Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D

This paper is concerned with the Cauchy problem of $2$D Klein-Gordon-Zakharov system with very low regularity initial data. We prove the bilinear estimates which are crucial to get the local in time well-posedness. The estimates are established by the Fourier restriction norm method. We utilize the bilinear Strichartz estimates and the nonlinear version of the classical Loomis-Whitney inequality which was applied to Zakharov system.


Introduction
We consider the Cauchy problem of the Klein-Gordon-Zakharov system: where u, n are real valued functions, 0 < c < 1. As a physical model, (1.1) describes the interaction of the Langmuir wave and the ion acoustic wave in a plasma. The condition 0 < c < 1, which plays an important role in the paper, comes from a physical phenomenon. See Bellan [4], Masmoudi and Nakanishi [13]. There are some works on the Cauchy problem of (1.1) in low regularity Sobolev spaces. For 3D, Ozawa, Tsutaya and Tsutsumi [14] proved that (1.1) is globally well-posed in the energy space In the case of c = 1, (1.1) is very similar to the Cauchy problem of the following quadratic derivative nonlinear wave equation. (

1.2)
For s > 0, the local well-posedness of (1.2) was obtained from the iteration argument by using the usual Strichartz estimates. As opposed to that, Lindblad showed that (1.2) is ill-posed for s ≤ 0, see [11]- [12].
In [14], Ozawa, Tsutaya and Tsutsumi showed that the difference between the propagation speeds of the two equations in (1.1) allows for a better result. That is, they applied the Fourier restriction norm method and obtained the local well-posedness of (1.1) in the energy space, and then by the energy conservation law, they extended solutions globally in time. By the similar argument, Tsugawa established that (1.1) is local well-posed in 2D for s ≥ −1/2. For 4 and higher dimensions, I. Kato [8] recently proved that (1.1) is locally well-posed at s = 1/4 when d = 4 and s = s c + 1/(d + 1) when d ≥ 5 where s c = d/2 − 2 is the critical exponent of (1.1). He also proved that if the initial data are radially symmetric then the small data globally well-posedness can be obtained at the scaling critical regularity for d ≥ 4. He utilized the U 2 , V 2 spaces introduced by Koch-Tataru [10]. We would like to emphasize that the above results hold under the condition 0 < c < 1. Our aim in this paper is to get the local well-posedness of (1.1) at very low regularity s in 2 dimensions. Hereafter we assume d = 2.
We make a comment on Theorem 1.1. Applying the iteration argument by the usual Strichartz estimates, we get the local well-posedness of (1.3) for −1/4 ≤ s. This suggests that if c = 1 the minimal regularity such that the well-posedness of (1.3) holds seems to be −1/4. Tsugawa [17] found that if we utilize the condition 0 < c < 1 in the same way as in [14] with minor modification, we can show that (1.3) is local well-posed only for s ≥ −1/2. We can say that the known arguments is not enough to get the well-posedness for s < −1/2 which is the most difficult case. To overcome this, we employ a new estimate which was introduced in [5], [3] and applied to the Zakharov system in [1] and [2]. See Proposition 4.4 below. The Zakharov system consists of a wave equation and a Schrödinger equation: We denote K ±,1 N,L by K ± N,L , the space X s, b ±, 1 by X s, b ± and its norm by · X s, b ± , and also X s, b ±, 1 by X s, b ± and its norm by · X s, b ± .
(2) It might be natural that we use τ ± c ξ instead of τ ± c|ξ| in the definition of K ±,c N,L . As was seen in [14], these two weights are equivalent and therefore X s, b ±,c does not depend on the choice of them in the definition of K ±,c N,L . Once Theorem 1.2 is verified, we can obtain Theorem 1.1 by the iteration argument given in [6] and many other papers. For example, see [9], [16]. Therefore we focus on the proof of Theorem 1.2 in this paper.
Next, we show the negative result for s < −3/4. Theorem 1.3. Let d = 2, 0 < c < 1 and s < − 3 4 . Then for any T > 0, the data-to-solution map (u 0 , u 1 , n 0 , n 1 ) → (u, n) of (1.1), as a map from the unit ball in Theorem 1.3 implies that the iteration argument, which is applied to the proof of Theorem 1.1, is no longer available for the case s < −3/4.
The paper is organized as follows. In Section 2, we introduce some fundamental estimates and property of the solution spaces as preliminary. In Section 3, we show (1.6) and (1.7) with ± 1 = ± 2 which is the easier case compared to ± 1 = ± 2 . In Section 4, we prove (1.6) and (1.7) with ± 1 = ± 2 , and complete the proof of Theorem 1.2. Lastly as Section 5, we show the negative result, Theorem 1.3.

Preliminaries
In this section, we introduce some estimates which will be utilized for the proof of Theorem 1.2. Throughout the paper, we use the following notations. A B means that there exists C > 0 such that A ≤ CB. Also, A ∼ B means A B and B A. It should be emphasized that the signs and ∼ frequently depend on 1 − c in the paper. Thus, the necessary condition of a time ingredient T in (1.1) to show Theorem 1.1 also depends on 1 − c. Since the aim of the paper is to show the local well-posedness, here we are not concerned with how the necessary condition of T for Theorem 1.1 changes as c approaches to 1. Let u = u(t, x). F t u, F x u denote the Fourier transform of u in time, space, respectively. F t, x u = u denotes the Fourier transform of u in space and time. We first observe that fundamental properties of X s, b ±, c . A simple calculation gives the following: for 0 < c ≤ 1 and s, b ∈ R. Next we define the angular decomposition of R 3 in frequency. For a dyadic number A ≥ 64 and an integer j ∈ [−A, A − 1], we define the sets {D A j } ⊂ R 3 as follows: For any function u : Lastly we introduce the useful two estimates which are called the bilinear Strichartz estimates. The first one holds true regardless of c. As opposed to that, the second one is given by using the condition 0 < c < 1. The first estimate is obtained by the same argument as in the proof of Theorem 2.1 in [15]. We omit the proof.
Proposition 2.2. Let 0 < c < 1. Then we have regardless of the choice of ± j .
Proof. Let A = 2 10 (1 − c) −1/2 . It follows from the finiteness of A and that we can replace g with χ D A j g in (2.2) for fixed j. After applying rotation in space, we may assume that j = 0. Also we can assume that there exists ξ ′ ∈ R 2 such that the support of χ D A j g is contained in the cylinder C N 012 min (ξ ′ ) := {(τ, ξ) ∈ R 3 | |ξ − ξ ′ | ≤ N 012 min }. We sketch the validity of the above assumption roughly. See [16] for more details. If N 2 ∼ N 012 min the above assumption is harmless obviously. Therefore we may assume that N 0 = N 012 min ≪ N 2 or N 1 = N 012 min ≪ N 2 . Since both are treated similarly, we here consider only the former case. Note that the condition N 0 ≪ N 2 means N 2 /2 ≤ N 1 ≤ 2N 2 , otherwise the left-hand side of (2.2) vanishes. We can choose the two sets where #k and #ℓ denote the numbers of k and ℓ, respectively. We see that for fixed k, independently of N 0 , N 1 , N 2 , there is only a finite number of ℓ which satisfy 4 and vice versa. This means that k and ℓ depend on each other. Once we obtain for fixed k, from Minkowski inequality and ℓ 2 almost orthogonality, we confirm ,τ , which verify the validity of the assumption. Hereafter, we call the above argument "ℓ 2 almost orthogonality".
We turn to the proof of (2.2).
In (1.6)-(1.7), replacing u and n with its complex conjugatesū andv respectively, we easily find that there is no difference between the case (± 0 , ± 1 , ± 2 ) and (∓ 0 , ∓ 1 , ∓ 2 ). Here ∓ j denotes a different sign to ± j . Therefore we assume ± 1 = − in (1.6)-(1.7) hereafter. By the dual argument and Plancherel theorem, we observe that where Similarly, (1.7) is verified by the following estimate.  where We now try to establish (3.1) and (3.2). First we assume that ± 2 = −. In this case, we can obtain (3.1) and (3.2) by using the bilinear estimates Propositions 2.1, 2.2 and the following estimate: , the following estimates hold:

5)
where Proof. For simplicity, we use f ±,c : Since the proof of (3.5) is analogous to that of (3.4), we establish only (3.4). From Lemma 3.1, it holds that L 012 max N 12 max . We decompose the proof into the three cases: First we consider the case (I). Considering that L 012 max N 12 max , we subdivide the cases further: (Ia) N 1 L 0 . We deduce from Hölder inequality and Proposition 2.1 that Similarly, from Hölder inequality and Proposition 2.1 we get For the case (II), we can show (3.4) in the same manner as above. We omit the proof. Lastly, we consider the case (III).
(IIIa) N 0 L 0 . We deduce from Hölder inequality and Proposition 2.1 that In this case, we need to utilize Proposition 2.2 instead of Proposition 2.1.
In this section, we establish (3.1) and (3.2) with ± 2 = +. Note that if one of the inequalities and we can verify (3.1) and (3.2) by the same proof as in the case ± 2 = −. To avoid redundancy, we omit the proof.
Then the following estimate holds; For the proof of the above proposition, we introduce the important estimate. See [2] for more general case.

Remark 3.
As was mentioned in [3], the condition of S * i (i) is used only to ensure the existence of a global representation of S i as a graph. In the proof of Proposition 4.3, the implicit function theorem and the other conditions may show the existence of such a graph. Thus we will not treat the condition (i) in the proof of Proposition 4.3.
Proof of Proposition 4.3. Let θ ± 0 ∈ (0, π) be defined as cos θ ± 0 = ±c. We divide the proof into the following two cases: π) is the smaller angle between ξ and ξ 1 . We here assume that A > 2 20 (1 − c) −2 . If A ≤ 2 20 (1 − c) −2 , the proposition is verified by the almost same proof as that for the case (II) below.
We first consider the case (I). The proof is very similar to that for ± 1 = ± 2 . We utilize the following two estimates.
First we decompose f by angular localization characteristic functions where A 1 is the minimal dyadic number which satisfies A 1 ≥ 2 20 (1 − c) −2 A and thickened circular localization characteristic functions where [s] denotes the maximal integer which is not greater than s ∈ R and S ξ 0 From the assumption (II), we see that the sum of (k, j) is ∼ A 3 4 . Therefore we only need to verify If we setc k = N −1 0 c k , inequality (4.7) reduces to Thus from the ℓ 2 almost orthogonality, we may assume that there exist ξ 0 1 , ξ 0 2 such that such that space variables of suppg 1 • φ + c1 and suppg 2 • φ − c2 are contained in the balls Bδ(ξ 0 1 ) and Bδ(ξ 0 2 ), respectively. By density and duality it suffices to show for continuousg 1 andg 2 that where S 1 , S 2 denote the following surfaces For any σ i ∈ S i , i = 1, 2, 3, there exist ξ 1 , ξ 2 , ξ such that ξ), and the unit normals n i on σ i are written as |ξ| . 13 We deduce from 1 |ξ| and (4.9) that the surfaces S 1 , S 2 , S ∓ 3 satisfy the following Hölder condition.
Next we show (III). Note that the unit normalsñ i onS i are written as follows.
(4.37) is equivalent to (4.38) Here we utilized the denotations f ±,c : For simplicity, we use where τ = τ 1 + τ 2 and ξ = ξ 1 + ξ 2 . By the decomposition of R 3 × R 3 where M 0 and M 1 are the minimal dyadic number which satisfies respectively we only need to show I(f ±,c , g −,A,j1 where g −,A,j1 and g +,A,j2 . We further simplify (I)-(IV). From ℓ 2 Cauchy-Schwarz inequality and L 012 max N 1 , it suffices to show that there exists 0 < ε ′ < 1 such that the following estimates hold; If −3/4 < s, (I) ′ is immediately established by using Proposition 4.7.
Next we prove (II) ′ . It follows from Proposition 4.7 that (III) ′ is verified as follows. By Lemma 4.2, we have N 0 L 012 max . For the sake of simplicity, we here consider the case of N 0 L 0 . The other cases can be proved similarly. We deduce from Proposition 2.1 and Hölder

20
This completes the proof of (IV) ′ .

Negative result
In this section, we establish Theorem 1.3. For convenience, we restate the Theorem 1.3.
Theorem 5.1. Let d = 2, 0 < c < 1 and s < − 3 4 . Then for any T > 0, the data-to-solution map (u 0 , u 1 , n 0 , n 1 ) → (u, n) of (1. Proof. By the same argument as in the proof of Theorem 1.4 in [7], it suffices to prove that for every C > 0, there exist real-valued functions u 0 ∈ H s+1 and n 0 ∈ H s such that  Thus we can choose 0 < t < T , N ≫ 1 such that the first term of (5.2) dominates the second term. This completes the proof of (5.1).