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International Journal of Partial Differential Equations
Volume 2013 (2013), Article ID 748761, 10 pages
http://dx.doi.org/10.1155/2013/748761
Research Article

An Initial Boundary Value Problem for the Zakharov Equation

1Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210008, China
2Department of Mathematics, Nanjing Xiaozhuang College, Nanjing 210008, China
3Department of Mathematics, Wellesley College, Wellesley, MA 02481, USA

Received 15 February 2013; Accepted 20 June 2013

Academic Editor: Nikolai A. Kudryashov

Copyright © 2013 Quankang Yang and Charles Bu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper studies an inhomogeneous initial boundary value problem for the one-dimensional Zakharov equation. Existence and uniqueness of the global strong solution are proved by Galerkin’s method and integral estimates.

1. Introduction

In this paper, we consider the following inhomogeneous initial boundary value problem for the Zakharov equations in one dimension:

Zakharov equations play an important role in the turbulence theory for plasma waves and resemble closely to the nonlinear Schrödinger equations. There has been extensive study both theoretically and numerically on these equations (e.g., see [117]). Most investigation have been focused on the Cauchy problem of the system of Zakharov equations (SZEs), sometimes with a homogeneous boundary condition. It is well known that Zakharov equations possess 1D soliton solutions. Some numerical experiments suggest that the solutions of 2D and 3D SZE may become singular in finite time [7, 18]. Global existence of solutions of the -dimensional SZE () has been only proved for small initial data [5, 6]. It has been shown that solutions with large initial data may blow up in finite time [8].

The system (1)-(2) above describes the interaction of a Langmuir wave and an ion acoustic wave in a plasma (Dendy [19] and Bellan [20]). Here, is an unknown complex vector-valued function that denotes a slowly varying envelope of a highly oscillatory electric field. Meanwhile, is an unknown real function that denotes the fluctuation in the ion density about its equilibrium value (see [19, 21]). We assume that , are given smooth functions.

Let be any function on with compact support satisfying , and we define Thus, the problem (1)–(4) is equivalent to

In Section 2, we obtain the existence of a local weak solution via Galerkin’s method and the principle of compactness. In Section 3, we derive estimates of higher-order derivatives of Galerkin’s approximate solution to obtain the existence and uniqueness of the local strong solution. In Section 3, we prove the existence and uniqueness of the global strong solution.

2. Existence of a Local Strong Solution

We first work on Galerkin’s approximation solution for the problem (6)–(9) by choosing the basic functions as follows:

The approximate solution for the problem (6)–(9) can be written as

According to Galerkin’s method, these undetermined coefficients must satisfy the following initial value problem for a system of ordinary differential equations: with , , and

To obtain existence of a local weak solution, we need the following lemmas.

Lemma 1. Assume that , then .

Proof. We multiply (13) by and sum up in to obtain Since , taing imaginary parts of (16) yields . Therefore, is a constant, and (16) is proved.
From Gagliardo-Nirenberg inequality [22], one has (if ) which will be used in the following estimates.

Lemma 2. Let , let , let , and let . Then, there is such that for any , where is a positive constant depending only on data and .

Proof. We multiply (13) by , multiply (14) by , and sum up in to get By taking the imaginary parts of (19), we get From (20), we have Combining (21) and (22), we find
We multiply (13) by and sum up to obtain Therefore, Differentiating (13) in , we have By taking the imaginary parts of (27), we obtain the following inequalities: From (23), we have Combining (28), (29), and (25), we get By Gagliardo-Nirenberg’s inequality with , we obtain the following estimates: From (25), (31), (32), and (33), we have with . Put the above inequalities into (30), and use Young’s inequality to obtain Hence, there exist such that for any . Here, only depends on data and but not .

Lemma 3. Under the conditions of Lemma 2, there exist a positive constant , such that for any , and is a positive constant that depends only on data and .

Proof. By Lemma 2 and (31), (34), and (35), we have

Lemma 4. Assume that , , , and , then there exist a positive constant such that for any , where is a positive constant that depends only on data and .

Proof. By differentiating (13) twice in , we have Multiply (40) by , and sum up in to obtain Taking the imaginary parts of (41), we have By Lemmas 2 and 3, we have Substituting (43) and (44) into (42), we obtain Differentiating (14) in , we have Multiplying (46) by and summing up in , we obtain That is where Differentiating (13) in , we have Multiplying (51) by and summing up the above formulas for form to , we obtain Taking the real parts of (52), we get Multiplying (26) by and summing up in , we obtain By the above lemmas, we have Substituting (55)–(58) into (54), we have Substituting (59) into (50), we get Substituting (49) and (60) into (48), we obtain
Combining (43), (45), (53), and (61), we have That is
Therefore, there exist a positive constant , such that for any , and is a positive constant that depends only on data and (not on ).
Multiplying (14) by and then summing up the above formulas for form to , we obtain That is where
This implies that By the above lemmas, we get Hence,
This completes the proof.

Lemma 5 (see [23, Lemma 4]). Let be Banach spaces. If the embedding is compact, then the following embeddings are also compact:(1),(2).

Theorem 6. Under the conditions of Lemma 4, then there exists a local strong solution for (6)–(9).

Proof. According to those estimates outlined in the above lemmas, we know that , , , and are all bounded uniformly in . By the principle of compactness, there exist subsequences of , , , and (we still denote these by the same letter) such that
Since , are uniformly bounded in , , we can choose subsequences of , (we still denote these by the same index) such that
Therefore, In fact, we have
Since
we get
On the other hand, since and for all , , we see that
As is dense in , this establishes the local strong solutions and completes the proof of Theorem 6.

3. Existence and Uniqueness of a Global Strong Solution

In the following, in order to obtain the global strong solution we will give a priori estimates for the solution which we get in Theorem 6.

Lemma 7. Assume that , then

Proof. Multiply both sides of (1) by , integrate the equation on (), and then take the real parts. It is easy to see that . This implies that is a conserved quantity in respect to time. Therefore, Lemma 7 is proved.

Lemma 8. Assume that , , , and ; then for , , one has where is a positive constant that depends only on data and .

Proof. From (7), we have Integrating (81) over , we obtain Taking the inner product of (6) and , we get Taking the real parts of (83), it follows that
Substituting (82) into (84), we have By (82), we get
Integrating by parts yields that
Combining (85), (86), and (87), we have
Integrating (88) over , we obtain where
Substituting (82) into (90), we have
Therefore, Since we have the following estimate after combining (89) and (92):
We may choose such that to obtain
By Groonwall’s lemma, we get for all , for any given .
This implies that
Taking the inner product of (6) and , we have
By (98), (99), and Lemma 2, we obtain
Taking the inner product of (7) and , we get
Differentiating (6) with respect to then taking the inner product of the result and , we have Taking the imaginary parts of (105), we obtain Combining (104) and (106), we get By Gronwall’s lemma, we obtain From (95), we get
This completes the proof of Lemma 8. So, we have the following theorem.

Theorem 9. The solution given by Theorem 6 exists for all time, that is, .

The proof of uniqueness is done via standard integral estimates, so, we omit it here.

Acknowledgments

This work is supported by Qing Lan and “333” Project of Jiangsu Province and the NSF of the Jiangsu Higher Education Committee of China (11KJA110001). It is supported by the Brachman Hoffman Small Grant and Wellesley College Faculty Award.

References

  1. G. Boling and S. Longjun, “The global existence and uniqueness of classical solutions of periodic initial boundary value problems of Zakharov equations,” Acta Mathematicae Applicatae Sinica, vol. 5, no. 2, pp. 310–324, 1982 (Chinese).
  2. G. Boling, “The initial boundary value problem for generalized Zakharov system,” Applied Mathematics-A Journal of Chinese Universities, vol. 9, no. 1, pp. 1–12, 1994 (Chinese).
  3. J. Colliander, “Wellposedness for Zakharov systems with generalized nonlinearity,” Journal of Differential Equations, vol. 148, no. 2, pp. 351–363, 1998. View at Scopus
  4. T. Ozawa, K. Tsutaya, and Y. Tsutsumi, “Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions,” Mathematische Annalen, vol. 313, no. 1, pp. 127–140, 1999. View at Scopus
  5. L. Yongsheng, “On the initial boundary value problem for two dimensional systems of Zakharov equations and of complex-Schrödinger-real-Boussinesq equations,” Journal of Partial Differential Equations, vol. 5, no. 2, pp. 81–93, 1992.
  6. C. E. Kenig, G. Ponce, and L. Vega, “On the zakharov and zakharov-schulman systems,” Journal of Functional Analysis, vol. 127, no. 1, pp. 204–234, 1995. View at Publisher · View at Google Scholar · View at Scopus
  7. G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Singular solutions of the Zakharov equations for Langmuir turbulence,” Physics of Fluids B, vol. 3, no. 4, pp. 969–980, 1991. View at Scopus
  8. L. Glangetas and F. Merle, “Existence of self-similar blow-up solutions for Zakhrov equation in dimension two. Part I,” Communications in Mathematical Physics, vol. 160, no. 1, pp. 173–215, 1994. View at Publisher · View at Google Scholar · View at Scopus
  9. G. Boling, “Initial boundary value problem for one class of system of multidimensional nonlinear Schrödinger-Boussinesq type equation,” Journal of Mathematical Research and Exposition, vol. 8, no. 1, pp. 61–71, 1988.
  10. G. Boling and S. Longjun, “The global solution of initial value problem for nonlinear Schrödinger-Boussinesq equation in 3-Dimensions,” Acta Mathematicae Applicatae Sinica, vol. 6, no. 1, pp. 11–21, 1990.
  11. Z. Zhou, “Spectral method for Zakharov equation with periodic boundary conditions,” Acta Mathematicae Applicatae Sinica, vol. 5, no. 3, pp. 279–288, 1989. View at Publisher · View at Google Scholar · View at Scopus
  12. G. Boling, “Global smooth solutions for the system of Zakharov equations in nonhonogeneous medium,” Northeastern Mathematical Journal, vol. 6, no. 4, pp. 379–390, 1990.
  13. G. Boling and Y. Guangwei, “Global smooth solution for the Klein-Gordon-Zakharov equations,” Journal of Mathematical Physics, vol. 36, no. 8, pp. 4119–4124, 1995. View at Scopus
  14. G. Boling, “On the initial boundary value problem for some more extensive Zakharov equations,” Journal of Mathematics, vol. 7, no. 3, pp. 269–275, 1987 (Chinese).
  15. L. Yongsheng and G. Boling, “Attractor of dissipative radially symmetric Zakharov equations outside a ball,” Acta Mathematicae Applicatae Sinica, vol. 27, pp. 803–818, 2004.
  16. G. Boling, “On global solution for a class of systems of multi-dimensional generalized Zakharov type equation,” Acta Mathematicae Applicatae Sinica, vol. 10, no. 4, pp. 419–433, 1994. View at Publisher · View at Google Scholar · View at Scopus
  17. V. Masselin, “A result on the blow-up rate for the Zakharov system in dimension 3,” SIAM Journal on Mathematical Analysis, vol. 33, no. 2, pp. 440–447, 2001. View at Scopus
  18. P. A. Robinson, D. L. Newman, and M. V. Goldman, “Three-dimensional strong langmuir turbulence and wave collapse,” Physical Review Letters, vol. 61, no. 6, pp. 702–705, 1988. View at Publisher · View at Google Scholar · View at Scopus
  19. R. O. Dendy, Plasma Dynamics, Oxford University Press, Oxford, UK, 1990.
  20. P. M. Bellan, Fundamentals of Plasmas Physics, Cambridge University Press, Cambridge, UK, 2006.
  21. V. E. Zakharov, “The collapse of langmuir waves,” Journal of Experimental and Theoretical Physics, vol. 35, pp. 908–914, 1972.
  22. L. Nirenberg, “On elliptic partial differential equations,” Annali della Scuola Normale Superiore di Pisa, vol. 13, pp. 115–162, 1959.
  23. J. Simon, “Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure,” SIAM Journal on Mathematical Analysis, vol. 21, no. 5, pp. 1093–1117, 1990.