A Note on C 2 Ill-posedness Results for the Zakharov System in Arbitrary Dimension

ABSTRACT This work is concerned with the Cauchy problem for a Zakharov system with initial data in Sobolev spaces H k ( ℝ d ) × H l ( ℝ d ) × H l - 1 ( ℝ d ) . We recall the well-posedness and ill-posedness results known to date and establish new ill-posedness results. We prove C 2 ill-posedness for some new indices ( k , l ) ∈ ℝ 2 . Moreover, our results are valid in arbitrary dimension. We believe that our detailed proofs are built on a methodical approach and can be adapted to obtain similar results for other systems and equations.


INTRODUCTION
This work is concerned with the Cauchy problem for the following Zakharov system (u, n, ∂ t n)| t=0 ∈ H k,l , where H k,l is a short notation for the Sobolev space H k (R d ; C)×H l (R d ; R)×H l−1 (R d ; R), (k, l) ∈ R 2 and ∆ is the laplacian operator for the spatial variable.
V. E. Zakharov introduced the system (Z) in [19] to describe the long wave Langmuir turbulence in a plasma. The function u represents the slowly varying envelope of the rapidly oscillating electric field and the function n denotes the deviation of the ion density from its mean value.
In this note we prove that, for any dimension d, the system (Z) is C 2 ill-posed in H k,l , for the indices (k, l) displayed in Figure 1 and Figure 2 (see Theorem 1.2 and Theorem 1.3 for the precise statements). The first C 2 ill-posedness result was proved by Tzvetkov in [18] for the KdV equation, improving the previous C 3 ill-posedness result of Bourgain found in [6]. We essentially follow the same ideas of [18], but our proofs are structured as in [9]. Two slightly different senses of C 2 ill-posedness are considered in our results (see also Remark 1). 1  Ginibre, Tsutsumi and Velo introduced in [11] a heuristic critical regularity for the system (Z), which is given by (k, l) = ( d/2 − 3/2 , d/2 − 2). In particular, our result in Theorem 1.2 with d = 3 (physical dimension) shows that the critical regularity (0, −1/2) is the endpoint for achieving well-posedness by fixed point procedure. We point out that local well-posedness at critical regularity is an open problem for d ≥ 3.
The system (Z) has been studied in several works. Bourgain and Colliander proved in [7] local well-posedness in the energy norm for d = 2, 3. They construct local solutions applying the contraction principle in X s,b spaces introduced in [5]. Local well-posedness in arbitrary dimension under weaker regularity assumptions was obtained in [11] by Ginibre, Tsutsumi and Velo. We recall the last result in the next theorem (see Figure 3). Theorem 1.1. (Ginibre, Tsutsumi and Velo [11]) Let d ≥ 1. The system (Z) is locally well-posed, provided (1.1) Now, we list the best results to date (as far as we know) for the system (Z).
For d = 2, Bejenaru, Herr, Holmer and Tataru in [2] proved l.w.p. for (k, l) = (0, −1/2) and Theorem 1.1 is the best result for the remaining indices k and l. Concerning ill-posedness, Theorem 1.2 (see Remark 1) and Theorem 1.3 are the best results.  For d ≥ 4, Kato and Tsugawa in [13] proved the global well-posedness of the Zakharov system for small data in the mixed inhomogeneous and homogeneous space . Global well-posedness for the Zakharov system is also studied in [16], [17], [8], [10], [15] and [1]. Now we start to state our results. First, we outline some definitions. Assume that the system (Z) is locally well-posed in the time interval [0, T ]. Then the solution mapping associated to the system (Z) is the following map Since Theorem 1.1 was obtained by means of contraction method, one can conclude the following: If (k, l) satisfies conditions (1.1) then for every fixed r > 0 there is a T = T (r, k, l) > 0 such that the solution mapping (1.2) is analitic (see Theorem. 3 in [3]). So, if the system (Z) is locally well-posed in H k,l and the solution mapping (1.2) fails to be m-times differentiable, then the usual contraction method can not be applied to prove the local well-posedness. In this case, we have a sense of ill-posedness and we say that the system (Z) is ill-posed by the method or simply Hereafter we call flow mapping associated to the system (Z) the following map We are now ready to enunciate our results. Our first theorem shows that, in any dimension, the regularity (k, l) = (0, −1/2) is the endpoint for achieving well-posedness by contraction method (see Figure 1).

Remark 1.
The sense of ill-posedness stated in Theorem 1.2 is slightly stronger than the sense stated in Theorem 1.3. Indeed, if the flow mapping (1.3) is not C 2 , neither is, a fortiori, the solution mapping (1.2). Thus, Theorem 1.2 slightly improves the ill-posedness results in [12] and [2], for d = 1 and d = 2, respectively, both establishing that the solution mapping (1.2) is not C 2 for l < −1/2 or l > 2k − 1/2 .
Remark 2. Theorem 1.3 establishes C 2 ill-posedness for new indices (k, l) (see Figure 2). For such indices, the difference of regularity between the initial data is large (i.e., l ≫ k or k ≫ l). Such result seems natural, due to coupling of the system via nonlinearities. Indeed, for instance, high regularity for u(t) is not expect when n(t) has low regularity, in view of (3.1). By the way, the C 2 ill-posedness for l < k − 2 is obtained by dealing with (3.1).

Remark 3.
In the periodic setting, Kishimoto proved in [14] the C 2 ill-posedness 3 of the These indices (k, l) are exactly the same of Theorems 1.2 and 1.3, excepting for admitting −1/2 ≤ l < 0. We point out that in [2] was proved, by means of contraction method, that the system (Z) is locally well-posed for d = 2, k = 0 and l = −1/2.
This paper is organized as follows. In Section 2, we introduce some notations to be used throughout the whole text. In Section 3, is presented a preliminary analysis which provides a methodical approach to our proofs, exposing the main ideas. In Section 4, we prove Theorem 1.2 and in Section 5, we prove Theorem 1.3.
• S (R d ) denotes the Schwartz space and S ′ (R d ) denotes the space of tempered distributions.
• f andf denote, respectively, the Fourier transform and the inverse Fourier transform of f ∈ S ′ (R d ).

PRELIMINARY ANALYSIS
The integral equations associated to the system (Z) with initial data (u, v, ∂ t n)| t=0 = (ϕ, ψ, φ ) are where {e it∆ } t∈R is the unitary group in H s (R d ) associated to the linear Schrödinger equation, given by e it∆ ϕ : Assume that the system (Z) is locally well-posed in H k,l , in the time interval [0, T ]. Suppose also that there exists t ∈ [0, T ] such that the flow mapping (1.3) is two times Fréchet differentiable at the origin in H k,l . Then, the second Fréchet derivative of S t at origin belongs to B, the normed space of bounded bilinear applications from H k,l × H k,l to H k,l . In particular, we have the following estimate for the second Gâteaux derivative of S t at origin  To overcome such difficulties, we made use of a sequence t N → 0, in consequence, we merely prove that estimate (3.5) is false, obtaining an ill-posedness result in a slightly weaker sense.
Remark 1. For the case l < −1/2 in Theorem 1.2, by a good choice of A N and B N , it is possible to obtain a "high + high = high" interaction in (3.10) providing "high" ⟨ξ ⟩ k ⟨ξ 1 ⟩ k ⟨ξ 2 ⟩ l , "low" σ + and "high" σ − , which yield good lower bounds for (3.12). But for the case k − l > 2 in Theorem 1.3, to obtain "high" ⟨ξ ⟩ k ⟨ξ 1 ⟩ k ⟨ξ 2 ⟩ l , the interaction must be of type "low + high = high", implying "high" σ + and "high" σ − , which do not provide lower bound for (3.12). Then we choose a sequence t N → 0, allowing us to obtain lower bounds directly from (3.11) L .

PROOF OF THEOREM ??
Assume that the solution mapping (1.2) is C 2 at the origin. Employing the same procedure that yields (3.11) from (3.4), one can conclude, from (3.5), the following estimate for bounded subsets A, B ⊂ R d