactness theory. Furthermore, we discuss the approximation limit of the global solution when the coefficient of the strong nonlinear term tends to zero.
1 Introduction
The Zakharov system, derived by Zakharov in 1972 [1], describes the interaction between Langmuir (dispersive) and ion acoustic (approximately nondispersive) waves in an unmagnetized plasma. The usual Zakharov system defined in the space \(\mathbb{R}^{d+1}\) is given by
where the wave fields \(\varepsilon(x,t)\) and \(n(x,t)\) are complex and real, respectively. It has become commonly accepted that the Zakharov system is a general model to govern interaction of dispersive and nondispersive waves.
The generalized Zakharov system has found a number of applications in various physical problems, such as interaction of intramolecular vibrations giving rise to Davydov solitons with acoustic disturbances [2], interaction of high- and low-frequency gravity disturbances in an atmosphere [3], and so on. In the past decades, the Zakharov system has been studied by many authors [4–13].
Gajewski and Zacharias [14] studied the following generalized Zakharov system and established the global existence for initial value problem:
where the parameters \(\beta>0\) and α are real numbers.
You and Ning [15] considered the existence and uniqueness of the global smooth solution for the initial value problem of the following generalized Zakharov equations in dimension two:
with initial data
where \(\varepsilon(x,t)=(\varepsilon_{1}(x,t), \varepsilon _{2}(x,t),\ldots,\varepsilon_{N}(x,t))\) is an N-dimensional complex-valued unknown functional vector, \(v(x,t)=(v_{1}(x,t), v_{2}(x,t))\) is a two-dimensional real-valued unknown functional vector, \(n(x,t)\) is a real-valued unknown function, \(x\in\mathbb {R}^{2}\), and \(\varphi(s)\) is a real function.
In the present paper, we study the following initial boundary value problem for generalized Zakharov equations:
with initial data
and boundary conditions
where the parameters \(p>0\), \(\beta>0\), α, and δ are real numbers, and \(\varphi(s)\) is a real function. Taking \(\delta=0\), \(\beta =0\), and \(\varphi^{\prime}(s)=\) Constant in this system, it becomes the classical Zakharov equation system. From a physical point of view, this system has stronger nonlinear excitation and interaction. It also can be considered as a further generalization of the generalized Zakharov system discussed in [14]. From the perspective of both mathematical research and physical applications, the problem is of great significance.
For convenience of the following contexts, we set some notations. For \(1\leq p\leq\infty\), we denote by \(L^{p}[0,L]\) the space of all pth-power integrable functions in \([0,L]\) equipped with norm \(\|\cdot \|_{L^{p}}\), and by \(H^{s,p}\) the Sobolev space with norm \(\|\cdot\| _{H^{s,p}}\). For \(p=2\), we write \(H^{s}\) instead of \(H^{s,2}\). \(C^{k}(R)\) is the space of k times continuously differentiable functions on R. If \(k=0\), then we write \(C(R)\) instead of \(C^{0}(R)\). Let \((f,g)=\int_{0}^{L}f(x)\overline{g(x)}\,dx\), where \(\overline {g(x)}\) denotes the complex conjugate function of \(g(x)\). The real and imaginary parts of a complex number A are denoted, respectively, by ReA and ImA. Throughout the paper, C is a generic constant, which may have different meanings in different places.
This paper is organized as follows. In Sect. 2, we establish a priori estimations for problem (1.1)–(1.5). In Sect. 3, we study the existence and uniqueness of global generalized solutions for problem (1.1)–(1.5). In Sect. 4, we discuss the regularity of global generalized solution for problem (1.1)–(1.5). In Sect. 5, we give the approximation limit of the global solution when the coefficient of the strong nonlinear term tends to zero.
2 A priori estimations for problem (1.1)–(1.5)
Lemma 2.1
Let\(\varepsilon_{0}\in L^{2}[0,L]\). Then for the solution of problem (1.1)–(1.5), we have
Proof
Taking the inner product of (1.1) and ε, we have
Since \(\operatorname{Im}(i\varepsilon_{t},\varepsilon)=\frac{1}{2}\frac {d}{dt}\|\varepsilon\|^{2}_{L^{2}}\), \(\operatorname{Im}(\varepsilon _{xx}+(\alpha-n)\varepsilon+\delta|\varepsilon|^{p}\varepsilon ,\varepsilon)=0\), and hence from (2.1) we get
that is,
 □
Lemma 2.2
Suppose that (1) \(\varepsilon_{0}\in H^{1}[0,L]\), \(v_{0}\in L^{2}[0,L]\), \(n_{0}\in L^{2}[0,L]\), and (2) \(\varphi(v)\in C(R)\). Then for the solution of problem (1.1)–(1.5), we have
Proof
Taking the inner product of (1.1) and \(-\varepsilon_{t}\), we get that
Since
we have
and hence from (2.2) we get
Taking the inner product of (1.2) and v, we have
Since
where
and hence from (2.4) we get
Thus
 □
Lemma 2.3
(Sobolev estimates)
-
(1)
Assuming that\(u\in L^{q}(R^{n})\), \(D^{m}u\in L^{r}(R^{n})\), \(1\leq q, r\leq\infty\), \(0\leq j< m\), we have the estimates
$$\bigl\Vert D^{j}u \bigr\Vert _{L^{p}}\leq C \bigl\Vert D^{m}u \bigr\Vert ^{\theta}_{L^{r}} \Vert u \Vert ^{1-\theta}_{L^{q}}, $$where
$$\frac{1}{p}=\frac{j}{n}+\theta\biggl(\frac{1}{r}- \frac{m}{n}\biggr)+(1-\theta)\frac {1}{q},\quad\frac{j}{m} \leq\theta< 1, $$andCis a positive constant depending only onn, m, j, q, r, andθ.
-
(2)
For\(\gamma>0\)and\(s\in Z^{+}\), we can get a constantC (it only depends onγands) such that
$$\begin{gathered} \biggl\Vert \frac{\partial^{k}u}{\partial x^{k}} \biggr\Vert _{L^{\infty}}\leq C \Vert u \Vert _{L^{2}}+\gamma \biggl\Vert \frac{\partial^{s}u}{\partial x^{s}} \biggr\Vert _{L^{2}},\quad k< s, \\ \biggl\Vert \frac{\partial^{k}u}{\partial x^{k}} \biggr\Vert _{L^{2}}\leq C \Vert u \Vert _{L^{2}}+\gamma \biggl\Vert \frac{\partial^{s}u}{\partial x^{s}} \biggr\Vert _{L^{2}},\quad k\leq s.\end{gathered} $$
Lemma 2.4
Suppose that the conditions of Lemma 2.2are satisfied and\(0< p<4\). Then for the solution of problem (1.1)–(1.5), we have
Proof
From Lemmas 2.1–2.3 and Young’s inequality we get
and hence
By (2.6) it follows that
 □
Corollary 2.1
Suppose that the conditions of Lemma 2.4are satisfied. Then we have
Proof
By Lemmas 2.3 and 2.4, the result of Corollary 2.1 is obvious. □
Lemma 2.5
Suppose that the conditions of Lemma 2.4are satisfied, and assume that (1) \(\varepsilon_{0}\in H^{2}[0,L]\), \(v_{0}\in H^{1}[0,L]\), \(n_{0}\in H^{1}[0,L]\), and (2) \(\varphi(v)\in C^{1}(R)\), \(|\varphi'(v)|\leq C(|v|^{q}+1)\), \(0\leq q\leq2\). Then for the solution of problem (1.1)–(1.5), we have
Proof
Differentiating (1.1) with respect to t, we get
Taking the inner product of (2.7) and \(\varepsilon_{t}\), it follows that
Since
and hence from (2.8), (1.3), and Corollary 2.1 we get
Taking the inner product of (1.2) and \(-v_{xx}\), it follows that
Since
and hence from (2.10) we get
and thus by Gronwall’s inequality we obtain
By (1.1), Lemmas 2.1, 2.3, and 2.4, Corollary 2.1, Young’s inequality, and (2.12) we obtain
By (1.3), (2.12), and (2.13) we obtain
The Lemma 2.5 is proved. □
Corollary 2.2
Suppose that the conditions of Lemma 2.5are satisfied. Then we have
Proof
By Lemmas 2.3 and 2.5 the result of Corollary 2.2 is obvious. □
Lemma 2.6
Suppose that the conditions of Lemma 2.5are satisfied, and assume that (1) \(\varepsilon_{0}\in H^{3}[0,L]\), \(v_{0}\in H^{2}[0,L]\), \(n_{0}\in H^{2}[0,L]\), and (2) \(\varphi(v)\in C^{2}(R)\). Then for the solution of problem (1.1)–(1.5), we have
Proof
Taking the inner product of (2.7) and \(-\varepsilon_{txx}\), it follows that
Since
from (2.14) we get
Taking the inner product of (1.2) and \(v_{xxxx}\), it follows that
Since
from (2.16) we get
By (2.15) and (2.17) we obtain
and thus by Gronwall’s inequality we obtain
By (1.1), Young’s inequality, and (2.18) we obtain
By (1.3), (2.18), and (2.19) we obtain
By (1.2) we obtain
By (2.20) and (2.21) we obtain
 □
Corollary 2.3
Suppose that the conditions of Lemma 2.6are satisfied. Then we have
Proof
By Lemmas 2.3 and 2.6 the result of Corollary 2.3 is obvious. □
Lemma 2.7
Suppose that (1) \(\varepsilon_{0}\in H^{l+2}[0,L]\), \(v_{0}\in H^{l+1}[0,L]\), \(n_{0}\in H^{l+1}[0,L]\), \(l\in Z^{+}\), (2) \(\varphi(v)\in C^{l+1}(R)\), \(|\varphi'(v)|\leq C(|v|^{q}+1)\), \(0\leq q\leq2 \), and (3) \(0< p<4\). Then for the solution of problem (1.1)–(1.5), we have
Proof
We prove this lemma by mathematical induction. By Lemma 2.6 the lemma is true for \(l=1\). Suppose it is is true for \(l=k\) (\(k\geq 1\)), that is,
Next, we will show that the lemma is true for \(l=k+1\).
Taking the inner product of (1.2) and \((-1)^{k+2}v_{x^{2k+4}}\), it follows that
Since
from (2.22) we get
By Gronwall’s inequality we obtain
and by Eqs. (1.2) and (1.3) we get
Taking the inner product of (2.7) and \((-1)^{k+1}\varepsilon ^{2(k+1)}_{t}\), it follows that
Since
from (2.25) we get
By Gronwall’s inequality we obtain
and by Eq. (1.1) we get
By (2.23), (2.24), (2.26), and (2.27) we get
 □
3 The existence and uniqueness of global generalized solutions for problem (1.1)–(1.5)
Definition 1
The set of functions \(\varepsilon(x,t)\in L^{\infty}(0,T; H^{3}[0,L])\cap W^{1,\infty}(0,T; H^{1}[0,L])\), \(v(x,t)\in L^{\infty }(0,T; H^{2}[0,L])\cap L^{2}(0,T; H^{3}[0,L])\cap W^{1,\infty}(0,T; L^{2}[0,L])\), and \(n(x,t)\in L^{\infty}(0,T; H^{2}[0,L])\cap W^{1,\infty}(0,T; H^{1}[0,L])\) is called the generalized solution of problem (1.1)–(1.5) if for any \(\omega\in L^{2}[0,L]\), the functions satisfy
Theorem 3.1
Suppose that the conditions of Lemma 2.6are satisfied. Then there exists a global generalized solution of the initial boundary value problem (1.1)–(1.5),
Proof
By using the Galerkin method we choose a basis \(\{{\omega_{j}(x)}\} \subseteq H^{2}[0,L]\cap H_{0}^{1}[0,L]\) consisting of the eigenfunctions of the problem
Then the approximate solution of problem (1.1)–(1.4) can be written as
According to Galerkin’s method, the undetermined coefficients \(\alpha _{jm}(t)\), \(\beta_{jm}(t)\), and \(\gamma_{jm}(t)\) need to satisfy the following initial value problem of ordinary differential equations:
where
Similarly to the proof of Lemmas 2.1, 2.4, 2.5, and 2.6, for the solution \(\varepsilon_{m}(x,t)\), \(v_{m}(x,t)\), \(n_{m}(x,t)\) of problem (3.9)–(3.12), we can establish the following estimate:
where the constant C is independent of m. By compact argument we can choose a subsequence \(\varepsilon_{\nu}(x,t)\), \(v_{\nu}(x,t)\), \(n_{\nu }(x,t)\) such that, as \(\nu\rightarrow\infty\),
where \(Q_{T}=[0,L]\times[0,T]\). Hence, taking \(m=\nu\rightarrow\infty\) in (3.9)–(3.13), by using the density of \(\omega _{j}(x)\) in \(L^{2}[0,L]\) we get the existence of a local generalized solution for problem (1.1)–(1.5). From the conditions of the theorem and a priori estimates in Sect. 2 we can get the existence of a global generalized solution for problem (1.1)–(1.5) by the continuation extension principle. □
Theorem 3.2
Suppose that the conditions of Theorem 3.1are satisfied. Then the global generalized solution of the initial boundary value problem (1.1)–(1.5) is unique, and
Proof
Suppose that there are two solutions \(\varepsilon_{1}\), \(n_{1}\), \(\varphi _{1}\) and \(\varepsilon_{2}\), \(n_{2}\), \(\varphi_{2}\). Let
with initial data
and boundary conditions
Taking the inner product of (3.14) and ε, it follows that
Since
By the Lagrange mean value theorem we get
Therefore
Hence from (3.19) we get
Taking the inner product of (3.15) and v, it follows that
Since
from (3.21) we get
Since
by (3.20), (3.22), and (3.23) we get
Taking the inner product of (3.15) and \(-v_{xx}\), it follows that
Since
By (3.24) and (3.26) we obtain
By using Gronwall’s inequality we obtain
and hence
Therefore the proof of Theorem 3.2 is completed. □
4 The regularity of global generalized solution for problem (1.1)–(1.5)
To get the regularity of the global generalized solution for problem (1.1)–(1.5), we need the following lemma and corollary.
Lemma 4.1
Suppose that the conditions of Lemma 2.6are satisfied, and assume that (1) \(\varepsilon_{0}\in H^{4}[0,L]\), \(v_{0}\in H^{3}[0,L]\), \(n_{0}\in H^{3}[0,L]\), and (2) \(\varphi(v)\in C^{3}(R)\). Then for the solution of problem (1.1)–(1.5), we have
Proof
Taking the inner product of (2.7) and \(-\varepsilon_{tx^{4}}\), it follows that
Since
from (4.1) we get
Taking the inner product of (1.2) and \(v_{x^{6}}\), it follows that
Since
from (4.3) we get
and thus by using Gronwall’s inequality we obtain
By (1.1) and Young’s inequality we obtain
By (1.3), (4.5), and (4.6) we obtain
By (1.2) and Lemma 2.6 we obtain
 □
Corollary 4.1
Suppose that the conditions of Lemma 4.1are satisfied. Then we have
Proof
By Lemmas 2.3 and 4.1 the result of Corollary 4.1 is obvious. □
Theorem 4.1
Suppose that the conditions of Lemma 4.1are satisfied. Then there exists a unique global generalized solution of the initial boundary value problem (1.1)–(1.5), and
Proof
By using Theorem 3.1, Lemma 4.1, and Corollary 4.1 we can easily get this theorem. □
Theorem 4.2
Suppose that the conditions of Lemma 4.1are satisfied. Then there exists a unique global classical solution of the boundary value problem (1.1)–(1.5).
Proof
By using Theorem 4.1 and the embedding theorems of Sobolev spaces we can easily get this theorem. □
Theorem 4.3
Suppose that the conditions of Lemma 2.7are satisfied. Then there exists a unique global smooth solution of the initial boundary value problem (1.1)–(1.5), and
Proof
By Lemma 2.7 and the embedding theorems of Sobolev spaces the result of Theorem 4.3 is obvious. □
5 Approximation of solution
We now suppose that the generalized solution of initial boundary value problem (1.1)–(1.5) is approximated by the generalized solution of the following problem:
with initial data
and boundary conditions
where the parameters \(p>0\), \(\beta>0\), and α are real numbers, and \(\varphi(s)\) is a real function.
Letting \(F(t,x)=\varepsilon(t,x)-\eta(t,x)\), \(G(t,x)=v(t,x)-u(t,x)\), \(H(t,x)=n(t,x)-m(t,x)\), we obtain
with initial data
and boundary conditions
where the parameters \(p>0\), \(\beta>0\), α, and δ are real numbers, and \(\varphi(s)\) is a real function.
Lemma 5.1
Suppose that the conditions of Theorem 3.1are satisfied. Then for the solution of problem (5.6)–(5.10), we have
Proof
Taking the inner product of (5.6) and F, it follows that
Since
from (5.11) we get
Taking the inner product of (5.7) and G, it follows that
Since
from (5.13) we get
Since
by (5.12), (5.14), and (5.15) we get
Differentiating (5.6) with respect to t, we get
Taking the inner product of (5.17) and \(G_{t}\), it follows that
Since
from (5.18) we get
Taking the inner product of (5.7) and \(-G_{xx}\), it follows that
Since
Taking the inner product of (5.11) and \(F_{xx}\), it follows that
Since
from (5.22), we get
By (5.16), (5.19), (5.21), and (5.23) we obtain
By using Gronwall’s inequality we obtain
By (5.8) we get
By (5.24) and (5.25) we obtain
 □
Lemma 5.2
Suppose that the conditions of Theorem 3.1are satisfied. Then for the solution of problem (5.6)–(5.10), we have
Proof
Taking the inner product of (5.17) and \(-F_{txx}\), it follows that
Since
from (5.27) we get
Taking the inner product of (5.7) and \(G_{x^{4}}\), it follows that
Since
from (5.29) we get
By (5.28) and (5.30) we obtain
and thus by Gronwall’s inequality we obtain
By (5.8) we obtain
By (5.6), Young’s inequality, and (5.32) we obtain
By (1.3), (5.32), and (5.33) we obtain
By (5.7) we obtain
By (5.34) and (5.35) we obtain
 □
By Lemmas 5.1 and 5.2 we obtain the following:
Theorem 5.1
Suppose that the conditions of Theorem 3.1are satisfied. If\((\varepsilon, v, n)\)is the solution to problem (1.1)–(1.5) and\((\eta, u, m)\)is the solution to problem (5.1)–(5.5), then
whereCis a positive constant.
References
Zakharov, V.E.: Collapse of Langmuir waves. Sov. Phys. JETP 35, 908–914 (1972)
Davydov, A.S.: Solitons in molecular systems. Phys. Scr. 20, 387–394 (1979)
Stenflo, L.: Nonlinear equations for acoustic gravity waves. Phys. Scr. 33, 156–158 (1986)
Ma, S., Chang, Q.: Stranger attractors on pseudospectral solutions for dissipative Zakharov equations. Acta Math. Sci. 24B, 321–336 (2004)
Pecher, H.: An improved local well-posedness result for the one-dimensional Zakharov system. J. Math. Anal. Appl. 342, 1440–1454 (2008)
Guo, B.L., Zhang, J.J., Pu, X.K.: On the existence and uniqueness of smooth solution for a generalized Zakharov equation. J. Math. Anal. Appl. 365, 238–253 (2010)
Linares, F., Matheus, C.: Well-posedness for the 1D Zakharov–Rubenchik system. Adv. Differ. Equ. 14, 261–288 (2009)
Linares, F., Saut, J.C.: The Cauchy problem for the 3D Zakharov–Kuznetsov equation. Discrete Contin. Dyn. Syst., Ser. A 24, 547–565 (2009)
Guo, B.L.: On some problems for a wide class of systems of Zakharov equations. In: Proceedings of DD-3, Changchun Symposium, pp. 395–415 (1982)
Guo, B.L., Shen, L.J.: The existence and uniqueness of the classical solution on the periodic initial problem for Zakharov equation. Acta Math. Appl. Sin. 5, 310–324 (1982)
Guo, B.L., Gan, Z.H., Kong, L.H., Zhang, J.J.: The Zakharov System and Its Soliton Solutions. Springer, Singapore (2016)
Zheng, X.X., Shang, Y.D., Di, H.F.: The time-periodic solutions to the modified Zakharov equations with a quantum correction. Mediterr. J. Math. 14, Article ID 152 (2017)
Wang, X.Q., Shang, Y.D.: On the global existence and small dissipation limit for generalized dissipative Zakharov system. Math. Methods Appl. Sci. 41, 3718–3749 (2018)
Gajewski, H., Zacharias, K.: Über Näherungsverfahren zur Lösung der nichtlinearen Schrödinger-Gleichung mit selbstkonsistentem Potential. Math. Nachr. 89, 71–85 (1979)
You, S.J., Ning, X.Q.: On global smooth solution for generalized Zakharov equations. Comput. Math. Appl. 72, 64–75 (2016)
Acknowledgements
The authors are grateful to the referees for useful remarks and suggestions that greatly improved the presentation of this manuscript.
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This work is supported by the National Natural Science Foundation of China (Grant 11401223, 11871172), the National Natural Science Foundation of Guangdong (Grant 2015A030313424), and the Science and Technology Program of Guangzhou (Grant 201607010005).
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Wang, X., Shang, Y. & Lei, C. Initial boundary value problem for generalized Zakharov equations with nonlinear function terms. Bound Value Probl 2020, 85 (2020). https://doi.org/10.1186/s13661-020-01383-8
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DOI: https://doi.org/10.1186/s13661-020-01383-8