On one dimensional Quantum Zakharov system

In this paper, we discuss the properties of one dimensional quantum Zakharov system which describes the nonlinear interaction between the quantum Langmuir and quantum ion-acoustic waves. The system with initial data $(E(0),n(0),\partial_t n(0))\in H^k\bigoplus H^l\bigoplus H^{l-2}$ is local well posedness in low regularity spaces. Especially, the low regularity result for $k$ satisfies $-3/4<k\leq -1/4$ is obtained by using the key observation that the convoluted phase function is convex and careful bilinear analysis. The result can not be obtained by using only Strichartz inequalities for"Schr\"{o}dinger"waves.


Introduction
The one-dimensional quantum Zakharov equations with initial conditions read i ∂E ∂t + ∂ 2 E ∂x 2 − ε 2 ∂ 4 E ∂x 4 = nE, (1.1a) ∂ 2 n ∂t 2 − ∂ 2 n ∂x 2 + ε 2 ∂ 4 n ∂x 4 = ∂ 2 |E| 2 ∂x 2 , (1.1b) E(0, x) = E 0 (x), n(0, x) = n 0 (x), ∂n ∂t (0, x) = n 1 (x), (1.1c) where the complex valued function E = E(t, x) is the envelope electric field and the real valued function n = n(t, x) is the plasma density fluctuation (measured from its equilibrium value). They are defined in R + t × R x . We assume E 0 ∈ H k (R), n 0 ∈ H l (R) and (−∆ + ε 2 ∆ 2 ) −1/2 n 1 ∈ H l (R) for the study of local well posedness. The dimensionless quantum parameter is the ratio between the ion plasmon energy and the electron thermal energy, where is Planck's constant divided by 2π, ω i is the ion plasma frequency, κ B is the Boltzmann constant and T e is the electron fluid temperature. The quantum Zakharov equations are obtained to describe the nonlinear interaction between highfrequency quantum Langmuir waves and the low-frequency quantum ion-acoustic waves [10,11]. The formal classical limit ε → 0 yields the original Zakharov equations: ∂n ∂t (0, x) = n 1 (x), (1.3c) which are one of the most important models in plasma physics [20,21]. It describe the interaction between high-frequency Langmuir waves and low-frequency ionacoustic wave. For the adiabatic limit of the Zakharov equations (1.3a)-(1.3b), one neglects the second order time derivative of the density fluctuation, ∂ 2 n ∂t 2 ≈ 0, then ∂ 2 ∂x 2 (n + |E| 2 ) = 0 which implies n = −|E| 2 and the resulting equation is the cubic nonlinear Schrödinger equation which is known to be completely integrable and is one of most important nonlinear partial differential equations. However, for the quantum Zakharov equations (1.1a)-(1.1b), the adiabatic limit will be − n + ε 2 ∂ 2 n ∂x 2 = |E| 2 , (1.5b) If we further take the limit ε → 0, the semiclassical limit, then n = −|E| 2 and the cubic nonlinear Schrödinger equation (1.4) will be recovered. Thus it is natural to consider (1.6) i ∂E ∂t as the quantum perturbation of the cubic nonlinear Schrödinger equation.
The main purpose of this paper is to study the local well-posedness of one dimensional quantum Zakharov system (1.1a)-(1.1b) with low regularity initial data. The local well-posedness of Cauchy problem for the Zakharov system in Euclidean space has been extensively studied. We do not intend to list all the references, one can see for example [16,6,8,4,7,3,2] and references therein. Unlike the Zakharov system, there are only few well-posedness results for the quantum Zakharov system. The current results are mainly focused on higher regularity spaces, for example [9]. In general, the well-posedness results of the Zakharov system as well as other dispersive equations with low regularity initial data can be established by the Strichartz inequalities. The key step is to derive non-linear estimates by extensive use of Strichartz inequalities which has its origin from Bourgain [5]. For one dimensional Zakharov system, Ginibre, Tsutsumi and Velo [8] established the local well-posedness result in low regularity spaces by adapting a method first proposed by Kenig, Ponce and Vega [14] to treat the Korteweg-de Vries equation which is a variant of Bourgain's method. Their method does not use Strichartz inequalities in the derivation of non-linear estimates, and relies instead on using Schwarz inequality cleverly followed by a direct estimation.
Similar to the method developed by Ginibre, Tsutumi and Velo [8] in studying one spatial dimensional Zakharov system, we combine the Strichartz and Schwarz inequalities to estimate the non-linear interactions. The challenge in the study of quantum Zakharov system is that the interactions of the non-linear part are much more complicated than that of the Zakharov system due to the appearance of the fourth order terms as well as the quantum parameter ε in (1.1a) and (1.1b). When two waves are close enough, their interactions can be treated by Strichartz inequalities as [8]. However, when two waves are away from each other, we have to use the Schwarz inequality instead. In [8], this part is not complicated and can be overcome by the change of variable argument. In our case, the bi-harmonic operator prohibits us to apply that method. The miracle here is that we can make use of the key observation that the convoluted phase function of "waves" is convex to get the quantitative estimates which describe the separation of waves through the bilinear analysis. It is worth noting that the lower regularity result for k satisfies −3/4 < k ≤ −1/4 can not be obtained by Strichartz inequalities.
These quantitative estimates allow us to get the well-posedness in low regularity spaces. We believe that this ingredient will be the key for studying the quantum Zakharov system in higher dimensional spaces and the other couple dispersive systems.
The main result of this paper is the following. Theorem 1.1. For any fixed ε ≤ 1, the quantum Zakharov system (1.1a)-(1.1b) with initial data (E 0 , n 0 , n 1 ) ∈ H k ⊕ H l ⊕ H l−2 is locally well posed provided (k, l) is in the set A defined by Also see Figure 1.
Remark 1.2. The condition ε ≤ 1 is for the convenience of discussion and the case ε → 0 is more interesting. The lowest regularity we obtained here is the pair (k, l) which is close to C = (−3/4, −3/4). It is not clear whether the pair (−3/4, −3/4) is optimal or not.
Remark 1.3. We can write the power 3 2 as 4−1 2 . Heuristically, 4 comes from fourth order term and 1 is necessary for non-linear estimates, while 2 in the denominator is due to us working on L 2 based spaces.
Remark 1.4. The dependence of time interval for well posedness on ε can be tracked explicitly as the C(ε) in lemma 3.5 and 3.7 are of order ε −1 , ε −2 respectively. They are from the estimates in section 4,5,6 where C(ε) are of order ε −1 in lemma 4.1, 4.2 and 4.3, of order ε −1/4 in lemma 5.2, 5.3, 6.1 and 6.2. Those orders can be checked in the proofs of lemmas and will not be emphasized later. The question of the singular limits of the Zakharov and related systems, the Klein-Gordon Zakharov system for example, has been studied extensively. Quite often, the limiting solution (when it exists) satisfies a completely different nonlinear partial differential equations. The nonlinear Schrödinger limit of the Zakharov system is one physical problem involving plasma frequency and ion sound speed effects where such a singular limit process is interesting. The earlier results are shown in [1,19] and the case when initial layer occurs was investigated by Ozawa and Tsutsumi in [17]. The readers are also referred to Masmaudi-Nakanishi [15] for a complete result where they were able to overcome the difficulty of the existence of a resonance frequency. The convergence of the quantum Zakharov system to the Zakharov system is also interesting, see [9] for the recent result. However, the convergence in the lower regularity developed in this paper is a challenging problem and it will be our main research project in the near future.
The rest of paper is organized as follows. In Section 2, we discuss the conservation laws and hydrodynamic limit (formally) of the quantum Zakharov system. In Section 3, we reduce the well-posedness of Cauchy problem of the quantum Zakharov system to three key estimates. These estimates will be proved in Section 4 by using the estimates built up in Sections 5 and 6. In Section 5, we build the estimate describing the interaction of two "fourth order wave". Finally, we prove the estimate describing the interaction of one "second order wave" and one "fourth order wave" in Section 6.
Notation The expression X Y means that X ≤ CY for some constant C depending on each occurrence. The notation X ≈ Y means that there exists two positive constants C 1 , C 2 such that C 1 Y ≤ X ≤ C 2 Y . We also use the bracket ξ = (1 + |ξ|) for the convenience. A constant C(b) means the constant C depends on b, and a constant C means that constant is a uniform constant. The notation B− appearing in section 4,5 and 6 means B − δ where δ is a small positive number which can be chosen arbitrarily close to zero.
We define the inner product in L 2 space by f (x), g(x) = f (x)g(x)dx. The Fourier transform and inverse Fourier transform are defined by respectively. The Sobolev norm is defined by Fourier transform as When we consider the time-space Fourier transform and its inverse, we use (t, x) to denote the time-space variables and (τ, ξ) to denote their Fourier counters.

Conservation law
The one-dimensional quantum Zakharov equations (1.1a)-(1.1b) is derivable from a variational principle [10], where the auxiliary variable u, satisfying ∂u ∂x = n, is introduced such that the density can be found. The variational derivative δS/δE * = δS/δE = 0 produce (1.1a) and its complex conjugate equation, respectively. But the equation for n is not straightforward, instead, taking the variational derivative δS/δu = 0 we have 4 = 0 which will reproduce (1.1b) after differentiation with respect to x. The Lagrangian formulation allows us to systematically derive conserved quantities by means of Noether's theorem, relating invariance, symmetries and conservation laws. The action (2.1) is trivially invariant under the phase transformation, i.e., gauge invariant, and thus quantum Zakharov equations admit the conservation law for the mass (or the number of high frequency quanta) Similarly, the action is invariant under time translation, and we have the conservation for the energy (or Hamiltonian) where V = −D −1 x n t . From (2.5), we see that if (E, n) ∈ H 2 ⊕ H 1 , one can control the energy H since | n|E| 2 dx| ≤ n H 1 E H 2 . Hence the local wellposedness result in Theorem 1.1 implies the global well-posedness of the Cauchy problem (1.1a)-(1.1b) in energy spaces (cf. [9]).
The conservative quantities are associated with the local conservation laws which can be obtained directly from the equations. We multiply (1.1a) by E * and its complex conjugate by E, and subtract the latter from the former to obtain The O(ǫ 2 ) term of (2.6) shows the quantum effect of the Langmuir wave. The conservation of mass comes from the imaginary part of (1.1a) and hence contains no contribution of n.
We multiply (1.1a) by ∂E * ∂x and its complex conjugate by ∂E ∂x , and then add them to obtain Next, we differentiate (1.1a) and its complex conjugate with respect to x, multiply the former by E * and the latter by E, and then add them together. This leads to the result If we subtract the above two equations and, then, simplify, it turns out that The momentum J is not conservative due to the coupling of the density |E| 2 and Multiply the second equation by 2A to obtain (2.14) which is equivalent to (2.6) by defining Formally letting ε → 0 in (2.6), (2.11) (or (2.12) after taking ∂ x ) and (1.3b) we have the hydrodynamical equations which are equivalent to the Zakharov equation (1.3a)-(1.3b) as long as the solutions are smooth.

Reduction
In order to solve equations (1.1a)-(1.1c), we first split n into positive and negative frequency parts according to can be transformed into the first order system First, we briefly review the Bourgain's method [5]. The presentation here is closely related to Ginibre, Tsutumi and Velo [8]. We want to solve the equation of the type where Φ is a real function (or a real symmetric matrix valued function) defined in R and f is a nonlinear function. In our case, u will be replaced by (E, n + , n − ) and Φ(ξ) will be a diagonal matrix with entries The Cauchy problem for (3.2) with initial data u(0) = u 0 is rewritten as the integral equation where U (t) = exp[−itΦ(−i∇)] is the unitary group that solves the linear equation and * R denotes the retarded convolution in time. In order to solve the Cauchy problem locally in time for some time interval [−T, T ], one introduces a time cut off in (3.3). Let β 1 ∈ C ∞ 0 be even, with 0 ≤ β 1 ≤ 1, β 1 (t) = 1 for −1 ≤ t ≤ 1 and β 1 (t) = 0 for |t| ≥ 2 and let β T = β 1 (t/T ) for 0 < T ≤ 1. Then one replace the integral equation (3.3) by the cut off equation since the solution of this equation is equal to the locally in time solution of (3.3).
The Banach spaces X = X s,b where to solve the equation (3.3) are defined as spaces of functions so that U (−t)u belongs to the Sobolev space H = H s,b , i.e.
By definition, we have And we have the estimate for β T (U * R f (u)) X s,b .
From above lemma we can build the local well posedness of the Cauchy problem in space X s,b by the standard contraction mapping argument if 1 − b + b ′ > 0 and non-linear estimates of the form hold for some power n depending on f . We also require b > 1 2 to ensure X s,b ⊂ C (R, H s ) and this completes the local well posedness in H s spaces. For more details, we refer the readers to [8] or [12] and references therein.
Within this framework, we can solve (3.1a)-(3.1c). Indeed, we define the operators where φ ε (ξ) = ξ 2 + ε 2 ξ 4 . The Cauchy problems for (3.1a)-(3.1c) can be solved by showing the mapping Ψ = (Ψ 0 , Ψ 1 ) is a contracting mapping in a suitable Banach space where Ψ 0 and Ψ 1 are defined respectively by Following (3.5), the norms E X k,b 1 and n X l,b ± we need are given by It is worth noting that the calculations of norms E X k,b 1 and n X l,b ± can be done by duality argument. Observe that E ∈ X k,b1 if and only if (1 + |ξ|) For our purpose, we will use the Strichartz inequality for the fourth order Schrödinger equation in our estimate. We define the non-homogeneous differentiation operator (3.9) The notation ξ ε α is called the symbol of differentiation operator D α ε . The Strichartz inequality we need is the following.
Similar to Lemma 2.3 of [8], we have the following emdedding which is implied by Strichartz inequality.
Returning to our main stream of discussion for the local well posedness of the Cauchy problem, we have to prove the following.
Proof. Here we sketch the proof. Recall that the set A is give by (1.7). The discussion before (3.7a) and (3.7b) indicates that we need the corresponding estimates of (3.6) in our setting, i.e., These non-linear estimates will be given by lemma 3.5 and lemma 3.7.
Remark 3.6. In order to get the optimal region of (k, l) in the above lemma as well as in the below lemma, we can for example take θ is a small positive number arbitrary close to zero. This choice also gives 1 > 0 which is need in Lemma 3.1 to ensure the power of T is positive. However we keep notation b, b ′ , b 1 , b ′ 1 in these two lemmas to track easily.
To estimate n ± E X k,−c 1 , we take its scalar product with a generic function E 1 in X −k,c1 , i.e.
We note that |ξ| ≤ 1 implies Hence we only have to show that The proof of (3.15) follows the same idea as Lemma 3.2 of [8] with modification. The slight difference here is that the Strichartz inequality of forth order Schrödinger equation has more regularity than that of Schrödinger equation. We should not need this fact for the proof of (3.15) but it will be used in the next lemma. Hence we write down the detail of the proof for the convenience of readers and for the later use.
Since S 1 is decreasing as a function of b, b 1 and c 1 , it is sufficient to prove the case b = 1 4 and b 1 = c 1 = 1 4 +. Then Therefore (3.15) is proved.
For simplicity of representation, let ζ = (τ, ξ) and ζ i = (τ i , ξ i ) and note ζ = ζ 1 −ζ 2 from (3.13). Now the integral S is of the from By the Schwarz inequality we have Hence, the proof is completed if we can show that C 1 (ε) is bounded which is done in Lemma 4.1.
Another non-linear estimate we need is the following.
, we take its scalar product with a generic function n ∈ X −l,c , i.e. Similarly As we should point out that the estimate remains unchanged if we put an absolute value sign to the integrand. We also note that Thus to estimate (3.22) is equal to estimate Therefore to prove (3.19) is equivalent to show In order to prove (3.24), we should discuss two cases, i.e. − 1 4 ≤ k and − 1 4 > k. Case 1.− 1 4 < k. There are two sub-cases; |ξ| ≤ 1 and |ξ| > 1. Case 1a. |ξ| ≤ 1.
The argument here is similar to case 1 in the previous lemma. It is clear that we only have to show This can be done by the same method as the case 1 of Lemma 3.5. Here we have The terms ξ i 1/4 in the first and third norms do not hurt. Since ξ i ≤ C(ε) (ξ i ) ε (recall (3.9) for notation ξ ε ) means (3.10) or (3.16) is equal to Case 1b. |ξ| > 1.
By applying argument as (3.17), we see that the proof of (3.24) is reduced to the boundedness of (3.28) The boundedness of C 2 (ε) is proved in Lemma 4.2.
To complete the proof, we have to prove that (3.24) holds for the missing part Figure 1) For this purpose, we consider the following. Integrating with respect to ζ 1 , ζ 2 instead of ζ 1 , ζ in (3.23) (Jacobian is 1) and applying argument as (3.17) yields For the convenience of further discussion, we relabel the variables and write them as In Lemma 4.3, we should prove that C 3 (ε) is bounded in the desired (k, l) region.
Remark 3.8. Note that condition k ≤ −1/4 is required in order to prove C 3 is bounded. (See the proof of case 1c. of Lemma 6.1.) Thus we have to discuss two cases when k < 0.

Proof of Theorem
In order to complete the proof of Theorem 1.1, we have to show that C 1 , C 2 and C 3 are bounded. In fact, we will prove C 1 is bounded in the following first lemma.
There exists a suitable θ > 0 and close enough to 0, such that if we let where Γ = ξ 2 + ε 2 ξ 4 and B = −b ′ 1 . Therefore, the problem is reduced to prove It is easy to see that for any τ we have where 3 2 − = 3 2 − δ and δ is a small positive number depending on θ. Therefore we conclude that (4.1) holds when 3 2 − ≥ |k| − l.
In the following lemma we will prove that C 2 (ε) is uniform bounded with respect to τ and ξ.
θ and apply Lemma 6.1 , then we have the following two cases for discussing with B = −b ′ .

4th order wave vs 4th order wave
In this section, we should discuss two lemmas which describe the non-linear interactions between two fourth order waves, i.e. τ 1 + ξ 4 1 and τ − τ 1 + (ξ 1 − ξ) 4 , when they are away each other (|ξ| > 1). For these two models, we are able to characterize the two waves interaction quantitatively. Lemma 5.2 characterizes one type of the interaction of two fourth order waves which is needed in the non-linear estimate of "Schrödinger part". On the other hand, Lemma 5.3 describes another type of interaction between two fourth order waves which is needed in the nonlinear estimate of "Wave part". In both lemmas we can see that the majority of interaction is again a fourth order wave Γ = ξ 2 + ε 2 ξ 4 .
We first introduce an elementary estimate.
Hence we are reduced to prove The key observation is that the following inequality Hence we are reduced to prove We rewrite the left hand side as From this representation, we see that it suffices to study the case when ξ is positive, that is ξ > 1. In the following we should discuss different cases according to the sign of τ . We note that it is sufficient to estimate the second integral due to symmetry with respect to τ . For simplicity of notation, we let f τ,ξ (η) = τ + 2(ξ + ε 2 ξ 3 )η + 4ε 2 ξη 3 .
Let R be the root of f τ,ξ (η) = 0. We consider the following decomposition We begin with the estimate of second integral. Observe that 1 + |f τ,ξ (R) + (ξ + ε 2 ξ 3 )r| < 1 + |f τ,ξ (R + r)| holds for 0 ≤ r ≤ ξ. Therefore Similarly for s ≥ 0, we have Now we turn to the estimate of the first integral of (5.6). Let P be the number such that P R = −τ . It is easy to see that By concavity of f τ,ξ (η), we have The last inequality follows from the fact that 0 < 1 − 2B < 1.
There are two sub-cases for discussing. In case |τ | ≤ Γ, we combine (5.8) and above to get In case |τ | > |Γ|, we use the fact Combining above inequalities and the relation P R = −τ , we have Hence the problem is reduced to prove Employing the inequality and estimate (5.3) in the Lemma 5.2, we can conclude (5.10).
Because Γ 1/4 ≤ 2ξ when ξ > 1, we can estimate the integral as From estimate (5.5), the first term of last line is bounded by We can estimate the last integral of (5.15) as case 2b. τ < 0. Let R be the root of f τ,ξ (η) = 0. It is easy to see We consider the following decomposition The numerator of integrand in the first two integrals in (5.18) is clearly bounded by max{R, ξ} −4k . From the estimates (5.6) and (5.7) we know these two integrals are bounded by Next, we note that for s ≥ 0 Hence, for the third integral of (5.18) we have Similar to (5.16), we have

2nd order wave vs 4th order wave
In this section we should prove lemma 6.1 which characterize the interaction of a second order wave τ 1 −τ ± φ ε (ξ 1 − ξ) and a fourth order wave τ 1 +φ ε (ξ 1 ) . The majority part of interaction possess the features of the second order wave as well as the fourth order wave. Compare to lemma 5.2 and 5.3, we see that its magnitude is of second order while the power of wave is determine by the fourth order wave.
With the same conditions but l ≥ 0 being replaced by l < 0 we have inside the bracket of second line of (6.1).

(6.2)
Proof. By Lemma 5.1, it suffices to prove that By the decomposition we can estimate the four integrals in the last line of above equation. ¿From the reduction above, we may assume that ξ ≥ 0. We can also see from the following that Λ + comes from the integral with τ + φ ε (ξ 1 − ξ) + φ ε (ξ 1 ) in the integrand while Λ − comes from the integral with τ − φ ε (ξ 1 + ξ) + φ ε (ξ 1 ) in the integrand. Thus we will use Λ instead of Λ + or Λ − in the following proof for the simplicity of notation.
Part I. First, we estimate the integral Before estimating the integral, please note that for l ≥ 0 , k < 0 and ξ 1 ≥ 0 , ξ ≥ 0, the inequality always holds. But this is not true when ξ 1 is close to ξ for l < 0 , k < 0 and ξ 1 ≥ 0 , ξ > 0. However we have for −1/2 < l < 0. This will be used to estimate (6.3). The last integral of (6.5) will be estimated in Lemma 6.2. Hence we only need to estimate the first integral in the right hand side of (6.5) where l could be negative or non-negative.
Equation (6.7) implies that the first derivative f ′ τ,ξ (ξ 1 ) is strictly increasing. Using this fact and f τ,ξ (A 2 ) = 0, one can see that each integral from the third to the last terms of (6.25) is smaller than the integration from A m to A 2 . The estimate of the first term of (6.25) is similar to the above. Let a 0 = A m and a i = 1 i+1 A m , i = 1, · · · , |Λ| − 1. From the convexity of f τ,ξ (ξ 1 ), we have Following the same arguments as before, we conclude that the estimate holds for this term.
In this case, we need more effort to locate the value of A m and m. However we do not need these information for our purpose. The boundedness of ξ is sufficient to get the desired result. By the discussion after (6.4) and (6.5), we only need to estimate since the last integral of (6.5) is bounded by C(ε).
With these information, we discuss the three cases τ +m ≥ m− φ ε (ξ), τ − φ ε (ξ) and φ ε (ξ) < τ < 2m − φ ε (ξ) respectively then we can prove the result by the same method as Case 1 of Part I. When 0 ≤ ξ ≤ 32ε −2 we can again get the desired result by the same method as Case 2 of Part I.
We omit the details since it is easy to follow the same method of Part I to obtain the desired result.
Finally we prove the supplemental lemma.