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Cauchy problem of the generalized Zakharov type system in \(\mathbf{R}^{2}\)
Journal of Inequalities and Applications volume 2017, Article number: 32 (2017)
Abstract
In this paper, we consider the initial value problem for a two-dimensional generalized Zakharov system with quantum effects. We prove the existence and uniqueness of global smooth solutions to the initial value problem in the Sobolev space through making a priori integral estimates and the Galerkin method.
1 Introduction
In the recent years, special interest has been devoted to quantum corrections to the Zakharov equations for Langmuir waves in a plasma [1]. By use of a quantum fluid approach, the following modified Zakharov equations are obtained:
where H is the dimensionless quantum parameter given by the ratio of the ion plasmon and electron thermal energies. For \(H=0\), this system was derived by Zakharov in [2] to model a Langmuir wave in plasma. The Zakharov system attracted many scientists’ wide interest and attention [3–14].
In this paper, we deal with the following generalized Zakharov system:
where \((E, n ): (x, t)\in\mathbf{R}^{2}\times\mathbf{R}\) and the initial data are taken to be
To study a smooth solution of the generalized Zakharov system, we transform it into the following form:
with initial data
Now we state the main results of the paper.
Theorem 1.1
Suppose that \(E_{0}(x)\in H^{l+4}(\mathbf{R}^{2})\), \(n_{0}(x)\in H^{l+2}(\mathbf{R}^{2})\), \(n_{1}(x)\in H^{l}(\mathbf{R}^{2})\), \(l\geq0\). Then there exists a unique global smooth solution of the initial value problem (3)-(5).
The obtained results may be useful for better understanding the nonlinear coupling between the ion-acoustic waves and the Langmuir waves in a two-dimensional space.
2 A priori estimates
Lemma 2.1
Suppose that \(E_{0}(x)\in L^{2}(\mathbf{R}^{2})\). Then, for the solution of problem (6)∼(9), we have
Proof
Taking the inner product of (6) and E, then taking the imaginary part, we have
Hence, we get
We thus get Lemma 2.1. □
Lemma 2.2
Sobolev’s estimations
Assume that \(u\in L^{q}(\mathbf{R}^{n})\), \(D^{m}u\in L^{r}(\mathbf{R}^{n})\), \(1\leq q,r\leq \infty, 0\leq j\leq m\), we have the estimations
where C is a positive constant, \(0\leq\frac{j}{m}\leq \alpha\leq1\),
Lemma 2.3
Suppose that \(E_{0}(x)\in H^{2}(\mathbf{R}^{2})\), \(n_{0}(x)\in H^{1}(\mathbf{R}^{2})\), \(\varphi_{0}(x)\in H^{1}(\mathbf{R}^{2})\). Then we have
where
Proof
Take the inner products of (6) and \(-E_{t}\). Since
thus it follows that
Letting
and noticing (10), we obtain
 □
Lemma 2.4
Suppose that \(E_{0}(x)\in H^{2}(\mathbf{R}^{2})\), \(n_{0}(x)\in H^{1}(\mathbf{R}^{2})\), \(\varphi_{0}(x)\in H^{1}(\mathbf{R}^{2})\). Then we have
Proof
By Hölder’s inequality, Young’s inequality and Lemma 2.2, it follows that
From Lemma 2.3 we get
Take the inner products of Eq. (8) and φ. It follows that
since
where
Hence, from Eq. (11) we get
Using Gronwall’s inequality, we obtain that
We thus get Lemma 2.4. □
Lemma 2.5
Suppose that \(E_{0}(x)\in H^{4}(\mathbf{R}^{2})\), \(n_{0}(x)\in H^{2}(\mathbf{R}^{2})\), \(\varphi_{0}(x)\in H^{2}(\mathbf{R}^{2})\). Then we have
Proof
Differentiating (6) with respect to t, then taking the inner products of the resulting equation and \(E_{t}\), we have
since
By Lemma 2.2, it follows that
thus from Eq. (12) we get
Differentiating Eq. (7) with respect to t, then taking the inner products of the resulting equation and \(n_{t}\), we have
since
Noting that
from Eq. (14) we get
By Gronwall’s inequality, it follows that
Take the inner products of Eq. (8) and \(\varphi_{t}\). It follows that
since
From Eq. (17) we get
Take the inner products of Eq. (6) and ΔE. It follows that
since
From Eq. (18) we get
From (6) we obtain
where
From (7) we obtain
We thus get Lemma 2.5. □
Lemma 2.6
Suppose that \(f_{1},f_{2}\in H^{s}(\Omega)\), \(\Omega\subseteq\mathbf{R}^{n}\). Then we have
where the constant \(C_{s}\) is independent of \(f_{1}\) and \(f_{2}\).
Lemma 2.7
Suppose that \(E_{0}(x)\in H^{m+4}(\mathbf{R}^{2})\), \(n_{0}(x)\in H^{m+2}(\mathbf{R}^{2})\), \(\varphi_{0}(x)\in H^{m+2}(\mathbf{R}^{2})\), \(m\geq0\). Then we have
Proof
Lemma 2.7 is true when \(m=0\) (Lemma 2.5). Suppose that Lemma 2.7 is true when \(m=k, (k\geq0)\). Take the inner products of (8) and \((-1)^{k+1}\Delta ^{k+3}\varphi\). It follows that
since
thus from Eq. (19) it follows that
By using Gronwall’s inequality, we have
Differentiating (6) with respect to t, then taking the inner products of the resulting equation and \((-1)^{k+1}\Delta ^{k+1}E_{t}\), we obtain
Since
thus from Eq. (21) we get
By using Gronwall’s inequality, we get
From (6) we obtain
Hence
This means Lemma 2.7 is true when \(m=k+1\). Thus Lemma 2.7 is proved completely. □
3 Existence and uniqueness of solution
Now, with these lemmas, we are able to prove Theorem 1.1. First we obtain the existence and uniqueness of the global generalized solution of problem (6)-(9).
Definition 3.1
The functions \(E\in L^{\infty}(0, T; H^{4})\cap W^{1,\infty}(0, T; L^{2})\), \(n\in L^{\infty}(0, T; H^{2})\cap W^{1,\infty}(0, T; L^{2})\) and \(\varphi \in L^{\infty}(0, T; H^{2})\cap W^{1,\infty}(0, T; L^{2})\) are called a generalized solution of problem (6)-(9) if for any \(\omega\in L^{2}\) they satisfy the integral equality
with initial data
Now, one can estimate the following theorem.
Theorem 3.1
Suppose that \(E_{0}(x)\in H^{l+4}(\mathbf{R}^{2})\), \(n_{0}(x)\in H^{l+2}(\mathbf{R}^{2})\), \(\varphi_{0}(x)\in H^{l+2}(\mathbf{R}^{2})\), \(l\geq0\). Then there exists a global smooth solution of the initial value problem (6)-(9).
Proof
By using the Galerkin method, choose the basic periodic functions \(\{\omega_{\kappa}(x)\}\) as follows:
The approximate solution of problem (6)-(9) can be written as
where
Ω is a two-dimensional cube with 2D in each direction, that is, \(\overline{\Omega}=\{x=(x_{1}, x_{2})\vert \vert x_{j}\vert \leq2D, j=1, 2\}\). According to Galerkin’s method, these undetermined coefficients \(\alpha^{l}_{\kappa}(t)\), \(\beta^{l}_{\kappa}(t)\) and \(\gamma^{l}_{\kappa }(t)\) need to satisfy the following initial value problem of the system of ordinary differential equations:
with initial data
where
Similarly to the proof of Lemmas 2.1-2.5, for the solution \(E^{l}(x,t)\), \(n^{l}(x,t)\), \(\varphi^{l}(x,t)\) of problem (23)-(26), we can establish the following estimates:
where the constants C are independent of l and D. By compact argument, some subsequence of \((E^{l}, n^{l}, \varphi^{l} )\), also labeled by l, has a weak limit \((E, n, \varphi )\). More precisely
and
By using Guo and Shen’s method [15], one can prove the existence of a local solution for the periodic initial problem (6)-(9). Similarly to Zhou and Guo’s proof [16], letting \(D\rightarrow\infty\), the existence of a local solution for the initial value problem (6)-(9) can be obtained. By the continuation extension principle, from the conditions of the theorem and a priori estimates in Section 2, we can get the existence of a global generalized solution for problem (6)-(9). By Lemma 2.7 and the Sobolev imbedding theorem, Theorem 3.1 is proved. □
Next, we prove the uniqueness of a solution for problem (6)-(9).
Theorem 3.2
Suppose that \(E_{0}(x)\in H^{l+4}(\mathbf{R}^{2})\), \(n_{0}(x)\in H^{l+2}(\mathbf{R}^{2})\), \(\varphi_{0}(x)\in H^{l+2}(\mathbf{R}^{2})\), \(l\geq0\). Then the global solution of the initial value problem (6)-(9) is unique.
Proof
Suppose that there are two solutions \(E_{1},n_{1},\varphi_{1}\) and \(E_{2},n_{2},\varphi_{2}\). Let
with initial data
Take the inner product of (27) and E. Since
thus we obtain
Take the inner product of (29) and φ. Since
thus we get
Hence from (31) and (32) we get
By using Gronwall’s inequality and noticing (30), we arrive at
Theorem 3.2 is proved. This completes the proof of Theorem 1.1. □
5 Conclusions
By a priori integral estimates and the Galerkin method, we have the following conclusion.
Suppose that \(E_{0}(x)\in H^{l+4}(\mathbf{R}^{2})\), \(n_{0}(x)\in H^{l+2}(\mathbf{R}^{2})\), \(n_{1}(x)\in H^{l}(\mathbf{R}^{2})\), \(l\geq0\). Then there exists a unique global smooth solution of the initial value problem (3)-(5).
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Acknowledgements
The authors would like to thank the National Natural Science Foundation of China (Grant No. 11501232), Research Foundation of Education Bureau of Hunan Province (Grant Nos. 15B185 and 16C1272) and Scientific Research Fund of Huaihua University (Grant No. HHUY2015-05) for the support.
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Authors’ contributions
SY carried out the existence studies and drafted the manuscript. XN carried out the uniqueness of the solution and helped to draft the manuscript. All authors read and approved the final manuscript.
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You, S., Ning, X. Cauchy problem of the generalized Zakharov type system in \(\mathbf{R}^{2}\) . J Inequal Appl 2017, 32 (2017). https://doi.org/10.1186/s13660-017-1306-2
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DOI: https://doi.org/10.1186/s13660-017-1306-2