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The fourth order nonlinear Schrodinger limit for quantum Zakharov system

This paper is concerned with the quantum Zakharov system. We prove that when the ionic speed of sound goes to infinity, the solution to the fourth order Schrodinger part of the quantum Zakharov system converges to the solution to quantum modified nonlinear Schrodinger eqaution.

Let us review the local and global well-posedness for the classical Zakharov system (1.2). We will mention the Cauchy problem of (1.2) in R d , d = 1, 2, 3 only. For the high dimensional case and the multi-dimensional torus case, see [11] and [24], respectively.
For space dimensions d = 1, 2, 3, Guo-Zhang-Guo [14] have proved the global well-posedness of (1. 1) and k ≥ 2, respectively. Especially, they proved the global well-posedness of the quantum Zakharov system (1.1) in the energy space However it is interesting that the classical Zakharov system (1.2) has the blow up solution in the energy space which is [12,13]. As pointed out in [14], this difference is caused by the strong dispersion stems from the quantum effect, namely the quantum effect stabilizes the solution.
In this paper, we consider the convergence of the solution to the quantum Zakharov system (1.1) as λ → ∞. Before stating the main result for (1.1), we review the convergence of the solution to the classical Zakharov system (1.2) as λ → ∞.
Let us turn to the quantum system (1.1). In analogy with the classical system (1.2), taking λ → ∞ for the second equation in (1.1) and together with the assumption n ε,∞ + (1 − ε 2 ∆) −1 |E ε,∞ | 2 vanishes at space infinity, we obtain the relation Substituting this relation into the first equation in (1.1), we see that E ε,∞ satisfies the quantum modified nonlinear Schrödinger type equation (1.5) The main purpose of this paper is to prove that the solution E ε,λ to the quantum Zakharov system converges to the solution E ε,∞ to the fourth order nonlinear Schrödinger type equation (1.5). We introduce several function spaces and notations. For non-negative integers m, n, the Sobolev space H m and the weighted Sobolev space H m,n are defined by where the k-th derivative ∇ k is defined by if k is even.
We also define the following notations Following Kishimoto-Maeda [25], we define the operator ∇ −1 via the Helmholtz decomposition. The homogeneous Sobolev spaceḢ −σ is defined bẏ Let m and M be integers throughout the paper. For the sake of convenience, we define where X, Y , and Z are Sobolev spaces. From now on, we drop the parameter ε of the solutions E ε,λ and n ε,λ . We also denote the space, for d = 1, 2, 3, Our main results are as follows.
If we further assume that for any λ ∈ [1, ∞), where the constant C depends on ε and T . ε,λ is solution to the fourth order wave equation ε,λ represents the initial layer for (1.1). Note that Q (0) 0,λ coincides with the initial layer for the Zakharov system (1.2), see [1,34,30].
Let us give an outline of the proofs for Theorems 1.1, 1.2 and 1.3. The proofs follow from the arguments due to Ozawa-Tsutsumi [30] and Ukai [35] with several modifications.
From (1.1) and (1. (1.13) where U ε (t) = exp(it∆ ε ) is an L 2 -unitary group generated by the differential operator i∆ ε . It is easy to evaluate the first term in the integral of (1.13) in H m x . To estimate the second term in the integral of (1.13) in H m x , we derive the relevant equation for Q λ : which implies that Q λ (t) can be written as (1.15) The difficulty to evaluate the first term of (1.15) comes from the lack of the explicit representation for the unitary group cos(λt √ −∆ ε ). For the classical Zakharov system (1.2), the explicit formula is utilized for the unitary group cos(λt √ −∆) and the Schrödinger part is localized near the origin, see [30]. To overcome this difficulty, we employ the method of stationary phase.
The key point is that the wave part of (1.1) has no stationary point near the origin. Therefore the wave part of (1.1) decays faster than (λt) −1 as λ → ∞, which guarantees that the interaction between Q λ and E λ are weak in the sense that the first term in (1.15) decays like λ −1 in H m as λ → ∞.
The plan of this paper is as follows. In Section 2, we prove the solvability and uniform estimates for the solutions to (1.1) and (1.5). In Section 3, we prove Theorems 1.1, 1.2 and 1.3 and finally in Section 4, we derive the interaction estimate between Q λ and E λ .
Throughout this paper, we use the notation A ∼ B to represent C 1 A ≤ B ≤ C 2 A for some constants C 1 and C 2 . We also use the notation A B to denote A ≤ CB for some constant C.

Preliminaries
In this section we consider the solvability and uniform estimates for the solutions to (1.1) and (1.5). For the quantum Zakharov system (1.1), we have Furthermore, for any T ∈ (0, ∞) and some We need the following lemmas to show Proposition 2.1.
The next lemma is due to Brézis-Gallouët [5] which is needed to prove Proposition 2.1 for d = 2.
Lemma 2.2 (Brézis-Gallouët inequality). We have the following inequalities. ( To prove Proposition 2.1 for d = 3, we employ the Strichartz estimates for the Schrödinger equations and the fouth-order Schrödinger equations We recall some known results. ) be a solution of (2.2). For any biharmonic pairs (q, r) and ( q, r), it satisfies where C depends only on q ′ , and r ′ . Besides, for any Schrödinger admissible pairs (q, r) and (a, b), and any s ≥ 0, we have Proof of Proposition 2.1. We prove this proposition by the induction argument on M . For the simplicity, we abbreviate (E λ , n λ ) to (E, n). By the density We first derive the L 2 bound of E. Taking the imaginary part of the inner product in L 2 x between the first equation of (1.1) and E, we have for any t ∈ (0, ∞). Next we derive the H 2 × H 1 bound for (E, n). Taking the real part of the inner product in L 2 x between the first equation of (1.1) and Applying the operator ∇ −1 to the second equation of (1.1) and taking the inner product in L 2 x between the resultant equation and ∇ −1 ∂ t n, we have d dt (2.9) Let holds for some constant C independent of λ ∈ [1, ∞). Indeed, Lemma 2.1 and the Young inequality yield (2.11) Furthermore from (2.11), we find where the constant C is independent of λ ∈ [1, ∞). Combining (2.10), (2.12) and (2.13), we have for any t ∈ [0, ∞). This proves H 2 × H 1 bound for (E, n). Finally we derive that for any integers M ≥ 3, (2.14) Applying ∇ M −2 to the first equation of (1.1) and taking the real part of the inner product in L 2 x between the resulting equation and ∂ t ∇ M −2 E, we obtain Similarly, applying ∇ M −3 to the second equation of (1.1) and taking the inner product in L 2 x between the resulting equation and ∂ t ∇ M −3 n, we find d dt In fact, the positive constant C M is chosen in a way such that From (2.15) and (2.16), we obtain where For I 1 and I 3 , we use the Sobolev inequality and the induction hypothesis (2.14) to obtain An integration by parts leads Again an integration by parts yields Now we discuss the cases d = 1, 2, 3 separately.

The Gronwall lemma yields
We first prove that then we show that The Strichartz estimates (2.6) and (2.25) give that T. T. (2.30) Therefore we obtain (2.26) for E.
Next we estimate N . Notice that N satisfies To apply the Strichartz estimates to (2.31) we need the following proposition.
Proposition 2.2. Suppose that (q, r) and ( q, r) are Schrödinger admissible. If u is a solution of for some data u 0 , h and time 0 < T < ∞, then
The similar argument as given in (2.48), we have d dt This shows that the inequality (2.43) holds for (L, K) = (ℓ, k). Collecting those estimates we obtain the inequality (2.42).
Next we consider the fourth order nonlinear Schrödinger type equation (1.5). For the existence and the uniform bound of solution for (1.5), we have the following. Furthermore, for any T ∈ (0, ∞) and some C > 0, E ∞ satisfies Proof of Proposition 2.5. The existence and uniqueness of solution to (1.5) follows from the combination of the Strichartz estimate for the unitary group U ε (t) and the contraction mapping principle, see [6] for instance. The global existence of solution in H 2 follows from the conservations of mass and Hamiltoniañ and We first evaluate J 1 . Since the operator I ε is bounded from H m x to H m−2 x , we obtain Combining the above inequalities and (2.42), we have where the constant C is independent of λ ∈ [1, ∞).

Estimates for Q λ
In this section we prove Lemmas 3.1 -3.5. We first denote some notations which will be used throughout this section. Then we set the functions where To estimate the integrals Q (k) λ (t, x 0 ) for k = 0, 1, 2, we first compute the partial derivatives of ϕ ± over the variable ξ and we get, for d = 1, We note the oscillatory integrals Q and repeating the integration by parts twice, we have (4.5) For d = 2, 3, the phase function is ϕ ± = x 0 · ξ ± |ξ| 1 + ε 2 |ξ| 2 whose second derivative ∆φ ± ∼ ±λt (d − 1)|ξ| −1 ξ + |ξ| ξ −1 is singular at the origin. This would require that the initial data lies inḢ −2 which is more than that is required in the Hamiltonian (1.3). To avoid the difficulty, we modify the identity (4.4) as follows: .

(4.6)
Combining the identity (4.6) and repeating the integration by parts twice, we have 7) where the functions g k for k = 0, 1, 2, will be specified later, .
Combining the above estimates, for 0 ≤ k ≤ m, we get |∇ k Q