High frequency approximation of solutions to Klein–Gordon equation in Orlicz framework

Abstract

In this paper, we investigate the behavior of solutions to 2D Klein–Gordon equation in the framework of Orlicz norm. The analysis we conducted in this article, which is based on profiles decompositions, emphasizes the distinguished role played by the $1$ -oscillating component of the sequence of the Cauchy data. This phenomenon is strikingly different from those obtained in previous works, such as in Bahouri and Gérard [American Journal of Mathematics 121 (1999), 131–175] and Merle and Vega [International Mathematics Research Notices 8 (1998), 399–425].

Keywords

Free Klein–Gordon equation Olicz space profile decomposition

1. Introduction

1.1. Setting of the problem

In this paper, we investigate the feature of the solutions to the 2D free Klein–Gordon equation: $\begin{matrix} (KG) \{\begin{matrix} □ v + v = 0 in R \times R^{2} \\ {(v, \partial_{t} v)}_{| t = 0} = (ψ, g) \in H_{rad}^{1} (R^{2}) \times L_{rad}^{2} (R^{2}), \end{matrix} \end{matrix}$ where $□ = \partial_{t}^{2} - △_{x}$ denotes the wave operator.

Let us emphasize that the solutions v to $(KG)$ satisfy the conservation of energy, in the sense that for all times t, we have $\begin{array}{rcl} (1) & E (v, t) & = & {‖ v (t, \cdot) ‖}_{L^{2}}^{2} + {‖ \nabla v (t, \cdot) ‖}_{L^{2}}^{2} + {‖ \partial_{t} v (t, \cdot) ‖}_{L^{2}}^{2} = E_{0} (v) = E_{0} (ψ, g) . \end{array}$

Our analysis in this work will be conducted in the framework of the Orlicz space $\tilde{L} (R^{2})$ , namely the space of measurable functions $u : R^{2} \to C$ for which there is a positive real number λ such that $\begin{matrix} \int_{R^{2}} (e^{\frac{| u (x) |^{2}}{λ^{2}}} - 1 - \frac{| u (x) |^{2}}{λ^{2}}) d x < \infty . \end{matrix}$ This study is motivated by the results achieved in [6,9], where it was highlighted that the behavior of solutions to $(KG)$ in Orlicz framework plays a crucial role in the study of the Klein–Gordon equation with exponential nonlinearity.

Given a bounded sequence $(ψ_{n}, g_{n})$ in $H_{rad}^{1} (R^{2}) \times L_{rad}^{2} (R^{2})$ , we focused in this article on the description of the sequences of solutions to the free Klein–Gordon equation: $\begin{matrix} (KG) \{\begin{matrix} □ v_{n} + v_{n} = 0 \\ {(v_{n}, \partial_{t} v_{n})}_{| t = 0} = (ψ_{n}, g_{n}) \in H_{rad}^{1} (R^{2}) \times L_{rad}^{2} (R^{2}), \end{matrix} \end{matrix}$ up to a small remainder term in Orlicz norm, uniformly with respect to time. This description is carried out by means of profile decompositions.

Structure theorems originate in the elliptic frame in the works of Brézis and Coron in [11] and Struwe in [23], then in the framework of evolution equations in the articles of Bahouri and Gérard [5] and Merle and Vega [20]. These profile decomposition techniques have since been successfully used in various contexts, whether to study the possible blow-up of solutions to nonlinear partial differential equations or the global wellposedness or the asymptotic completeness. Among others, one can mention [10,12,16,17]. Note that in all the articles mentioned above, the qualitative study of the solutions was achieved with respects norms which are invariant under scaling, as for instance the Strichartz norms.

The novelty here is that the qualitative study of sequences of solutions to $(KG)$ is addressed with respect to the Orlicz norm, which as will be discussed in Section 1.2 does not enjoy scaling invariance property.

The approach we adopted in this article, which is based on profiles decompositions, highlights the distinguished role played by the $1$ -oscillating component1

¹
We designate by $1$ the scale in which all the terms are equal to the number 1.

of the sequence of the Cauchy data, according to the vocabulary of [14] (see also Definition 2.1 in this paper). This phenomenon is strikingly different from those obtained in previous works, where all the oscillating components play the same role, up to a change of scale (see for instance [5,20]). As we shall see in the sequel, two types of elements involve in our analysis. Actually, these elements have different behaviors in the frequency space: the first type generated by the

1

-oscillating component in spectrally localized while the second type produced by the unrelated component to the scale

1

(see Definition 2.1 in this paper) is spread in frequency. The major difficulty in this work has been to develop a strategy to extract elements of different kinds.

1.2. Orlicz spaces

In order to state our results in a clear way, let us start by introducing the so-called Orlicz spaces on $R^{d}$ and some related basic facts.

Definition 1.1.
Let $ϕ : R^{+} \to R^{+}$ be a convex increasing function such that $\begin{matrix} ϕ (0) = 0 = lim_{s \to 0^{+}} ϕ (s), lim_{s \to \infty} ϕ (s) = \infty . \end{matrix}$ We say that a measurable function $u : R^{d} \to C$ belongs to $L^{ϕ} (R^{d})$ if there exists $λ > 0$ such that $\begin{matrix} \int_{R^{d}} ϕ (\frac{| u (x) |}{λ}) d x < \infty . \end{matrix}$ We denote then $\begin{matrix} (2) & ‖ u ‖_{L^{ϕ} (R^{d})} = inf {λ > 0, \int_{R^{d}} ϕ (\frac{| u (x) |}{λ}) d x ⩽ 1} . \end{matrix}$

Orlicz spaces involve in our purpose by the following Sobolev embeddings $\begin{matrix} (3) & H^{1} (R^{2}) ↪ L^{ϕ_{p}} (R^{2}), \end{matrix}$ where $L^{ϕ_{p}} (R^{2})$ denotes the Orlicz space associated to the function $\begin{matrix} ϕ_{p} (s) = e^{s^{2}} - \sum_{k = 0}^{p - 1} \frac{s^{2 k}}{k!} . \end{matrix}$ These embeddings derive immediately from the following Trudinger–Moser type inequalities (see [1,21,22,24]): $\begin{matrix} (4) & sup_{‖ u ‖_{H^{1} (R^{2})} ⩽ 1} \int_{R^{2}} (e^{4 π | u |^{2}} - 1) d x : = κ < \infty . \end{matrix}$ In the sequel, we shall denote by $\tilde{L} (R^{2})$ the Orlicz space associated to the function $ϕ_{2} (s) = e^{s^{2}} - 1 - s^{2}$ endowed with the norm $‖ \cdot ‖_{\tilde{L} (R^{2})}$ where the number 1 in (2) is replaced by the constant κ defined in (4). Thus, the Sobolev embedding of $H^{1} (R^{2})$ into $\tilde{L} (R^{2})$ writes: $\begin{matrix} (5) & ‖ u ‖_{\tilde{L} (R^{2})} ⩽ \frac{1}{\sqrt{4 π}} ‖ u ‖_{H^{1} (R^{2})} . \end{matrix}$
Remark 1.2.
It is easy to check that the Orlicz norm $‖ \cdot ‖_{\tilde{L} (R^{2})}$ is invariant under translation and oscillation but not under scaling.

Since the works of Lions in [18,19], it is known that the Sobolev embedding $H^{1} (R^{2}) ↪ \tilde{L} (R^{2})$ is non compact at least for two reasons. The first reason is the lack of compactness at infinity. A typical example is given by $u_{n} (x) = φ (x + x_{n})$ , where $0 \neq φ \in D$ and $‖ x_{n} ‖ \to \infty$ . The second reason is of concentration-type as shown by the example by Moser (see [21]) defined as follows: $\begin{matrix} f_{α_{n}} (x) : = \sqrt{\frac{α_{n}}{2 π}} L (\frac{- log | x |}{α_{n}}), \end{matrix}$ where $\begin{matrix} L (t) = \{\begin{matrix} 0 & if t ⩽ 0, \\ t & if 0 ⩽ t ⩽ 1, \\ 1 & if t ⩾ 1, \end{matrix} \end{matrix}$ and ${(α_{n})}_{n \in N}$ is a sequence of positive real numbers tending to infinity. Let us emphasize that it was established in [2] that the sequences ${(u_{n})}_{n \in N}$ and ${(f_{α_{n}})}_{n \in N}$ defined above are different. With the vocabulary of Gérard in [14] (see also Definition 2.1 in this paper) the first one is $1$ -oscillating, while the second one in spread in frequency. This fact is behind the distinguished role played by the $1$ -oscillating component of the sequence of the Cauchy data.

It will be useful for our purpose to recall the characterization of the lack of compactness of the Sobolev embedding (5) in the radial framework (for more details, consult [6,9]). Let us then start by introducing some objects:
Definition 1.3.
We shall designate by scale any sequence $\underline{α} : = {(α_{n})}_{n \in N}$ of positive real numbers going to infinity, and we shall say that two scales $\underline{α}$ and $\underline{β}$ are orthogonal if $\begin{matrix} | log (\frac{β_{n}}{α_{n}}) | \overset{n \to \infty}{⟶} \infty . \end{matrix}$ We shall designate by profile any element of the set $\begin{matrix} P : = {ψ \in L^{2} (R, e^{- 2 s} d s); ψ^{'} \in L^{2} (R), ψ_{|] - \infty, 0]} = 0} . \end{matrix}$

The description of the lack of compactness of the Sobolev embedding of $H_{rad}^{1} (R^{2})$ into $\tilde{L} (R^{2})$ assumes the following form: Theorem 1.4.
Let ${(u_{n})}_{n \in N}$ be a bounded sequence in $H_{rad}^{1} (R^{2})$ such that $\begin{array}{l} (6) & u_{n} ⇀ 0 and \\ (7) & \underset{n \to \infty}{lim sup} ‖ u_{n} ‖_{\tilde{L} (R^{2})} = A_{0} > 0 . \end{array}$ Then, there exist a sequence ${({\underline{α}}^{(j)})}_{j ⩾ 1}$ of pairwise orthogonal scales in the sense of Definition 1.3 and a sequence of profiles ${(ψ^{(j)})}_{j ⩾ 1}$ in $P$ such that, up to a subsequence extraction, we have for all $ℓ ⩾ 1$ , $\begin{matrix} (8) & u_{n} (x) = \sum_{j = 1}^{ℓ} \sqrt{\frac{α_{n}^{(j)}}{2 π}} ψ^{(j)} (\frac{- log | x |}{α_{n}^{(j)}}) + r_{n}^{(ℓ)} (x), \underset{n \to \infty}{lim sup} {‖ r_{n}^{(ℓ)} ‖}_{\tilde{L} (R^{2})} \overset{ℓ \to \infty}{⟶} 0 . \end{matrix}$ Moreover, we have the following stability estimates $\begin{matrix} (9) & ‖ \nabla u_{n} ‖_{L^{2} (R^{2})}^{2} = \sum_{j = 1}^{ℓ} {‖ {ψ^{(j)}}^{'} ‖}_{L^{2} (R)}^{2} + {‖ \nabla r_{n}^{(ℓ)} ‖}_{L^{2} (R^{2})}^{2} + \circ (1), n \to \infty . \end{matrix}$
Remarks 1.5.

Let us note that the elements $\begin{matrix} w_{n}^{(j)} : = \sqrt{\frac{α_{n}^{(j)}}{2 π}} ψ^{(j)} (\frac{- log | \cdot |}{α_{n}^{(j)}}), \end{matrix}$ called generalization of the example by Moser, satisfy (see [6] for further details) $\begin{array}{l} (10) & {‖ \nabla w_{n}^{(j)} ‖}_{L^{2} (R^{2})} = {‖ {ψ^{(j)}}^{'} ‖}_{L^{2} (R)}, lim_{n \to \infty} {‖ w_{n}^{(j)} ‖}_{L^{2} (R^{2})} = 0 and \\ (11) & lim_{n \to \infty} {‖ w_{n}^{(j)} ‖}_{\tilde{L} (R^{2})} = \frac{1}{\sqrt{4 π}} max_{s > 0} \frac{| ψ^{(j)} (s) |}{\sqrt{s}} . \end{array}$

Let us also recall that it was established in [2] that the concentration elements $w_{n}^{(j)}$ are spread in frequency and satisfy $\begin{matrix} (12) & {‖ w_{n}^{(j)} ‖}_{{\dot{B}}_{2, \infty}^{1} (R^{2})} \overset{n \to \infty}{⟶} 0 . \end{matrix}$

1.3. Statement of the result

As mentioned above, our main goal in this paper is to describe the sequences solutions to the free Klein–Gordon equation, up to a small term in $L^{\infty} (R, \tilde{L} (R^{2}))$ . Our result states as follows:

Theorem 1.6.
Let ${(ψ_{n}, g_{n})}_{n ⩾ 0}$ be a bounded sequence in $H_{rad}^{1} (R^{2}) \times L_{rad}^{2} (R^{2})$ , and $v_{n}$ be the solution to the free Klein–Gordon equation $(KG)$ with $v_{n_{| t = 0}} = ψ_{n}$ , $\partial_{t} v_{n_{| t = 0}} = g_{n}$ . Then there exist a sequence ${(w^{(k)})}_{k ⩾ 0}$ of functions in $L_{rad}^{2} (R^{2})$ , a sequence ${({\tilde{w}}^{(k)})}_{k ⩾ 0}$ of functions in $H_{rad}^{1} (R^{2})$ , two sequences of profiles ${(ψ^{(j)})}_{j ⩾ 1}$ and ${({\tilde{ψ}}^{(j)})}_{j ⩾ 1}$ in $P$ , a sequence ${({\underline{α}}^{(j)})}_{j ⩾ 1}$ of scales in the sense of Definition 1.3 and two sequences ${({\underline{t}}^{(j)})}_{j ⩾ 1}$ and ${({\underline{\tilde{t}}}^{(k)})}_{k ⩾ 0}$ of reals sequences such that $\begin{matrix} (13) & \begin{matrix} \forall j \neq i, | log (α_{n}^{(j)} / α_{n}^{(i)}) | \overset{n \to \infty}{⟶} \infty or α_{n}^{(j)} = α_{n}^{(i)} and \\ - \frac{log | t_{n}^{(j)} - t_{n}^{(i)} |}{α_{n}^{(j)}} \overset{n \to \infty}{⟶} a \in [- \infty, + \infty [, \end{matrix} \end{matrix}$ with in the case when $a \in] 0, + \infty [$ , $(ψ^{(j)} + {\tilde{ψ}}^{(j)}) (s)$ or $(ψ^{(i)} + {\tilde{ψ}}^{(i)}) (s)$ null for $s < a$ , $\begin{matrix} (14) & for any k \neq m, | {\tilde{t}}_{n}^{(k)} - {\tilde{t}}_{n}^{(m)} | \to \infty, n \to \infty, \end{matrix}$ and, up to subsequence extraction, we have for all $ℓ ⩾ 1$ 2
²
We used the classical notation $⟨ D ⟩ = {(1 + | D |^{2})}^{\frac{1}{2}}$ .

$\begin{array}{rcl} v_{n} (t, \cdot) & = & v (t, \cdot) + \sum_{k = 0}^{ℓ} U (t - {\tilde{t}}_{n}^{(k)}) ({\tilde{w}}^{(k)}, w^{(k)}) \\ + \sum_{j = 1}^{ℓ} U (t - t_{n}^{(j)}) \sqrt{\frac{α_{n}^{(j)}}{2 π}} (ψ^{(j)} (\frac{- log | \cdot |}{α_{n}^{(j)}}), ⟨ D ⟩ {\tilde{ψ}}^{(j)} (\frac{- log | \cdot |}{α_{n}^{(j)}})) + r_{n}^{(ℓ)} (t, \cdot), \end{array}$ where v is the weak limit of the sequence $(v_{n})$ , $U (t)$ denotes the unitary group associated to the free Klein–Gordon equation, and with ${lim sup}_{n \to \infty} ‖ r_{n}^{(ℓ)} ‖_{L^{\infty} (R, \tilde{L} (R^{2}))} \overset{ℓ \to \infty}{⟶} 0$ . Moreover, we have the following stability estimates as n tends to infinity: $\begin{array}{rcl} E_{0} (v_{n}) & = & E_{0} (v) + \sum_{k = 0}^{ℓ} ({‖ {\tilde{w}}^{(k)} ‖}_{H^{1} (R^{2})}^{2} + {‖ w^{(k)} ‖}_{L^{2} (R^{2})}^{2}) \\ (15) & + \sum_{j = 1}^{ℓ} ({‖ {ψ^{(j)}}^{'} ‖}_{L^{2} (R)}^{2} + {‖ {\tilde{ψ}}^{{(j)}^{'}} ‖}_{L^{2} (R)}^{2}) + E_{0} (r_{n}^{(ℓ)}) + \circ (1) . \end{array}$
Remarks 1.7.

All along this paper, we shall note under the above notations: $\begin{array}{l} (16) & F_{n}^{(j)} (t, \cdot) : = U (t - t_{n}^{(j)}) \sqrt{\frac{α_{n}^{(j)}}{2 π}} (ψ^{(j)} (\frac{- log | \cdot |}{α_{n}^{(j)}}), ⟨ D ⟩ {\tilde{ψ}}^{(j)} (\frac{- log | \cdot |}{α_{n}^{(j)}})) and \\ (17) & G_{n}^{(k)} (t, \cdot) : = U (t - {\tilde{t}}_{n}^{(k)}) ({\tilde{w}}^{(k)}, w^{(k)}) . \end{array}$

Our result is about the general structure of sequences of solutions to the free Klein–Gordon equation. Roughly speaking, Theorem 1.6 asserts that, under the boundedness of the Cauchy data in $H_{rad}^{1} (R^{2}) \times L_{rad}^{2} (R^{2})$ , any such sequence is the sum of four terms: its weak limit v, a superposition of the concentrating waves $F_{n}^{(j)}$ , a superposition of the concentrating waves $G_{n}^{(k)}$ and a small remainder in $L^{\infty} (R, \tilde{L} (R^{2}))$ .

Clearly $\begin{array}{l} {‖ F_{n}^{(j)} ‖}_{L^{\infty} (R, \tilde{L} (R^{2}))} ⩾ \frac{1}{\sqrt{4 π}} max_{s > 0} \frac{| ψ^{(j)} (s) |}{\sqrt{s}} and \\ {‖ G_{n}^{(k)} ‖}_{L^{\infty} (R, \tilde{L} (R^{2}))} ⩾ {‖ {\tilde{w}}^{(k)} ‖}_{\tilde{L} (R^{2})} . \end{array}$

The elements $F_{n}^{(j)}$ and $G_{n}^{(k)}$ have different behaviors in frequency space. According to Remarks 1.5, $F_{n}^{(j)}$ is spread in frequency, while it is easy to see that the concentration elements $G_{n}^{(k)}$ are asymptotically concentrated in frequency in rings of size 1.

The orthogonality hypothesis (13) asserts that the interaction between the elementary concentrations $F_{n}^{(j)}$ is negligible in the energy space. Roughly speaking, condition (13) requires in the case where the concentrating times $t_{n}^{(i)}$ and $t_{n}^{(j)}$ are not sufficiently distant, namely in the case where $\frac{- log | t_{n}^{(i)} - t_{n}^{(j)} |}{α_{n}^{(j)}} \overset{n \to \infty}{⟶} a > 0$ , the vanishing of $(ψ^{(j)} + {\tilde{ψ}}^{(j)}) (s)$ or $(ψ^{(i)} + {\tilde{ψ}}^{(i)}) (s)$ for $s < a$ .

In this paper, we restrict ourselves to radially symmetric solutions. Our approach here uses in a crucial way the radial setting and particularly the compactness of the embedding $H_{rad}^{1} (R^{2})$ into $L^{4} (R^{2})$ as well as the fact that we deal with bounded functions far away from the origin. The study of the general frame is much harder as shown by the articles [7] and [8] respectively making use of capacity arguments and log-oscillating notion.

The case of the wave equation has already been investigated in [5] in the framework of Strichartz norm. Our result is in line with the paper of Bahouri-Majdoub-Masmoudi [6] where the Orlicz norm plays a decisive role. In [6] the feature of the nonlinear Klein-Gordon equation with exponential nonlinearity was investigated by adopting the approach of P. Gérard in [13] which consists in comparing the evolution of oscillations and concentration effects displayed by sequences of solutions of the nonlinear Klein-Gordon equation and solutions of the linear Klein-Gordon equation.

Note finally that, in view of the dispersion effect for the Klein–Gordon equation, the main contribution in Orlicz norm for the concentration waves $F_{n}^{(j)}$ and $G_{n}^{(k)}$ occurs around the concentrating times.

1.4. Layout of the paper

The paper is organized as follows: in Section 2, we introduce the background materiel that will be needed in the proof of our results, namely the notions of being oscillating with respect to a scale and of being unrelated to any scale. The effective construction of the decomposition is addressed in Section 3. Finally, we mention that the letter C will be used to denote a constant which may vary from line to line. We also use $A ≲ B$ to denote an estimate of the form $A ⩽ C B$ . For simplicity, we shall also still denote by $(u_{n})$ any subsequence of $(u_{n})$ and designate by $\circ (1)$ any sequence which tends to 0 as n goes to infinity.

2. Oscillatory components of bounded sequences in $L^{2} (R^{d})$

In this section, we introduce the notions of being oscillating with respect to a scale and of being unrelated to any scale, and some results that will be used to demonstrate Theorem 1.6 (see [14], for further details).

Definition 2.1.
Let $f : = {(f_{n})}_{n ⩾ 0}$ be a bounded sequence in $L^{2} (R^{d})$ and $\underline{h} : = {(h_{n})}_{n ⩾ 0}$ be a sequence of positive real numbers3
³
Where $\hat{u}$ denotes the Fourier transform of u defined by $\hat{u} (ξ) = \int_{R^{2}} e^{- i x \cdot ξ} u (x) d x$ .

The sequence f is said $\underline{h}$ -oscillating if $\begin{matrix} (18) & \underset{n \to \infty}{lim sup} (\int_{h_{n} | ξ | ⩽ \frac{1}{R}} {| \hat{f_{n}} (ξ) |}^{2} d ξ + \int_{h_{n} | ξ | ⩾ R} {| \hat{f_{n}} (ξ) |}^{2} d ξ) \overset{R \to \infty}{⟶} 0 . \end{matrix}$

The sequence f is said unrelated to the scale $\underline{h}$ if for any reals $b > a > 0$ $\begin{matrix} (19) & \int_{a ⩽ h_{n} | ξ | ⩽ b} {| \hat{f_{n}} (ξ) |}^{2} d ξ \overset{n \to \infty}{⟶} 0 . \end{matrix}$

The first result (Proposition 2.5 in [14]) that we shall recall here concerns the decomposition of an arbitrary sequence with respect to a given scale:
Proposition 2.2.
Let $\underline{h} = {(h_{n})}_{n ⩾ 0}$ be a sequence of positive real numbers and ${(f_{n})}_{n ⩾ 0}$ be a bounded sequence in $L^{2} (R^{d})$ . Then, up to a subsequence extraction, there is a bounded sequence ${(g_{n})}_{n ⩾ 0}$ in $L^{2} (R^{d})$ such that
${(g_{n})}_{n ⩾ 0}$ is $\underline{h}$ -oscillating;

${(f_{n} - g_{n})}_{n ⩾ 0}$ is unrelated to the scale $\underline{h}$ .

The second result (Theorem 2.9 in [14]) generalizes Proposition 2.2 and states as follows: Proposition 2.3.
Let ${(f_{n})}_{n ⩾ 0}$ be a bounded sequence in $L^{2} (R^{d})$ . Then, there exist a sequence of scales ${({\underline{h}}^{(j)})}_{j ⩾ 1}$ and a sequence of bounded sequences ${(g^{(j)})}_{j ⩾ 1}$ in $L^{2} (R^{d})$ such that:
if $j \neq k$ , then $| log (h_{n}^{(j)} / h_{n}^{(k)}) | \overset{n \to \infty}{⟶} \infty$ ,

for all j, $g^{(j)}$ is ${\underline{h}}^{(j)}$ -oscillating,

up to a subsequence extraction, we have for all $ℓ ⩾ 1$ $\begin{matrix} (20) & f_{n} (x) = \sum_{j = 1}^{ℓ} g_{n}^{(j)} (x) + r_{n}^{(ℓ)} (x), \underset{n \to \infty}{lim sup} {‖ r_{n}^{(ℓ)} ‖}_{{\dot{B}}_{2, \infty}^{0} (R^{d})} \overset{ℓ \to \infty}{⟶} 0, \end{matrix}$ where ${\dot{B}}_{2, \infty}^{0} (R^{d})$ stands for the usual homogeneous Besov space (see for instance [ 4 ] for a detailed exposition on Besov spaces).

Furthermore for any ℓ, ${(r_{n}^{(ℓ)})}_{n ⩾ 0}$ is unrelated to the scales $({\underline{h}}^{(j)})$ , for $j = 1, \dots, ℓ$ , and $\begin{matrix} (21) & ‖ f_{n} ‖_{L^{2} (R^{d})}^{2} = \sum_{j = 1}^{ℓ} {‖ g_{n}^{(j)} ‖}_{L^{2} (R^{d})}^{2} + {‖ r_{n}^{(ℓ)} ‖}_{L^{2} (R^{d})}^{2} + \circ (1), n \to \infty . \end{matrix}$

3. Proof of Theorem 1.6

3.1. Strategy of proof

The proof of the structure theorem relies on a diagonal subsequence extraction process. Roughly speaking, it is done in three steps. In the first step, using Proposition 2.2 due to P. Gérard we decompose, up to a subsequence extraction, the sequence of Cauchy data $(ψ_{n}, g_{n})$ on three parts: its weak limit, its 1-oscillating component and a remainder term which is unrelated to the scale $1$ . Thereafter, we deal separately the evolution of each part over time under the free Klein–Gordon equation. In the second step, we investigate the sequence of solutions to $(KG)$ associated to the $1$ -oscillating component. The third step is devoted to the decomposition of the sequence of solutions to $(KG)$ generated by the unrelated component to the scale $1$ . The key point consists to establish that we deal for this part with a sequence converging to zero in $L^{\infty} (R, L^{4} (R^{2}))$ .

3.2. Decomposition of the Cauchy data

Since the sequence ${(ψ_{n})}_{n ⩾ 0}$ is bounded in $H_{rad}^{1} (R^{2})$ , there is a function $ψ^{(0)} \in H_{rad}^{1} (R^{2})$ such that, up to a subsequence extraction $\begin{array}{rcl} (22) & ψ_{n} \overset{n \to \infty}{⟶} ψ^{(0)} in H^{1} (R^{2}) . \end{array}$ By virtue of the compactness of the embedding of $H_{rad}^{1} (R^{2})$ into $L^{4} (R^{2})$ (see for example [15]), $\begin{matrix} (23) & ‖ {\tilde{ψ}}_{n} ‖_{L^{4} (R^{2})} \to 0 as n \to \infty, \end{matrix}$ where we denote $ψ_{n} - ψ^{(0)}$ by ${\tilde{ψ}}_{n}$ . Thanks to the following well-known radial estimate $\begin{matrix} (24) & | u (x) | ⩽ \frac{C_{p}}{| x |^{\frac{2}{2 + p}}} ‖ u ‖_{L^{p} (R^{2})}^{\frac{p}{p + 2}} ‖ \nabla u ‖_{L^{2} (R^{2})}^{\frac{2}{p + 2}}, \end{matrix}$ available for any u in $H_{rad}^{1} (R^{2})$ and any $2 ⩽ p < \infty$ (see for instance [6] for a detailed proof), we deduce that the $L^{\infty}$ -norm of ${\tilde{ψ}}_{n}$ tends to 0 far away from the origin.

Now, using Proposition 2.2, we extract the $1$ -oscillating component of the sequence ${(⟨ D ⟩ {\tilde{ψ}}_{n})}_{n ⩾ 0}$ 4

⁴
We recall that $F (⟨ D ⟩ f) (ξ) = ⟨ ξ ⟩ F (f) (ξ)$ , where $F$ denotes the Fourier transform.

that will be noted

{(⟨ D ⟩ ψ_{n}^{(1)})}_{n ⩾ 0}

. This allows us to decompose

{\tilde{ψ}}_{n}

on two parts:

\begin{matrix} {\tilde{ψ}}_{n} (x) = ψ_{n}^{(1)} (x) + ψ_{n}^{(2)} (x), \end{matrix}

where

{(⟨ D ⟩ ψ_{n}^{(2)})}_{n ⩾ 0}

is unrelated to the scale

1

. According to orthogonality arguments, we get

\begin{matrix} \int_{R^{2}} ⟨ D ⟩ ψ_{n}^{(1)} (x) \overline{⟨ D ⟩ ψ_{n}^{(2)}} (x) d x \overset{n \to \infty}{⟶} 0 . \end{matrix}

This gives rise to

\begin{array}{rcl} {‖ ⟨ D ⟩ {\tilde{ψ}}_{n} ‖}_{L^{2} (R^{2})}^{2} & = & {‖ ⟨ D ⟩ ψ_{n}^{(1)} ‖}_{L^{2} (R^{2})}^{2} + {‖ ⟨ D ⟩ ψ_{n}^{(2)} ‖}_{L^{2} (R^{2})}^{2} + \circ (1), \end{array}

as n tends to infinity, and so

\begin{array}{rcl} ‖ ψ_{n} ‖_{H^{1} (R^{2})}^{2} & = & {‖ ψ^{(0)} ‖}_{H^{1} (R^{2})}^{2} + {‖ ψ_{n}^{(1)} ‖}_{H^{1} (R^{2})}^{2} + {‖ ψ_{n}^{(2)} ‖}_{H^{1} (R^{2})}^{2} + \circ (1), as n \to \infty . \end{array}

Along the same lines, since the sequence

{(g_{n})}_{n ⩾ 0}

is bounded in

L_{rad}^{2} (R^{2})

, there is a function

g^{(0)}

L_{rad}^{2} (R^{2})

such that, up to a subsequence extraction

\begin{matrix} (25) & g_{n} \overset{n \to \infty}{⟶} g^{(0)} in L^{2} (R^{2}) . \end{matrix}

Denoting

g_{n} - g^{(0)}

{\tilde{g}}_{n}

and using again Proposition 2.2, we extract the

1

-oscillating component of the sequence

{({\tilde{g}}_{n})}_{n ⩾ 0}

that will be noted

{(g_{n}^{(1)})}_{n ⩾ 0}

, which implies that

({\tilde{g}}_{n})

reads as

\begin{matrix} {\tilde{g}}_{n} = g_{n}^{(1)} + g_{n}^{(2)}, \end{matrix}

with

(g_{n}^{(2)})

unrelated to the scale

1

. Arguing as for

ψ_{n}

, we deduce that

\begin{array}{rcl} ‖ g_{n} ‖_{L^{2} (R^{2})}^{2} & = & {‖ g^{(0)} ‖}_{L^{2} (R^{2})}^{2} + {‖ g_{n}^{(1)} ‖}_{L^{2} (R^{2})}^{2} + {‖ g_{n}^{(2)} ‖}_{L^{2} (R^{2})}^{2} + \circ (1), as n \to \infty . \end{array}

In what follows, we shall denote respectively by

v_{n}^{(1)}

and

v_{n}^{(2)}

the sequences of solutions to

(KG)

associated to the Cauchy data (

ψ_{n}^{(1)}, g_{n}^{(1)}

) and (

ψ_{n}^{(2)}, g_{n}^{(2)}

), and treat separately their evolution.

3.3. Decomposition of the sequence

{(v_{n}^{(1)})}_{n ⩾ 0}

Proposition 3.1.
With the above notations, there exist a sequence ${(w^{(k)})}_{k ⩾ 0}$ of functions in $L_{rad}^{2} (R^{2})$ , a sequence ${({\tilde{w}}^{(k)})}_{k ⩾ 0}$ of functions in $H_{rad}^{1} (R^{2})$ , and a sequence ${({\underline{\tilde{t}}}^{(k)})}_{k ⩾ 1}$ of sequences of reals numbers such that $\begin{matrix} (26) & for any k \neq m, | {\tilde{t}}_{n}^{(k)} - {\tilde{t}}_{n}^{(m)} | \to \infty, as n \to \infty, \end{matrix}$ and, up to the extraction of a subsequence, we have for all integer $ℓ ⩾ 1$ and any time t in $R$ $\begin{array}{rcl} (27) & v_{n}^{(1)} (t, \cdot) & = & \sum_{k = 1}^{ℓ} U (t - {\tilde{t}}_{n}^{(k)}) ({\tilde{w}}^{(k)}, w^{(k)}) + {\tilde{r}}_{n}^{(ℓ)} (t, \cdot), \end{array}$ where ${lim sup}_{n \to \infty} ‖ {\tilde{r}}_{n}^{(ℓ)} ‖_{L^{\infty} (R, \tilde{L} (R^{2}))} \overset{ℓ \to \infty}{⟶} 0$ .

Moreover, we have the following orthogonality equality as n tends to infinity $\begin{array}{rcl} (28) & E_{0} (v_{n}^{(1)}) & = & \sum_{k = 1}^{ℓ} ({‖ {\tilde{w}}^{(k)} ‖}_{H^{1} (R^{2})}^{2} + {‖ w^{(k)} ‖}_{L^{2} (R^{2})}^{2}) + E_{0} ({\tilde{r}}_{n}^{(ℓ)}) + \circ (1) . \end{array}$
Proof.
To establish Proposition 3.1, we shall follow the approach adopted in [5] except for the smallness of the remainder term in $L^{\infty} (R, \tilde{L} (R^{2}))$ . To this end, let us begin by introducing some notations as in [5]. Denote by $ν ({\underline{v}}^{(1)})$ the set of solutions $U (t) (\tilde{w}, w)$ to $(KG)$ such that, for some sequence ${({\tilde{t}}_{n})}_{n ⩾ 0}$ of reals numbers, we have, up to a subsequence extraction, $\begin{matrix} v_{n}^{(1)} (t + {\tilde{t}}_{n}, \cdot) \overset{n \to \infty}{⇀} U (t) (\tilde{w}, w) weakly . \end{matrix}$ Setting $\begin{matrix} η ({\underline{v}}^{(1)}) = sup {E_{0} {(\tilde{w}, w)}^{\frac{1}{2}}; U (t) (\tilde{w}, w) \in ν ({\underline{v}}^{(1)})}, \end{matrix}$ let us firstly observe that $\begin{matrix} (29) & η ({\underline{v}}^{(1)}) ⩽ \underset{n \to \infty}{lim sup} E_{0} {(v_{n}^{(1)})}^{\frac{1}{2}} . \end{matrix}$ Indeed, if $U (t) (\tilde{w}, w) \in ν ({\underline{v}}^{(1)})$ , then by definition there exists some sequence ${({\tilde{t}}_{n})}_{n ⩾ 0}$ of reals numbers such that, up to a subsequence extraction $\begin{matrix} v_{n}^{(1)} (t + {\tilde{t}}_{n}, y) \overset{n \to \infty}{⇀} U (t) (\tilde{w}, w) (y) weakly . \end{matrix}$ This implies that $\begin{array}{rcl} E_{0} (\tilde{w}, w) & = & lim_{n \to \infty} (\int_{R^{2}} v_{n}^{(1)} ({\tilde{t}}_{n}, y) \tilde{w} (y) d y + \int_{R^{2}} \nabla v_{n}^{(1)} ({\tilde{t}}_{n}, y) \nabla \tilde{w} (y) d y \\ + \int_{R^{2}} \partial_{t} v_{n}^{(1)} ({\tilde{t}}_{n}, y) w (y) d y) . \end{array}$ Using Cauchy–Schwarz inequality, we deduce that $\begin{matrix} E_{0} (\tilde{w}, w) ⩽ \underset{n \to \infty}{lim sup} \frac{E_{0} (v_{n}^{(1)})}{2} + \frac{E_{0} (\tilde{w}, w)}{2}, \end{matrix}$ which ensures that $\begin{matrix} E_{0} (\tilde{w}, w) ⩽ \underset{n \to \infty}{lim sup} E_{0} (v_{n}^{(1)}), \end{matrix}$ and ends the proof of Assertion (29). The first step in the proof of Proposition 3.1 consists to show the following lemma: Lemma 3.2.
Under the above notation, there exist a sequence ${(w^{(k)})}_{k ⩾ 0}$ of functions in $L_{rad}^{2} (R^{2})$ , a sequence ${({\tilde{w}}^{(k)})}_{k ⩾ 0}$ of functions in $H_{rad}^{1} (R^{2})$ and a family ${({\tilde{\underline{t}}}^{(k)})}_{k ⩾ 1}$ of sequences of reals numbers such that, up to subsequence extraction, $\begin{matrix} (30) & for any k \neq m, | {\tilde{t}}_{n}^{(k)} - {\tilde{t}}_{n}^{(m)} | \to \infty, as n \to \infty, \end{matrix}$ and, up to a subsequence extraction, we have for all integer $ℓ ⩾ 1$ $\begin{array}{rcl} (31) & v_{n}^{(1)} (t, \cdot) & = & \sum_{k = 1}^{ℓ} U (t - {\tilde{t}}_{n}^{(k)}) ({\tilde{w}}^{(k)}, w^{(k)}) + {\tilde{r}}_{n}^{(ℓ)} (t, \cdot), \end{array}$ where $η ({\tilde{r}}_{n}^{(ℓ)}) \overset{ℓ \to \infty}{⟶} 0$ and, as n tends to infinity $\begin{array}{rcl} (32) & E_{0} (v_{n}^{(1)}) & = & \sum_{k = 1}^{ℓ} ({‖ {\tilde{w}}^{(k)} ‖}_{H^{1} (R^{2})}^{2} + {‖ w^{(k)} ‖}_{L^{2} (R^{2})}^{2}) + E_{0} ({\tilde{r}}_{n}^{(ℓ)}) + \circ (1) . \end{array}$
Proof.
If $η ({\underline{v}}^{(1)}) = 0$ , then $U (t) ({\tilde{w}}^{(k)}, w^{(k)}) = 0$ for all k, and we are done. Otherwise, there exists $U (t) ({\tilde{w}}^{(1)}, w^{(1)}) \in ν ({\underline{v}}^{(1)})$ such that $\begin{matrix} (33) & E_{0} {({\tilde{w}}^{(1)}, w^{(1)})}^{\frac{1}{2}} ⩾ \frac{1}{2} η ({\underline{v}}^{(1)}) > 0 . \end{matrix}$ Moreover by definition, there exists a sequence ${({\tilde{t}}_{n}^{(1)})}_{n ⩾ 0}$ such that, up to a subsequence extraction, $\begin{matrix} v_{n}^{(1)} (t + {\tilde{t}}_{n}^{(1)}, y) \overset{n \to \infty}{⇀} U (t) ({\tilde{w}}^{(1)}, w^{(1)}) (y) weakly . \end{matrix}$ Therefore setting $\begin{matrix} (34) & {\tilde{r}}_{n}^{(1)} (t, y) : = v_{n}^{(1)} (t, y) - U (t - {\tilde{t}}_{n}^{(1)}) ({\tilde{w}}^{(1)}, w^{(1)}) (y), \end{matrix}$ we deduce that $\begin{matrix} (35) & E_{0} (v_{n}^{(1)}) = E_{0} ({\tilde{w}}^{(1)}, w^{(1)}) + E_{0} ({\tilde{r}}_{n}^{(1)}) + \circ (1), n \to \infty . \end{matrix}$ If $η ({\underline{\tilde{r}}}^{(1)}) = 0$ , we stop the process. If not, we apply the above argument to the sequence ${\underline{\tilde{r}}}^{(1)}$ and thus there exist $U (t) ({\tilde{w}}^{(2)}, w^{(2)}) \in ν ({\underline{\tilde{r}}}^{(1)})$ and ${({\tilde{t}}_{n}^{(2)})}_{n ⩾ 0}$ a sequence of reals numbers such that, up to a subsequence extraction, $\begin{matrix} E_{0} {({\tilde{w}}^{(2)}, w^{(2)})}^{\frac{1}{2}} ⩾ \frac{1}{2} η ({\underline{\tilde{r}}}^{(1)}) > 0, \end{matrix}$ and $\begin{matrix} {\tilde{r}}_{n}^{(1)} (t + {\tilde{t}}_{n}^{(2)}, y) \overset{n \to \infty}{⇀} U (t) ({\tilde{w}}^{(2)}, w^{(2)}) (y) weakly . \end{matrix}$ Setting $\begin{matrix} {\tilde{r}}_{n}^{(2)} (t, y) : = {\tilde{r}}_{n}^{(1)} (t, y) - U (t - {\tilde{t}}_{n}^{(2)}) ({\tilde{w}}^{(2)}, w^{(2)}) (y), \end{matrix}$ we infer that $\begin{matrix} (36) & E_{0} ({\tilde{r}}_{n}^{(1)}) = E_{0} ({\tilde{w}}^{(2)}, w^{(2)}) + E_{0} ({\tilde{r}}_{n}^{(2)}) + \circ (1), n \to \infty . \end{matrix}$ In conclusion, we have $\begin{array}{rcl} v_{n}^{(1)} (t, \cdot) & = & U (t - {\tilde{t}}_{n}^{(1)}) ({\tilde{w}}^{(1)}, w^{(1)}) + U (t - {\tilde{t}}_{n}^{(2)}) ({\tilde{w}}^{(2)}, w^{(2)}) + {\tilde{r}}_{n}^{(2)} (t, \cdot), \end{array}$ with $\begin{matrix} E_{0} (v_{n}^{(1)}) = E_{0} ({\tilde{w}}^{(1)}, w^{(1)}) + E_{0} ({\tilde{w}}^{(2)}, w^{(2)}) + E_{0} ({\tilde{r}}_{n}^{(2)}) + \circ (1), n \to \infty . \end{matrix}$ We claim that $| {\tilde{t}}_{n}^{(1)} - {\tilde{t}}_{n}^{(2)} | \to \infty$ , as n tends to infinity. Otherwise, there exists a finite real number $t_{0}$ such that, up to a subsequence extraction, $\begin{matrix} (37) & lim_{n \to \infty} ({\tilde{t}}_{n}^{(2)} - {\tilde{t}}_{n}^{(1)}) = t_{0} . \end{matrix}$ This implies that ${\tilde{r}}_{n}^{(1)} (t + {\tilde{t}}_{n}^{(2)}, y) \overset{n \to \infty}{⟶} 0$ . Since by hypothesis ${\tilde{r}}_{n}^{(2)} (t + {\tilde{t}}_{n}^{(2)}, y) \overset{n \to \infty}{⇀} 0$ and $\begin{matrix} U (t) ({\tilde{w}}^{(2)}, w^{(2)}) = {\tilde{r}}_{n}^{(1)} (t + {\tilde{t}}_{n}^{(2)}, \cdot) - {\tilde{r}}_{n}^{(2)} (t + {\tilde{t}}_{n}^{(2)}, \cdot), \end{matrix}$ we deduce that $U (t) ({\tilde{w}}^{(2)}, w^{(2)}) = 0$ and $η ({\underline{\tilde{r}}}^{(1)}) = 0$ , which is in contradiction with the fact $η ({\underline{\tilde{r}}}^{(1)}) > 0$ .

At iteration ℓ, there exit $U (t) ({\tilde{φ}}^{(ℓ)}, φ^{(ℓ)}) \in ν ({\underline{\tilde{r}}}^{(ℓ - 1)})$ and a sequence ${({\tilde{t}}_{n}^{(ℓ)})}_{n ⩾ 0}$ such that, up to a subsequence extraction, $\begin{array}{l} \forall j \neq k, | {\tilde{t}}_{n}^{(j)} - {\tilde{t}}_{n}^{(k)} | \overset{n \to \infty}{⟶} \infty, \\ (38) & v_{n}^{(1)} (t, \cdot) = \sum_{k = 1}^{ℓ} U (t - {\tilde{t}}_{n}^{(k)}) ({\tilde{w}}^{(k)}, w^{(k)}) + {\tilde{r}}_{n}^{(ℓ)} (t, \cdot), with \\ (39) & E_{0} (v_{n}^{(1)}) = \sum_{k = 1}^{ℓ} E_{0} ({\tilde{w}}^{(k)}, w^{(k)}) + E_{0} ({\tilde{r}}_{n}^{(ℓ)}) + \circ (1), n \to \infty . \end{array}$ Decomposition (39) ensures that the series $\sum_{k ⩾ 1} E_{0} ({\tilde{w}}^{(k)}, w^{(k)})$ converges. Since by construction, we have $\begin{matrix} (40) & E_{0} ({\tilde{w}}^{(ℓ + 1)}, w^{(ℓ + 1)}) ⩾ \frac{1}{2} η ({\underline{\tilde{r}}}^{(ℓ)}), \end{matrix}$ we deduce that $η ({\underline{\tilde{r}}}^{(ℓ)}) \overset{ℓ \to \infty}{⟶} 0$ , which ends the proof of the lemma. □

To end the proof of Proposition 3.1, it remains to show that $\begin{matrix} (41) & \underset{n \to \infty}{lim sup} {‖ {\tilde{r}}_{n}^{(ℓ)} ‖}_{L^{\infty} (R, \tilde{L} (R^{2}))} \overset{ℓ \to \infty}{⟶} 0 . \end{matrix}$ Recall that in view of Lemma 3.2, the remainder term ${({\tilde{r}}_{n}^{(ℓ)})}_{n ⩾ 0}$ verifies the following properties: Property 1.
For every $ℓ ⩾ 1$ , the sequence ${({\tilde{r}}_{n}^{(ℓ)})}_{n ⩾ 0}$ is uniformly (with respect to n and ℓ) bounded in energy space.
Property 2.
$\begin{matrix} \underset{n \to \infty}{lim sup} \int_{{| ξ | ⩽ \frac{1}{R}} \cup {| ξ | ⩾ R}} ((1 + | ξ |^{2}) {| \hat{{\tilde{r}}_{n}^{(ℓ)}} (0, ξ) |}^{2} + {| \hat{\partial_{t} {\tilde{r}}_{n}^{(ℓ)}} (ξ, 0) |}^{2}) d ξ \overset{R \to \infty}{⟶} 0 . \end{matrix}$
Property 3.
$η ({\underline{\tilde{r}}}^{(ℓ)}) \overset{ℓ \to \infty}{⟶} 0$ .

In order to establish Estimate (41), we shall proceed by contradiction by assuming that ${lim sup}_{n \to \infty} ‖ {\tilde{r}}_{n}^{(ℓ)} ‖_{L^{\infty} (R, \tilde{L} (R^{2}))} \overset{ℓ \to \infty}{⟶} ϵ_{0} > 0$ . Thus there exist subsequences ${(n_{k})}_{k ⩾ 0}$ , ${(ℓ_{k})}_{k ⩾ 0}$ and ${(t_{k})}_{k ⩾ 0}$ such that for k large enough $\begin{matrix} (42) & {‖ {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, \cdot) ‖}_{\tilde{L} (R^{2})} ⩾ \frac{ϵ_{0}}{2} . \end{matrix}$ Invoking Sobolev embedding (5) together with Property 2, we infer that there exists a real number $R_{0}$ such that for k large enough $\begin{matrix} (43) & {‖ (Id - X_{R_{0}} (D)) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, \cdot) ‖}_{\tilde{L} (R^{2})} ⩽ \frac{ϵ_{0}}{8}, \end{matrix}$ where $\hat{X_{R_{0}} (D) u (ξ)} : = 1_{[\frac{1}{R_{0}}, R_{0}]} (| ξ |) \hat{u} (ξ)$ .

Now $R_{0}$ being fixed so that (43) occurs, let us prove that for k sufficiently large $\begin{matrix} (44) & {‖ X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, \cdot) ‖}_{\tilde{L} (R^{2})} ⩽ \frac{ϵ_{0}}{8} . \end{matrix}$ Since by construction, we know that the function $X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, \cdot)$ is spectrally localized in the set ${\frac{1}{R_{0}} ⩽ | ξ | ⩽ R_{0}}$ , we deduce that for any integer m, we have $\begin{array}{rcl} {‖ X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, \cdot) ‖}_{H^{m} (R^{2})}^{2} & ⩽ & C_{R_{0}, m} \int_{\frac{1}{R_{0}} ⩽ | ξ | ⩽ R_{0}} (1 + | ξ |^{2}) {| \hat{{\tilde{r}}_{n_{k}}^{(ℓ_{k})}} (t_{k}, ξ) |}^{2} d ξ ≲ E_{0} ({\tilde{r}}_{n_{k}}^{(ℓ_{k})}) . \end{array}$ In view of Property 1, this ensures that the sequence ${(χ_{R_{0}} (D) r_{n_{k}}^{(ℓ_{k}), 1} (t_{k}, \cdot))}_{k ⩾ 0}$ is uniformly bounded (on k) in $H^{m} (R^{2})$ . Taking advantage of Property 3, we deduce that, up to the extraction of a subsequence, $\begin{matrix} (45) & X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, \cdot) \overset{k \to \infty}{⇀} 0 in H^{m} (R^{2}) . \end{matrix}$ Now to establish (44), let us for a given positive real number λ investigate the integral $\begin{matrix} \int_{R^{2}} (e^{| \frac{X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, x)}{λ} |^{2}} - 1 - {| \frac{X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, x)}{λ} |}^{2}) d x . \end{matrix}$ On the one hand, denoting for $δ > 0$ , $\begin{array}{rcl} H_{δ} & : = & \int_{| x | ⩾ δ} (e^{| \frac{X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, x)}{λ} |^{2}} - 1 - {| \frac{X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, x)}{λ} |}^{2}) d x, \end{array}$ we get $\begin{array}{rcl} H_{δ} & ≲ & {‖ X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, \cdot) ‖}_{L^{4} (| x | ⩾ δ)}^{4} \\ ≲ & {‖ X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, \cdot) ‖}_{L^{2} (| x | ⩾ δ)}^{2} {‖ X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}) ‖}_{L^{\infty} (| x | ⩾ δ)}^{2}, \end{array}$ which in view of the radial estimate (24) and the energy conservation (1), implies that $\begin{array}{rcl} H_{δ} & ≲ & \frac{1}{δ} {‖ X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, \cdot) ‖}_{L^{2} (R^{2})}^{3} {‖ \nabla (X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, \cdot)) ‖}_{L^{2} (R^{2})} \\ ≲ & \frac{1}{δ} . \end{array}$ Thus $ε > 0$ being fixed, there exists a positive real number $δ_{0}$ such that $\begin{array}{rcl} (46) & H_{δ_{0}} & ⩽ & \frac{ε}{2} . \end{array}$ On the other hand, arguing as above we find that $\begin{array}{rcl} K_{δ_{0}} & : = & \int_{| x | ⩽ δ_{0}} (e^{| \frac{X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, x)}{λ} |^{2}} - 1 - {| \frac{X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, x)}{λ} |}^{2}) d x \\ ≲ & {‖ X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, \cdot) ‖}_{L^{2} (| x | ⩽ δ_{0})}^{2} {‖ X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, \cdot) ‖}_{L^{\infty} (| x | ⩾ δ_{0})}^{2}, \end{array}$ knowing that in view of (45), we have $‖ X_{R_{0}} (D) {\tilde{r}}_{n_{k}}^{(ℓ_{k})} (t_{k}, \cdot) ‖_{L^{\infty} (| x | ⩽ δ_{0})} \overset{k \to \infty}{⟶} 0$ .

This implies that for k large enough $\begin{matrix} K_{δ_{0}} ⩽ \frac{ε}{2}, \end{matrix}$ which according to (46) achieves the proof of the desired result. □
3.4. Decomposition of the sequence ${(v_{n}^{(2)})}_{n ⩾ 0}$

This section is devoted to the decomposition of the sequence ${(v_{n}^{(2)})}_{n ⩾ 0}$ . The heart of the matter consists to prove that it converges to 0 in $L^{\infty} (R, L^{4} (R^{2}))$ .

3.4.1. Convergence of the sequence ${(v_{n}^{(2)})}_{n ⩾ 0}$ to 0 in $L^{\infty} (R, L^{4} (R^{2}))$

The goal of this paragraph is to establish the following proposition:

Proposition 3.3.
The sequence ${(v_{n}^{(2)})}_{n ⩾ 0}$ of solutions to $(KG)$ converges to 0 in $L^{\infty} (R, L^{4} (R^{2}))$ .
Proof.
In view of the continuity of the Fourier transform $\begin{matrix} (47) & F : L^{\frac{4}{3}} (R^{2}) \to L^{4} (R^{2}), \end{matrix}$ it suffices to prove that $\begin{matrix} (48) & {‖ \hat{v_{n}^{(2)}} ‖}_{L^{\infty} (R, L^{\frac{4}{3}} (R^{2}))} \overset{n \to \infty}{⟶} 0 . \end{matrix}$ Since $v_{n}^{2}$ is the solution to $(KG)$ associated to the Cauchy data $(ψ_{n}^{(2)}, g_{n}^{(2)})$ , we have $\begin{matrix} (49) & \hat{v_{n}^{(2)}} (t, ξ) = \hat{ψ_{n}^{(2)}} (ξ) cos (t ⟨ ξ ⟩) + \hat{g_{n}^{(2)}} (ξ) \frac{sin (t ⟨ ξ ⟩)}{⟨ ξ ⟩} . \end{matrix}$ According to Proposition 2.3 due to P. Gérard, there exist a sequence of orthogonal scales ${(h_{n}^{(j)})}_{n ⩾ 0}$ and a family ${(M_{n}^{(j)}, N_{n}^{(j)})}_{n ⩾ 0}$ of sequences $h_{n}^{(j)}$ -oscillatory such that, up to a subsequence extraction, we have $\begin{matrix} (⟨ D ⟩ ψ_{n}^{(2)}, g_{n}^{(2)}) = \sum_{j = 1}^{ℓ} (M_{n}^{(j)}, N_{n}^{(j)}) + (r_{n}^{1, (ℓ)}, r_{n}^{2, (ℓ)}), \end{matrix}$ where ${(r_{n}^{1, (ℓ)}, r_{n}^{2, (ℓ)})}_{n ⩾ 0}$ is unrelated to the scale $h_{n}^{(j)}$ , for $1 ⩽ j ⩽ ℓ$ , and $\begin{matrix} \underset{n \to \infty}{lim sup} ({‖ r_{n}^{1, (ℓ)} ‖}_{{\dot{B}}_{2, \infty}^{0} (R^{2})} + {‖ r_{n}^{2, (ℓ)} ‖}_{{\dot{B}}_{2, \infty}^{0} (R^{2})}) \overset{ℓ \to \infty}{⟶} 0 . \end{matrix}$ Moreover, since the sequence ${(⟨ D ⟩ ψ_{n}^{(2)}, g_{n}^{(2)})}_{n ⩾ 0}$ is unrelated to the scale 1, there are two possibilities for the scales $h_{n}^{(j)}$ : either they tend to infinity or to zero.

Now, denoting by $\begin{matrix} \hat{w_{n}^{(j)}} (t, ξ) : = \hat{M_{n}^{(j)}} (ξ) \frac{cos (t ⟨ ξ ⟩)}{⟨ ξ ⟩} + \hat{N_{n}^{(j)}} (ξ) \frac{sin (t ⟨ ξ ⟩)}{⟨ ξ ⟩}, \end{matrix}$ let start by proving that $\begin{matrix} (50) & lim_{n \to \infty} {‖ \hat{w_{n}^{(j)}} ‖}_{L^{\infty} (R, L^{\frac{4}{3}} (R^{2}))} = 0 . \end{matrix}$ On the one hand, in the case where $h_{n}^{(j)} \overset{n \to \infty}{⟶} 0$ , we have for any fixed $R > 0$ $\begin{array}{rcl} \int_{R^{2}} {| \hat{M_{n}^{(j)}} (ξ) \frac{cos (t ⟨ ξ ⟩)}{⟨ ξ ⟩} |}^{\frac{4}{3}} d ξ & = & A_{n, R}^{(j)} + B_{n, R}^{(j)}, \end{array}$ with $\begin{matrix} A_{n, R}^{(j)} = \int_{{\frac{1}{R} ⩽ h_{n}^{(j)} | ξ | ⩽ R}} {| \hat{M_{n}^{(j)}} (ξ) \frac{cos (t ⟨ ξ ⟩)}{⟨ ξ ⟩} |}^{\frac{4}{3}} d ξ . \end{matrix}$ By hypothesis, we have ${⟨ D ⟩}^{- 1} M_{n}^{(j)} \in H_{rad}^{1} (R^{2})$ . Thus, in view of Fourier–Plancherel formula, the functions $\frac{ρ^{\frac{1}{2}}}{⟨ ρ ⟩} \hat{M_{n}^{(j)}} (ρ)$ and $\frac{ρ^{\frac{3}{2}}}{⟨ ρ ⟩} \hat{M_{n}^{(j)}} (ρ)$ belong to $L^{2} (R_{+})$ , where we noted $| ξ | = ρ$ . Applying Hölder inequality, this gives rise to $\begin{array}{rcl} A_{n, R}^{(j)} & ⩽ & 2 π \int_{{\frac{1}{R} ⩽ h_{n}^{(j)} ρ ⩽ R}} {| \hat{M_{n}^{(j)}} (ρ) \frac{ρ^{\frac{3}{2}}}{⟨ ρ ⟩} |}^{\frac{4}{3}} \frac{d ρ}{ρ} \\ ⩽ & C_{R} {({‖ \nabla {⟨ D ⟩}^{- 1} M_{n}^{(j)} ‖}_{L^{2} (R^{2})}^{2} h_{n}^{(j)})}^{\frac{2}{3}} \overset{n \to \infty}{⟶} 0 . \end{array}$ Besides $\begin{array}{rcl} B_{n, R}^{(j)} & ⩽ & 2 π (\int_{{h_{n}^{(j)} ρ ⩽ \frac{1}{R}}} {| \hat{M_{n}^{(j)}} (ρ) ρ^{\frac{1}{2}} |}^{\frac{4}{3}} \frac{ρ^{\frac{1}{3}}}{{(1 + ρ^{2})}^{\frac{2}{3}}} d ρ + \int_{{h_{n}^{(j)} ρ ⩾ R}} {| \hat{M_{n}^{(j)}} (ρ) ρ^{\frac{1}{2}} |}^{\frac{4}{3}} \frac{ρ^{\frac{1}{3}}}{{(1 + ρ^{2})}^{\frac{2}{3}}} d ρ) \\ ≲ & {(\int_{{h_{n}^{(j)} | ξ | ⩽ \frac{1}{R}}} {| \hat{M_{n}^{(j)}} (ξ) |}^{2} d ξ)}^{\frac{2}{3}} + {(\int_{{h_{n}^{(j)} | ξ | ⩾ R}} {| \hat{M_{n}^{(j)}} (ξ) |}^{2} d ξ)}^{\frac{2}{3}} . \end{array}$ Using the fact that the sequence $(M_{n}^{(j)})$ is $h_{n}^{(j)}$ -oscillating, we deduce that ${lim sup}_{n \to \infty} B_{n, R}^{(j)} \overset{R \to \infty}{⟶} 0$ , which implies that $\begin{matrix} \int_{R^{2}} {| \hat{M_{n}^{(j)}} (ξ) \frac{cos (t ⟨ ξ ⟩)}{⟨ ξ ⟩} |}^{\frac{4}{3}} d ξ \overset{n \to \infty}{⟶} 0 . \end{matrix}$ Along the same lines, we prove that $\begin{matrix} \int_{R^{2}} | {(\hat{N_{n}^{(j)}} (ξ) \frac{sin (t ⟨ ξ ⟩)}{⟨ ξ ⟩} |}^{\frac{4}{3}} d ξ \overset{n \to \infty}{⟶} 0 . \end{matrix}$ This ends the proof of Estimate (50) in the case when $h_{n}^{(j)}$ tends to zero.

Now in the case when $h_{n}^{(j)} \overset{n \to \infty}{⟶} \infty$ , and under the above notations, the same reasoning leads to $\begin{array}{rcl} A_{n, R}^{(j)} & ⩽ & C_{R} {(\frac{‖ {⟨ D ⟩}^{- 1} M_{n}^{(j)} ‖_{L^{2} (R^{2})}^{2}}{h_{n}^{(j)}})}^{\frac{2}{3}} \overset{n \to \infty}{⟶} 0, \end{array}$ and $\begin{array}{rcl} B_{n, R}^{(j)} & ≲ & {(\int_{{h_{n}^{(j)} | ξ | ⩽ \frac{1}{R}}} {| \hat{M_{n}^{(j)}} (ξ) |}^{2} d ξ)}^{\frac{2}{3}} + {(\int_{{h_{n}^{(j)} | ξ | ⩾ R}} {| \hat{M_{n}^{(j)}} (ξ) |}^{2} d ξ)}^{\frac{2}{3}}, \end{array}$ which easily ends the proof of Estimate (50).

Finally, since $\begin{matrix} v_{n}^{(2)} = \sum_{j = 1}^{ℓ} w_{n}^{(j)} + R_{n}^{(ℓ)}, \end{matrix}$ with $\hat{R_{n}^{(ℓ)}} (t, ξ) = \hat{r_{n}^{1, (ℓ)}} (ξ) \frac{cos (t ⟨ ξ ⟩)}{⟨ ξ ⟩} + \hat{r_{n}^{2, (ℓ)}} (ξ) \frac{sin (t ⟨ ξ ⟩)}{⟨ ξ ⟩}$ , we are reduced to demonstrate that $\begin{matrix} (51) & \underset{n \to \infty}{lim sup} {‖ \hat{R_{n}^{(ℓ)}} ‖}_{L^{\infty} (R, L^{\frac{4}{3}} (R^{2}))} \overset{ℓ \to \infty}{⟶} 0 . \end{matrix}$ Obviously, we have $\begin{array}{rcl} \int_{R^{2}} {| \hat{r_{n}^{1, (ℓ)}} (ξ) \frac{cos (t ⟨ ξ ⟩)}{⟨ ξ ⟩} |}^{\frac{4}{3}} d ξ & ⩽ & \sum_{q \in Z} \int_{{2^{q} ⩽ | ξ | ⩽ 2^{q + 1}}} {| \frac{\hat{r_{n}^{1, (ℓ)}} (ξ)}{⟨ ξ ⟩} |}^{\frac{4}{3}} d ξ, \end{array}$ which, by virtue of Hölder inequality, leads to $\begin{array}{rcl} \int_{R^{2}} {| \hat{r_{n}^{1, (ℓ)}} (ξ) \frac{cos (t ⟨ ξ ⟩)}{⟨ ξ ⟩} |}^{\frac{4}{3}} d ξ & ≲ & \sum_{q \in Z} \frac{2^{\frac{2 q}{3}}}{{(1 + 2^{2 q})}^{\frac{2}{3}}} {(\int_{{2^{q} ⩽ | ξ | ⩽ 2^{q + 1}}} {| \hat{r_{n}^{1, (ℓ)}} (ξ) |}^{2} d ξ)}^{\frac{2}{3}} \\ ≲ & {‖ r_{n}^{1, (ℓ)} ‖}_{{\dot{B}}_{2, \infty}^{0} (R^{2})}^{\frac{4}{3}} . \end{array}$ Knowing that ${lim sup}_{n \to \infty} ‖ r_{n}^{1, (ℓ)} ‖_{{\dot{B}}_{2, \infty}^{0} (R^{2})} \overset{ℓ \to \infty}{⟶} 0$ , we find that $\begin{matrix} (52) & \underset{n \to \infty}{lim sup} {‖ \hat{r_{n}^{1, (ℓ)}} (ξ) \frac{cos (t ⟨ ξ ⟩)}{⟨ ξ ⟩} ‖}_{L^{\infty} (R, L^{\frac{4}{3}} (R^{2}))} \overset{ℓ \to \infty}{⟶} 0 . \end{matrix}$ Along the same lines, we get $\begin{matrix} (53) & \underset{n \to \infty}{lim sup} {‖ \hat{r_{n}^{2, (ℓ)}} (ξ) \frac{sin (t ⟨ ξ ⟩)}{⟨ ξ ⟩} ‖}_{L^{\infty} (R, L^{\frac{4}{3}} (R^{2}))} \overset{ℓ \to \infty}{⟶} 0, \end{matrix}$ which achieves the proof of (51) and thus of Proposition 3.3. □

3.4.2. Decomposition of the sequence ${(v_{n}^{(2)})}_{n ⩾ 0}$ in the framework of Orlicz norm

Proposition 3.4.

Let ${(v_{n}^{(2)})}_{n ⩾ 0}$ be the sequence of solutions to $(KG)$ associated to the Cauchy data ${(ψ_{n}^{(2)}, g_{n}^{(2)})}_{n ⩾ 0}$ . Then there exist two sequences of profiles ${(ψ^{(j)})}_{j ⩾ 1}$ and ${({\tilde{ψ}}^{(j)})}_{j ⩾ 1}$ in $P$ , a sequence ${({\underline{α}}^{(j)})}_{j ⩾ 1}$ of scales in the sense of Definition 1.3 , a sequence ${({\underline{t}}^{(j)})}_{j ⩾ 1}$ of reals sequences such that $\begin{array}{l} \forall j \neq i, | log (α_{n}^{(j)} / α_{n}^{(i)}) | \overset{n \to \infty}{⟶} \infty or α_{n}^{(j)} = α_{n}^{(i)} and \\ - \frac{log | t_{n}^{(j)} - t_{n}^{(i)} |}{α_{n}^{(j)}} \overset{n \to \infty}{⟶} a \in [- \infty, + \infty [, \end{array}$ with in the case when $a \in] 0, + \infty [$ , $(ψ^{(j)} + {\tilde{ψ}}^{(j)}) (s)$ or $(ψ^{(i)} + {\tilde{ψ}}^{(i)}) (s)$ null for $s < a$ and, up to subsequence extraction, we have for all integer $ℓ ⩾ 1$ $\begin{matrix} (54) & v_{n}^{(2)} (t, \cdot) = \sum_{j = 1}^{ℓ} F_{n}^{(j)} (t, \cdot) + R_{n}^{(ℓ)} (t, \cdot), \end{matrix}$ where $F_{n}^{(j)}$ is given by ( 16 ), and with ${lim sup}_{n \to \infty} ‖ R_{n}^{(ℓ)} ‖_{L^{\infty} (R, \tilde{L} (R^{2}))} \overset{ℓ \to \infty}{⟶} 0$ .

Moreover, we have the following orthogonality equality, as n tends to infinity $\begin{array}{rcl} (55) & E_{0} (v_{n}^{(2)}) & = & \sum_{j = 1}^{ℓ} {‖ {ψ^{(j)}}^{'} ‖}_{L^{2} (R)}^{2} + \sum_{j = 1}^{ℓ} {‖ {\tilde{ψ}}^{{(j)}^{'}} ‖}_{L^{2} (R)}^{2} + E_{0} (R_{n}^{(ℓ)}) + \circ (1) . \end{array}$

Proof.

To establish Proposition 3.4, we shall follow the strategy developed in [6] which uses in a crucial way the radial setting, and namely the fact that we deal with bounded functions far away from the origin. For that purpose, let us set $K_{n} (t, s) = v_{n}^{(2)} (t, e^{- s})$ . Using the radial estimate (24) with $p = 4$ , the conservation law (1) and Proposition 3.3, we obtain for any real number M $\begin{matrix} (56) & ‖ K_{n} ‖_{L^{\infty} (R, L^{\infty} (] - \infty, M [))} \to 0, n \to \infty . \end{matrix}$ Now denoting by $\begin{matrix} (57) & \underset{n \to \infty}{lim sup} {‖ v_{n}^{(2)} ‖}_{L^{\infty} (R, \tilde{L} (R^{2}))} : = A_{0} > 0, \end{matrix}$ we have the following result: Proposition 3.5.

Under the above notations, the following property holds $\begin{matrix} (58) & sup_{s ⩾ 0, t \in R} (4 {| \frac{K_{n} (t, s)}{A_{0}} |}^{2} - s) \to \infty, n \to \infty . \end{matrix}$

Proof.

We proceed by contradiction assuming that, up to a subsequence extraction, there is a positive constant C such that $\begin{matrix} (59) & sup_{s ⩾ 0, n \in N, t \in R} (4 {| \frac{K_{n} (t, s)}{A_{0}} |}^{2} - s) ⩽ C < \infty . \end{matrix}$ Designating by $ϵ_{n} = ‖ v_{n}^{(2)} ‖_{L^{\infty} (R, L^{4} (R^{2}))}$ , this gives rise to $\begin{array}{rcl} \int_{| x | < ϵ_{n}} (e^{4 | \frac{v_{n}^{(2)} (t, x)}{A_{0}} |^{2}} - 1 - 4 {| \frac{v_{n}^{(2)} (t, x)}{A_{0}} |}^{2}) d x & ⩽ & 2 π \int_{0}^{ϵ_{n}} e^{4 | \frac{v_{n}^{(2)} (t, r)}{A_{0}} |^{2}} r d r \\ (60) & ⩽ & 2 π e^{C} ϵ_{n} . \end{array}$ Taking advantage again of the radial estimate (24) and the conservation law (1), we infer that $\begin{array}{l} \int_{ϵ_{n} ⩽ | x | < 1} (e^{4 | \frac{v_{n}^{(2)} (t, x)}{A_{0}} |^{2}} - 1 - 4 {| \frac{v_{n}^{(2)} (t, x)}{A_{0}} |}^{2}) d x & ⩽ 2 π \int_{ϵ_{n}}^{1} (e^{\frac{4 C^{2} ‖ v_{n}^{(2)} ‖_{L^{\infty} (R, L^{4})}^{\frac{4}{3}} E_{0}^{\frac{1}{3}} (v_{n}^{(2)})}{r^{\frac{2}{3}} A_{0}^{2}}} - 1) r d r \\ ⩽ 2 π \int_{ϵ_{n}}^{1} (e^{\frac{4 C^{2} ϵ_{n}^{\frac{2}{3}} E_{0}^{\frac{1}{3}} (v_{n}^{(2)})}{A_{0}^{2}}} - 1) r d r, \end{array}$ which ensures that $\begin{array}{rcl} (61) & \int_{ϵ_{n} ⩽ | x | < 1} (e^{4 | \frac{v_{n}^{(2)} (t, x)}{A_{0}} |^{2}} - 1 - 4 {| \frac{v_{n}^{(2)} (t, x)}{A_{0}} |}^{2}) d x & ≲ & ϵ_{n}^{\frac{2}{3}} . \end{array}$ Finally, using Proposition 3.3 and the simple fact that for any positive M, there exists a finite constant $C_{M}$ such that $\begin{matrix} sup_{| t | ⩽ M} (\frac{e^{t^{2}} - 1 - t^{2}}{t^{4}}) < C_{M}, \end{matrix}$ we get $\begin{matrix} (62) & \int_{| x | ⩾ 1} (e^{4 | \frac{v_{n}^{(2)} (t, x)}{A_{0}} |^{2}} - 1 - 4 {| \frac{v_{n}^{(2)} (t, x)}{A_{0}} |}^{2}) d x ≲ {‖ v_{n}^{(2)} ‖}_{L^{\infty} (R, L^{4} (R^{2}))}^{4} \overset{n \to \infty}{⟶} 0 . \end{matrix}$ Invoking (60), (61) and (62), we deduce that $\begin{matrix} \int_{R^{2}} (e^{4 | \frac{v_{n}^{(2)} (t, x)}{A_{0}} |^{2}} - 1 - 4 {| \frac{v_{n}^{(2)} (t, x)}{A_{0}} |}^{2}) d x \to 0, n \to \infty, \end{matrix}$ uniformly with respect to $t \in R$ . Consequently $\begin{matrix} \underset{n \to \infty}{lim sup} {‖ v_{n}^{(2)} ‖}_{L^{\infty} (R, \tilde{L} (R^{2}))} ⩽ \frac{A_{0}}{2}, \end{matrix}$ which is in contradiction with Hypothesis (57). □

An immediate consequence of the previous proposition is the following Corollary whose proof is similar to that of Corollaries 2.4 and 2.5 in [6]. Corollary 3.6.

There exists a sequence ${(t_{n}^{(1)}, α_{n}^{(1)})}_{n ⩾ 0}$ in $R \times R$ with ${(α_{n}^{(1)})}_{n ⩾ 0}$ tending to infinity such that $\begin{matrix} (63) & 4 {| \frac{K_{n} (t_{n}^{(1)}, α_{n}^{(1)})}{A_{0}} |}^{2} - α_{n}^{(1)} \to \infty, \end{matrix}$ and there exists a positive constant C such that $\begin{matrix} \frac{A_{0}}{2} \sqrt{α_{n}^{(1)}} ⩽ | K_{n} (t_{n}^{(1)}, α_{n}^{(1)}) | ⩽ C \sqrt{α_{n}^{(1)}} + \circ (1), \end{matrix}$ with $C = {lim sup}_{n \to \infty} \frac{E_{0}^{\frac{1}{2}} (v_{n}^{(2)})}{\sqrt{2 π}}$ .

Since $v_{n}^{(2)}$ solves the free Klein–Gordon equation $(KG)$ with $v_{n_{| t = 0}}^{(2)} = ψ_{n}^{(2)}$ , $\partial_{t} v_{n_{| t = 0}}^{(2)} = g_{n}^{(2)}$ , it assumes the following form: $\begin{matrix} v_{n}^{(2)} (t, \cdot) = v_{1, n}^{(2)} (t, \cdot) + v_{2, n}^{(2)} (t, \cdot), \end{matrix}$ with $v_{1, n}^{(2)} (t, \cdot) = cos (t ⟨ D ⟩) ψ_{n}^{(2)}$ and $v_{2, n}^{(2)} (t, \cdot) = \frac{sin (t ⟨ D ⟩)}{⟨ D ⟩} g_{n}^{(2)}$ .

Now, setting $\begin{matrix} ϕ_{1, n} (y) = \sqrt{\frac{2 π}{α_{n}^{(1)}}} K_{1, n} (t_{n}^{(1)}, α_{n}^{(1)} y) and ϕ_{2, n} (y) = \sqrt{\frac{2 π}{α_{n}^{(1)}}} K_{2, n} (t_{n}^{(1)}, α_{n}^{(1)} y), \end{matrix}$ where $K_{1, n} (t_{n}^{(1)}, s) = v_{1, n}^{(2)} (t_{n}^{(1)}, e^{- s})$ and $K_{2, n} (t_{n}^{(1)}, s) = v_{2, n}^{(2)} (t_{n}^{(1)}, e^{- s})$ , we have the following key lemma: Lemma 3.7.

Under notations of Corollary 3.6 , there exists a positive constant C such that $\begin{matrix} (64) & \frac{A_{0}}{2} \sqrt{2 π} ⩽ | ϕ_{n} (1) | ⩽ C + \circ (1), \end{matrix}$ where $ϕ_{n} = ϕ_{1, n} + ϕ_{2, n}$ . Moreover, there exist two profiles $ψ^{(1)}$ and ${\tilde{ψ}}^{(1)}$ in $P$ such that, up to a subsequence extraction $\begin{matrix} ϕ_{1, n}^{'} ⇀ ψ^{{(1)}^{'}}, ϕ_{2, n}^{'} ⇀ {\tilde{ψ}}^{{(1)}^{'}} in L^{2} (R) and {‖ ψ^{{(1)}^{'}} + {\tilde{ψ}}^{{(1)}^{'}} ‖}_{L^{2} (R)} ⩾ \frac{\sqrt{2 π}}{2} A_{0} . \end{matrix}$

Proof.

The proof of Lemma 3.7 is in the same spirit as that of Lemma 2.6 in [6]. We sketch it here for the convenience of the reader. Firstly note that Assertion (64) derives immediately from Corollary 3.6. To establish the second assertion, let us start by observing that since $\begin{matrix} {‖ ϕ_{1, n}^{'} ‖}_{L^{2} (R)} = {‖ \nabla v_{1, n}^{(2)} (t_{n}^{(1)}, \cdot) ‖}_{L^{2} (R^{2})} ⩽ E_{0}^{\frac{1}{2}} (v_{1, n}^{(2)}) = {‖ ψ_{n}^{(2)} ‖}_{H^{1} (R^{2})} \end{matrix}$ and $\begin{matrix} {‖ ϕ_{2, n}^{'} ‖}_{L^{2} (R)} = {‖ \nabla v_{2, n}^{(2)} (t_{n}^{(1)}, \cdot) ‖}_{L^{2} (R^{2})} ⩽ E_{0}^{\frac{1}{2}} (v_{2, n}^{(2)}) = {‖ g_{n}^{(2)} ‖}_{L^{2} (R^{2})}, \end{matrix}$ then the sequences ${(ϕ_{1, n}^{'})}_{n ⩾ 0}$ and ${(ϕ_{2, n}^{'})}_{n ⩾ 0}$ are bounded in $L^{2} (R)$ . Consequently, up to a subsequence extraction, ${(ϕ_{1, n}^{'})}_{n ⩾ 0}$ and ${(ϕ_{2, n}^{'})}_{n ⩾ 0}$ converge weakly in $L^{2} (R)$ respectively to g and $\tilde{g}$ . Let us then introduce the functions $\begin{matrix} ψ^{(1)} (s) : = \int_{0}^{s} g (τ) d τ and {\tilde{ψ}}^{(1)} (s) : = \int_{0}^{s} \tilde{g} (τ) d τ . \end{matrix}$ Arguing as in the proof of Lemma 2.6 in [6], we show that the functions $ψ^{(1)}$ and ${\tilde{ψ}}^{(1)}$ belong to the set $P$ , which ends the proof of the first part of the second assertion. It remains to establish the second part, namely that $\begin{matrix} {‖ ψ^{{(1)}^{'}} + {\tilde{ψ}}^{{(1)}^{'}} ‖}_{L^{2} (R)} ⩾ \frac{\sqrt{2 π}}{2} A_{0} . \end{matrix}$ According to the obvious inequality $\begin{matrix} | ψ^{(1)} (1) + {\tilde{ψ}}^{(1)} (1) | = | \int_{0}^{1} {(ψ^{(1)} + {\tilde{ψ}}^{(1)})}^{'} (τ) d τ | ⩽ {‖ {ψ^{(1)}}^{'} + {\tilde{ψ}}^{{(1)}^{'}} ‖}_{L^{2} (R)}, \end{matrix}$ and the fact that in view of (64), $\begin{matrix} | ψ^{(1)} (1) + {\tilde{ψ}}^{(1)} (1) | ⩾ \frac{\sqrt{2 π}}{2} A_{0}, \end{matrix}$ we end up with the result. □

Consider $\begin{matrix} (65) & R_{n}^{(1)} (t, x) = v_{n}^{(2)} (t, x) - U (t - t_{n}^{(1)}) \sqrt{\frac{α_{n}^{(1)}}{2 π}} (ψ^{(1)} (\frac{- log | \cdot |}{α_{n}^{(1)}}), ⟨ D ⟩ {\tilde{ψ}}^{(1)} (\frac{- log | \cdot |}{α_{n}^{(1)}})) . \end{matrix}$ It can be easily seen, as in the proof of Lemma 2.6 in [6], that the sequence ${(R_{n}^{(1)})}_{n ⩾ 0}$ satisfies the same assumptions as the sequence ${(v_{n}^{(2)})}_{n ⩾ 0}$ , and that $\begin{matrix} (66) & E_{0} (v_{n}^{(2)}) = {‖ {ψ^{(1)}}^{'} ‖}_{L^{2} (R)}^{2} + {‖ {\tilde{ψ}}^{{(1)}^{'}} ‖}_{L^{2} (R)}^{2} + E_{0} (R_{n}^{(1)}) + \circ (1), as n \to \infty . \end{matrix}$ This leads to the crucial estimate $\begin{array}{rcl} \underset{n \to \infty}{lim sup} E_{0} (R_{n}^{(1)}) & = & \underset{n \to \infty}{lim sup} E_{0} (v_{n}^{(2)}) - {‖ {ψ^{(1)}}^{'} ‖}_{L^{2} (R)}^{2} - {‖ {\tilde{ψ}}^{{(1)}^{'}} ‖}_{L^{2} (R)}^{2} \\ ⩽ & \underset{n \to \infty}{lim sup} E_{0} (v_{n}^{(2)}) - \frac{1}{2} {‖ {ψ^{(1)}}^{'} + {\tilde{ψ}}^{{(1)}^{'}} ‖}_{L^{2} (R)}^{2} \\ ⩽ & C - \frac{π}{4} A_{0}^{2}, \end{array}$ with $C = {lim sup}_{n \to \infty} E_{0} (v_{n}^{(2)})$ .

Let us now define $A_{1} = {lim sup}_{n \to \infty} ‖ R_{n}^{(1)} ‖_{L^{\infty} (R, \tilde{L} (R^{2}))}$ . If $A_{1} = 0$ , we stop the process. If not, we apply the above arguments to $R_{n}^{(1)}$ , which ensures the existence of a scale $(α_{n}^{(2)})$ , of two profiles $ψ^{(2)}$ and ${\tilde{ψ}}^{(2)}$ and of a real sequence $(t_{n}^{(2)})$ , such that $\begin{matrix} R_{n}^{(1)} (t, x) = U (t - t_{n}^{(2)}) \sqrt{\frac{α_{n}^{(2)}}{2 π}} (ψ^{(2)} (\frac{- log | \cdot |}{α_{n}^{(2)}}), ⟨ D ⟩ {\tilde{ψ}}^{(2)} (\frac{- log | \cdot |}{α_{n}^{(2)}})) + R_{n}^{(2)} (t, x), \end{matrix}$ with $\begin{matrix} {‖ {ψ^{(2)}}^{'} + {\tilde{ψ}}^{{(2)}^{'}} ‖}_{L^{2} (R)} ⩾ \frac{\sqrt{2 π}}{2} A_{1} . \end{matrix}$ Moreover, we have $\begin{array}{rcl} (67) & E_{0} (R_{n}^{(1)}) & = & {‖ {ψ^{(2)}}^{'} ‖}_{L^{2} (R)}^{2} + {‖ {\tilde{ψ}}^{{(2)}^{'}} ‖}_{L^{2} (R)}^{2} + E_{0} (R_{n}^{(2)}) + \circ (1), as n \to \infty \end{array}$ and $\begin{matrix} \underset{n \to \infty}{lim sup} E_{0} (R_{n}^{(2)}) ⩽ C - \frac{π}{4} A_{0}^{2} - \frac{π}{4} A_{1}^{2} . \end{matrix}$ In addition, we claim that $(α_{n}^{(1)}, t_{n}^{(1)}, ψ^{(1)}, {\tilde{ψ}}^{(1)})$ and $(α_{n}^{(2)}, t_{n}^{(2)}, ψ^{(2)}, {\tilde{ψ}}^{(2)})$ satisfy the orthogonality condition (13). In fact, if $| log (α_{n}^{(1)} / α_{n}^{(2)}) | \overset{n \to \infty}{⟶} \infty$ , we are done. If not, we can suppose that $α_{n}^{(1)} = α_{n}^{(2)}$ and, up to a subsequence extraction, that $\begin{matrix} - \frac{log | t_{n}^{(2)} - t_{n}^{(1)} |}{α_{n}^{(2)}} ⟶ a \in [- \infty, + \infty] . \end{matrix}$ We claim, in the case when $a > - \infty$ that ${(ψ^{(2)} + {\tilde{ψ}}^{(2)})}^{'} (y) = 0$ for all $y < a$ . Indeed, if not, then there exists $0 < \tilde{a} < a$ such that $\begin{matrix} (68) & \int_{- \infty}^{\tilde{a}} {| {(ψ^{(2)} + {\tilde{ψ}}^{(2)})}^{'} (y) |}^{2} d y = δ > 0 . \end{matrix}$ On the one hand, if $ε > 0$ is selected so that $\tilde{a} + ε < a$ , then $\begin{matrix} (69) & - \frac{log | t_{n}^{(2)} - t_{n}^{(1)} |}{α_{n}^{(2)}} ⩾ \tilde{a} + ε, for n large enough . \end{matrix}$ On the other hand, by construction we have $\begin{matrix} \int_{- \infty}^{\tilde{a}} {| {(ψ^{(2)} + {\tilde{ψ}}^{(2)})}^{'} (y) |}^{2} d y = lim_{n \to \infty} I_{n}, \end{matrix}$ with $\begin{matrix} I_{n} = \int_{- \infty}^{\infty} \sqrt{\frac{2 π}{α_{n}^{(2)}}} {(R_{n}^{(1)} (t_{n}^{(2)}, e^{- α_{n}^{(2)} y}))}^{'} \overline{{(ψ^{(2)} + {\tilde{ψ}}^{(2)})}^{'}} (y) 1_{[0, \tilde{a}]} (y) d y . \end{matrix}$ Write $\begin{matrix} I_{n} = \int_{- \infty}^{\infty} \sqrt{\frac{2 π}{α_{n}^{(2)}}} {(R_{n}^{(1)} (t_{n}^{(2)}, e^{- α_{n}^{(2)} y}) - R_{n}^{(1)} (t_{n}^{(1)}, e^{- α_{n}^{(2)} y}))}^{'} \overline{{(ψ^{(2)} + {\tilde{ψ}}^{(2)})}^{'}} (y) 1_{[0, \tilde{a}]} (y) d y + R_{n}, \end{matrix}$ with $\begin{matrix} R_{n} = \int_{- \infty}^{\infty} \sqrt{\frac{2 π}{α_{n}^{(2)}}} {(R_{n}^{(1)} (t_{n}^{(1)}, e^{- α_{n}^{(2)} y}))}^{'} \overline{{(ψ^{(2)} + {\tilde{ψ}}^{(2)})}^{'}} (y) 1_{[0, \tilde{a}]} (y) d y . \end{matrix}$ Taking advantage of the fact that the sequence ${(\sqrt{\frac{2 π}{α_{n}^{(2)}}} {(R_{n}^{(1)} (t_{n}^{(1)}, e^{- α_{n}^{(2)} \cdot}))}^{'})}_{n ⩾ 0}$ converges weakly to 0 in $L^{2} (R)$ , we deduce that $R_{n}$ tends to 0 as n goes to infinity.

Thus, performing the change of variable $y = - \frac{log | x |}{α_{n}^{(2)}}$ , we get $\begin{array}{l} I_{n} = {\tilde{I}}_{n} + \circ (1), with \\ {\tilde{I}}_{n} = \int_{R^{2}} (\nabla R_{n}^{(1)} (t_{n}^{(2)}, x) - \nabla R_{n}^{(1)} (t_{n}^{(1)}, x)) \cdot \nabla \sqrt{\frac{α_{n}^{(2)}}{2 π}} \overline{(ψ_{♭}^{(2)} + {\tilde{ψ}}_{♭}^{(2)})} (\frac{- log | x |}{α_{n}^{(2)}}) d x, \end{array}$ where we noted $(ψ_{♭}^{(2)} + {\tilde{ψ}}_{♭}^{(2)}) (s) : = \int_{0}^{s} {(ψ^{(2)} + {\tilde{ψ}}^{(2)})}^{'} (y) 1_{[0, \tilde{a}]} (y) d y$ .

Clearly, in view of (66), $R_{n}^{(1)}$ solves the free Klein–Gordon equation $(KG)$ , with bounded Cauchy data $(f_{1, n}, f_{2, n})$ in $H_{rad}^{1} (R^{2}) \times L_{rad}^{2} (R^{2})$ . This ensures that $\begin{matrix} R_{n}^{(1)} (t, \cdot) = cos (t ⟨ D ⟩) f_{1, n} + \frac{sin (t ⟨ D ⟩)}{⟨ D ⟩} f_{2, n} . \end{matrix}$ Combining Cauchy–Schwarz inequality together with Fourier–Plancherel formula, we deduce that $\begin{matrix} | {\tilde{I}}_{n} | ≲ {‖ (f_{1, n}, f_{2, n}) ‖}_{H^{1} (R^{2}) \times L^{2} (R^{2})} {‖ (t_{n}^{(1)} - t_{n}^{(2)}) F (⟨ D ⟩ \nabla w_{n}^{(2)}) ‖}_{L^{2} (R^{2})}, \end{matrix}$ where $w_{n}^{(2)} (x) : = \sqrt{\frac{α_{n}^{(2)}}{2 π}} \overline{(ψ_{♭}^{(2)} + {\tilde{ψ}}_{♭}^{(2)})} (\frac{- log | x |}{α_{n}^{(2)}})$ .

According to the fact that the sequence ${(f_{1, n}, f_{2, n})}_{n ⩾ 0}$ is bounded in $H_{rad}^{1} (R^{2}) \times L_{rad}^{2} (R^{2})$ , this leads to $\begin{matrix} | {\tilde{I}}_{n} | ≲ {‖ (t_{n}^{(1)} - t_{n}^{(2)}) F (⟨ D ⟩ \nabla w_{n}^{(2)}) ‖}_{L^{2} (R^{2})} . \end{matrix}$ Since ${(ψ_{♭}^{(2)} + {\tilde{ψ}}_{♭}^{(2)})}^{'}$ is supported in $[0, \tilde{a}]$ , we have (see [3]) $\begin{matrix} (70) & | \hat{\nabla w_{n}^{(2)}} (ξ) | ≲ \frac{1_{[1, e^{\tilde{a} α_{n}^{(2)}}]} (| ξ |)}{\sqrt{α_{n}^{(2)}} | ξ |} + {\hat{t}}_{n} (ξ), \end{matrix}$ with $‖ t_{n} ‖_{L^{2} (R^{2})} \overset{n \to \infty}{⟶} 0$ . This implies that $\begin{matrix} | {\tilde{I}}_{n} |^{2} ≲ \int_{1 ⩽ | ξ | ⩽ e^{\tilde{a} α_{n}^{(2)}}} {| t_{n}^{(1)} - t_{n}^{(2)} |}^{2} (1 + | ξ |^{2}) | ξ |^{2} {| {\hat{w}}_{n}^{(2)} (ξ) |}^{2} d ξ + \circ (1) . \end{matrix}$ Taking advantage of Hypothesis (69), we get $\begin{matrix} | {\tilde{I}}_{n} |^{2} ≲ e^{- 2 α_{n}^{(2)} ε} {‖ \nabla w_{n}^{(2)} ‖}_{L^{2} (R^{2})}^{2} + \circ (1) . \end{matrix}$ This implies that $I_{n} \overset{n \to \infty}{⟶} 0$ which contradicts Claim (68), and then ends the proof of the orthogonality property (13). Besides since by construction $ψ^{(2)} + {\tilde{ψ}}^{(2)}$ is not null, we deduce that necessarily $a < + \infty$ .

Finally, iterating the above process, we get at step ℓ $\begin{matrix} v_{n}^{(2)} (t, x) = \sum_{j = 1}^{ℓ} U (t - t_{n}^{(j)}) \sqrt{\frac{α_{n}^{(j)}}{2 π}} (ψ^{(j)} (\frac{- log | \cdot |}{α_{n}^{(j)}}), ⟨ D ⟩ {\tilde{ψ}}^{(j)} (\frac{- log | \cdot |}{α_{n}^{(j)}})) + R_{n}^{(ℓ)} (t, x), \end{matrix}$ with $\begin{matrix} \underset{n \to \infty}{lim sup} E_{0} (R_{n}^{(ℓ)}) ⩽ C - \frac{π}{4} A_{0}^{2} - \frac{π}{4} A_{1}^{2} - \dots - \frac{π}{4} A_{ℓ - 1}^{2}, \end{matrix}$ which implies that $A_{ℓ} \to 0$ as $ℓ \to \infty$ and achieves the proof of Proposition 3.4. □

Observing that $\begin{matrix} v_{n} (t, \cdot) = v (t, \cdot) + v_{n}^{(1)} (t, \cdot) + v_{n}^{(2)} (t, \cdot), \end{matrix}$ we end up with the proof of Theorem 1.6, according to Propositions 3.1 and 3.4.

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High frequency approximation of solutions to Klein–Gordon equation in Orlicz framework

Abstract

Keywords

1. Introduction

1.1. Setting of the problem

1 We designate by 1 the scale in which all the terms are equal to the number 1.

2. Oscillatory components of bounded sequences in L 2 ( R d )

3.1. Strategy of proof

3.2. Decomposition of the Cauchy data

4 We recall that F ( ⟨ D ⟩ f ) ( ξ ) = ⟨ ξ ⟩ F ( f ) ( ξ ) , where F denotes the Fourier transform.

3.4.1. Convergence of the sequence ( v n ( 2 ) ) n ⩾ 0 to 0 in L ∞ ( R , L 4 ( R 2 ) )

References

¹
We designate by $1$ the scale in which all the terms are equal to the number 1.

2. Oscillatory components of bounded sequences in $L^{2} (R^{d})$

⁴
We recall that $F (⟨ D ⟩ f) (ξ) = ⟨ ξ ⟩ F (f) (ξ)$ , where $F$ denotes the Fourier transform.

3.4.1. Convergence of the sequence ${(v_{n}^{(2)})}_{n ⩾ 0}$ to 0 in $L^{\infty} (R, L^{4} (R^{2}))$