Research Article | Open Access
Quankang Yang, Charles Bu, "An Initial Boundary Value Problem for the Zakharov Equation", International Journal of Partial Differential Equations, vol. 2013, Article ID 748761, 10 pages, 2013. https://doi.org/10.1155/2013/748761
An Initial Boundary Value Problem for the Zakharov Equation
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.
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 [1–17]). 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 .
The system (1)-(2) above describes the interaction of a Langmuir wave and an ion acoustic wave in a plasma (Dendy  and Bellan ). 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.
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
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 , 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 .
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
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).
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
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