1 Introduction

By linearizing Maxwell's equations and using the ion and electric hydrodynamical approximations, Zakharov^[1] derived the Zakharov equations in 1972. The Zakharov equations play important roles in the strong turbulence theory for plasma waves, and have received much attention from many mathematicians and physicists.

By use of a quantum fluid approach, Garcia et al.^[2] obtained the following modified Zakharov equations in one dimension:

where H is the dimensionless quantum parameter given by the ratio of the ion plasmon energy to the electron thermal energy. The equations have been studied by many authors. Guo et al.^[3] studied the quantum Zakharov system, and obtained the global well-posedness. Fang et al.^[4] obtained the existence of the weak global solution for quantum Zakharov equations. Guo et al.^[5] studied the asymptotic behaviors of the solutions for dissipative quantum Zakharov equations. You and Guo^[6] studied the asymptotic behavior of the solutions.

By means of the usual two-time scale method applied to the fully three-dimensional (3D) quantum hydrodynamic model, the quantum Zakharov equations in three spatial dimensions is derived^[7]. In a non-dimensional form, the equations governing the electric field E and the number density n read

(1)

(2)

Equation (2) originates from the following hydrodynamic system:

(3)

which governs the ion sound waves. The parameter α is the square ratio of the light speed to the electron Fermi velocity, which is usually large. The coefficient Γ measures the influence of the quantum effects, and is usually very small^[8]. Some results of the mathematical aspects have been obtained. You^[9] established the existence of the generalized solution for this quantum Zakharov equation.

The aim of this paper is to study the existence of the global solution for the following initial value problem for the system (1) and (2) :

(5)

where

We eliminate the term ▽×(▽× E) in Eq. (1) by applying the divergence on it, i.e.,

(6)

Let E=▽φ. With the quantum Zakharov system (1) and (2) , we have

(7)

(8)

(9)

The initial data are

(10)

Now, we state the main results of the paper.

Theorem 1 Assume that , and , where m≥ 1. Then, there exists a unique global smooth solution for the problem (1) -(5) such that

For the sake of convenience, we introduce some notations. For 1 ≤q ≤∞, we denote the space of all qth power integrable functions in , equipped with norm or simply , and the Banach space with norm . If p=2, we write instead of . We use C to represent the various constants depending on the initial data. When Gronwall's inequality is used, the constants usually depend on the time, the initial data, and Γ in this manuscript. Otherwise, the constants are independent of the time.

The paper is organized as follows. In Section 2, we establish some a priori estimates. In Section 3, we prove the global existence and uniqueness of the solution for the problem (1) -(5) .

2 A priori estimates

The proof of Theorem 1 depends on the following lemma.

Lemma 1 Let . Then, for the solution for the problem (7) -(10) , we have

Proof Taking the inner product of Eq. (7) and φ and taking the imaginary part, we have

Therefore, we get

We thus get Lemma 1.

Lemma 2 (Gagliardo-Nirenberg inequlity^[10]) Assume that

we have

where C is a positive constant, , and

Lemma 3 Suppose that , and . Then, we have

Proof Taking the inner product of Eq. (7) and φ_t and taking the real part, we have

Note that

We have

Therefore,

(11)

where

Using the Hölder inequality, we have

Using the Gagliardo-Nirenberg inequality, we write

It follows that

(12)

Using Eqs. (11) and (12) , we get Lemma 3.

Lemma 4 Suppose that , and . Then, we have

Proof Taking the derivative of Eq. (8) with respect to t and taking the inner product of the obtained equation and n_t, we have

and

The above successive inequalities result from

Thus, we get

Thanks to Gronwall's inequality, we derive

(13)

Take the derivative of Eq. (7) with respect to t, and take the inner product of the obtained equation and φ_t. Then, taking the imaginary part, we have

Therefore,

With Gronwall's inequality, we get

(14)

Taking the inner product of Eq. (7) and △φ, we have

Therefore, we have

(15)

From Eqs. (13) -(15) , we get Lemma 4.

Lemma 5 Suppose that , and . Then, we have

Proof Take the derivative of Eq. (7) with respect to t, and take the inner product of the obtained equation and △φ_t. Then, taking the imaginary part, we have

The above successive inequalities result from

Therefore,

(16)

Take the derivative of Eq. (9) with respect to t, and take the inner product of the obtained equation and V_t. It follows that

Then, we have

(17)

Combining Eqs. (16) and (17) , we obtain

With Gronwall's inequality, we have

(18)

Taking the inner product of Eq. (9) and ▽³n, we have

Therefore, we have

(19)

Taking the inner product of Eq. (7) and △³φ, we have

Then, we obtain

(20)

From Eqs. (18) -(20) , we get Lemma 5.

Lemma 6 Suppose that , and m≥ 1. Then, we have

(21)

Proof The Lemma will be proven by the induction for m. Obviously, when m=1, Eq. (21) is true (see Lemma 5) . Now, suppose that Eq. (21) holds for m=k(k≥ 1) , i.e.,

We shall prove that Eq. (21) holds for m=k+1.

Take the derivative of Eq. (7) with respect to t, and take the inner product of the obtained equation and △^k+1φ_t. Then, taking the imaginary part, we have

Therefore,

(22)

Take the derivative of Eq. (9) with respect to t, and take the inner product of the obtained equation and △^k V_t. It follows that

Then, we have

(23)

Combining Eqs. (22) and (23) , we have

With Gronwall's inequality, we have

(24)

Taking the inner product of Eq. (9) and ▽^2k+3 n, we have

Then, we get

(25)

Taking the inner product of Eq. (7) and △^k+3φ, we have

Then, we get

(26)

From Eqs. (24) -(26) , we can see that Eq. (21) is true for m=k+1. The proof of Lemma 6 is completed.

3 Existence and uniqueness of solution

In this section, we formulate the proof of Theorem 1. First, we obtain the existence and uniqueness of the global generalized solution for the problem (7) -(10) .

Definition 1 The functions , and are called the generalized solution for the problem (7) -(10) if for any ω∈ L² they satisfy

with the initial data

Now, one can estimate the following theorem.

Theorem 2 Suppose that , and , where m≥ 1. Then, the problem (7) -(10) has a global smooth solution.

Proof With Galerkin's method, we choose the basic periodic functions {ω_κ(x)} as follows:

The approximate solution for the problem (7) -(10) can be written as follows:

where Ω is a 3D cube with 2L in each direction. According to Galerkin's method, these undetermined coefficients , and need to satisfy the following initial value problem of the system of ordinary differential equations:

(27)

(28)

(29)

with the initial data

(30)

where

Similar to the proofs of Lemmas 1-5, for the solution ϕ^l(x, t), n^l(x, t), , and V^l(x, t) for the problem (27) -(30) , we establish the following estimates:

(31)

where C is independent of l and L. By compact argument, some subsequence of (ϕ^l, n^l, V^l), also labeled by l, has a weak limit (φ, n, V). More precisely,

(32)

(33)

(34)

and

(35)

(36)

(37)

With Guo and Shen's method^[11], we can prove the existence of the local solution for the periodic initial problem (7) -(10) . Similar to Zhou and Guo's proof^[12], letting L→∞, the existence of the local solution for the initial value problem (7) -(10) can be obtained. With the continuation extension principle, from the conditions of the theorem and a priori estimates in Section 2, we can get the existence of the global generalized solution for the problem (7) -(10) . With Lemma 6 and the Sobolev imbedding theorem, we can conclude that Theorem 2 is proved.

Next, we prove the uniqueness of the solution for the problem (7) -(10) .

Theorem 3 Suppose that , and , where m≥ 1. Then, the global solution for the initial value problem (7) -(10) is unique.

Proof Suppose that there are two solutions E₁, n₁, V₁ and E₂, n₂, V₂. Denote

Then, E, n, and V satisfy the following system:

(38)

(39)

(40)

with the initial data

(41)

Taking the inner product of Eq. (38) and φ and taking the imaginary part, we have

Then, we have

(42)

Taking the inner product of Eq. (40) and V, we have

Then, we have

(43)

Take the derivative of Eq. (39) with respect to t, and then take the inner product of the obtained equation and n_t. It follows that

Then, we have

and

(44)

Taking the inner product of Eq. (38) and φ_t and taking the real part, we have

Then,

(45)

Take the derivative of Eq. (38) with respect to t, and take the inner product of the obtained equation and φ_t. Then, taking the imaginary part, we have

Therefore,

(46)

Combining Eqs. (42) -(46) , we have

With Gronwall's inequality and Eq. (41) , we have

Theorem 3 is proven. This completes the proof of Theorem 1.