Multilinear Weighted Estimates and Quantum Zakharov System

We consider the compact case of one-dimensional quantum Zakharov system, as an initial-value problem with periodic boundary conditions. We apply the Bourgain norm method to show low regularity local well-posedness for a certain class of Sobolev exponents that are sharp up to the boundary, under the condition that Schr\"odinger Sobolev regularity is non-negative. Using the conservation law and energy method, we show global well-posedness for sufficiently regular initial data, without any smallness assumption. Lastly we show the semi-classical limit as $\epsilon \to 0$ on a compact time interval, whereas the quantum perturbation proves to be singular on an infinite time interval.


Introduction.
We consider the well-posedness and the semi-classical limit of the compact one-dimensional quantum Zakharov system (QZS). Thus we assume the periodic boundary condition , (u(x, 0), n(x, 0), ∂ t n(x, 0)) = (u 0 , n 0 , n 1 ) ∈ H s,l := H s (T) × H l (T) × H l−2 (T), (1.1) where u is complex-valued, n is real-valued, T = R 2πZ , T > 0 is the time-of-existence (to be determined), and α, β > 0, s, l ∈ R. When ǫ = 0, QZS is well-known as the classical Zakharov system (ZS), a pair of nonlinear PDE developed to model the interaction of Langmuir turbulence waves and ion-acoustic waves. Here u(x, t) denotes the slowly-varying envelope of electric field, and n(x, t) represents an ion-acoustic wave that models the density fluctuation of ions [19]. A thrust of interest in rigorously studying the quantum effects unexplained by ZS came from the physics community [7]. There the quantum effect is characterized by a fourth-order perturbation with a quantum parameter ǫ > 0 that is non-negligible when either the ion-plasma frequency is high or the electrons temperature is low; for more background in the physics of this model, see [10,15].
Our goal is to understand the effect of quantum perturbations, represented by the biharmonic operator. We will do this in the context of well-posedness theory, thereby extending results of [16]. We show that the biharmonic operator provides an extra degree of smoothing that nullifies the distinction between resonance ( β α ∈ Z) and non-resonance ( β α / ∈ Z), something which played a central role in [16]. More precisely, we show that the regions of Sobolev exponent pairs (s, l) ∈ R 2 yielding well-posedness for ǫ = 0 (which depend on β α ∈ Z or β α / ∈ Z), are no longer different when ǫ > 0. We apply the Bourgain norm method to show that if ZS is well-posed in a certain Sobolev space of initial data, then so is QZS. Under the condition s ≥ 0, we show that our application of Bourgain norm method yields a region of Sobolev exponents for the local well-posedness that is sharp up to the boundary. With the more precise statement given in Section 4, we state our main result. We define the region Ω L ⊆ R 2 by Ω L := {s ≥ 0, −1 ≤ l < 2s + 1, −2 < s − l ≤ 2} . Although the QZS model is relatively new, the method of multilinear weighted estimates via Fourier transform and the Cauchy-Schwarz inequality has been used successfully by many. These include (but are not limited to) Bourgain, Kenig-Ponce-Vega, and Ginibre-Tsutsumi-Velo [1,13,12,8], in applications to various dispersive equations such as KdV, nonlinear Schrödinger equation with various nonlinearities, and ZS on R d . Additionally, Tao [17] investigated an alternative approach based on orthogonality and dyadic decompositions.
Typically the task of proving boundedness for certain multilinear operators reduces to spacetime Lebesguetype estimates in Fourier space, which can be a challenge on periodic spatial domains where satisfactory Strichartz estimates are not available. Despite this difficulty, see [5,14,2] for various applications of Bourgain norm methods to ZS on periodic domains. On R d , as opposed to the compact case, it is generally expected that there is a wider range of Sobolev exponents for a well-posedness theory, with the full range of Strichartz estimates at one's disposal; for more recent work on QZS on R, see [6,3,11].
The QZS defines a Hamiltonian PDE with an energy functional H defined on H 2,1 ; see Section 2 for an explicit representation of this. We show, via the conservation law and an energy method, that the local flow obtained from Theorem 1.1 is global whenever initial data are sufficiently regular, with finite energy. Here the difficulty is proving persistence of regularity, given that any initial data with a finite energy has a global solution in H 2,1 , for which we derive an explicit growth rate of Sobolev norms. While our energy method for QZS provides an exponential bound on growth in time, see [5] for results on polynomial growth rates for the classical ZS on T.
We expect, however, that the above QZS local flow can be uniquely extended to a global flow, from scaling-invariance perspectives suggested in [8], and provide here a heuristic argument for this. Assuming for the moment that α = β = ǫ = 1, suppose the long-time behavior of the solution is governed by the simplified system (i∂ t − ∆ 2 )u = un, (∂ tt + ∆ 2 )n = ∆(|u| 2 ). (1.4) Assuming further that both n(x, t), ∂ t n(x, t) have mean zero for all t ∈ R, consider a change of variable N ± = n ± i∆ −1 ∂ t n, under which from the previous equation yields (1.5) If we add the assumption, as in [8], that the higher order biharmonic operator dominates the scaling property of N ± , we can neglect the ∓∆ in the N ± equation.
and hence the pair of critical Sobolev exponents is In the last part of the paper, we consider the semi-classical limit of QZS to ZS as ǫ → 0. Under the ǫ-perturbation, we expect the qualitative behavior of solutions to differ from that of the unperturbed system, and hence singular perturbation theory lies at the core of the analysis of QZS. As is well known, similar issues arise in the WKB method, multiscale analysis, and boundary layer theory; see [4] for an application of singular perturbation theory to ODE in the context of renormalization group and normal form method. Here we extend the results of Guo-Zhang-Guo [9] to show that the solutions behave continuously as ǫ → 0 on a compact time interval. Although their work is on R d and for integer Sobolev exponents, an analogue of their argument works on T as well, and extends to non-integer exponents. On the other hand, we provide a simple example that illustrates that the biharmonic operator ǫ 2 ∆ 2 , for any ǫ > 0, is a singular perturbation on an infinite time interval. Here we address a subtlety based on the fact that QZS generates a flow on H s,l whereas the classical ZS does so on H s,l 0 := H s (T) × H l (T) × H l−1 (T). To overcome this apparent discontinuity of solution space, we need to uniformly bound the solution in various norms, with bounds independent of ǫ > 0.
We briefly outline the organization of the paper. In Section 2, we introduce important notations and invoke the Lagrangian formalism of (1.1). In Section 3, we summarize a set of linear estimates that are used throughout the paper. In Section 4, nonlinear estimates are proved and applied to yield local well-posedness of (1.1); in particular, we prove the more precise statement of theorem 1.1. In Section 5, we extend local solutions to global solutions for a fixed ǫ > 0 and consider the ǫ → 0 problem. Throughout the paper, α, β > 0 are fixed and the adiabatic limit β → ∞ is not considered.

Background.
As is conventional, we first define Fourier transform of f ∈ L 2 (T) and the inverse transform of F ∈ l 2 (Z): We use x = (1 + x 2 ) 1 2 and define a family of Sobolev spaces W s,p ,Ẇ s,p (inhomogeneous and homogeneous, respectively) with s ∈ R, p ∈ (1, ∞) as and of particular importance is when p = 2 for which we write H s ,Ḣ s as is usual, and the norms are defined via Fourier multipliers: Whenever we take the direct sum of normed spaces, we will define the product norm to be the sum of the components, for instance, (u 0 , n 0 , n 1 ) H s,l = u 0 H s + n 0 H l + n 1 H l−2 .
As a consequence of the invariance of (1.1) under u(x, t) → e iθ u(x, t) and time-translation, mass and energy are conserved: We can assume that n 0 , n 1 have zero means. If n 0 , n 1 = 0, then we can consider the change of variable u(x, t); n(x, t) → n(x, t) − t 2π which can be directly checked to satisfy (1.1) with zero means in the new variable. By integrating the second equation of (1.1) over space, one obtains d 2 dt 2 T n = 0, and therefore the mean zero condition on n 0 , n 1 allows us to make sense of ∂ t n Ḣ−1 in the energy functional. We will use this idea extensively to obtain global solutions.
One expects a Hamiltonian system to have its Lagrangian counterpart via Legendre transform. Define where u is a complex field, u, the conjugate field of u, and ν, a real field where we impose n := ∂ x ν. The action functional S corresponding to L is defined in the usual way as follows: where ∂ µ denotes higher derivatives. To look for the critical points of S, we impose where we only require continuity in t. We wish to obtain a strong solution (u, n, ∂ t n) to (1.1) and by this we mean (u, n, ∂ t n) ∈ C([0, T ], H s,l ) for some T > 0 that satisfies the Duhamel's principle To obtain low-regularity well-posedness, we define the modified Bourgain norm adapted to the linear operators of interest. Take a complex-valued f ∈ C ∞ c (T × R) and define from which we define X s,b S and X l,b W as the closure of C ∞ c (T × R) with respect to the norms introduced above, respectively. We refer to expressions such as τ + αk 2 + ǫ 2 k 4 and |τ | − β|k| ǫk as dispersive weights. As in literature, we refer to these spaces as Bourgain spaces. As usual, f denotes the spacetime Fourier transform and whenever it is clear, we use f to denote either the spatial Fourier transform or the spacetime transform.
we are interested in the endpoint case b = 1 2 where the continuous embedding into C(R, H s ) fails. Motivated by the Fourier inversion theorem, we augment the norm and consider from which we can recover the desired continuous embedding, that is, and similarly for Y l W . To control the Duhamel term coming from the nonlinearities, we consider the companion spaces to Y s S , Y l W : To obtain solutions for small time, we further define the time-restricted space for T > 0 where such restriction for other Bourgain spaces can be defined analogously.
Though not necessary, we assume the perturbation parameter ǫ ≤ 1 to make the exposition clearer.
Proof. The first line of inequalities follows from the unitarity of Schrödinger operator; see [18,Lemma 2.8].
A similar argument can be used to show the other inequalities.
Proof. The first inequality is standard in literature; see [18,Proposition 2.12]. The second and third are proved similarly where The following estimates allow us to extract a (small) positive time factor, which is applied to obtain local well-posedness. Then Proof. The first inequality follows from [18,Lemma 2.11]. A similar argument can be used to show the second inequality.

Lemma 4.2.
For all e 1 > 1 4 , e 2 > 1 3 , proof of lemma 4.2. The second inequality can be proven in a similar way as to [5, lemma 3(c)]. For the first inequality, there exists c > 0 independent of k, k 1 such that Hence the term ǫ(k − k 1 ) in the summation can be replaced with |ǫ(k − k 1 )|. Then where the constant is independent of k, τ by an argument similar to [5, lemma 3(c)].
where the last inequality is by (4.6).
From (4.5), we have where If |k| ≫ 1, we argue as in lemma 4.4 to obtain from which, we estimate by lemma 4.2 and l ≥ −2.
In this region, the lower bound of the dispersive weight is similar to (4.23).
, the corresponding L ∞ L 1 estimate reduces to the current case by an appropriate change of variable. We modify the examples of spacetime functions given in [16] to give a converse statement for proposition 4.1. From the next result, it is deduced that s ≥ −1 + ρ 2 is necessary for proposition 4.1 to hold. It is of interest to find out whether proposition 4.1 holds for s ∈ [−1 + ρ 2 , 0) when ρ ∈ [0, 1]. The proof of the following proposition is presented in the appendix.
5 Global well-posedness and semi-classical limit.
Here our goal is to extend the local solutions obtained in the previous section. For simplicity, fix α = β = 1. While Guo-Zhang-Guo [9] used the energy method and a compactness argument to derive global wellposedness results on R d for d = 1, 2, 3 for initial data with integer Sobolev regularity, we extend their results to the compact domain T for all initial data in certain fractional Sobolev spaces with improved bounds. We show Theorem 5.1. If (u 0 , n 0 , n 1 ) ∈ Ω G , then the unique local solution obtained in theorem 4.1 can be extended to a global solution. More precisely, there exists (u, n, ∂ t n) ∈ C loc ([0, ∞), H s,l ) that satisfies (1.1) such that for all T > 0, (u, n, ∂ t n) is a unique solution in Y s S,T × Y l W,T × Y l−2 W,T .
To fully exploit the conservation law (2.4), we assume that n 0 , n 1 are of mean zero. Recall that both n and ∂ t n, assumed to be sufficiently regular, are of mean zero whenever they are defined, and thus H[u, n, ∂ t n](t) = H[u, n, ∂ t n](0) = H 0 < ∞. In fact, the nonlinear part of energy is bounded above by the linear part by Gagliardo-Nirenberg inequality. Once we establish a global solution for mean zero data, then we invoke the change of variable (2.5) to conclude that such global extension holds without the mean zero assumption.
It is crucial to obtain a uniform bound on (u ǫ , n ǫ , ∂ t n ǫ ) that depends only on R, T (see the next lemma), after which one can adopt the proof of [9, theorem 1.3] to prove remark 5.1. We observe that the argument in [9] in studying the ǫ → 0 problem on R d applies to T as well with certain subtleties, which we clarify in the concluding remarks. Now we state a useful Sobolev space inequality: which uniquely determines q ∈ [2, ∞), and therefore validates (5.3).