# Global existence of high-frequency solutions to a semi-linear wave equation
with a null structure

Arthur Touati

###### Abstract

We study the propagation of a compactly supported high-frequency wave through
a semi-linear wave equation with a null structure. We prove that the self-
interaction of the wave creates harmonics which remain close to the light-cone
in the evolution. By defining a well-chosen ansatz, we describe precisely
those harmonics. Moreover, by applying the vector field method to the equation
for the remainder in the ansatz, we prove that the solution exists globally.
The interaction between the dispersive decay of waves and their high-frequency
behaviour is the main difficulty, and the latter is not compensated by
smallness of the initial data, allowing us to consider the high-frequency
limit where the wavelength tends to 0.

###### Contents

  1. 1 Introduction
    1. 1.1 Presentation of the result
    2. 1.2 Strategy of proof
      1. 1.2.1 Global existence for non-linear wave equations
      2. 1.2.2 The high-frequency hierarchy
    3. 1.3 A toy model for general relativity
      1. 1.3.1 The Einstein vacuum equations in wave coordinates
      2. 1.3.2 Burnett’s conjecture
  2. 2 Preliminaries
    1. 2.1 Notations and function spaces
      1. 2.1.1 Coordinates and derivatives
      2. 2.1.2 Null forms
      3. 2.1.3 Function spaces
    2. 2.2 Minkowski vector fields
    3. 2.3 Useful estimates
      1. 2.3.1 Weighted Klainerman-Sobolev inequality
      2. 2.3.2 Weighted energy estimate
  3. 3 Main result
  4. 4 Construction of the ansatz
    1. 4.1 High-frequency expansion of the semi-linear wave equation
      1. 4.1.1 Expansion of the wave operator
      2. 4.1.2 Expansion of the null form
    2. 4.2 The reduced system
  5. 5 Global existence for the reduced system
    1. 5.1 The background wave equation
    2. 5.2 The high-frequency hierarchy
      1. 5.2.1 The transport equation for the oscillating terms
      2. 5.2.2 The wave equation for the non-oscillating terms
    3. 5.3 The equation for the remainder
      1. 5.3.1 Bootstrap assumptions and first consequences
      2. 5.3.2 Decay estimates
      3. 5.3.3 Energy estimates
  6. A Commutators
  7. B The transport equations
    1. B.1 The reduced transport equation
    2. B.2 The transport operator $\mathscr{L}$

##  1 Introduction

###  1.1 Presentation of the result

In this article we are interested in proving the global existence of high-
frequency solutions to the following semi-linear wave system

$\displaystyle\Box\Phi=Q(\partial\Phi,\partial\Phi)$ (1.1)

on $\mathbb{R}_{+}\times\mathbb{R}^{3}$ where $\Box=-\partial_{t}^{2}+\Delta$
is the standard d’Alembertian operator and $Q$ has the null structure (see
Definition 2.1). The initial data we consider are highly oscillatory in the
radial direction. This corresponds to a large data regime, which is to be
compared with the well-known small data regime, see the discussion below. Let
us give a rough version of the theorem proved in this article (see Theorem 3.1
for a precise version):

######  Theorem 1.1.

Consider sufficiently regular functions $(F_{0},\varphi_{0},\varphi_{1})$ of
size $\varepsilon$ on $\mathbb{R}^{3}$ and let $\lambda>0$. If $\varepsilon$
is small enough (independent of $\lambda$), there exists a global solution
$\Phi_{\lambda}$ to $\Box u=Q(\partial u,\partial u)$ of the form:

$\Phi_{\lambda}=\varphi+\lambda
F\cos\left(\frac{t-r}{\lambda}\right)+\lambda^{2}F_{\lambda}$ (1.2)

where

  * •

$\varphi$ satisfies $\Box\varphi=Q(\partial\varphi,\partial\varphi)$ and
$(\varphi,\partial_{t}\varphi)_{|t=0}=(\varphi_{0},\varphi_{1})$,

  * •

$F$ satisfies
$\left(\partial_{t}+\partial_{r}+\frac{1}{r}\right)F=F\partial\varphi$ and
$F_{|t=0}=F_{0}$,

  * •

$F_{\lambda}$ satisfies $|F_{\lambda}|\lesssim\varepsilon$.

In the next section we will review in detail our strategy of proof, for now
let us just mention that the global existence of solutions to (1.1) on
$\mathbb{R}^{3+1}$ with any (regular and small enough) initial data is a
classical result of Klainerman (see [13]), which introduced the concept of
null quadratic forms precisely for this purpose, and Christodoulou (see [6])
which used conformal compactification. Our proof relies crucially on the
vector field method Klainerman introduced, but our theorem is not a
consequence of his result for the following reasons:

  * •

Theorem 1.1 can be summarized as follows: take some oscillating initial data
containing terms like $\lambda\cos\left(\frac{r}{\lambda}\right)$, then the
unique global solution arrising from those initial data actually presents the
same oscillating behaviour. Unlike Klainerman, we not only prove global
existence but propagate the shape of the high-frequency ansatz.

  * •

Another crucial aspects of Theorem 1.1 is that the smallness threshold is
independent of $\lambda$. One interesting aspects of high-frequency solutions
of the form (1.2) is that their $H^{s}$ norm is large for any $s>1$, if
$\lambda$ is close to 0. Indeed, by differentiating (1.2) strictly more than
once we obtain negative powers of $\lambda$. Therefore, if one wants to apply
Klainerman’s result with a fixed but small $\lambda$ one needs to
counterbalance the size of $\frac{1}{\lambda}$ with the smallness of the
initial data (concretely by assuming something like $\varepsilon\ll\lambda$),
see Remark 3.1. Therefore, our work gives an example of how one can relax the
smallness assumption of Klainerman’s result.

Our motivations for studying solutions of the form (1.2) to non-linear wave
equations like (1.1) come from Burnett’s conjecture in general relativity and
are presented at the end of this introduction (see Section 1.3). However,
highly oscillating solution to hyperbolic equations have been intensely
studied in the field of geometric optics. In 1957, Lax laid the foundations of
this field in his article [14], where he proved that linear hyperbolic
equations admit WKB solutions. Named after the physicists Wentzel, Kramers and
Brillouin, WKB solutions were first introduced to understand the behaviour of
quantum systems in a semi-classical regime, the main idea being to write some
part of the solution as an asymptotic expansion in terms of the small
parameter $\hbar$. In geometric optics, the small parameter is a wavelength.

In this article, we prove global existence of a highly oscillating solutions
of a non-linear wave equation. The construction of high-frequency ansatz for
non-linear systems has been first conducted by Choquet-Bruhat in [3], where
she applies her result to deduce the ill-posedness of the Cauchy problem in
$C^{1}$ for the systems she considers. In [11], Joly and Rauch extend the
approximation procedure to higher space dimension. A complete overview of
geometric optics can be found in the book of Rauch [16].

In the limit $\lambda\to 0$, solutions of the form (1.2) explode in any
$W^{k,p}$ if $k>0$. Therefore, Theorem 1.1 allows us to consider large initial
data for which we are still able to prove global existence. In [17], Wang and
Yu also consider the large data regime by studying the short pulse ansatz,
inspired by Christodoulou and his study of black holes formation.

###  1.2 Strategy of proof

Let us present our strategy of proof, which can be viewed as a high-frequency
adaptation of Klainerman’s vector field method.

####  1.2.1 Global existence for non-linear wave equations

In spatial dimension strictly larger than 3, the standard decay of waves is
enough to ensure the global existence under the small initial data assumption
of solutions to wave equations of the form

$\displaystyle\Box u=(\partial u)^{2}.$ (1.3)

However, since the counter example of John (see [10]), it is known that
solutions to (1.3) on $\mathbb{R}^{3+1}$ don’t necessarily exist for all time,
even in the small initial data regime. The globality of solutions depends on
the actual structure of the quadratic non-linearity. In 1984, Klainerman
introduced the famous null condition (see [13] and Definition 2.1). The null
condition can be summarized as follows: in every product $\partial u\partial
u$ at least one of the derivatives is tangent to the light cone, and therefore
enjoys better decay. This restriction on the non-linearity in (1.3) is
sufficient (though not necessary) to obtain the existence of global solutions
for small initial data.

In order to prove global existence of a solution, one powerful method is the
vector field method, which goes back to Klainerman (see [12]). Instead of
using standard derivatives $\partial_{\alpha}$ to derive energy estimates, the
heart of the method consists in using special vector fields linked to the
symmetries of the Minkowski spacetime (see (2.3)). These vector fields are
interesting for two reasons: they enjoy nice commutation properties with the
wave operator and a more refined version of the classic Sobolev embedding
$H^{s}\subset L^{\infty}$ for $s>\frac{3}{2}$ holds when replacing the
standard derivatives in the $H^{s}$ norm by the Minkowski vector fields (the
so-called Klainerman-Sobolev inequality, see Proposition 2.2). This refined
Sobolev embedding comes with decay rates in the null coordinates $t+r$ and
$t-r$, which is the key argument to prove global existence.

This method can be improved by using a weighted energy estimate with an
increasing function of $q=t-r$ as a weight. This weight implies the presence
of an additional term on the LHS of the usual energy estimates (see Lemma
2.1), allowing to better control some of the derivatives of the solution (in
our case the derivatives tangent to the light cone). This is known as the
ghost weight method and was first introduced by Alinhac in [1].

####  1.2.2 The high-frequency hierarchy

The main particularity of our result is the use of an high-frequency ansatz
with a precise description of the oscillating terms, displaying what we call a
half-chessboard shape. Let $\Phi_{\lambda}$ be the solution of (1.1) admitting
the following formal high-frequency expansion

$\displaystyle\Phi_{\lambda}(t,x)\sim\sum_{k\geq
0}\lambda^{k}\Phi^{(k)}\left(t,x,\frac{t-r}{\lambda}\right).$ (1.4)

By half-chessboard shape, we mean that each $\Phi^{(i)}(t,x,\theta)$ contains
several oscillating terms in the $\theta$ variable according to the following
pattern:

$\displaystyle\Phi^{(0)}$ $\displaystyle=1$ $\displaystyle\Phi^{(1)}$
$\displaystyle=\cos(\theta)$ $\displaystyle\Phi^{(2)}$
$\displaystyle=\sin(\theta)+\cos(2\theta)$ (1.5) $\displaystyle\Phi^{(3)}$
$\displaystyle=\cos(\theta)+\sin(2\theta)+\cos(3\theta)$
$\displaystyle\quad\vdots$

######  Remark 1.1.

In (1.5), we only wrote down the oscillating terms, i.e the dependence of
$\Phi^{(i)}$ on its third variable, and by $\Phi^{(0)}=1$ we mean that it does
not actually depends on the third variable, i.e it is not oscillating.

The main challenge in defining a pattern of oscillation for each order is to
capture the creation of harmonics due to the non-linear part of (1.1). It is
not trivial that the half-chessboard shape is ”stable” under quadratic non-
linearity of the form $(\partial\Phi_{\lambda})^{2}$, i.e that those terms
only create harmonics already present in the ansatz for $\Phi_{\lambda}$. This
will be rigourously proved in Section 4, where we present the exact form of
the ansatz for $\Phi_{\lambda}$ (see (4.1)). The non-linear interactions also
create non-oscillating terms which can’t be absorbed by the ansatz, unless we
add to (1.5) some non-oscillating fields. To lighten this introduction we
choose not to show them yet.

From a differential point of view, plugging oscillating terms
$\lambda^{k}\cos\left(\frac{u}{\lambda}\right)F$ (with $u$ a phase to be
chosen later) in the wave operator gives

$\displaystyle\Box\left(\lambda^{k}\cos\left(\frac{u}{\lambda}\right)F\right)$
$\displaystyle=-\lambda^{k-2}\left(m^{\alpha\beta}\partial_{\alpha}u\partial_{\beta}u\right)\cos\left(\frac{u}{\lambda}\right)F$
(1.6)
$\displaystyle\quad-\lambda^{k-1}\sin\left(\frac{u}{\lambda}\right)\left(2m^{\alpha\beta}\partial_{\alpha}u\partial_{\beta}+\Box
u\right)F$
$\displaystyle\quad+\lambda^{k}\cos\left(\frac{u}{\lambda}\right)\Box F$

where
$m^{\alpha\beta}\partial_{\alpha}u\partial_{\beta}u=-(\partial_{t}u)^{2}+|\nabla
u|^{2}$. We recover what is called the geometric optics approximation: in
order to have
$\Box\left(\lambda^{k}\cos\left(\frac{u}{\lambda}\right)F\right)$ at the same
order as $\lambda^{k}\cos\left(\frac{u}{\lambda}\right)F$ we need

  * •

$m^{\alpha\beta}\partial_{\alpha}u\partial_{\beta}u=0$, i.e $u$ to be a
solution of the eikonal equation, or said differently we need the spacetime
gradient of $u$, denoted $\partial u$, to be a null vector field for the
Minkowski metric. In the rest of the article we will choose $u=t-r$.

  * •

$\left(2m^{\alpha\beta}\partial_{\alpha}u\partial_{\beta}+\Box u\right)F=0$,
i.e $F$ to be transported along the rays of $u$. If $u=t-r$, this rewrites

$\displaystyle\left(\partial_{t}+\partial_{r}+\frac{1}{r}\right)F=0.$

The fact that $u$ satisfies the eikonal equation has another consequence,
related to our pattern of oscillation (1.5). Indeed since $Q$ is a null form
and $\partial u$ is a null vector the following term vanishes

$\displaystyle\left(\sin\left(\frac{u}{\lambda}\right)F^{(1)}\right)^{2}Q(\partial
u,\partial u)$ (1.7)

where $F^{(1)}$ is the coefficient of $\Phi^{(1)}$. The presence of this term
would totally break the half-chessboard shape since we would need $\Phi^{(1)}$
(as well as $\Phi^{(k)}$ for $k\geq 2$ because of the triangular structure) to
contain all frequencies (see Remark 4.10).

The conclusion of this discussion is that solving (1.1) is equivalent to
solving a triangular hierarchy of linear transport equations along the rays of
$u$ for each coefficients in the high-frequency expansion of $\Phi_{\lambda}$,
as well as a semi-linear wave equation for the remainder $h$, which is added
to the formal expansion (1.4) to make it an exact solution of (1.1). Our
ansatz is therefore of the following schematic form

$\displaystyle\Phi_{\lambda}(t,x)=\sum_{k=0}^{K}\lambda^{k}\Phi^{(k)}\left(t,x,\frac{t-r}{\lambda}\right)+h_{\lambda}$
(1.8)

where $h_{\lambda}=O\left(\lambda^{K}\right)$ solves an equation of the form

$\displaystyle\Box h_{\lambda}$ $\displaystyle=Q(\partial h_{\lambda},\partial
h_{\lambda})+\lambda^{K}\Phi^{(\cdot)}$ (1.9)

where $Q$ is the null form and $\lambda^{K}\Phi^{(\cdot)}$ denotes terms of
orders $\lambda^{K}$ depending polynomially on the high-frequency waves
$\Phi^{(k)}$ and their derivatives. Note that we deliberately forgot a lot of
terms in (1.9), since from the point of view of the dependence in $\lambda$
the quadratic terms in $\partial h_{\lambda}$ are the most dangerous.

In (1.8), the degree of precision of the expansion defining $\Phi_{\lambda}$
is given by the integer $K$. Our work shows that we need $K\geq 4$ to close
the bootstrap procedure for (1.9). This is due to the method used to prove
global existence to (1.9), i.e the vector field method. Indeed, we want
$h_{\lambda}$ to satisfy bootstrap assumptions of the form $\left\|\partial
Z^{I}h_{\lambda}\right\|_{L^{2}}\lesssim\lambda^{K-|I|}$ (where $Z$ denotes
any Minkowskian vector fields, see (2.3)). This mimics a high-frequency
behaviour while $h_{\lambda}$ is not oscillating. In order to improve these
bootstrap assumptions, we will recover decay from the Klainerman-Sobolev
inequality. This inequality allows one to bound $|f|$ by
$\left\|Z^{3}f\right\|_{L^{2}}$ with appropriate decay rates, which we omit in
this introduction (see Proposition 2.2). The bootstrap assumptions for
$h_{\lambda}$ therefore imply that $|\partial
h_{\lambda}|\lesssim\lambda^{K-4}$. This explains the lower bound on the
degree of precision of the ansatz $K\geq 4$.

As a final remark on our strategy of proof, let us explain how we get decay
for the high-frequency waves $\Phi^{(k)}$, which appear as source terms in
(1.9). As explained above, the high-frequency waves solve a hierarchy of
transport equation of the form

$\displaystyle\left(\partial_{t}+\partial_{r}+\frac{1}{r}\right)f=g$ (1.10)

with a RHS depending on the lower order high-frequency waves (thus respecting
the triangular structure of the system). This equation rewrites
$(\partial_{t}+\partial_{r})(rf)=rg$, from which we get $|f|\lesssim r^{-1}$
if the RHS satisfies $|g|\lesssim r^{-3}$. See Proposition 5.2 for a precise
statement. In the hierarchy of transport equations, the most dangerous term is
$\Box\Phi^{(k)}$, i.e comes from the third line in (1.6). Schematically if we
only consider this term we have

$\displaystyle\left(\partial_{t}+\partial_{r}+\frac{1}{r}\right)\Phi^{(k+1)}$
$\displaystyle=\Box\Phi^{(k)}.$

Therefore, in order to apply Proposition 5.2 we need
$\left|\Box\Phi^{(k)}\right|\lesssim r^{-3}$. We can’t get this decay by
commuting $\Box$ with the transport operator
$\partial_{t}+\partial_{r}+\frac{1}{r}$ since it would only give
$\left|\Box\Phi^{(k)}\right|\lesssim r^{-1}$. Instead we compute and estimate
$\Box\Phi^{(k)}$ directly from $\partial_{t}+\partial_{r}+\frac{1}{r}$ in
Lemma A.2. See Remark 5.2 for a more detailled discussion of this procedure.

###  1.3 A toy model for general relativity

Here we present our principal motivation for studying the global existence of
high-frequency solutions to (1.1).

####  1.3.1 The Einstein vacuum equations in wave coordinates

The Einstein equations are the main equations of the general relativity
theory, they spell out the link between the curvature of the spacetime and the
matter and energy it contains. On a Lorentzian manifold $(\mathcal{M},g)$ and
in the vacuum case they can be written in an elegant and compact form:

$\mathrm{Ric}(g)=0,$ (1.11)

where $\mathrm{Ric}(g)$ is the Ricci tensor of the metric $g$. In order to
define a initial value problem for this equation, we need to specify a
coordinate system on $\mathcal{M}$. Noticed by Einstein in 1916 and used by
Choquet-Bruhat to show local existence of the Einstein system in the 1950’s,
the wave coordinates define the most common coordinate system used in the
study of the Einstein equations. In this coordinate system, the equation
(1.11) becomes

$\Box_{g}g_{\alpha\beta}=P_{\alpha\beta}(\partial g,\partial g),$ (1.12)

where $\Box_{g}$ is the wave operator associated to $g$ and $P_{\alpha\beta}$
is a quadratic non-linearity. Thus, the Einstein vacuum equations are recast
as a quasi-linear hyperbolic system for the metric coefficients, which allows
one to define and study a Cauchy problem (see [8] for the first proof of local
well-posedness using wave coordinates).

One of the major question in general relativity is the question of the
stability of particular solutions, such as the Minkowski spacetime, which,
thanks to (1.12) reduces to a hyperbolic long-time existence problem. As it
was proved by Choquet-Bruhat in [4], the non-linearity in (1.12) does not
satisfy the null condition presented in Section 1.2. Note that the Einstein
vacuum equations do satisfy a null condition through a dynamical gauge choice,
see the seminal work [7]. However, the Einstein vacuum equations in wave gauge
satisfy what Lindblad and Rodnianski call a weak null condition, allowing them
to prove the non-linear stability of the Minkowski spacetime in this gauge in
[15].

Therefore, studying the semi-linear equation $\Box u=Q(\partial u,\partial u)$
where $Q$ is a null form is a toy model for the Einstein vacuum equations: it
is a semi-linear equation instead of a quasi-linear equation, with the
quadratic part being a null form instead of merely satisfying the weak null
condition.

####  1.3.2 Burnett’s conjecture

The Einstein vacuum equations are highly non-linear, which a priori allows
backreaction to occur. This phenomenon can be described as follows: if a
sequence of metrics $(g_{\lambda})_{\lambda}$ converges in some sense to a
background metric $g_{0}$, and if for all $\lambda$ we have
$\mathrm{Ric}(g_{\lambda})=0$, do we have $\mathrm{Ric}(g_{0})=0$ ? Said
differently, can we pass to the limit in the Ricci tensor? Identifying the
potential limit matter models occuring because of backreaction is called the
Burnett conjecture, and was introduced by Burnett in [2].

If the convergence $g_{\lambda}\to g_{0}$ happens in a very strong topology
then the answer is obviously positive, but if the convergence is weak enough,
we might obtain non-zero contributions from quadratic terms like
$\partial(g_{\lambda}-g_{0})\partial(g_{\lambda}-g_{0})$. In the framework
chosen by Burnett, the convergence is the following: $g_{\lambda}$ converges
strongly in $L^{\infty}$ and $\partial g_{\lambda}$ converges weakly in
$L^{2}$. High-frequency ansatz of the form (1.4)-(1.5) precisely enjoy this
type of convergence, and have been used to tackle Burnett’s conjecture locally
in time by Huneau and Luk in [9]. This fully motivates the study of high-
frequency solutions to (1.1).

######  Remark 1.2.

Even though the high-frequency solution $\Phi_{\lambda}$ converges in the
Burnett sense to $\Phi^{(0)}$, no backreaction occurs in the case of the
equation (1.1), meaning that the background limit solution $\Phi^{(0)}$ also
satisfy (1.1) (see also (BG) in the reduced system). As Choquet-Bruhat noticed
in [4], this is linked to the null condition satisfied by (1.1). Indeed,
backreaction could only occur through the non-oscillating contributions of the
squared sinus in (1.7), but as we saw (1.7) vanishes because $Q$ is a null
form.

The absence of backreaction for (1.1) takes nothing away from the study of
high-frequency solutions to this equation, as it is a first step towards
proving the stability of high-frequency perturbation of some background metric
$g_{0}$ close to Minkowski for the equation (1.12), which would require to
incorporate the ideas of [15] to the high-frequency setting presented in this
article.

##  2 Preliminaries

In this section, we describe our geometric and analytic setting, and introduce
the key estimates used in this article.

###  2.1 Notations and function spaces

####  2.1.1 Coordinates and derivatives

We work in $\mathbb{R}_{+}\times\mathbb{R}^{3}$ with the usual global
coordinates system $(t=x_{0},x_{1},x_{2},x_{3})$. We also introduce the null
coordinates

$s=t+r\quad\text{and}\quad q=r-t.$

For $t\geq 0$ we use the standard notation
$\Sigma_{t}\vcentcolon=\\{t\\}\times\mathbb{R}^{3}$. We also define the
following subset of $\mathbb{R}_{+}\times\mathbb{R}^{3}$ for some $R>0$:

$\displaystyle A^{R}$
$\displaystyle\vcentcolon=\left\\{(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{3}~{}\middle|~{}R^{-1}\leq
q\leq R\right\\},$ $\displaystyle B^{R}$
$\displaystyle\vcentcolon=\left\\{(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{3}~{}\middle|~{}q\leq
R\right\\},$

and we set $A_{0}^{R}\vcentcolon=A^{R}\cap\Sigma_{0}$. If
$x=(x_{1},x_{2},x_{3})$ and $r=|x|$, recall that
$\partial_{r}=\sum_{i=1,2,3}\omega_{i}\partial_{i}$ where
$\omega_{i}=\frac{x_{i}}{r}$. If
$f:\mathbb{R}_{+}\times\mathbb{R}^{3}\longrightarrow\mathbb{R}$ is a scalar
function, then we define its spacetime gradient by the vector $\partial
f=(\partial_{t}f,\partial_{1}f,\partial_{2}f,\partial_{3}f)$. The ”good”
derivatives are defined to be

$\overline{\partial}_{0}=\frac{1}{2}(\partial_{t}+\partial_{r})\quad\text{and}\quad\overline{\partial}_{i}=\partial_{i}-\frac{x_{i}}{r}\partial_{r}.$

As we will see, they enjoy better decay than the other derivatives. In this
article we consider $\mathbb{R}^{d}$-valued functions for some $d\geq 1$, i.e
vectorial functions of the form $u=(u_{1},\dots,u_{d})$. For such a function
we define

$\displaystyle|\partial u|=\sum_{\begin{subarray}{c}\alpha=0,\dots,3\\\
i=1,\dots,d\end{subarray}}|\partial_{\alpha}u_{i}|\quad\text{and}\quad|\overline{\partial}u|=\sum_{\begin{subarray}{c}\alpha=0,\dots,3\\\
i=1,\dots,d\end{subarray}}|\overline{\partial}_{\alpha}u_{i}|.$

####  2.1.2 Null forms

Now we define the class of quadratic non-linearties we consider in this
article, i.e the classical null forms introduced by Klainerman.

######  Definition 2.1.

A quadratic form $Q$ is said to be a null quadratic form (also called null
form in the sequel) if it is a linear combination of $Q_{0}$ and
$Q_{\alpha\beta}$ where

$Q_{\alpha\beta}(X,Y)=X_{\alpha}Y_{\beta}-X_{\beta}Y_{\alpha}\quad\text{and}\quad
Q_{0}(X,Y)=-X_{0}Y_{0}+\delta^{ij}X_{i}Y_{j}$

for two vector fields $X,Y$ on $\mathbb{R}_{+}\times\mathbb{R}^{3}$, and
$\alpha,\beta=0,1,2,3$.

One can show that the null forms defined in this way are precisely the
quadratic forms on the tangent bundle of Minkowski spacetime vanishing when
contracted twice with the same null vector field.

Since the spacetime gradient of a scalar function is a vector field on
$\mathbb{R}_{+}\times\mathbb{R}^{3}$, this definition allows us to consider
$Q(\partial u,\partial v)$ for $u,v$ two scalar functions. However in this
article, we would like to consider vectorial functions. Therefore if
$u:\mathbb{R}_{+}\times\mathbb{R}^{3}\longrightarrow\mathbb{R}^{d}$ with
$u=(u_{1},\dots,u_{d})$ and
$v:\mathbb{R}_{+}\times\mathbb{R}^{3}\longrightarrow\mathbb{R}^{d^{\prime}}$
with $v=(v_{1},\dots,v_{d^{\prime}})$, and if $Q_{k\ell}$ are null forms, we
define

$\displaystyle Q(\partial u,\partial
v)\vcentcolon=\sum_{\begin{subarray}{c}k=1,\dots,d\\\
\ell=1,\dots,d^{\prime}\end{subarray}}Q_{k\ell}(\partial u_{k},\partial
v_{\ell}).$ (2.1)

With this notation, we can write down properly the system studied in this
article. For $d\geq 1$, we want to solve the system

$\displaystyle\Box u_{i}=Q_{i}(\partial u,\partial u)$

where $u:\mathbb{R}_{+}\times\mathbb{R}^{3}\longrightarrow\mathbb{R}^{d}$ is a
vectorial function, i.e $u=(u_{1},\dots,u_{d})$, and where $Q_{i}$ is as in
(2.1) for $i=1,\dots,d$. Though the Einstein equations motivates the study of
a system of wave equations in this article, this plays no particular role here
and from now on (with the exception of the statement of the main result) we
will drop the indexes relative to $\mathbb{R}^{d}$ and simply use the
condensed notation

$\displaystyle\Box u=Q(\partial u,\partial u).$

The null forms are in some sense the best quadratic non-linearity we can hope
for in the sense that in each product $\partial u\partial u$, at least one of
the derivatives is a good one, meaning that:

$|Q(\partial u,\partial v)|\lesssim|\partial
u||\overline{\partial}v|+|\overline{\partial}u||\partial v|.$ (2.2)

####  2.1.3 Function spaces

In terms of function spaces, we use the usual $L^{p}$, $C^{m}$ and Sobolev
spaces defined on $\mathbb{R}^{3}$, together with weighted Sobolev spaces,
defined by

######  Definition 2.2.

Let $m\in\mathbb{N}$, $1<p<\infty$, $\delta\in\mathbb{R}$. The weighted
Sobolev space $W^{m,p}_{\delta}$ is the completion of $C^{\infty}_{0}$ under
the norm

$\|u\|_{W^{m,p}_{\delta}}=\sum_{|\beta|\leq m}\left\|\langle
x\rangle^{\delta+\beta}\nabla^{\beta}u\right\|_{L^{p}}.$

We will use the notation $H^{m}_{\delta}=W^{m,2}_{\delta}$. The weighted
Hölder space $C^{m}_{\delta}$ is the completion of $C^{m}$ under the norm

$\|u\|_{C^{m}_{\delta}}=\sum_{|\beta|\leq m}\left\|\langle
x\rangle^{\delta+\beta}\nabla^{\beta}u\right\|_{L^{\infty}}.$

Note that we defined scalar functions spaces. However by setting

$\displaystyle\left\|u\right\|_{X}=\sum_{i=1,\dots,d}\left\|u_{i}\right\|_{X}$

for $u=(u_{1},\dots,u_{d})$ and $X$ any previously defined space, we extend
those spaces and the corresponding norms to $\mathbb{R}^{d}$-valued functions.
A weighted equivalent of the Sobolev embedding holds for those spaces (see [5]
for a proof):

######  Proposition 2.1.

Let $s,m\in\mathbb{N}$. If $s\geq 2$ and $\beta\leq\delta+\frac{3}{2}$, then
we have the continuous embedding

$H^{s+m}_{\delta}\xhookrightarrow{}C^{m}_{\beta}.$

###  2.2 Minkowski vector fields

We consider the set of Minkowski vector fields

$\left\\{\partial_{\alpha},S=t\partial_{t}+r\partial_{r},\Omega_{\alpha\beta}=x_{\alpha}\partial_{\beta}-x_{\beta}\partial_{\alpha}\right\\}$
(2.3)

and denote by $Z^{I}$ any product of $|I|$ elements of this set, where $I$ is
an 11-dimensional integer. Note that since $x_{0}=-x^{0}$, we have
$\Omega_{0i}=-t\partial_{i}-x_{i}\partial_{t}$. These vector fields allow us
to recover the usual derivatives $\partial$ while gaining a weight, whose
nature explain why the derivatives $\overline{\partial}$ are called ”good”
derivatives:

$|\partial
u|\lesssim\frac{1}{1+|q|}\sum_{|I|=1}|Z^{I}u|\quad\text{and}\quad|\overline{\partial}u|\lesssim\frac{1}{1+s}\sum_{|I|=1}|Z^{I}u|.$
(2.4)

The Minkowski vector fields enjoy good commutativity properties with the wave
operator:

$[\Box,Z]=0$

except for $S$, which satisfies $[\Box,S]=2\Box$. This implies that

$|\Box Z^{I}u|\lesssim\sum_{|I^{\prime}|\leq|I|}|Z^{I^{\prime}}\Box u|.$ (2.5)

The Minkowski vector fields behave nicely with the null forms, in the sense
that the following holds:

$|Z^{I}Q(\partial u,\partial
v)|\lesssim\sum_{|I_{1}|+|I_{2}|\leq|I|}|Q(\partial Z^{I_{1}}u,\partial
Z^{I_{2}}v)|$ (2.6)

###  2.3 Useful estimates

In this section we collect different estimates crucially used in this paper.

####  2.3.1 Weighted Klainerman-Sobolev inequality

We start by an estimate which allows us to bound the $L^{\infty}$ norm of a
function knowing the $L^{2}$ norm of some of its derivatives with weights.

######  Proposition 2.2.

Consider the weight

$w(q)\vcentcolon=\left\\{\begin{array}[]{c}1+(1+|q|)^{-\alpha}\quad\text{for}\quad
q<0\\\ 1+(1+|q|)^{\beta}\quad\text{for}\quad q>0\end{array}\right.$

with $1<\beta<3$ and $\alpha>0$. Let $f$ be compactly supported, we have

$|w^{\frac{1}{2}}(q)f|\lesssim\frac{1}{(1+s)\sqrt{1+|q|}}\sum_{|I|\leq
3}\left\|w^{\frac{1}{2}}(q)Z^{I}f\right\|_{L^{2}}.$ (2.7)

This is the so-called weighted Klainerman-Sobolev inequality, for which a
proof can be found in [15]. The assumption on $f$ in the previous proposition
can be relaxed: the proposition holds as long as the RHS of (2.7) is finite.

####  2.3.2 Weighted energy estimate

As is standard, our strategy of proof relies on energy estimates for the wave
operator $\Box$.

######  Lemma 2.1.

Let $w(q)$ be a non-decreasing weight function. Then

$\int_{\Sigma_{t}}|\partial\varphi|^{2}w(q)\mathrm{d}x+\int_{0}^{t}\int_{\Sigma_{\tau}}|\overline{\partial}\varphi|^{2}w^{\prime}(q)\mathrm{d}x\mathrm{d}\tau\lesssim\int_{\Sigma_{0}}|\partial\varphi|^{2}w(q)\mathrm{d}x+\int_{0}^{t}\int_{\Sigma_{\tau}}|\Box\varphi\partial_{t}\varphi|w(q)\mathrm{d}x\mathrm{d}\tau.$

This lemma is the semi-linear equivalent of Lemma 6.1 in [15]. It gives a
weighted energy estimate, with a weight depending only on $q$. This is called
the ghost weight method and was first introduced by Alinhac in [1]. Depending
on the choice of the weight, it gives us extra decay for solutions of a wave
equation in the region $q>0$, i.e the exterior of the light-cone. The other
advantage of this method is the presence of a space-time integral on the LHS
involving only the ”good” derivatives of the solution.

The usual energy estimate for the wave operator is obtained when $w(q)\equiv
1$, and one of the standard application of this case is the finite speed of
propagation for solutions of semi-linear wave equation.

######  Lemma 2.2.

We define

$C_{t_{0},x_{0}}=\left\\{(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{3}~{}\middle|~{}0\leq
t\leq t_{0}\quad\text{and}\quad|x-x_{0}|\leq t_{0}-t\right\\}$

and $B(x_{0},t_{0})$ is the ball in $\mathbb{R}^{3}$ centered at $x_{0}$ of
radius $t_{0}$. Let $P:\mathbb{R}^{3+1}\to\mathbb{R}$ a smooth function such
that

$|P(X)|\lesssim|X|+|X|^{2}.$

Let $(t_{0},x_{0})\in\mathbb{R}_{+}\times\mathbb{R}^{3}$ and
$u:\mathbb{R}^{3+1}\to\mathbb{R}$ a solution of $\Box u=P(\partial u)$ in
$C_{t_{0},x_{0}}$. If $(u,\partial_{t}u)$ initially vanishes on
$B(x_{0},t_{0})$, then $u_{|C_{t_{0},x_{0}}}=0$.

This lemma will be used to localize the different terms in the decomposition
of our solution (see (4.1)).

##  3 Main result

We now state the main result of this article.

######  Theorem 3.1.

Let $d\geq 1$, $N\geq 26$, $R>0$, $-\frac{1}{2}<\delta<\frac{1}{2}$, $F_{0}\in
C^{N-4}(\mathbb{R}^{3},\mathbb{R}^{d})$ supported in $A_{0}^{R}$ and
$(\varphi_{0},\varphi_{1})\in
H^{N+1}_{\delta}(\mathbb{R}^{3},\mathbb{R}^{d})\times
H^{N}_{\delta+1}(\mathbb{R}^{3},\mathbb{R}^{d})$ such that

$\|F_{0}\|_{C^{N-4}}+\left\|\varphi_{0}\right\|_{H^{N+1}_{\delta}}+\left\|\varphi_{1}\right\|_{H^{N}_{\delta+1}}\leq\varepsilon.$

There exists $\varepsilon_{0}=\varepsilon_{0}(N,R,\delta)>0$ such that if
$\varepsilon\leq\varepsilon_{0}$ and $\lambda\in(0,1]$ there exists a unique
global solution
$\Phi_{\lambda}:\mathbb{R}_{+}\times\mathbb{R}^{3}\longrightarrow\mathbb{R}^{d}$
to

$\left\\{\begin{array}[]{rcl}\Box\Phi_{\lambda,i}&=&Q_{i}(\partial\Phi_{\lambda},\partial\Phi_{\lambda})\\\
\Phi_{\lambda|t=0}&=&\varphi_{0}+\lambda
F_{0}\cos\left(\frac{r}{\lambda}\right)\\\
\partial_{t}\Phi_{\lambda|t=0}&=&\varphi_{1}+F_{0}\sin\left(\frac{r}{\lambda}\right)+\lambda\tilde{F}_{0,\lambda}\end{array}\right.$

where $Q_{i}$ is a null form and $\tilde{F}_{0,\lambda}$ is supported in
$A^{R}_{0}$ depending only on $(\varphi_{0},\varphi_{1},F_{0})$ and which
satisfies
$\left\|\tilde{F}_{0,\lambda}\right\|_{L^{\infty}}\lesssim\varepsilon$. The
function $\Phi_{\lambda}$ admits the decomposition

$\Phi_{\lambda}=\varphi+\lambda
F\cos\left(\frac{t-r}{\lambda}\right)+\lambda^{2}\tilde{F}_{\lambda},$

where

  * •

$\varphi\in H^{N+1}(\mathbb{R}_{+}\times\mathbb{R}^{3},\mathbb{R}^{d})$
satisfies $\Box\varphi_{i}=Q_{i}(\partial\varphi,\partial\varphi)$ and
$(\varphi,\partial_{t}\varphi)_{|t=0}=(\varphi_{0},\varphi_{1})$,

  * •

$F\in C^{N-4}(\mathbb{R}_{+}\times\mathbb{R}^{3},\mathbb{R}^{d})$ is supported
in $A^{R}$ and satisfies

$\displaystyle\left(\partial_{t}+\partial_{r}+\frac{1}{r}\right)F_{i}=\frac{1}{2}\sum_{k,\ell=1,\dots,d}F_{k}Q_{i}(\partial(t-r),\partial\varphi_{\ell})$

and $F_{|t=0}=F_{0}$,

  * •

$\tilde{F}_{\lambda}$ is supported in $B^{R}$ and it is such that
$|\tilde{F}_{\lambda}|\lesssim\varepsilon$ and
$|\partial\tilde{F}_{\lambda}|\lesssim\frac{\varepsilon}{\lambda}$.

######  Remark 3.1.

The main feature of this theorem is that the smallness constant
$\varepsilon_{0}$ does not depend on $\lambda$, which is allowed to take any
value in $(0,1]$. In addition to the nature of the decomposition of
$\Phi_{\lambda}$, this is at the heart of the high-frequency nature of the
solution we construct: if we let $\lambda$ tends to 0, the high order norms of
$\Phi_{\lambda}$ (precisely the $W^{\infty,p}$ norm for $p>1$) blows up but
this is not compensated by letting $\varepsilon_{0}$ go to 0 as well.
Therefore, the limit of $\Phi_{\lambda}$ as $\lambda$ tends to 0 is
meaningful, and we have:

$\displaystyle\Phi_{\lambda}$ $\displaystyle\to\varphi,\quad\text{uniformly in
$L^{\infty}$,}$ $\displaystyle\partial\Phi_{\lambda}$
$\displaystyle\rightharpoonup\partial\varphi,\quad\text{weakly in $L^{2}$.}$

##  4 Construction of the ansatz

The heart of the proof of Theorem 3.1 is to construct a high-frequency ansatz
for the solution $\Phi_{\lambda}$. The main purpose of this ansatz is to
capture the creation of higher-order harmonics through the quadratic non-
linearity. Those harmonics may also include non-oscillating terms. As
explained in the introduction, we choose an ansatz with a half-chessboard
shape. More precisely we consider a solution $\Phi_{\lambda}$ of the form

$\Phi_{\lambda}=\varphi+\sum_{2\leq k\leq
K-1}\lambda^{k}\psi^{(k)}+\sum_{1\leq i\leq k\leq
K}\lambda^{k}F^{(k,i)}\mathbb{T}_{\lambda}^{(k,i)}+h_{\lambda},$ (4.1)

where $K\geq 2$ is an integer and where we use the following notation for and
$k,i\in\mathbb{Z}$:

$\mathbb{T}_{\lambda}^{(k,i)}=\left\\{{\begin{array}[]{cc}\sin\left(\frac{i(t-r)}{\lambda}\right)&\textrm{
if }k+i\textrm{ is odd,}\\\ \cos\left(\frac{i(t-r)}{\lambda}\right)&\textrm{
if }k+i\textrm{ is even.}\end{array}}\right.$ (4.2)

Besides from being at the very heart of what we call an half-chessboard shaped
ansatz, this notation is useful to capture the behaviours of the standard
trigonometric functions regarding products and derivation (see Lemma 4.1). A
bit of vocabulary:

  * •

$\varphi$ is called the background solution,

  * •

the functions $\psi^{(\ell)}$ are called the non-oscillating terms of the
ansatz, whereas $F^{(k,i)}\mathbb{T}_{\lambda}^{(k,i)}$ are the oscillating
terms,

  * •

the function $h_{\lambda}$ is the remainder of the ansatz and is non-
oscillating, however it mimics a high-frequency behaviour, i.e it loses one
power of $\lambda$ for each derivatives, a fact clearly seen on the schematic
estimates it will satisfy:

$\left\|\partial^{m}h_{\lambda}\right\|_{L^{2}}\lesssim\lambda^{K+1-m}\quad\text{for}\quad
m\geq 1.$

The degree of precision of the ansatz (4.1) is given by the integer $K$. Our
work is valid for all values of $K\geq 2$, but shows that global existence as
a solution of (1.1) requires $K\geq 4$. As explained in the introduction, this
is due to the Klainerman-Sobolev inequality.

######  Remark 4.1.

Since we adopt the standard convention that sums over empty sets vanish, there
are no $\psi^{(\ell)}$ functions in the ansatz if $K=2$. However, one major
interest of our result is precisely the freedom we have on $K$: even though we
are interested in a lower bound on $K$ to ensure the global existence of
$\Phi_{\lambda}$, our proof also shows that we can describe the propagation of
high-frequency waves with an arbitrary large precision.

######  Remark 4.2.

In the rest of the article we will use the following convention:
$\psi^{(1)}=0$ and $F^{(k,i)}=0$ if $i>k$ or $k=0$. This will be useful in the
sequel not to be too bothered with complicated sums.

Our strategy of proof is to derive formally from
$\Box\Phi_{\lambda}=Q(\partial\Phi_{\lambda},\partial\Phi_{\lambda})$ a system
of equations for the unknowns $(\varphi,\psi^{(\ell)},F^{(k,i)},h_{\lambda})$
and prove global existence for this system, called in the rest of this article
the reduced system. Let us start by stating a technical lemma summarizing some
basic properties of the trigonometric functions $\mathbb{T}_{\lambda}^{(k,i)}$
regarding products, the wave operator and the null forms.

######  Remark 4.3.

In the next lemma and in the rest of the article (including the technical next
section), we won’t write the numerical constants which appear in the
computations, meaning for example that an equality like (4.3) has to be
understood in the following sense: there exists real constants
$A=A(k,\ell,i,j)$ and $B=B(k,\ell,i,j)$ such that

$\mathbb{T}_{\lambda}^{(k,i)}\mathbb{T}_{\lambda}^{(\ell,j)}=A\mathbb{T}_{\lambda}^{(k+\ell,i+j)}+B\mathbb{T}_{\lambda}^{(k-\ell,i-j)}.$

Though these numerical constants can depend on every indexes appearing in
(4.1), they are not allowed to depend on $\lambda$, meaning that we will
always keep trace of $\lambda$ in the formal computations.

######  Lemma 4.1.

Let $k,\ell,i,j\in\mathbb{Z}$, $f$ and $g$ any function and $Q$ any null form.
We have $\mathbb{T}_{\lambda}^{(k,i)}=\mathbb{T}_{\lambda}^{(|k|,|i|)}$ and

$\mathbb{T}_{\lambda}^{(k,i)}\mathbb{T}_{\lambda}^{(\ell,j)}=\mathbb{T}_{\lambda}^{(k+\ell,i+j)}+\mathbb{T}_{\lambda}^{(k+\ell,i-j)},$
(4.3) $\Box\left(g\mathbb{T}_{\lambda}^{(k,i)}\right)=\Box
g\mathbb{T}_{\lambda}^{(k,i)}+\frac{1}{\lambda}\mathscr{L}g\mathbb{T}_{\lambda}^{(k-1,i)},$
(4.4) $Q\left(\partial\left(f\mathbb{T}_{\lambda}^{(k,i)}\right),\partial
g\right)=Q(\partial f,\partial
g)\mathbb{T}_{\lambda}^{(k,i)}+\frac{1}{\lambda}f\omega\overline{\partial}g\mathbb{T}_{\lambda}^{(k-1,i)},$
(4.5) $\displaystyle
Q\left(\partial\left(f\mathbb{T}_{\lambda}^{(k,i)}\right),\partial\left(g\mathbb{T}_{\lambda}^{(\ell,j)}\right)\right)$
$\displaystyle=Q(\partial f,\partial
g)\mathbb{T}_{\lambda}^{(k,i)}\mathbb{T}_{\lambda}^{(\ell,j)}$ (4.6)
$\displaystyle\quad+\frac{1}{\lambda}f\omega\overline{\partial}g\mathbb{T}_{\lambda}^{(k-1,i)}\mathbb{T}_{\lambda}^{(\ell,j)}+\frac{1}{\lambda}g\omega\overline{\partial}f\mathbb{T}_{\lambda}^{(k,i)}\mathbb{T}_{\lambda}^{(\ell-1,j)},$

where $\mathscr{L}$ is the following transport operator

$\mathscr{L}\vcentcolon=\partial_{t}+\partial_{r}+\frac{1}{r}$

and where $\omega\overline{\partial}$ stands for a linear combination of
$\omega_{i}\overline{\partial}_{\alpha}$ for $i=1,2,3$ and $\alpha=0,1,2,3$.

###### Proof.

The proof of (4.3) is a direct rewriting with our notation (4.2) of the usual
linearization formulas

$\displaystyle 2\cos(a)\cos(b)$ $\displaystyle=\cos(a-b)+\cos(a+b),$
$\displaystyle 2\sin(a)\cos(b)$ $\displaystyle=\sin(a-b)+\sin(a+b),$
$\displaystyle 2\sin(a)\sin(b)$ $\displaystyle=\cos(a-b)-\cos(a+b).$

The proof of (4.4) follows from a direct computation, and even though $\Box$
is a second order operator, we don’t lose two powers of $\lambda$ since the
phase $t-r$ satisfies the eikonal equation for the Minkowski metric, i.e

$\displaystyle m^{\alpha\beta}\partial_{\alpha}(t-r)\partial_{\beta}(t-r)=0.$
(4.7)

The proof of (4.5) follows from the computations

$\displaystyle Q_{0}(\partial(t-r),\partial h)$
$\displaystyle=-2\overline{\partial}_{0}h,$ $\displaystyle
Q_{0i}(\partial(t-r),\partial h)$
$\displaystyle=\overline{\partial}_{i}h+2\omega_{i}\overline{\partial}_{0}h,$
$\displaystyle Q_{ij}(\partial(t-r),\partial h)$
$\displaystyle=\omega_{j}\overline{\partial}_{i}h-\omega_{i}\overline{\partial}_{j}h,$

(which holds for any scalar function $h$) and the definition of a null form as
in Definition 2.1.

The formula (4.6) follows directly from (4.5). ∎

######  Remark 4.4.

Note that even though a null form only involves first order derivatives, we
would expect some $\frac{1}{\lambda^{2}}$ terms in (4.6) since $Q(\partial
u,\partial v)$ is a quadratic expression. It is not the case since null forms
are precisely the quadratic forms vanishing on null vector fields in Minkowski
space, and (4.7) precisely means that the space-time gradient of $t-r$ is a
null vector field.

######  Remark 4.5.

Since $|\partial^{|\alpha|}\omega_{i}|\lesssim 1$ in the region
$\\{r\geq\frac{1}{R}\\}$ for any multi-index $\alpha$ and since formulas (4.5)
and (4.6) will be applied with $f$ supported in $A^{R}$, the $\omega$ factors
in (4.5) and (4.6) are irrelevant in our estimates and we omit them in the
sequel.

###  4.1 High-frequency expansion of the semi-linear wave equation

In this section, we plug the ansatz (4.1) into the equation

$\displaystyle\Box\Phi_{\lambda}=Q(\partial\Phi_{\lambda},\partial\Phi_{\lambda}).$
(4.8)

The computations are rather tedious, and we present them in the form of two
lemmas, one for the LHS of (4.8) and one for its RHS.

####  4.1.1 Expansion of the wave operator

We start with the RHS of (4.8).

######  Lemma 4.2.

If $\Phi_{\lambda}$ is defined by (4.1) we have

$\displaystyle\Box\Phi_{\lambda}$ $\displaystyle=\sum_{0\leq p\leq
K}\lambda^{p}\left(\Box\Phi_{\lambda}\right)^{(p)}$

with

$\displaystyle\left(\Box\Phi_{\lambda}\right)^{(0)}$
$\displaystyle=\Box\varphi+\mathscr{L}F^{(1,1)}\mathbb{T}_{\lambda}^{(0,1)}$
(4.9) $\displaystyle\left(\Box\Phi_{\lambda}\right)^{(K)}$
$\displaystyle=\sum_{1\leq i\leq K}\Box
F^{(K,i)}\mathbb{T}_{\lambda}^{(K,i)}+\frac{1}{\lambda^{K}}\Box h_{\lambda}$
(4.10)

and

$\displaystyle\left(\Box\Phi_{\lambda}\right)^{(p)}$
$\displaystyle=\sum_{1\leq i\leq p+1}\left(\mathscr{L}F^{(p+1,i)}+\Box
F^{(p,i)}\right)\mathbb{T}_{\lambda}^{(p,i)}+\Box\psi^{(p)}$ (4.11)

for $1\leq p\leq K-1$.

The proof of this lemma is left to the reader and is an application of (4.4)
together with the convention stated in Remark 4.2.

####  4.1.2 Expansion of the null form

######  Lemma 4.3.

If $\Phi_{\lambda}$ is defined by (4.1) we have

$\displaystyle Q(\partial\Phi_{\lambda},\partial\Phi_{\lambda})$
$\displaystyle=\sum_{0\leq k\leq K-1}\lambda^{k}Q^{(k)}+\lambda^{K}Q^{(\geq
K)}$

with

$\displaystyle Q^{(0)}$
$\displaystyle=Q(\partial\varphi,\partial\varphi)+F^{(1,1)}\overline{\partial}\varphi\mathbb{T}_{\lambda}^{(0,1)},$
(4.12) $\displaystyle Q^{(\geq K)}$
$\displaystyle=\frac{1}{\lambda^{K}}\mathcal{N}(h_{\lambda})+Q(\partial
F,\partial\varphi)\mathbb{T}_{\lambda}$ (4.13)
$\displaystyle\quad+\lambda^{+}\left(F\overline{\partial}F\mathbb{T}_{\lambda}+Q(\partial
F,\partial F)\mathbb{T}_{\lambda}+Q(\partial
F,\partial\psi)\mathbb{T}_{\lambda}+F\overline{\partial}\psi\mathbb{T}_{\lambda}+Q(\partial\psi,\partial\psi)\right),$

and

$\displaystyle Q^{(p)}$
$\displaystyle=Q(\partial\psi^{(p)},\partial\varphi)+\sum_{2\leq k\leq
p-2}Q(\partial\psi^{(k)},\partial\psi^{(p-k)})$ (4.14)
$\displaystyle\;+\sum_{1\leq i\leq
p+1}\left(F^{(p+1,i)}\overline{\partial}\varphi+Q(\partial
F^{(p,i)},\partial\varphi)\right)\mathbb{T}_{\lambda}^{(p,i)}$
$\displaystyle\;+\sum_{\begin{subarray}{c}1\leq i\leq k\leq p\\\ 1\leq j\leq
p-k+1\end{subarray}}\left(Q(\partial F^{(k,i)},\partial
F^{(p-k,j)})+F^{(k,i)}\overline{\partial}F^{(p-k+1,j)}+F^{(p-k+1,j)}\overline{\partial}F^{(k,i)}\right)\left(\mathbb{T}_{\lambda}^{(p,i+j)}+\mathbb{T}_{\lambda}^{(p,i-j)}\right)$
$\displaystyle\;+\sum_{\begin{subarray}{c}0\leq k\leq p-1\\\ 1\leq i\leq
k+1\end{subarray}}\mathbb{T}_{\lambda}^{(k,i)}\left(F^{(k+1,i)}\overline{\partial}\psi^{(p-k)}+Q(\partial
F^{(k,i)},\partial\psi^{(p-k)})\right)$

for $1\leq p\leq K-1$, and where $\mathcal{N}(h_{\lambda})$ is defined in
(4.16).

######  Remark 4.6.

In this lemma, we use the following notations for clarity:

  * •

$F$ or $\psi$ denotes linear combinations of terms of the families $F^{(k,i)}$
or $\psi^{(\ell)}$,

  * •

$\mathbb{T}_{\lambda}$ denotes linear combinations of terms of the family
$\mathbb{T}^{(k,i)}_{\lambda}$,

  * •

in front of a linear combination, the symbol $\lambda^{+}$ means that the
coefficients of the linear combination contains some positive powers of
$\lambda$.

###### Proof.

Using (4.5)-(4.6), we obtain by direct expansion

$\displaystyle Q(\partial\Phi_{\lambda},\partial\Phi_{\lambda})$
$\displaystyle=Q(\partial\varphi,\partial\varphi)+Q(\partial
h_{\lambda},\partial h_{\lambda})+Q(\partial
h_{\lambda},\partial\varphi)+\lambda^{+}Q(\partial(F\mathbb{T}_{\lambda}),\partial
h_{\lambda})+\lambda^{+}Q(\partial\psi,\partial h_{\lambda})$
$\displaystyle\quad+\sum_{1\leq i\leq k\leq K}\lambda^{k}Q(\partial
F^{(k,i)},\partial\varphi)\mathbb{T}_{\lambda}^{(k,i)}+\sum_{1\leq i\leq k\leq
K}\lambda^{k-1}F^{(k,i)}\overline{\partial}\varphi\mathbb{T}_{\lambda}^{(k-1,i)}$
$\displaystyle\quad+\sum_{2\leq k\leq
K-1}\lambda^{k}Q(\partial\psi^{(k)},\partial\varphi)$
$\displaystyle\quad+\sum_{\begin{subarray}{c}1\leq k,\ell\leq K\\\ 1\leq i\leq
k\\\ 1\leq
j\leq\ell\end{subarray}}\lambda^{k+\ell-1}\left(F^{(k,i)}\overline{\partial}F^{(\ell,j)}+F^{(\ell,j)}\overline{\partial}F^{(k,i)}\right)\left(\mathbb{T}_{\lambda}^{(k+\ell-1,i+j)}+\mathbb{T}_{\lambda}^{(k+\ell-1,i-j)}\right)$
(4.15) $\displaystyle\quad+\sum_{\begin{subarray}{c}1\leq k,\ell\leq K\\\
1\leq i\leq k\\\ 1\leq j\leq\ell\end{subarray}}\lambda^{k+\ell}Q(\partial
F^{(k,i)},\partial
F^{(\ell,j)})\left(\mathbb{T}_{\lambda}^{(k+\ell,i+j)}+\mathbb{T}_{\lambda}^{(k+\ell,i-j)}\right)$
$\displaystyle\quad+\sum_{\begin{subarray}{c}1\leq i\leq k\leq K\\\
2\leq\ell\leq K-1\end{subarray}}\lambda^{k+\ell}Q(\partial
F^{(k,i)},\partial\psi^{(\ell)})\mathbb{T}_{\lambda}^{(k,i)}+\sum_{\begin{subarray}{c}1\leq
i\leq k\leq K\\\ 2\leq\ell\leq
K-1\end{subarray}}\lambda^{k+\ell-1}F^{(k,i)}\overline{\partial}\psi^{(\ell)}\mathbb{T}_{\lambda}^{(k-1,i)}$
$\displaystyle\quad+\sum_{2\leq k,\ell\leq
K-1}\lambda^{k+\ell}Q(\partial\psi^{(k)},\partial\psi^{(\ell)})$

Note that in order to simplify the terms $F\overline{\partial}F$ we used (4.3)
and its consequence

$\displaystyle\mathbb{T}_{\lambda}^{(k-1,i)}\mathbb{T}_{\lambda}^{(\ell,j)}=\mathbb{T}_{\lambda}^{(k,i)}\mathbb{T}_{\lambda}^{(\ell-1,j)}=\mathbb{T}_{\lambda}^{(k+\ell-1,i+j)}+\mathbb{T}_{\lambda}^{(k+\ell-1,i-j)}.$

Now we want to regroup terms by their $\lambda$ powers. For this we use the
following facts

$\displaystyle\sum_{1\leq k,\ell\leq K}\lambda^{k+\ell}a_{k,\ell}$
$\displaystyle=\sum_{2\leq p\leq K-1}\lambda^{p}\sum_{1\leq k\leq
p-1}a_{k,p-k}+O\left(\lambda^{K}\right)$
$\displaystyle\sum_{\begin{subarray}{c}1\leq k\leq K\\\ 2\leq\ell\leq
K-1\end{subarray}}\lambda^{k+\ell}a_{k,\ell}$ $\displaystyle=\sum_{3\leq p\leq
K-1}\lambda^{p}\sum_{1\leq k\leq p-2}a_{k,p-k}+O\left(\lambda^{K}\right)$
$\displaystyle\sum_{2\leq k,\ell\leq K-1}\lambda^{k+\ell}a_{k,\ell}$
$\displaystyle=\sum_{4\leq p\leq K-1}\lambda^{p}\sum_{2\leq k\leq
p-2}a_{k,p-k}+O\left(\lambda^{K}\right)$

which holds for every families $(a_{k,\ell})_{(k,\ell)\in\mathbb{N}^{2}}$.
Therefore if $1\leq p\leq K-1$ we have

$\displaystyle Q^{(p)}$ $\displaystyle=\sum_{1\leq i\leq p}Q(\partial
F^{(p,i)},\partial\varphi)\mathbb{T}_{\lambda}^{(p,i)}+\sum_{1\leq i\leq
p+1}F^{(p+1,i)}\overline{\partial}\varphi\mathbb{T}_{\lambda}^{(p,i)}+Q(\partial\psi^{(p)},\partial\varphi)$
$\displaystyle\quad+\sum_{\begin{subarray}{c}1\leq i\leq k\leq p-1\\\ 1\leq
j\leq p-k\end{subarray}}Q(\partial F^{(k,i)},\partial
F^{(p-k,j)})\left(\mathbb{T}_{\lambda}^{(p,i+j)}+\mathbb{T}_{\lambda}^{(p,i-j)}\right)$
$\displaystyle\quad+\sum_{1\leq i\leq k\leq p-1}Q(\partial
F^{(k,i)},\partial\psi^{(p-k)})\mathbb{T}_{\lambda}^{(k,i)}+\sum_{2\leq k\leq
p-2}Q(\partial\psi^{(k)},\partial\psi^{(p-k)})$
$\displaystyle\quad+\sum_{\begin{subarray}{c}1\leq i\leq k\leq p\\\ 1\leq
j\leq
p-k+1\end{subarray}}\left(F^{(k,i)}\overline{\partial}F^{(p-k+1,j)}+F^{(p-k+1,j)}\overline{\partial}F^{(k,i)}\right)\left(\mathbb{T}_{\lambda}^{(p,i+j)}+\mathbb{T}_{\lambda}^{(p,i-j)}\right)$
$\displaystyle\quad+\sum_{1\leq i\leq k\leq
p-1}F^{(k,i)}\overline{\partial}\psi^{(p-k+1)}\mathbb{T}_{\lambda}^{(k-1,i)}$

where we used several times our convention on $\psi^{(1)}$ and on sums over
empty sets. If we regroup terms according to their oscillating behaviour in
this expression we obtain (4.14).

It remains to compute $Q^{(0)}$ and $Q^{(\geq K)}$. For $Q^{(0)}$, we simply
look at (4.15). For $Q^{(\geq K)}$, we don’t need to be precise in terms of
the oscillating behaviour and the hierarchy since $\Box h_{\lambda}$ will
absorb everything. This explains the expression of the lemma, with the
following definition:

$\displaystyle\mathcal{N}(h_{\lambda})=Q(\partial h_{\lambda},\partial
h_{\lambda})+Q(\partial
h_{\lambda},\partial\varphi)+\lambda^{+}Q(\partial(F\mathbb{T}_{\lambda}),\partial
h_{\lambda})+\lambda^{+}Q(\partial\psi,\partial h_{\lambda}).$ (4.16)

∎

###  4.2 The reduced system

As the two previous lemmas show, solving

$\Box\Phi_{\lambda}=Q(\partial\Phi_{\lambda},\partial\Phi_{\lambda})$

is equivalent to solving the following high-frequency hierarchy:

$\displaystyle(\Box\Phi_{\lambda})^{(p)}=Q^{(p)}$ (4.17)

for $0\leq p\leq K-1$ and

$\displaystyle(\Box\Phi_{\lambda})^{(K)}=Q^{(\geq K)}.$ (4.18)

Since $h_{\lambda}$ is not-oscillating, (4.18) simply rewrites as a semi-
linear wave equation with oscillating coefficients:

$\displaystyle\Box h_{\lambda}$
$\displaystyle=\mathcal{N}(h_{\lambda})+\lambda^{K}Q(\partial
F,\partial\varphi)\mathbb{T}_{\lambda}+\lambda^{K}\Box F\mathbb{T}_{\lambda}$
$\displaystyle\quad+\lambda^{K}\lambda^{+}\left(F\overline{\partial}F\mathbb{T}_{\lambda}+Q(\partial
F,\partial F)\mathbb{T}_{\lambda}+Q(\partial
F,\partial\psi)\mathbb{T}_{\lambda}+F\overline{\partial}\psi\mathbb{T}_{\lambda}+Q(\partial\psi,\partial\psi)\right)$

For the remaining equations of order $\lambda^{p}$ for $0\leq p\leq K-1$, note
that both the LHS and RHS contain non-oscillating and oscillating terms, which
we have to identify precisely.

For $p=0$ it is quite straightforward: the non-oscillating part can be
absorbed by the background equation

$\displaystyle\Box\varphi=Q(\partial\varphi,\partial\varphi)$

and the part oscillating like $\mathbb{T}_{\lambda}^{(0,1)}$ is absorbed
thanks to a transport equation for $F^{(1,1)}$:

$\displaystyle\mathscr{L}F^{(1,1)}=F^{(1,1)}\overline{\partial}\varphi.$

If $1\leq p\leq K-1$, the distribution between oscillating and non-oscillating
terms is less clear, especially since the interaction between the different
$F^{(k,i)}$ leads to the creation of non-oscillating terms. This is a
consequence of the presence of $\mathbb{T}_{\lambda}^{(p,i-j)}$ in (4.14),
which is non-oscillating if and only if $p$ is even and $i=j$. These created
non-oscillating terms are the motivation behind the terms $\psi^{(\ell)}$,
indeed they are absorbed by $\Box\psi^{(\ell)}$ present in (4.11). Therefore,
since $p$ has to be even so that $\mathbb{T}_{\lambda}^{(p,i-j)}$ is non-
oscillating, we only need $\psi^{(\ell)}$ for $\ell$ even and we take
$\psi^{(\ell)}=0$ for $\ell$ odd.

This has the following nice consequence: in the last term of $Q^{(p)}$ (for
$1\leq p\leq K-1$)

$\displaystyle\sum_{\begin{subarray}{c}0\leq k\leq p-1\\\ 1\leq i\leq
k+1\end{subarray}}\mathbb{T}_{\lambda}^{(k,i)}\left(F^{(k+1,i)}\overline{\partial}\psi^{(p-k)}+Q(\partial
F^{(k,i)},\partial\psi^{(p-k)})\right)$

we now impose that $p-k$ is even, which implies
$\mathbb{T}_{\lambda}^{(k,i)}=\mathbb{T}_{\lambda}^{(p,i)}$ (since $p$ and $k$
have the same parity). Therefore, the oscillating terms in $Q^{(p)}$ are all
of the form $\mathbb{T}_{\lambda}^{(p,i)}$ for $1\leq i\leq p+1$ and we can
rewrite $Q^{(p)}$ in the following more condensed form

$\displaystyle Q^{(p)}$ $\displaystyle=\sum_{1\leq i\leq
p+1}Q^{(p,i)}\mathbb{T}_{\lambda}^{(p,i)}+\pi Q^{(p)}$ (4.19)

where

$\displaystyle Q^{(p,i)}$
$\displaystyle=F^{(p+1,i)}\overline{\partial}\varphi+Q(\partial F^{(\leq
p)},\partial\varphi)+Q(\partial F^{(\leq p)},\partial F^{(\leq p)})$
$\displaystyle\quad+F^{(\leq p)}\overline{\partial}F^{(\leq p)}+F^{(\leq
p)}\overline{\partial}\psi^{(\leq p)}+Q(\partial F^{(\leq
p)},\partial\psi^{(\leq p)})$

and

$\displaystyle\pi
Q^{(p)}=Q(\partial\psi^{(p)},\partial\varphi)+Q(\partial\psi^{(\leq
p-2)},\partial\psi^{(\leq p-2)})+Q(\partial F^{(\leq p)},\partial F^{(\leq
p)})+F^{(\leq p)}\overline{\partial}F^{(\leq p)}.$

######  Remark 4.7.

Here, we used the useful notation $\psi^{(\leq i)}$ to denote any $\psi^{(j)}$
for $j\leq i$. Samewise, $F^{(\leq k)}$ denotes any $F^{(\ell,i)}$ for $1\leq
i\leq\ell\leq k$.

Recall that our previous remark on the presence of non-oscillating terms
basically means that $\pi Q^{(p)}=0$ if $p$ is odd. Thanks to the
decomposition (4.19), we see that both sides of (4.17) oscillate in a similar
manner, and we can assume the following equations on $\psi^{(p)}$ and
$F^{(p+1,i)}$:

$\displaystyle\mathscr{L}F^{(p+1,i)}+\Box
F^{(p,i)}=Q^{(p,i)}\quad\text{and}\quad\Box\psi^{(p)}=\pi Q^{(p)}.$

By collecting the results of this discussion, we define the reduced system for
the unknowns $(\varphi,\psi^{(\ell)},F^{(k,i)},h_{\lambda})$ to be

$\displaystyle\Box\varphi$ $\displaystyle=Q(\partial\varphi,\partial\varphi),$
(BG) $\displaystyle\mathscr{L}F^{(k,i)}$ $\displaystyle=\Box
F^{(k-1)}+F^{(k,i)}\overline{\partial}\varphi+Q(\partial F^{(\leq
k-1)},\partial\varphi)+Q(\partial F^{(\leq k-1)},\partial F^{(\leq k-1)})$
(T$(k,i)$) $\displaystyle\quad+F^{(\leq k-1)}\overline{\partial}F^{(\leq
k-1)}+F^{(\leq k-1)}\overline{\partial}\psi^{(\leq k-1)}+Q(\partial F^{(\leq
k-1)},\partial\psi^{(\leq k-1)}),$ $\displaystyle\Box\psi^{(k)}$
$\displaystyle=Q(\partial\psi^{(k)},\partial\varphi)+Q(\partial\psi^{(\leq
k-2)},\partial\psi^{(\leq k-2)})+Q(\partial F^{(\leq k)},\partial F^{(\leq
k)})+F^{(\leq k)}\overline{\partial}F^{(\leq k)},$ (W$(k)$) $\displaystyle\Box
h_{\lambda}$ $\displaystyle=\mathcal{N}(h_{\lambda})+\lambda^{K}Q(\partial
F,\partial\varphi)\mathbb{T}_{\lambda}+\lambda^{K}\Box F\mathbb{T}_{\lambda}$
(W)
$\displaystyle\quad+\lambda^{K}\lambda^{+}\left(F\overline{\partial}F\mathbb{T}_{\lambda}+Q(\partial
F,\partial F)\mathbb{T}_{\lambda}+Q(\partial
F,\partial\psi)\mathbb{T}_{\lambda}+F\overline{\partial}\psi\mathbb{T}_{\lambda}+Q(\partial\psi,\partial\psi)\right).$

The initial data for $(\varphi,\psi^{(\ell)},F^{(k,i)},h_{\lambda})$ are given
by

$\displaystyle(\varphi,\partial_{t}\varphi)_{|t=0}$
$\displaystyle=(\varphi_{0},\varphi_{1}),$ $\displaystyle
F^{(k,i)}_{\;\;\;\quad|t=0}$
$\displaystyle=\left\\{\begin{array}[]{ll}F_{0}&\;\;\text{if $(k,i)=(1,1)$}\\\
0&\;\;\text{if $2\leq k\leq K$,}\\\ \end{array}\right.$ (4.22)
$\displaystyle(\psi^{(k)},\partial_{t}\psi^{(k)})_{|t=0}$
$\displaystyle=(0,0),$
$\displaystyle(h_{\lambda},\partial_{t}h_{\lambda})_{|t=0}$
$\displaystyle=(0,0).$

We denote by $S$ the system of equation (BG)-(T$(k,i)$)-(W$(k)$)-(W) where $k$
and $i$ take all the possible values such that $F^{(k,i)}$ or $\psi^{(k)}$ are
well-defined.

######  Remark 4.8.

Note that in this system, we follow again the convention never to write
explicitly the numerical constants appearing in the expansion. But since they
appear in Theorem 3.1, let us recall the exact equations satisfied by
$\varphi$ and $F^{(1,1)}$:

$\Box\varphi_{i}=Q_{i}(\partial\varphi,\partial\varphi)\quad\text{and}\quad\mathscr{L}F^{(1,1)}_{i}=\frac{1}{2}\sum_{k,\ell=1,\dots,d}F^{(1,1)}_{k}Q_{i}(\partial(t-r),\partial\varphi_{\ell}),$

where $Q_{i}$ are the null forms of the main system for $\Phi_{\lambda}$.

######  Remark 4.9.

As we explained in the discussion before the definition of the reduced system,
the functions $\psi^{(k)}$ only exist for even $k$ but for the sake of
clarity, it seems useful to forget this and consider that we want to solve the
equations (W$(k)$) for all values of $k$ between 2 and $K-1$.

We see that there is a coupling between (T$(k,i)$) and (W$(k)$) but with a
particular triangular structure: the RHS of (T$(k,i)$) involves only
$F^{(\ell+1,j)}$ and $\psi^{(\ell)}$ for $\ell\leq k-1$, and the RHS of
(W$(k)$) involves only $F^{(\ell,j)}$ and $\psi^{(\ell)}$ for $\ell\leq k$.
This is a consequence of our choice of ansatz (4.1) and it allows us to solve
this system with the following strategy:

  1. 1.

We first solve (BG), this is done in Section 5.1.

  2. 2.

We solve (T(1,1)), which initiates a strong induction argument:

    1. (a)

if we know $F^{(\ell,j)}$ and $\psi^{(\ell)}$ for $1\leq j\leq\ell\leq k$, we
can solve $(\textbf{T}(k+1,i))$ for $1\leq i\leq k+1$…

    2. (b)

… and then solve $(\textbf{W}(k+1))$.

This is done in Section 5.2.

  3. 3.

Finally in Section 5.3 we solve (W).

Following this strategy, we prove the following theorem about the reduced
system $S$:

######  Theorem 4.1.

Let $K\geq 4$, $N\geq 6+5K$, $R>0$, $-\frac{1}{2}<\delta<\frac{1}{2}$,
$0<\alpha<1$, $(\varphi_{0},\varphi_{1})\in H^{N+1}_{\delta}\times
H^{N}_{\delta+1}$ and $F_{0}\in C^{N-4}$ with support in $A^{R}_{0}$ such that

$\|F_{0}\|_{C^{N-4}}+\left\|\varphi_{0}\right\|_{H^{N+1}_{\delta}}+\left\|\varphi_{1}\right\|_{H^{N}_{\delta+1}}\leq\varepsilon.$
(4.23)

There exists $\varepsilon_{0}=\varepsilon_{0}(K,N,R,\delta)>0$ such that if
$\varepsilon\leq\varepsilon_{0}$ and $0<\lambda\leq 1$ there exists a global
solution $(\varphi,\psi^{(\ell)},F^{(k,i)},h_{\lambda})$ to the reduced system
$S$ with initial data as in (4.2). Moreover, the following estimates hold:

  * •

$\varphi\in H^{N+1}$ satisfies

$\displaystyle\left\|w^{\frac{1}{2}}\partial Z^{I}\varphi\right\|_{L^{2}}$
$\displaystyle\lesssim\varepsilon\qquad\text{for}\quad|I|\leq N,$

with

$w(q)\vcentcolon=\left\\{\begin{array}[]{c}1+(1+|q|)^{-\alpha}\quad\text{for}\quad
q<0\\\ 1+(1+|q|)^{2(\delta+1)}\quad\text{for}\quad q>0\end{array}\right.,$

  * •

$F^{(k,i)}\in C^{N+4-5K}$ is supported in $A^{R}$ and satisfies

$|Z^{I}F^{(k,i)}|\lesssim\frac{\varepsilon}{(1+r)},\qquad\text{for}\quad|I|\leq
N+4-5K,$

  * •

$\psi^{(\ell)}\in H^{N+9-5K}$ is supported in $B^{R}$ and satisfies

$\left\|\partial
Z^{I}\psi^{(\ell)}\right\|_{L^{2}}\lesssim\varepsilon,\qquad\text{for}\quad|I|\leq
N+8-5K,$

  * •

$h_{\lambda}\in H^{N+3-5K}$ is supported in $B^{R}$ and satisfies

$\left\|\partial
Z^{I}h_{\lambda}\right\|_{L^{2}}\lesssim\varepsilon\lambda^{K-|I|},\qquad\text{for}\quad|I|\leq
N+2-5K.$

Note that in this article, $\left\|f\right\|_{L^{2}}$ denotes the $L^{2}$ norm
on a $t=constant$ slice. The rest of this article is devoted to the proof of
this theorem, but we can already see that it implies our main result, i.e
Theorem 3.1.

###### Proof of Theorem 3.1.

The reduced system is defined such that if an ansatz of the form (4.1) solves
it, then $\Phi_{\lambda}$ solves
$\Box\Phi_{\lambda}=Q(\partial\Phi_{\lambda},\partial\Phi_{\lambda})$.
Therefore, Theorem 4.1 implies Theorem 3.1, setting

$\displaystyle\tilde{F}_{0,\lambda}$
$\displaystyle\vcentcolon=\frac{1}{\lambda}\sum_{\begin{subarray}{c}2\leq
k\leq K\\\ 1\leq i\leq
k\end{subarray}}\lambda^{k}\partial_{t}F^{(k,i)}_{\quad\;\;|t=0}\mathbb{T}_{\lambda\quad|t=0}^{(k,i)},$
$\displaystyle F$ $\displaystyle\vcentcolon=F^{(1,1)},$
$\displaystyle\tilde{F}_{\lambda}$
$\displaystyle\vcentcolon=\frac{1}{\lambda^{2}}\left(\sum_{2\leq k\leq
K-1}\lambda^{k}\psi^{(k)}+\sum_{\begin{subarray}{c}2\leq k\leq K\\\ 1\leq
i\leq
k\end{subarray}}\lambda^{k}F^{(k,i)}\mathbb{T}_{\lambda}^{(k,i)}+h_{\lambda}\right).$

As mentioned in Theorem 3.1, we want $\tilde{F}_{0,\lambda}$ to depend only on
$(\varphi_{0},\varphi_{1},F_{0})$. This holds since the quantities
$\partial_{t}F^{(k,i)}_{\quad\;\;|t=0}$ can be recover through (T$(k,i)$)
since $\partial_{t}=\mathscr{L}-\partial_{r}-\frac{1}{r}$. Because of the
triangular structure of the reduced system, it can be seen quite easily that
$\partial_{t}F^{(k,i)}_{\quad\;\;|t=0}$ only depends on
$(\varphi_{0},\varphi_{1},F_{0})$ and their spatial derivatives. ∎

######  Remark 4.10.

According to Klainerman’s result, considering quadratic non-linearities with a
null structure ensures global existence for the non-linear wave equation in
space dimension 3. But from a high-frequency perspective, null forms are also
of prime interest. Indeed, if $Q(\partial u,\partial v)$ were not a null form,
the expansion of $Q(\partial(f\mathbb{T}),\partial(g\mathbb{T}))$ would
contain a term of the form $fgQ(\partial(t-r),\partial(t-r))$. Looking at the
decomposition of $Q(\partial\Phi_{\lambda},\partial\Phi_{\lambda})$, we see
that the equation on $\Phi^{(1)}$ would be of the form

$\displaystyle\mathscr{L}\partial_{\theta}\Phi^{(1)}=\left(\partial_{\theta}\Phi^{(1)}\right)^{2}+\cdots$
(4.24)

where $\partial_{\theta}$ corresponds to derivatives with respect to the third
variable of $\Phi^{(1)}$ (see (1.4) for the formal definition of
$\Phi^{(1)}$). The non-linear equation (4.24) requires $\Phi^{(1)}$ to be
described by a ”full” Fourier series, i.e

$\displaystyle\Phi^{(1)}(t,x,\theta)=\sum_{\ell\in\mathbb{Z}}F^{(1,\ell)}(t,x)\exp\left(i\ell\theta\right)$

with an infinite amount of $F^{(1,\ell)}$ non-zero. Though the equations for
the higher order terms (i.e $\Phi^{(i)}$ for $i\geq 1$) would remain linear
equations, the triangular structure and the presence of $\Phi^{(1)}$ as source
term would also imply that all the $\Phi^{(i)}$ are described by full Fourier
series. Therefore the null structure allows us to consider a ”simple” ansatz,
the so-called half-chessboard ansatz.

######  Remark 4.11.

The weight $w$ is parametrized by its exponents in the exterior region $q>0$
and in the interior region $q<0$. The exterior exponent $2(\delta+1)$ is
chosen so that the bound $\left\|w^{\frac{1}{2}}\partial
Z^{I}\varphi\right\|_{L^{2}}\lesssim\varepsilon$ holds initially and follows
from the smallness assumptions (4.23). The constraint on $\delta$ then implies
that $1<2(\delta+1)<3$, which will allow us to apply Proposition 2.2. The
interior exponent $\alpha$ is chosen so that there exists $0<\nu\leq 1$ and
$2-\nu>\alpha+1$ (take for example $\nu=\frac{1-\alpha}{2}$), the latter
implying

$\displaystyle\frac{w}{(1+|q|)^{2-\nu}}\lesssim w^{\prime}.$ (4.25)

##  5 Global existence for the reduced system

In this section, we prove Theorem 4.1 following the strategy outlined
previously.

###  5.1 The background wave equation

We first study the global existence for (BG). From the celebrated work of
Klainermann on null forms, we know that a global solution exists. However,
since the resolution of (W) will be a sort of high-frequency equivalent of
this one, it seems helpful to the author to write down the complete argument
leading to the global existence for $\eqref{equation sur ffi}$. The proof is a
bootstrap argument, based on both $L^{\infty}$ and $L^{2}$ estimates. As
mentioned in the introduction, we use Alinhac ghost weight method. Since the
equation we solve satisfy the null condition, this is not mandatory. However,
the ghost weight method leads to a very efficient proof.

######  Proposition 5.1.

There exists $\varepsilon_{0}>0$ such that, if
$\varepsilon\leq\varepsilon_{0}$, (BG) admits a unique global solution
$\varphi$. Moreover, $\varphi$ satisfies

$\displaystyle\left\|w_{1}^{\frac{1}{2}}\partial Z^{I}\varphi\right\|_{L^{2}}$
$\displaystyle\lesssim\varepsilon\qquad\text{for}\quad|I|\leq N.$ (5.1)

###### Proof.

We define $T_{0}$ to be the maximal time of existence of a solution $\varphi$,
which, by the local in time theory, is positive and satisfies the following
blow-up criterion:

$T_{0}<+\infty\iff\lim_{t\to T_{0}^{-}}\sum_{|I|\leq
N}\left\|w_{1}^{\frac{1}{2}}\partial Z^{I}\varphi\right\|_{L^{2}}(t)=+\infty.$

Let $C_{0}\geq 1$ be a constant to be choosen later. We define $T<T_{0}$ to be
the maximal time such that the inequality

$\displaystyle\left\|w_{1}^{\frac{1}{2}}\partial
Z^{I}\varphi\right\|_{L^{2}}(t)$ $\displaystyle\leq
C_{0}\varepsilon\qquad\qquad\qquad\quad\text{for}\quad|I|\leq N,$ (5.2)

hold for $0\leq t\leq T$. By the assumptions on $\varphi_{0}$ and
$\varphi_{1}$ and the Proposition 2.1, we have $T>0$, for $C_{0}$ large
enough.

We start by deriving decay for $Z^{I}\varphi$ from (5.2). If $|I|\leq N-3$ we
can apply the Klainerman-Sobolev inequality stated in Proposition 2.2 (since
$1<2(\delta+1)<3$ and $\alpha>0$), use the bootstrap assumption (5.2) and
obtain

$\displaystyle\left|w_{1}^{\frac{1}{2}}\partial Z^{I}\varphi\right|$
$\displaystyle\lesssim\frac{C_{0}\varepsilon}{(1+s)\sqrt{1+|q|}}.$

Using the behaviour of the weight $w_{1}$ in the exterior and interior region
this implies

$\left|\partial
Z^{I}\varphi\right|\lesssim\left\\{\begin{array}[]{c}\frac{C_{0}\varepsilon}{(1+s)\sqrt{1+|q|}}\quad\text{for}\quad
q<0\\\
\frac{C_{0}\varepsilon}{(1+s)(1+|q|)^{\delta+\frac{3}{2}}}\quad\text{for}\quad
q>0\end{array}\right..$ (5.3)

In order to derive decay for $Z^{I}\varphi$, we integrate (5.3) along the
constant $s$ lines. Recall that the assumptions (4.23) and the weighted
Sobolev embedding stated in Proposition 2.1 implies that

$\left|\partial
Z^{I}\varphi\right|\lesssim\frac{C_{0}\varepsilon}{(1+r)^{\delta+\frac{3}{2}}}$

on the initial hypersurface. Note that $r=s$ if $t=0$. Now, let $q\geq 0$ and
integrate (5.3) from the initial hypersurface along the constant $s$ lines,
this gives

$\displaystyle\left|Z^{I}\varphi(q\geq 0)\right|$
$\displaystyle\lesssim\frac{C_{0}\varepsilon}{(1+s)^{\delta+\frac{3}{2}}}+\int_{q}^{s}\frac{C_{0}\varepsilon\;\mathrm{d}q^{\prime}}{(1+s)(1+|q^{\prime}|)^{\delta+\frac{3}{2}}}$
$\displaystyle\lesssim\frac{C_{0}\varepsilon}{(1+s)^{\delta+\frac{3}{2}}}+\frac{C_{0}\varepsilon}{(1+s)(1+|q|)^{\delta+\frac{1}{2}}}$
$\displaystyle\lesssim\frac{C_{0}\varepsilon}{(1+s)(1+|q|)^{\delta+\frac{1}{2}}}$

where we also used the initial spatial decay and $|q|\leq s$. In particular,
we obtain the following decay along the null cone
$\left|Z^{I}\varphi(q=0)\right|\lesssim\frac{C_{0}\varepsilon}{1+s}$. Now, let
$q<0$ and integrate (5.3) from the null cone along the constant $s$ lines,
this gives

$\displaystyle\left|Z^{I}\varphi(q<0)\right|$
$\displaystyle\lesssim\frac{C_{0}\varepsilon}{1+s}+\int_{q}^{0}\frac{C_{0}\varepsilon\;\mathrm{d}q^{\prime}}{(1+s)\sqrt{1+|q^{\prime}|}}\lesssim
C_{0}\varepsilon\frac{\sqrt{1+|q|}}{1+s}.$

Since $\delta+\frac{1}{2}>0$, the worst estimate is the interior one and
overall we have proved that

$|Z^{I}\varphi|\lesssim\frac{C_{0}\varepsilon\sqrt{1+|q|}}{(1+s)},\qquad\text{for}\quad|I|\leq
N-3.$ (5.4)

We then improve the inequality (5.2), using the weighted energy estimate given
by Lemma 2.1. We set

$\displaystyle E(t)$ $\displaystyle\vcentcolon=\sum_{|I|\leq
N}\left\|w_{1}^{\frac{1}{2}}\partial Z^{I}\varphi\right\|^{2}_{L^{2}}(t),$
$\displaystyle S(t)$ $\displaystyle\vcentcolon=\int_{0}^{t}\sum_{|I|\leq
N}\left\|(w^{\prime}_{1})^{\frac{1}{2}}\overline{\partial}Z^{I}\varphi\right\|_{L^{2}}^{2}(\tau)\mathrm{d}\tau.$

We use Lemma 2.1, sum for $|I|\leq N$ and thanks to (2.5) and (2.6) we obtain
for $t\in[0,T]$:

$\displaystyle E(t)+$ $\displaystyle S(t)\lesssim
E(0)+\sum_{|I_{1}|+|I_{2}|\leq N}\int_{0}^{t}\left\|w_{1}\partial
Z^{I_{1}}\varphi\overline{\partial}Z^{I_{2}}\varphi\partial_{t}Z^{I}\varphi\right\|_{L^{1}}(\tau)\mathrm{d}\tau.$
(5.5)

Because of the assumption $(\varphi_{0},\varphi_{1})\in H^{N+1}_{\delta}\times
H^{N}_{\delta+1}$ and the choice of exponent in the exterior region for the
weight $w_{1}$, we have $E(0)\lesssim\varepsilon^{2}$. Since
$|I_{1}|+|I_{2}|\leq N$, it must holds that $|I_{2}|\leq\frac{N}{2}$ or
$|I_{1}|\leq\frac{N}{2}$. We examine the two cases separately:

  * •

If $|I_{1}|\leq\frac{N}{2}$, then

$|\partial
Z^{I_{1}}\varphi|\lesssim\frac{1}{1+|q|}|Z^{I_{1}+1}\varphi|\lesssim\frac{C_{0}\varepsilon}{(1+s)^{\frac{1}{2}+\frac{\nu}{2}}(1+|q|)^{1-\frac{\nu}{2}}}$

where we used (2.4) and (5.4) and where $\nu$ is as in the Remark 4.11. Using
$2ab\leq a^{2}+b^{2}$ and (5.2) we obtain:

$\displaystyle\left\|w_{1}\partial
Z^{I_{1}}\varphi\overline{\partial}Z^{I_{2}}\varphi\partial_{t}Z^{I}\varphi\right\|_{L^{1}}(\tau)$
$\displaystyle\qquad\qquad\lesssim\frac{C_{0}\varepsilon}{(1+\tau)^{1+\nu}}\left\|w_{1}^{\frac{1}{2}}\partial_{t}Z^{I}\varphi\right\|_{L^{2}}^{2}(\tau)+C_{0}\varepsilon\left\|\frac{w_{1}^{\frac{1}{2}}}{(1+|q|)^{1-\frac{\nu}{2}}}\overline{\partial}Z^{I_{2}}\varphi\right\|_{L^{2}}^{2}(\tau)$
$\displaystyle\qquad\qquad\lesssim\frac{C_{0}^{3}\varepsilon^{3}}{(1+\tau)^{1+\nu}}+C_{0}\varepsilon\left\|\left(w^{\prime}_{1}\right)^{\frac{1}{2}}\overline{\partial}Z^{I_{2}}\varphi\right\|_{L^{2}}^{2}(\tau).$

The last step is valid because of (4.25).

  * •

If $|I_{2}|\leq\frac{N}{2}$, then

$|\overline{\partial}Z^{I_{2}}\varphi|\lesssim\frac{1}{1+s}|Z^{I_{2}+1}\varphi|\lesssim\frac{C_{0}\varepsilon}{(1+s)^{\frac{3}{2}}},$

where we agin used (2.4) and (5.4). Using the Cauchy-Schwarz inequality and
(5.2) we obtain directly:

$\displaystyle\left\|w_{1}\partial
Z^{I_{1}}\varphi\overline{\partial}Z^{I_{2}}\varphi\partial_{t}Z^{I}\varphi\right\|_{L^{1}}(\tau)$
$\displaystyle\lesssim\frac{C_{0}\varepsilon}{(1+\tau)^{\frac{3}{2}}}\left\|w_{1}^{\frac{1}{2}}\partial
Z^{I_{1}}\varphi\right\|_{L^{2}}\left\|w_{1}^{\frac{1}{2}}\partial_{t}Z^{I}\varphi\right\|_{L^{2}}\lesssim\frac{C_{0}^{3}\varepsilon^{3}}{(1+\tau)^{\frac{3}{2}}}$

Therefore, since $t\longmapsto(1+\tau)^{-1-\nu}$ and
$t\longmapsto(1+\tau)^{-\frac{3}{2}}$ are both integrable over
$\mathbb{R}_{+}$ we obtain for $t\in[0,T]$:

$\displaystyle\sum_{|I_{1}|+|I_{2}|\leq N}\int_{0}^{t}\left\|w_{1}\partial
Z^{I_{1}}\varphi\overline{\partial}Z^{I_{2}}\varphi\partial_{t}Z^{I}\varphi\right\|_{L^{1}}(\tau)\mathrm{d}\tau\lesssim
C_{0}^{3}\varepsilon^{3}+C_{0}\varepsilon S(t).$

Absorbing $C_{0}\varepsilon S(t)$into the LHS of (5.5) we have proved that

$\displaystyle\left\|w_{1}^{\frac{1}{2}}\partial Z^{I}\varphi\right\|_{L^{2}}$
$\displaystyle\leq
C\varepsilon+C\sqrt{C_{0}\varepsilon}C_{0}\varepsilon\qquad\text{for}\quad|I|\leq
N,$

with $C>0$ a numerical constant. We now choose $C_{0}$ such that
$C\leq\frac{C_{0}}{4}$ and $\varepsilon_{0}$ such that
$CC_{0}\varepsilon\leq\frac{C_{0}}{4}$ and
$C\sqrt{C_{0}\varepsilon}\leq\frac{1}{4}$. Thus, we proved that the inequality
(5.2) hold with a constant $\frac{C_{0}}{2}$ instead of $C_{0}$, which
contradict the maximality of $T$, thus proving that $T=T_{0}$. But if
$T=T_{0}$, it implies that the energy is bounded up to the time $T_{0}$,
implying that $T_{0}=+\infty$, which concludes the proof. ∎

###  5.2 The high-frequency hierarchy

In this section, we solve the coupled transport equations (T$(k,i)$) and wave
equations (W$(k)$) with a strong induction argument which heavily relies on
the triangular structure of the reduced system.

The equations (T$(k,i)$) are of the form

$\mathscr{L}f=f\mu+g.$ (5.6)

Everything we need to know about (5.6) is contained in the following
proposition, whose proof is postponed to Appendix B.

######  Proposition 5.2.

Let $M\in\mathbb{N}$, $f_{0}:\mathbb{R}^{3}\to\mathbb{R}$ a $C^{M}$ function
supported in $A_{0}$ such that

$\left\|f_{0}\right\|_{C^{M}}\lesssim\varepsilon.$

Moreover, let $\mu,g:\mathbb{R}_{+}\times\mathbb{R}^{3}\to\mathbb{R}$ two
$C^{M}$ functions with $g$ supported in $A^{R}$ and such that

$\displaystyle(1+r)^{2}|Z^{I}\mu|+(1+r)^{3}|Z^{I}g|$
$\displaystyle\lesssim\varepsilon\qquad\text{for}\quad|I|\leq M.$

There exists a unique global solution $f\in C^{M}$ to (5.6) supported in
$A^{R}$ and satisfying

$|Z^{I}f|\lesssim\frac{\varepsilon}{1+r}\qquad\text{for}\quad|I|\leq M.$

With Proposition 5.2 we are now ready to solve (T$(k,i)$) and (W$(k)$). Since
we will prove that the functions $F^{(k,i)}$ are supported in $A^{R}$, we will
be only interested in the decay in $r$, which gives decay in $s$. We will also
forget about $q$ and use many times the fact that in the region $A^{R}$, $r$
and $t$ are equivalent. We prove the following proposition:

######  Proposition 5.3.

Let $1\leq i\leq k\leq K$ and $2\leq\ell\leq K-1$. There exists $F^{(k,i)}\in
C^{N_{k}}$ supported in $A^{R}$ solving (T$(k,i)$) and $\psi^{(\ell)}\in
H^{M_{\ell}}$ supported in $B^{R}$ solving (W$(k)$).Moreover, they satisfy the
following estimates:

$\displaystyle|Z^{I}F^{(k,i)}|$
$\displaystyle\lesssim\frac{\varepsilon}{1+r},\quad\qquad\text{for}\quad|I|\leq
N_{k},$ (5.7) $\displaystyle|Z^{I}\Box F^{(k,i)}|$
$\displaystyle\lesssim\frac{\varepsilon}{(1+r)^{3}},\qquad\text{for}\quad|I|\leq
N_{k}-2,$ (5.8) $\displaystyle\left\|\partial
Z^{I}\psi^{(\ell)}\right\|_{L^{2}}$
$\displaystyle\lesssim\varepsilon,\quad\qquad\qquad\text{for}\quad|I|\leq
M_{\ell}.$ (5.9)

where $N_{k}=N+4-5k$ for $k\geq 2$ and $N_{1}=N-4$ and $M_{\ell}=N+3-5\ell$.

######  Remark 5.1.

Since $\psi^{(1)}$ is not defined, the limiting term in the equation
$\mathbf{T}(2,i)$ is $\Box F^{(1,1)}$, whereas if $k\geq 3$ then the limiting
term in the equation $\mathbf{T}(k,i)$ is $\partial\psi^{(k-1)}$ since we lose
three derivatives using the Klainerman-Sobolev inequality to obtain decay from
$L^{2}$ estimates. This explains why $N_{1}-N_{2}\neq N_{k}-N_{k+1}$ for all
$k\geq 2$.

To prove Proposition 5.3, we proceed by strong induction on the value of $k$.
More precisely, we will show that if the estimates (5.7)-(5.8)-(5.9) holds for
$1\leq k\leq k^{\prime}$ and $2\leq\ell\leq k^{\prime}$ for $k^{\prime}$ some
integer satisfying $2\leq k^{\prime}\leq K-2$, then they also hold for
$k,\ell=k^{\prime}+1$. Note that the borderline cases $k=1$ and $k=K$, for
which $\psi^{(k)}$ is not defined, are proved in a similar manner so we don’t
write down their specific proofs.

Before we start, we derive decay for solutions of wave equation, i.e
$\psi^{(\ell)}$ for $\ell\leq k^{\prime}$ and $\varphi$. This follows from the
weighted Klainerman-Sobolev inequality of Proposition 2.2 and the exact
argument has already been given in the proof of Proposition 5.1. Following the
same strategy, we obtain from (5.1) and (5.9):

$\displaystyle|Z^{I}\psi^{(\ell)}|$
$\displaystyle\lesssim\varepsilon\frac{\sqrt{1+|q|}}{1+s},\qquad\text{for}\quad|I|\leq
M_{\ell}-3\quad\text{and}\quad\ell\leq k^{\prime},$ (5.10)
$\displaystyle|Z^{I}\varphi|$
$\displaystyle\lesssim\varepsilon\frac{\sqrt{1+|q|}}{1+s},\qquad\text{for}\quad|I|\leq
N-3.$ (5.11)

####  5.2.1 The transport equation for the oscillating terms

We start by solving $(\textbf{T}(k^{\prime}+1,i))$, which we recall:

$\displaystyle\mathscr{L}F^{(k^{\prime}+1,i)}$ $\displaystyle=\Box
F^{(k^{\prime})}+F^{(k^{\prime}+1,i)}\overline{\partial}\varphi+Q(\partial
F^{(\leq k^{\prime})},\partial\varphi)+Q(\partial F^{(\leq
k^{\prime})},\partial F^{(\leq k^{\prime})})$ $\displaystyle\quad+F^{(\leq
k^{\prime})}\overline{\partial}F^{(\leq k^{\prime})}+F^{(\leq
k^{\prime})}\overline{\partial}\psi^{(\leq k^{\prime})}+Q(\partial F^{(\leq
k^{\prime})},\partial\psi^{(\leq k^{\prime})}).$

In sake of clarity we define $G^{(k^{\prime}+1,i)}$ such that
$\mathscr{L}F^{(k^{\prime}+1,i)}=F^{(k^{\prime}+1,i)}\overline{\partial}\varphi+G^{(k^{\prime}+1,i)}$.
Note that the expression of $G^{(k^{\prime}+1,i)}$ involves only
$F^{(\ell,j)}$ for $\ell\leq k^{\prime}$ and $\varphi^{(\ell)}$ for $\ell\leq
k^{\prime}$ so we can estimate it using (5.7), (5.8) and (5.9). This is done
in the following lemma.

######  Lemma 5.1.

$G^{(k^{\prime}+1,i)}$ is supported in $A^{R}$ and the following estimate
holds

$|Z^{I}G^{(k^{\prime}+1,i)}|\lesssim\frac{\varepsilon}{(1+r)^{3}},\qquad\text{for}\quad|I|\leq
N-1-5k^{\prime},$

###### Proof.

Formally we have

$\displaystyle G^{(k^{\prime}+1,i)}$ $\displaystyle=\Box
F^{(k^{\prime})}+Q(\partial F^{(\leq k^{\prime})},\partial\varphi)+Q(\partial
F^{(\leq k^{\prime})},\partial F^{(\leq k^{\prime})})+Q(\partial F^{(\leq
k^{\prime})},\partial\psi^{(\leq k^{\prime})}).$

Let us count how many times each terms can be differentiated:

  * •

we want to estimate $\partial\psi^{(\leq k^{\prime})}$ using (5.10), we can
differentiate this term at most $M_{k^{\prime}}-4$ times,

  * •

the terms $\partial F^{(\leq k^{\prime})}$ are better than $\Box
F^{(k^{\prime})}$ which thanks to (5.8) can be differentiated at most
$N_{k^{\prime}}-2$ times.

Since $M_{k^{\prime}}-4<N_{k^{\prime}}-2$, we can differentiate
$G^{(k^{\prime}+1,i)}$ at most $M_{k^{\prime}}-4$ times, i.e at most
$N-1-5k^{\prime}$ times. Now, let $|I|\leq N-1-5k^{\prime}$, we have:

$\displaystyle|Z^{I}G^{(k^{\prime}+1,i)}|$ $\displaystyle\lesssim|Z^{I}\Box
F^{(k^{\prime},i)}|+\sum_{|I_{1}|+|I_{2}|\leq|I|+1}\frac{|Z^{I_{1}}F^{(\leq
k^{\prime})}||Z^{I_{2}}F^{(\leq k^{\prime})}|}{1+r}$
$\displaystyle\qquad+\sum_{|I_{1}|+|I_{2}|\leq|I|+1}\frac{|Z^{I_{1}+1}F^{(\leq
k^{\prime})}||Z^{I_{2}+1}\varphi|}{1+r}+\sum_{|I_{1}|+|I_{2}|\leq|I|+1}\frac{|Z^{I_{1}}F^{(\leq
k^{\prime})}||Z^{I_{2}}\psi^{(\leq k^{\prime})}|}{1+r}$
$\displaystyle\lesssim\frac{\varepsilon}{(1+r)^{3}},$

where we used (5.7), (5.8), (5.10) and (5.11). ∎

######  Lemma 5.2.

There exists a unique solution $F^{(k^{\prime}+1,i)}\in C^{N-1-5k^{\prime}}$
to $(\textbf{T}(k^{\prime}+1,i))$ supported in $A^{R}$ such that

$\displaystyle|Z^{I}F^{(k^{\prime}+1,i)}|$
$\displaystyle\lesssim\frac{\varepsilon}{(1+r)},\qquad\text{for}\quad|I|\leq
N-1-5k^{\prime},$ (5.12) $\displaystyle|Z^{I}\Box F^{(k^{\prime}+1,i)}|$
$\displaystyle\lesssim\frac{\varepsilon}{(1+r)^{3}},\qquad\text{for}\quad|I|\leq
N-3-5k^{\prime}.$ (5.13)

###### Proof.

We apply Proposition 5.2 with $M=N-1-5k^{\prime}$, $g=G^{(k^{\prime}+1,i)}$
and $\mu=\overline{\partial}\varphi$ which satisfy the required estimates
thanks to Lemma 5.1 and to

$\left|Z^{I}\overline{\partial}\varphi\right|\lesssim|\overline{\partial}Z^{I}\varphi|+\sum_{|J|\leq|I|}\frac{|Z^{J}\varphi|}{1+s}\lesssim\sum_{|J|\leq|I|+1}\frac{|Z^{J}\varphi|}{1+s}\lesssim\frac{\varepsilon}{(1+r)^{2}}$
(5.14)

in the region $A^{R}$ for $|I|\leq N-4$ and where we use (5.11). This
concludes the proof of (5.12). Let us now turn to the proof of (5.13). Because
of Lemma A.2, we have

$\Box
F^{(k^{\prime}+1,i)}=\delta^{ij}\overline{\partial}_{i}\overline{\partial}_{j}F^{(k^{\prime}+1,i)}+\underline{L}(F^{(k^{\prime}+1,i)}\overline{\partial}\varphi)+\frac{1}{r}F^{(k^{\prime}+1,i)}\overline{\partial}\varphi+\underline{L}G^{(k^{\prime}+1,i)}+\frac{1}{r}G^{(k^{\prime}+1,i)},$

where $\underline{L}=\partial_{t}-\partial_{r}$. We estimate this expression
directly, for $I$ a multi-index such that $|I|\leq N-3-5k^{\prime}$:

$\displaystyle|Z^{I}\Box F^{(k^{\prime}+1,i)}|$
$\displaystyle\lesssim|Z^{I}\overline{\partial}^{2}F^{(k^{\prime}+1,i)}|+|Z^{I}\partial(F^{(k^{\prime}+1,i)}\overline{\partial}\varphi)|+|Z^{I}(r^{-1}F^{(k^{\prime}+1,i)}\overline{\partial}\varphi)|$
$\displaystyle\qquad+|Z^{I}\partial
G^{(k^{\prime}+1,i)}|+|Z^{I}(r^{-1}G^{(k^{\prime}+1,i)})|$
$\displaystyle\lesssim\sum_{|J|\leq|I|+2}\frac{|Z^{J}F^{(k^{\prime}+1,i)}|}{(1+r)^{2}}+\sum_{|I_{1}|+|I_{2}|\leq|I|+1}|Z^{I_{1}}F^{(k^{\prime}+1,i)}||Z^{I_{2}}\overline{\partial}\varphi|+\sum_{|J|\leq|I|+1}|Z^{J}G^{(k^{\prime}+1,i)}|$
$\displaystyle\lesssim\frac{\varepsilon}{(1+r)^{3}}$

where we used commutation property of $Z^{I}$ and $\overline{\partial}$ as in
(5.14), the Leibniz rule and $|Z^{K}(r^{-1})|\lesssim r^{-1}$ for every multi-
index $K$. Using (5.12), (5.14) and Lemma 5.1 we conclude the proof of (5.13).
∎

######  Remark 5.2.

Note that in the previous proof, we used Lemma A.2, which basically shows that
if we have a good control on $\mathscr{L}f$, then $\Box f$ decays better than
two derivatives of $f$, since it behaves like two ”good” derivatives of $f$,
i.e $\overline{\partial}^{2}f$. This prevents us from estimating $\Box f$ by
commuting $\Box$ and $\mathscr{L}$. The decay obtained through this method
would not be sufficient since

$[\mathscr{L},\Box]f=-\frac{2}{r}\delta^{ij}\partial_{i}\overline{\partial}_{j}f+\text{l.o.t}$

where ”l.o.t” stands for lower order terms. This would imply that if $f$
solves (5.6), then $\Box f$ solves

$(L-\mu)(r\Box f)=\partial\overline{\partial}f+\text{l.o.t}.$

Since a normal derivative doesn’t give extra decay in $A^{R}$, we would only
get $\Box f\sim\frac{1}{r}$ while we need $\frac{1}{r^{3}}$, given Proposition
5.2.

####  5.2.2 The wave equation for the non-oscillating terms

We now solve $(\textbf{W}(k^{\prime}+1))$:

$\displaystyle\Box\psi^{(k^{\prime}+1)}$
$\displaystyle=Q(\partial\psi^{(k^{\prime}+1)},\partial\varphi)+Q(\partial\psi^{(\leq
k^{\prime}-1)},\partial\psi^{(\leq k^{\prime}-1)})$
$\displaystyle\quad+Q(\partial F^{(\leq k^{\prime}+1)},\partial F^{(\leq
k^{\prime}+1)})+F^{(\leq k^{\prime}+1)}\overline{\partial}F^{(\leq
k^{\prime}+1)}.$

We define $H^{(k^{\prime}+1)}$ such that
$\Box\psi^{(k^{\prime}+1)}=Q(\partial\psi^{(k^{\prime}+1)},\partial\varphi)+H^{(k^{\prime}+1)}$.
Note that $H^{(k^{\prime}+1)}$ involves $\psi^{(\ell)}$ for $\ell\leq
k^{\prime}-1$ and $F^{(\ell,i)}$ for $\ell\leq k^{\prime}+1$, thus respecting
the triangular structure. We estimate $H^{(k^{\prime}+1)}$ in the following
lemma.

######  Lemma 5.3.

$H^{(k^{\prime}+1)}$ is supported in $B^{R}$ and the following estimate holds

$|Z^{I}H^{(k^{\prime}+1)}|\lesssim\frac{\varepsilon^{2}}{(1+s)^{3}}\qquad\text{for}\quad|I|\leq
N-2-5k^{\prime}.$ (5.15)

###### Proof.

Formally we have

$\displaystyle H^{(k^{\prime}+1)}$ $\displaystyle=Q(\partial\psi^{(\leq
k^{\prime}-1)},\partial\psi^{(\leq k^{\prime}-1)})+Q(\partial F^{(\leq
k^{\prime}+1)},\partial F^{(\leq k^{\prime}+1)})+F^{(\leq
k^{\prime}+1)}\overline{\partial}F^{(\leq k^{\prime}+1)}.$

Let us count how many times each terms can be differentiated:

  * •

we want to estimate $\partial\psi^{(\leq k^{\prime}-1)}$ using (5.10), we can
differentiate this term at most $M_{k^{\prime}-1}-4$ times,

  * •

we want to estimate $\partial F^{(\leq k^{\prime}+1)}$ using (5.12) for
$\partial F^{(k^{\prime}+1)}$ or (5.7) for $\partial F^{(\leq k^{\prime})}$ so
we can differentiate these terms at most $N-2-5k^{\prime}$.

Since $N-2-5k^{\prime}<M_{k^{\prime}-1}-4$, we can differentiate
$H^{(k^{\prime}+1)}$ at most $N-2-5k^{\prime}$. Now, let $|I|\leq
N-2-5k^{\prime}$, we estimate $Z^{I}H^{(k^{\prime}+1)}$ using (5.7), (5.12)
and (5.10):

$\displaystyle|Z^{I}H^{(2k+2)}|$
$\displaystyle\lesssim\sum_{|I_{1}|+|I_{2}|\leq|I|}\left(\frac{|Z^{I_{1}+1}F^{(\leq
k^{\prime}+1)}||Z^{I_{2}+1}F^{(\leq
k^{\prime}+1)}|}{1+r}+\frac{|Z^{I_{1}+1}\psi^{(\leq
k^{\prime}-1)}||Z^{I_{2}+1}\psi^{(\leq k^{\prime}-1)}|}{(1+s)(1+|q|)}\right)$
$\displaystyle\lesssim\frac{\varepsilon^{2}}{(1+s)^{3}}.$

∎

######  Lemma 5.4.

If $\varepsilon$ is small enough, $(\textbf{W}(k^{\prime}+1))$ admits a unique
global solution $\psi^{(k^{\prime}+1)}\in H^{N-2-5k^{\prime}}$ supported in
$B^{R}$ and satisfying

$\displaystyle\left\|\partial
Z^{I}\psi^{(k^{\prime}+1)}\right\|_{L^{2}}\lesssim\varepsilon\qquad\text{for}\quad|I|\leq
N-2-5k^{\prime}.$

###### Proof.

In order to prove Lemma 5.4 we use a continuity argument. We let $T_{0}$ the
maximal time of existence of a solution $\psi^{(k^{\prime}+1)}$ to
$(\textbf{W}(k^{\prime}+1))$, which satisfies

$T_{0}<+\infty\iff\lim_{t\to T_{0}^{-}}\sum_{|I|\leq
N-2-5k^{\prime}}\left\|w^{\frac{1}{2}}\partial
Z^{I}\psi^{(k^{\prime}+1)}\right\|_{L^{2}}(t)=+\infty$

where $w$ is defined in Theorem 4.1. Because of the support properties of
$H^{(k^{\prime}+1)}$ and Lemma 2.2, the function $\psi^{(k^{\prime}+1)}$ is
supported in $B_{R}$ (therefore the exterior exponent of $w$ plays no role
here). We make the following bootstrap assumption:

$\left\|w^{\frac{1}{2}}\partial
Z^{I}\psi^{(k^{\prime}+1)}\right\|_{L^{2}}(t)\leq\varepsilon\qquad\text{for}\quad|I|\leq
N-2-5k^{\prime}.$ (5.16)

Since the initial data for $\psi^{(k^{\prime}+1)}$ vanish, we can assume that
(5.16) holds on some time interval $[0,T]$ with $T>0$. Moreover, we assume
that $T$ is the maximal time so that (5.16) holds. We define

$\displaystyle E_{k^{\prime}+1}(t)$ $\displaystyle\vcentcolon=\sum_{|I|\leq
N-2-5k^{\prime}}\left\|w^{\frac{1}{2}}\partial
Z^{I}\psi^{(k^{\prime}+1)}\right\|^{2}_{L^{2}}(t),$ $\displaystyle
S_{k^{\prime}+1}(t)$ $\displaystyle\vcentcolon=\int_{0}^{t}\sum_{|I|\leq
N-2-5k^{\prime}}\left\|(w^{\prime})^{\frac{1}{2}}\overline{\partial}Z^{I}\psi^{(k^{\prime}+1)}\right\|_{L^{2}}^{2}(\tau)\mathrm{d}\tau.$

We apply the weighted energy estimate of Lemma 2.1, the Cauchy-Schwarz
inequality and our bootstrap assumption (5.16) to obtain (recall that the
initial data for $\psi^{(k^{\prime}+1)}$ vanish):

$\displaystyle E_{k^{\prime}+1}(t)+S_{k^{\prime}+1}(t)$
$\displaystyle\lesssim\sum_{|I|\leq N-2-5k^{\prime}}\int_{0}^{t}\left\|w\Box
Z^{I}\psi^{(k^{\prime}+1)}\partial
Z^{I}\psi^{(k^{\prime}+1)}\right\|_{L^{1}(B_{R}\cap\Sigma_{\tau})}\mathrm{d}\tau$
$\displaystyle\lesssim\sum_{|I|\leq
N-2-5k^{\prime}}\int_{0}^{t}\left\|w^{\frac{1}{2}}\Box
Z^{I}\psi^{(k^{\prime}+1)}\right\|_{L^{2}(B_{R}\cap\Sigma_{\tau})}\left\|w^{\frac{1}{2}}\partial
Z^{I}\psi^{(k^{\prime}+1)}\right\|_{L^{2}}\mathrm{d}\tau.$ (5.17)

Now if $|I|\leq N-2-5k^{\prime}$, using (2.5) and (2.6) we obtain:

$\displaystyle\left\|w^{\frac{1}{2}}\Box
Z^{I}\psi^{(k^{\prime}+1)}\right\|_{L^{2}(B_{R}\cap\Sigma_{\tau})}$
$\displaystyle\lesssim\sum_{|J_{1}|+|J_{2}|\leq|I|}\left\|w^{\frac{1}{2}}\partial
Z^{J_{1}}\psi^{(k^{\prime}+1)}\overline{\partial}Z^{J_{2}}\varphi\right\|_{L^{2}(B^{R}\cap\Sigma_{\tau})}$
$\displaystyle\quad+\sum_{|J_{1}|+|J_{2}|\leq|I|}\left\|w^{\frac{1}{2}}\overline{\partial}Z^{J_{1}}\psi^{(k^{\prime}+1)}\partial
Z^{J_{2}}\varphi\right\|_{L^{2}(B^{R}\cap\Sigma_{\tau})}$
$\displaystyle\quad+\sum_{|J|\leq|I|}\left\|w^{\frac{1}{2}}Z^{J}H^{(k^{\prime}+1)}\right\|_{L^{2}(B^{R}\cap\Sigma_{\tau})}$
$\displaystyle=\vcentcolon A+B+C.$

We start with the easiest term, i.e $C$, for which we use (5.15) and the fact
that $w$ is bounded on $B_{R}$:

$\displaystyle C$
$\displaystyle\lesssim\varepsilon^{2}\left(\int_{0}^{\tau+R}\frac{\mathrm{d}r}{(1+r+\tau)^{4}}\right)^{\frac{1}{2}}\lesssim\frac{\varepsilon^{2}}{(1+\tau)^{\frac{3}{2}}}$

which implies $C\lesssim\varepsilon^{2}$. For $A$, we use (5.11) and $|q|\leq
s$ to get

$|\overline{\partial}Z^{J_{2}}\varphi|\lesssim\frac{\varepsilon}{(1+s)^{\frac{3}{2}}}$

for all $|J_{2}|\leq N-2-5k^{\prime}$.This gives

$\displaystyle A$ $\displaystyle\lesssim\varepsilon\sum_{|I|\leq
N-2-5k^{\prime}}\frac{1}{(1+\tau)^{\frac{3}{2}}}\left\|w^{\frac{1}{2}}\partial
Z^{I}\psi^{(k^{\prime}+1)}\right\|_{L^{2}}\lesssim\frac{\varepsilon^{2}}{(1+\tau)^{\frac{3}{2}}}$

where we also used (5.16). Let us now look at $B$, by first using (5.11) to
obtain

$|\partial
Z^{J_{2}}\varphi|\lesssim\frac{\varepsilon}{(1+s)^{\frac{1}{2}+\frac{\nu}{2}}(1+|q|)^{1-\frac{\nu}{2}}}$

for $\nu$ defined at the beginning of this proof. Using (4.25) this gives

$\displaystyle B$
$\displaystyle\lesssim\frac{\varepsilon}{(1+\tau)^{\frac{1}{2}+\frac{\nu}{2}}}\sum_{|J|\leq|I|}\left\|\frac{w^{\frac{1}{2}}}{(1+|q|)^{1-\frac{\nu}{2}}}\overline{\partial}Z^{J}\psi^{(k^{\prime}+1)}\right\|_{L^{2}(B^{R}\cap\Sigma_{\tau})}$
$\displaystyle\lesssim\frac{\varepsilon}{(1+\tau)^{\frac{1}{2}+\frac{\nu}{2}}}\sum_{|J|\leq|I|}\left\|(w^{\prime})^{\frac{1}{2}}\overline{\partial}Z^{J}\psi^{(k^{\prime}+1)}\right\|_{L^{2}(B^{R}\cap\Sigma_{\tau})}.$

Putting our estimates on $A$, $B$ and $C$ together we obtain from (5.17):

$\displaystyle E_{k^{\prime}+1}(t)+S_{k^{\prime}+1}(t)$
$\displaystyle\lesssim\sum_{|I|\leq
N-2-5k^{\prime}}\int_{0}^{t}(A+C)\left\|w^{\frac{1}{2}}\partial
Z^{I}\psi^{(k^{\prime}+1)}\right\|_{L^{2}}\mathrm{d}\tau+\sum_{|I|\leq
N-2-5k^{\prime}}\int_{0}^{t}B\left\|w^{\frac{1}{2}}\partial
Z^{I}\psi^{(k^{\prime}+1)}\right\|_{L^{2}}\mathrm{d}\tau$
$\displaystyle\lesssim\varepsilon^{3}\int_{0}^{t}\frac{\mathrm{d}\tau}{(1+\tau)^{\frac{3}{2}}}$
$\displaystyle\quad+\sum_{|J|\leq|I|\leq
N-2-5k^{\prime}}\int_{0}^{t}\frac{\varepsilon}{(1+\tau)^{\frac{1}{2}+\frac{\nu}{2}}}\left\|(w^{\prime})^{\frac{1}{2}}\overline{\partial}Z^{J}\psi^{(k^{\prime}+1)}\right\|_{L^{2}(B^{R}\cap\Sigma_{\tau})}\left\|w^{\frac{1}{2}}\partial
Z^{I}\psi^{(k^{\prime}+1)}\right\|_{L^{2}}\mathrm{d}\tau$
$\displaystyle\lesssim\varepsilon^{3}\int_{0}^{t}\frac{\mathrm{d}\tau}{(1+\tau)^{\frac{3}{2}}}$
$\displaystyle\quad+\sum_{|I|\leq
N-2-5k^{\prime}}\left(\varepsilon\int_{0}^{t}\left\|(w^{\prime})^{\frac{1}{2}}\overline{\partial}Z^{I}\psi^{(k^{\prime}+1)}\right\|_{L^{2}(B^{R}\cap\Sigma_{\tau})}^{2}\mathrm{d}\tau+\varepsilon\int_{0}^{t}\frac{1}{(1+\tau)^{1+\nu}}\left\|w^{\frac{1}{2}}\partial
Z^{I}\psi^{(k^{\prime}+1)}\right\|_{L^{2}}^{2}\mathrm{d}\tau\right)$
$\displaystyle\lesssim\varepsilon^{3}\int_{0}^{t}\frac{\mathrm{d}\tau}{(1+\tau)^{\frac{3}{2}}}+\varepsilon
S_{k^{\prime}+1}(t)+\varepsilon^{3}\int_{0}^{t}\frac{\mathrm{d}\tau}{(1+\tau)^{1+\nu}}$

where for the integral involving $A+C$ we use (5.16) and for the integral
involving $B$ we used first $2ab\leq a^{2}+b^{2}$ and (5.16). Therefore, we
have proved that there exists a constant $C>0$ such that for all $t\in[0,T]$

$E_{k^{\prime}+1}(t)+(1-C\varepsilon)S_{k^{\prime}+1}(t)\leq
C\varepsilon^{3}.$

We now choose $\varepsilon$ so that $\varepsilon\leq\frac{1}{2C}$. This
implies that for all $t\in[0,T]$

$E_{k^{\prime}+1}(t)\leq\frac{1}{2}\varepsilon^{2}.$

Therefore, we improved (5.16), thus proving that actually $T=T_{0}$. This
breaks the blow-up criterion, proving that $T_{0}=+\infty$. Note that since
$\psi^{(k^{\prime}+1)}$ is supported in $B_{R}$ the ghost weight $w$ is
bounded and we can change (5.16) to the estimate of the lemma. ∎

This concludes the induction and thus the proof of Proposition 5.3.

###  5.3 The equation for the remainder

The proof of the global existence for (W) follows the same structure as the
proof of Proposition 5.1.

####  5.3.1 Bootstrap assumptions and first consequences

We use again the ghost weight $w$ defined in Theorem 4.1. We let $T_{0}$ be
the maximal time of existence of a solution $h_{\lambda}$, which satisfies the
following blow-up criterion:

$T_{0}<+\infty\iff\lim_{t\to T_{0}^{-}}\sum_{|I|\leq
N^{\prime}}\left\|w^{\frac{1}{2}}\partial
Z^{I}h_{\lambda}\right\|_{L^{2}}(t)=+\infty,$

where $N^{\prime}$ is some large enough integer. In terms of number of
derivation, the limiting term in the equation for $h_{\lambda}$ is $\Box
F^{(K,i)}$, which we can differentiate at most $N+2-5K$ thanks to (5.8).
Therefore we set $N^{\prime}=N+2-5K$, which satisfies $N^{\prime}\geq 8$. We
define two rescaled energies:

$\displaystyle\mathcal{E}_{\lambda}(t)$
$\displaystyle\vcentcolon=\sum_{|I|\leq
N^{\prime}}\lambda^{-2K+2|I|}\left\|w^{\frac{1}{2}}\partial
Z^{I}h_{\lambda}\right\|_{L^{2}}^{2}(t),$
$\displaystyle\mathcal{S}_{\lambda}(t)$
$\displaystyle\vcentcolon=\sum_{|I|\leq
N^{\prime}}\lambda^{-2K+2|I|}\int_{0}^{t}\left\|(w^{\prime})^{\frac{1}{2}}\overline{\partial}Z^{I}h_{\lambda}\right\|_{L^{2}}^{2}(\tau)\mathrm{d}\tau.$

We make the following bootstrap assumptions on $h_{\lambda}$:

$\displaystyle\sup_{t\in[0,T]}\mathcal{E}_{\lambda}(t)\leq
C_{2}^{2}\varepsilon^{2}.$ (5.18)

Here $C_{2}$ is a constant to be choosen later. We let $T<T_{0}$ be the
maximal time such that these estimates hold on $[0,T]$. Since
$(h_{\lambda},\partial_{t}h_{\lambda})_{|t=0}=0$, this bootstrap assumption is
satisfied at $t=0$ and $T>0$. Note that (5.18) immediately implies that for
every multi-index $|I|\leq N^{\prime}$ and for every $t\in[0,T]$ we have

$\left\|w^{\frac{1}{2}}\partial Z^{I}h_{\lambda}\right\|_{L^{2}}(t)\leq
C_{2}\varepsilon\lambda^{K-|I|}.$ (5.19)

Recall (W):

$\displaystyle\Box h_{\lambda}$
$\displaystyle=\mathcal{N}(h_{\lambda})+\lambda^{K}Q(\partial
F,\partial\varphi)\mathbb{T}_{\lambda}+\lambda^{K}\Box F\mathbb{T}_{\lambda}$
$\displaystyle\quad+\lambda^{K}\lambda^{+}\left(F\overline{\partial}F\mathbb{T}_{\lambda}+Q(\partial
F,\partial F)\mathbb{T}_{\lambda}+Q(\partial
F,\partial\psi)\mathbb{T}_{\lambda}+F\overline{\partial}\psi\mathbb{T}_{\lambda}+Q(\partial\psi,\partial\psi)\right)$

To capture the different support properties, we define

$\displaystyle\mathbf{A}_{\lambda}$ $\displaystyle=\lambda^{K}Q(\partial
F,\partial\varphi)\mathbb{T}_{\lambda}+\lambda^{K}\Box
F\mathbb{T}_{\lambda}+\lambda^{K}\lambda^{+}\left(F\overline{\partial}F\mathbb{T}_{\lambda}+Q(\partial
F,\partial F)\mathbb{T}_{\lambda}+Q(\partial
F,\partial\psi)\mathbb{T}_{\lambda}+F\overline{\partial}\psi\mathbb{T}_{\lambda}\right)$
$\displaystyle\mathbf{B}_{\lambda}$
$\displaystyle=\lambda^{K}\lambda^{+}Q(\partial\psi,\partial\psi)$

such that

$\displaystyle\Box h_{\lambda}$
$\displaystyle=\mathcal{N}(h_{\lambda})+\mathbf{A}_{\lambda}+\mathbf{B}_{\lambda}.$
(5.20)

Thanks to the vanishing of the initial data of $h_{\lambda}$ and the fact that
the RHS of (5.20) is supported in $B_{R}$, we can apply Lemma 2.2 to prove
that $h_{\lambda}$ is also supported in $B_{R}$.

####  5.3.2 Decay estimates

We start by deriving some decay for $h_{\lambda}$ and the non-homogeneous
terms in $\Box h_{\lambda}$. This again follows from the weighted Klainerman-
Sobolev inequality and we don’t repeat the argument. However, we state it in a
lemma to emphasize the appearance of the factor $\lambda^{K-3-|I|}$.

######  Lemma 5.5.

The following estimate holds on $[0,T]$

$|Z^{I}h_{\lambda}|\lesssim\frac{C_{2}\varepsilon\sqrt{1+|q|}}{1+s}\lambda^{K-3-|I|}\qquad\text{for}\quad|I|\leq
N^{\prime}-3.$ (5.21)

The following lemma estimates the non-homogeneous terms in $\Box h_{\lambda}$.

######  Lemma 5.6.

The function $\mathbf{A}_{\lambda}$ are supported in $A^{R}$ and satisfies

$|Z^{I}\mathbf{A}_{\lambda}|\lesssim\frac{\varepsilon\lambda^{K-|I|}}{(1+s)^{3}},\qquad\text{for}\quad|I|\leq
N^{\prime}.$ (5.22)

The function $\mathbf{B}_{\lambda}$ is supported in $B^{R}$ and satisfies

$|Z^{I}\mathbf{B}_{\lambda}|\lesssim\frac{\varepsilon^{2}}{(1+s)^{3}}\lambda^{K+1}\qquad\text{for}\quad|I|\leq
N^{\prime}.$ (5.23)

###### Proof.

Since

$|Z^{I}\mathbb{T}_{\lambda}^{(\alpha,\beta)}|\leq
C_{I,\alpha,\beta}\lambda^{-|I|},$ (5.24)

for some $C_{I,\alpha,\beta}>0$, we can directly estimate
$\mathbf{A}_{\lambda}$ using its expression:

$\displaystyle|Z^{I}\mathbf{A}_{\lambda}|$
$\displaystyle\lesssim\sum_{|I_{1}|+|I_{2}|\leq|I|}\lambda^{K-|I_{2}|}|Z^{I_{1}}\Box
F|$
$\displaystyle\quad+\sum_{|I_{1}|+|I_{2}|+|I_{3}|\leq|I|}\lambda^{K-|I_{3}|}\frac{|Z^{I_{1}+1}F|}{1+r}\left(|Z^{I_{2}+1}F|+|Z^{I_{2}+1}\psi|+|Z^{I_{2}+1}\varphi|\right)$
$\displaystyle\lesssim\frac{\varepsilon}{(1+s)^{3}}\sum_{|J|\leq|I|}\lambda^{K-|J|}$

where we used (5.7), (5.8),(5.10) and (5.24). For $\mathbf{B}_{\lambda}$ we
recall (5.10), which implies that for all $2\leq\ell\leq K-1$

$|Z^{I}\psi^{(\ell)}|\lesssim\varepsilon\frac{\sqrt{1+|q|}}{1+s},\qquad\text{for}\quad|I|\leq
M_{K-1}-3.\\\ $

Recalling the expression of $\mathbf{B}_{\lambda}$ concludes the proof:

$|Z^{I}\mathbf{B}_{\lambda}|\lesssim\lambda^{K+1}\sum_{|I_{1}|+|I_{2}|\leq|I|}\frac{|Z^{I_{1}+1}\psi||Z^{I_{2}+1}\psi|}{(1+s)(1+|q|)}\lesssim\frac{\varepsilon^{2}}{(1+s)^{3}}\lambda^{K+1}$

since $N^{\prime}+1\leq M_{K-1}-3$. ∎

####  5.3.3 Energy estimates

We conclude the bootstrap by improving (5.18).

######  Proposition 5.4.

For large enough $C_{2}$ and small enough $\varepsilon$ (depending on
$C_{2}$), we have for every $t\in[0,T]$

$\mathcal{E}_{\lambda}(t)\leq\frac{1}{2}C_{2}^{2}\varepsilon^{2}.$

###### Proof.

Applying Lemma 2.1 to $Z^{I}h_{\lambda}$ for every multi-index $|I|\leq
N^{\prime}$, multiplying what we obtain by $\lambda^{-2K+2|I|}$ and summing
over $I$ we get for every $t\in[0,T]$ (recall that the initial data for
$h_{\lambda}$ and $\partial_{t}h_{\lambda}$ vanish):

$\displaystyle\mathcal{E}_{\lambda}(t)+\mathcal{S}_{\lambda}(t)$
$\displaystyle\lesssim\sum_{|I|\leq
N^{\prime}}\int_{0}^{t}\lambda^{-2K+2|I|}\left\|w\Box Z^{I}h_{\lambda}\partial
Z^{I}h_{\lambda}\right\|_{L^{1}}\mathrm{d}\tau$
$\displaystyle\lesssim\sum_{|I|\leq
N^{\prime}}\int_{0}^{t}\lambda^{-2K+2|I|}\left\|w^{\frac{1}{2}}\Box
Z^{I}h_{\lambda}\right\|_{L^{2}}\left\|w^{\frac{1}{2}}\partial
Z^{I}h_{\lambda}\right\|_{L^{2}}\mathrm{d}\tau$ (5.25)

where we used the Cauchy-Schwarz inequality. Thanks to (5.20) and (2.5) we get
for $|I|\leq N^{\prime}$:

$\displaystyle\left\|w^{\frac{1}{2}}\Box Z^{I}h_{\lambda}\right\|_{L^{2}}$
$\displaystyle\lesssim\sum_{|J|\leq|I|}\left(\left\|w^{\frac{1}{2}}Z^{J}\mathcal{N}(h_{\lambda})\right\|_{L^{2}(B_{R}\cap\Sigma_{\tau})}+\left\|Z^{J}\mathbf{A}_{\lambda}\right\|_{L^{2}(A_{R}\cap\Sigma_{\tau})}+\left\|Z^{J}\mathbf{B}_{\lambda}\right\|_{L^{2}(B_{R}\cap\Sigma_{\tau})}\right).$

Thanks to Lemma 5.6 we have

$\displaystyle\left\|Z^{J}\mathbf{A}_{\lambda}\right\|_{L^{2}(A_{R}\cap\Sigma_{\tau})}+\left\|Z^{J}\mathbf{B}_{\lambda}\right\|_{L^{2}(B_{R}\cap\Sigma_{\tau})}$
$\displaystyle\lesssim\varepsilon\lambda^{K-|J|}\left(\int_{0}^{\tau+R}\frac{\mathrm{d}r}{(1+r+\tau)^{4}}\right)^{\frac{1}{2}}\lesssim\frac{\varepsilon\lambda^{K-|J|}}{(1+\tau)^{\frac{3}{2}}}.$
(5.26)

We now look at $\mathcal{N}(h_{\lambda})$ using (4.16):

$\displaystyle\left\|w^{\frac{1}{2}}Z^{J}\mathcal{N}(h_{\lambda})\right\|_{L^{2}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}|J_{1}|+|J_{2}|\leq|J|\\\
\Upsilon\in\\{\varphi,\psi,\lambda^{+}F\mathbb{T}_{\lambda}\\}\\\
j=1,2\end{subarray}}\left(N_{1}^{J_{1},J_{2}}+N^{J_{1},J_{2}}_{\Upsilon,j}\right)$

with

$\displaystyle N_{1}^{J_{1},J_{2}}$
$\displaystyle=\left\|w^{\frac{1}{2}}\partial
Z^{J_{1}}h_{\lambda}\overline{\partial}Z^{J_{2}}h_{\lambda}\right\|_{L^{2}},$
$\displaystyle N_{\Upsilon,1}^{J_{1},J_{2}}$
$\displaystyle=\left\|w^{\frac{1}{2}}\partial
Z^{J_{1}}h_{\lambda}\overline{\partial}Z^{J_{2}}\Upsilon\right\|_{L^{2}},$
$\displaystyle N_{\Upsilon,2}^{J_{1},J_{2}}$
$\displaystyle=\left\|w^{\frac{1}{2}}\partial
Z^{J_{1}}\Upsilon\overline{\partial}Z^{J_{2}}h_{\lambda}\right\|_{L^{2}}.$

We start by $N_{1}^{J_{1},J_{2}}$ and we distinguish two cases: since
$|J_{1}|+|J_{2}|\leq N^{\prime}$ we have either
$|J_{1}|\leq\frac{N^{\prime}}{2}$ or $|J_{2}|\leq\frac{N^{\prime}}{2}$.

  * •

If $|J_{1}|\leq\frac{N^{\prime}}{2}$, then (5.21) implies (since
$N^{\prime}\geq 8\Longrightarrow\frac{N^{\prime}}{2}+1\leq N^{\prime}-3$)

$\displaystyle|\partial
Z^{J_{1}}h_{\lambda}|\lesssim\frac{\varepsilon\lambda^{K-4-|J_{1}|}}{(1+s)^{\frac{1}{2}+\frac{\nu}{2}}(1+|q|)^{1-\frac{\nu}{2}}}$

where $\nu$ is as in (4.25). Using (4.25) we obtain

$\displaystyle N_{1}^{J_{1},J_{2}}$
$\displaystyle\lesssim\frac{\varepsilon\lambda^{K-4-|J_{1}|}}{(1+\tau)^{\frac{1}{2}+\frac{\nu}{2}}}\left\|(w^{\prime})^{\frac{1}{2}}\overline{\partial}Z^{J_{2}}h_{\lambda}\right\|_{L^{2}}.$
(5.27)

  * •

If $|J_{2}|\leq\frac{N^{\prime}}{2}$, then (5.21) implies

$\displaystyle\overline{\partial}Z^{J_{2}}h_{\lambda}\lesssim\frac{\varepsilon\lambda^{K-4-|J_{2}|}}{(1+s)^{\frac{3}{2}}}.$

Therefore using (5.19) we obtain

$\displaystyle N_{1}^{J_{1},J_{2}}$
$\displaystyle\lesssim\frac{\varepsilon\lambda^{K-4-|J_{2}|}}{(1+\tau)^{\frac{3}{2}}}\left\|w^{\frac{1}{2}}\partial
Z^{J_{1}}h_{\lambda}\right\|_{L^{2}}\lesssim\frac{\varepsilon^{2}\lambda^{2K-4-|J_{1}|-|J_{2}|}}{(1+\tau)^{\frac{3}{2}}}.$
(5.28)

We now turn to $N_{\Upsilon,1}^{J_{1},J_{2}}$ for
$\Upsilon\in\\{\varphi,\psi,\lambda^{+}F\mathbb{T}_{\lambda}\\}$. Using
(5.10), (5.11) and (5.7) we obtain

$\displaystyle|\overline{\partial}Z^{J_{2}}\Upsilon|\lesssim\frac{\varepsilon\lambda^{-|J_{2}|}}{(1+s)^{\frac{3}{2}}}$

where we used the presence of $\lambda^{+}$ to regain one power of $\lambda$.
Using then (5.19) this implies

$\displaystyle
N_{\Upsilon,1}^{J_{1},J_{2}}\lesssim\frac{\varepsilon\lambda^{-|J_{2}|}}{(1+\tau)^{\frac{3}{2}}}\left\|w^{\frac{1}{2}}\partial
Z^{J_{1}}h_{\lambda}\right\|_{L^{2}}\lesssim\frac{\varepsilon^{2}\lambda^{K-|J_{1}|-|J_{2}|}}{(1+\tau)^{\frac{3}{2}}}.$
(5.29)

Finally, let us look at $N_{\Upsilon,2}^{J_{1},J_{2}}$ for
$\Upsilon\in\\{\varphi,\psi,\lambda^{+}F\mathbb{T}_{\lambda}\\}$. Using again
(5.10), (5.11) and (5.7) we obtain

$\displaystyle|\partial
Z^{J_{1}}\Upsilon|\lesssim\frac{\varepsilon\lambda^{-|J_{1}|}}{(1+s)^{\frac{1}{2}+\frac{\nu}{2}}(1+|q|)^{1-\frac{\nu}{2}}}$

where $\nu$ is as in (4.25). Therefore using (4.25) we obtain

$\displaystyle
N_{\Upsilon,2}^{J_{1},J_{2}}\lesssim\frac{\varepsilon\lambda^{-|J_{1}|}}{(1+\tau)^{\frac{1}{2}+\frac{\nu}{2}}}\left\|(w^{\prime})^{\frac{1}{2}}\overline{\partial}Z^{J_{2}}h_{\lambda}\right\|_{L^{2}}.$
(5.30)

We now put (5.26)-(5.27)-(5.28)-(5.29)-(5.30) together into (5.25) to obtain

$\displaystyle\mathcal{E}_{\lambda}(t)+\mathcal{S}_{\lambda}(t)$
$\displaystyle\lesssim\sum_{|J|\leq|I|\leq
N^{\prime}}\int_{0}^{t}\frac{\varepsilon\lambda^{-K+2|I|-|J|}}{(1+\tau)^{\frac{3}{2}}}\left\|w^{\frac{1}{2}}\partial
Z^{I}h_{\lambda}\right\|_{L^{2}}\mathrm{d}\tau$
$\displaystyle\quad+\sum_{|J_{1}|+|J_{2}|\leq|I|\leq
N^{\prime}}\int_{0}^{t}\frac{\varepsilon^{2}\lambda^{-K+2|I|-|J_{1}|-|J_{2}|}}{(1+\tau)^{\frac{3}{2}}}\left\|w^{\frac{1}{2}}\partial
Z^{I}h_{\lambda}\right\|_{L^{2}}\mathrm{d}\tau$
$\displaystyle\quad+\sum_{|J_{1}|+|J_{2}|\leq|I|\leq
N^{\prime}}\int_{0}^{t}\frac{\varepsilon\lambda^{-2K+2|I|-|J_{1}|}}{(1+\tau)^{\frac{1}{2}+\frac{\nu}{2}}}\left\|(w^{\prime})^{\frac{1}{2}}\overline{\partial}Z^{J_{2}}h_{\lambda}\right\|_{L^{2}}\left\|w^{\frac{1}{2}}\partial
Z^{I}h_{\lambda}\right\|_{L^{2}}\mathrm{d}\tau$

where the first line corresponds to (5.26), the second to (5.28) and (5.29)
and the third to (5.27) and (5.30). Note that we used $K\geq 4$ to bound
$\lambda^{K-4}$ by 1 in (5.27) and (5.28). For the first two integrals, we
simply use (5.19). For the third integral we use $2ab\leq a^{2}+b^{2}$ and
split the $\lambda$ powers carefully:

$\displaystyle\int_{0}^{t}$
$\displaystyle\frac{\varepsilon\lambda^{-2K+2|I|-|J_{1}|}}{(1+\tau)^{\frac{1}{2}+\frac{\nu}{2}}}\left\|(w^{\prime})^{\frac{1}{2}}\overline{\partial}Z^{J_{2}}h_{\lambda}\right\|_{L^{2}}\left\|w^{\frac{1}{2}}\partial
Z^{I}h_{\lambda}\right\|_{L^{2}}$
$\displaystyle\lesssim\int_{0}^{t}\varepsilon\lambda^{-2K+2|J_{2}|}\left\|(w^{\prime})^{\frac{1}{2}}\overline{\partial}Z^{J_{2}}h_{\lambda}\right\|_{L^{2}}^{2}+\int_{0}^{t}\frac{\varepsilon\lambda^{-2K+4|I|-2|J_{1}|-2|J_{2}|}}{(1+\tau)^{1+\nu}}\left\|w^{\frac{1}{2}}\partial
Z^{I}h_{\lambda}\right\|_{L^{2}}^{2}$
$\displaystyle\lesssim\varepsilon\mathcal{S}_{\lambda}(t)+\varepsilon^{3}\lambda^{2(|I|-|J_{1}|-|J_{2}|)}$

Since $|J_{1}|+|J_{2}|\leq|I|$ and $|J|\leq|I|$ we have proved that there
exists a constant $C>0$ such that

$\displaystyle\mathcal{E}_{\lambda}(t)+(1-C\varepsilon)\mathcal{S}_{\lambda}(t)\leq
CC_{2}\varepsilon^{2}+C\varepsilon^{3}$

We now choose $C_{2}\geq 4C$ and $\varepsilon\leq\frac{C_{2}^{2}}{4C}$ so that
for every $t\in[0,T]$,
$\mathcal{E}_{\lambda}(t)\leq\frac{C_{2}^{2}}{2}\varepsilon^{2}$.

∎

The result of Proposition 5.4 contradicts the maximality of $T$ and thus
$T=T_{0}$. But now $\mathcal{E}_{\lambda}(t)$ is bounded up to $T_{0}$ by
$C_{2}^{2}\varepsilon^{2}$, which proves that $T_{0}=+\infty$. This concludes
the proof of Theorem 4.1.

##  Appendix A Commutators

In this first appendix, we estimate $[Z,\overline{\partial}]$ and
$[Z,\mathscr{L}]$. The later will be used in the second appendix, where we
solve equation (5.6).

######  Lemma A.1.

For every $Z\in\mathcal{Z}$, we have

$[Z,\overline{\partial}_{i}]=p^{\ell}_{i}\overline{\partial}_{\ell}+\sum_{Z^{\prime}\in\mathcal{Z}}\frac{p_{i,Z^{\prime}}}{r}Z^{\prime},$

for some functions $p^{\ell}_{i}$ and $p_{i,Z^{\prime}}$ such that for every
$Z^{\prime}\in\mathcal{Z}$, multi-index $K$ and $\ell\in\\{1,2,3\\}$ there
exists $C^{K,\ell}_{Z}$ such that

$|Z^{K}p^{\ell}_{i}|+|Z^{K}p_{i,Z^{\prime}}|\leq C^{K,I,\ell}_{J}$

in the region $A^{R}$.

###### Proof.

It follows from direct computations, using the formula
$\overline{\partial}_{i}=\partial_{i}-\frac{x_{i}}{r}\partial_{r}$:

$\displaystyle[\overline{\partial}_{i},\partial_{t}]$ $\displaystyle=0,$
$\displaystyle[\overline{\partial}_{i},\partial_{j}]$
$\displaystyle=\frac{\omega_{i}}{r}\overline{\partial}_{j}+\frac{1}{r}(\delta_{ij}-\omega_{i}\omega_{j})\partial_{r},$
$\displaystyle[\overline{\partial}_{i},S]$
$\displaystyle=\overline{\partial}_{i},$
$\displaystyle[\overline{\partial}_{i},\Omega_{jk}]$
$\displaystyle=\delta_{ik}\overline{\partial}_{j}-\delta_{ij}\overline{\partial}_{k},$
$\displaystyle[\overline{\partial}_{i},\Omega_{0j}]$
$\displaystyle=\frac{t\omega_{i}}{r}\overline{\partial}_{j}+\frac{\omega_{k}}{r}(\delta_{ij}-\omega_{i}\omega_{j})\Omega_{0k}.$

∎

The following proposition estimates the commutator between the Minkowski
vector fields and the transport operator $\mathscr{L}$.

######  Proposition A.1.

For every multi-index $I$ we have

$[\mathscr{L},Z^{I}]=\sum_{|J|\leq|I|-1}\left(\frac{a^{I,\ell}_{J}}{r}\overline{\partial}_{\ell}Z^{J}+b^{I}_{J}Z^{J}\mathscr{L}+\frac{c^{I}_{J}}{r^{2}}Z^{J}\right),$

for some functions $a^{I,\ell}_{J}$, $b^{I}_{J}$ and $c^{I}_{J}$ such that for
every multi-index $I$, $J$ and $K$ with $|J|+1\leq|I|$ and every
$\ell\in\\{1,2,3\\}$ there exists $C^{K,I,\ell}_{J}>0$ such that

$|Z^{K}a^{I,\ell}_{J}|+|Z^{K}b^{I}_{J}|+|Z^{K}c^{I}_{J}|\leq C^{K,I,\ell}_{J}$

in the region $A^{R}$.

###### Proof.

We prove this by strong induction on the value of $|I|$. Some direct
computations show that

$\displaystyle[\mathscr{L},\partial_{t}]=[\mathscr{L},\Omega_{ij}]$
$\displaystyle=0,$ $\displaystyle[\mathscr{L},\partial_{i}]$
$\displaystyle=-\frac{1}{r}\overline{\partial}_{i}+\frac{\omega_{i}}{r^{2}}\mathrm{Id},$
$\displaystyle[\mathscr{L},\Omega_{0i}]$
$\displaystyle=\frac{t-r}{r}\overline{\partial}_{i}-\omega_{i}\mathscr{L}+\frac{\omega_{i}(r-t)}{r^{2}}\mathrm{Id},$
$\displaystyle[\mathscr{L},S]$ $\displaystyle=\mathscr{L},$

which proves the proposition if $|I|=1$. We use the notation
$[\mathscr{L},Z]=\frac{a^{Z,\ell}_{0}}{r}\overline{\partial}_{\ell}+b^{Z}_{0}\mathscr{L}+\frac{c^{Z}_{0}}{r^{2}}\mathrm{Id}$.
Now let $|I|$ be a multi-index with $|I|\leq N-1$ and $Z$ be any minkowskian
vector field, we have

$\displaystyle[\mathscr{L},ZZ^{I}]$
$\displaystyle=Z[\mathscr{L},Z^{I}]+[\mathscr{L},Z]Z^{I}$
$\displaystyle=\sum_{|J|\leq|I|-1}\left(Z\left(\frac{a^{I,\ell}_{J}}{r}\right)\overline{\partial}_{\ell}Z^{J}+Z\left(b^{I}_{J}\right)Z^{J}\mathscr{L}+Z\left(\frac{c^{I}_{J}}{r^{2}}\right)Z^{J}\right)+\sum_{|J|\leq|I|}\left(b^{I}_{J}Z^{J}\mathscr{L}+\frac{c^{I}_{J}}{r^{2}}Z^{J}\right)$
$\displaystyle\qquad+\sum_{|J|\leq|I|-1}\frac{a^{I,\ell}_{J}}{r}Z\overline{\partial}_{\ell}Z^{J}+\frac{a^{Z,\ell}_{0}}{r}\overline{\partial}_{\ell}Z^{I}+b^{Z}_{0}\mathscr{L}Z^{I}+\frac{c^{Z}_{0}}{r^{2}}Z^{I}.$

Using the assumptions made on the functions $a^{I,\ell}_{J}$, $b^{I}_{J}$ and
$c^{I}_{J}$, the fact that $|Z^{K}(r^{-1})|\lesssim r^{-1}$ for every multi-
index $K$, we see that the first sum contains only good terms, meaning that
they have the form we want. It is also the case for the terms in the second
sum and for the terms
$\frac{a^{Z,\ell}_{0}}{r}\overline{\partial}_{\ell}Z^{I}$ and
$\frac{c^{Z}_{0}}{r^{2}}Z^{I}$. To conclude the proof, it remains to deal with
the third sum and $b^{Z}_{0}\mathscr{L}Z^{I}$:

  * •

for $b^{Z}_{0}\mathscr{L}Z^{I}$, we simply write
$b^{Z}_{0}\mathscr{L}Z^{I}=b^{Z}_{0}Z^{I}\mathscr{L}+b^{Z}_{0}[\mathscr{L},Z^{I}]$.
Because of the assumptions made on $b^{Z}_{0}$, it proves that the term
$b^{Z}_{0}\mathscr{L}Z^{I}$ has the desired form.

  * •

for the third sum, we use Lemma A.1:

$\displaystyle\sum_{|J|\leq|I|-1}\frac{a^{I,\ell}_{J}}{r}Z\overline{\partial}_{\ell}Z^{J}$
$\displaystyle=\sum_{|J|\leq|I|}\frac{a^{I,\ell}_{J}}{r}\overline{\partial}_{\ell}Z^{J}+\sum_{|J|\leq|I|-1}\frac{a^{I,\ell}_{J}}{r}[Z,\overline{\partial}_{\ell}]Z^{J}$
$\displaystyle=\sum_{|J|\leq|I|}\frac{a^{I,\ell}_{J}}{r}\overline{\partial}_{\ell}Z^{J}+\sum_{|J|\leq|I|-1}\frac{a^{I,\ell}_{J}p_{\ell}^{k}}{r}\overline{\partial}_{k}Z^{J}+\sum_{|J|\leq|I|-1}\frac{a^{I,\ell}_{J}p_{\ell,Z^{\prime}}}{r^{2}}Z^{\prime}Z^{J},$

which is of the desired form.

This concludes the proof of the proposition. ∎

The following Lemma estimates $\Box f$ if we know $\mathscr{L}f$.

######  Lemma A.2.

If $f$ and $g$ are such that $\mathscr{L}f=g$, then

$\Box
f=\delta^{ij}\overline{\partial}_{i}\overline{\partial}_{j}f-\underline{L}g-\frac{g}{r}.$

###### Proof.

Applying separately $-\partial_{t}+\partial_{r}$ to $\mathscr{L}f=g$ we obtain

$-\partial_{t}^{2}f+\partial_{r}^{2}f-\frac{1}{r}\left(\partial_{t}f-\partial_{r}f+\frac{f}{r}\right)=-\underline{L}g.$

We recognize on the LHS of the previous equation the operator $\mathscr{L}$
applied to $f$ so that :

$-\partial_{t}^{2}f+\partial_{r}^{2}f+\frac{2}{r}\partial_{r}f=-\underline{L}g-\frac{g}{r}.$

Recalling that
$\Delta=\partial_{r}^{2}+\frac{2}{r}\partial_{r}+\delta^{ij}\overline{\partial}_{i}\overline{\partial}_{j}$
concludes the proof. ∎

##  Appendix B The transport equations

The goal of this appendix is to prove Proposition 5.2.

###  B.1 The reduced transport equation

We start by considering the following reduced transport equation

$(L+\eta)f=g$ (B.1)

where

  * •

$g:\mathbb{R}_{+}\times\mathbb{R}^{3}\longmapsto\mathbb{R}$ is a continuous
function supported in $A^{R}$,

  * •

$\eta:\mathbb{R}_{+}\times\mathbb{R}^{3}\longmapsto\mathbb{R}$ is a continuous
function such that $|\eta|\lesssim\frac{\varepsilon}{(1+s)^{2}}$.

We start by constructing a well-chosen solution to $(L-\eta)\phi=0$ which is
close to 1.

######  Definition B.1.

Let $\chi:\mathbb{R}^{3}\to[0,1]$ be a smooth function supported in
$\\{\frac{1}{2R}\leq r\leq 2R\\}$ and such that $\chi_{|A_{0}}\equiv 1$. Using
polar coordinates we define $\hat{\chi}$ a cut-off function around $A^{R}$ in
$\mathbb{R}_{+}\times\mathbb{R}^{3}$ by

$\hat{\chi}(t,r,\omega)=\chi(r-t,\omega).$

######  Lemma B.1.

There exists a function $\beta$ vanishing outside of $\\{\frac{1}{2R}\leq
q\leq 2R\\}$ which satisfies

$|\beta|\lesssim\varepsilon\qquad\text{and}\qquad
L(1+\beta)=(1+\beta)\eta\quad\text{in $A^{R}$}.$

###### Proof.

We simply define $\beta$ using polar coordinates

$\beta(t,r,\omega)\vcentcolon=\exp\left(\hat{\chi}(t,r,\omega)\int_{0}^{t}\eta(t^{\prime},r-t+t^{\prime},\omega)\mathrm{d}t^{\prime}\right)-1.$

Thanks to our assumption on $\eta$, $\beta$ satisfies the bound
$|\beta|\lesssim\varepsilon$. Thus, if $\varepsilon$ is small enough, we can
bound $1+\beta$ from below by $\frac{1}{2}$, and therefore $\ln(1+\beta)$ is
well-defined and satisfies clearly $L(\ln(1+\beta))=\eta$ in $A^{R}$, which
implies $L(1+\beta)=(1+\beta)\eta$. ∎

It is now easy to solve (B.1):

######  Lemma B.2.

Let $f_{0}:\mathbb{R}^{3}\to\mathbb{R}$ supported in $A_{0}$ and
$g:\mathbb{R}_{+}\times\mathbb{R}^{3}\to\mathbb{R}$ supported in $A^{R}$.
There exists a unique global solution $f$ of (B.1) with initial data $f_{0}$.
This solution is supported in $A^{R}$ and satisfies

$|f|\lesssim\sup_{(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{3}}\left(|f_{0}(x)|+(1+|x|)^{2}|g(t,x)|\right).$

###### Proof.

The uniqueness follows easily from an energy estimate related to the operator
$L+\eta$. To construct a solution, we use polar coordinates and set

$\displaystyle f(t,r,\omega)\vcentcolon$
$\displaystyle=\frac{1}{1+\beta(t,r,\omega)}\Bigg{(}(1+\beta(0,r-t,\omega))f_{0}(r-t,\omega)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\int_{0}^{t}(1+\beta(s,r-t+s,\omega))g(s,r-t+s,\omega)\mathrm{d}s\Bigg{)}.$

By construction, we have $f_{|t=0}=f_{0}$ and $L((1+\beta)f)=(1+\beta)g$, and
$f$ is supported in $A^{R}$. Let $(t,x)\in A^{R}$, since
$L(1+\beta)=(1+\beta)\eta$ in $A^{R}$ we have:

$(L+\eta)f(t,x)=\frac{1}{1+\beta(t,x)}L((1+\beta)f)(t,x)=g(t,x).$

This proves the existence of a solution. The estimate on $|f|$ comes from the
expression of $f$, the estimate $\frac{1}{2}\leq 1+\beta\leq 2$ and the
support property of $g$ (note that in $A^{R}$, $r$ and $t$ are equivalent). ∎

###  B.2 The transport operator $\mathscr{L}$

Using the result of the previous section, we now solve the equation

$\mathscr{L}f=f\mu+g$ (B.2)

where

$(1+r)^{2}|Z^{I}\mu|+(1+r)^{3}|Z^{I}g|\lesssim\varepsilon$ (B.3)

for $|I|\leq M$, where $M$ is an integer. We consider initial data $f_{0}$
supported in $A_{0}$ and satisfying
$\left\|f_{0}\right\|_{C^{M}}\lesssim\varepsilon$. Our strategy is to first
deduce some a priori estimates.

We first take advantage of $[\mathscr{L},\Omega]=0$:

######  Lemma B.3.

If $f$ is a smooth and global in time solution of (B.2) initially supported in
$A_{0}$, then

$|\Omega^{J}f|\lesssim\frac{\varepsilon}{1+r}\qquad\text{for}\quad|J|\leq M.$
(B.4)

###### Proof.

We prove this lemma by strong induction on the value of $|J|$. If $|J|=0$, we
just rewrite (B.2) as

$(L-\mu)(rf)=rg$

and apply Lemma B.2 using (B.3). Assume now that the estimate (B.4) holds for
all value of $|J|\leq k$ with $k\leq M-1$ and let $J$ be a multi-index such
that $|J|=k+1$. Since $[\mathscr{L},\Omega]=0$, the equation for $\Omega^{J}f$
is

$\mathscr{L}\Omega^{J}f=\Omega^{J}f\mu+\sum_{\begin{subarray}{c}J_{1}+J_{2}=J\\\
|J_{1}|\leq k\end{subarray}}\Omega^{J_{1}}f\Omega^{J_{2}}\mu+\Omega^{J}g$

which rewrites

$(L-\mu)(r\Omega^{J}f)=\sum_{\begin{subarray}{c}J_{1}+J_{2}=J\\\ |J_{1}|\leq
k\end{subarray}}r\Omega^{J_{1}}f\Omega^{J_{2}}\mu+r\Omega^{J}g.$

Because of the induction hypothesis and of (B.3), the RHS of this equation is
bounded by $\frac{\varepsilon}{(1+r)^{2}}$. Thus, we can apply Lemma B.2 and
conclude the proof. ∎

Note that in the previous lemma, in order to apply Lemma B.2, we have to prove
that the initial data for $r\Omega^{J}f$ are bounded. Since the vector fields
$\Omega$ involve only spatial derivations, this boundedness property follows
from $f$ being supported in $A_{0}$ and smooth. In the next lemma however, we
fully use Proposition A.1 and estimate $\Omega^{J}Z^{I}f$, again with the help
of Lemma B.2. This requires to bound the initial data for $r\Omega^{J}Z^{I}f$,
which may involve some time derivative. To find $\partial^{k}_{t}f_{|t=0}$, we
simply use the equation (B.2) and $[\mathscr{L},\partial_{t}]=0$ to derive the
equation for $\partial^{k-1}_{t}f$ which gives

$\partial^{k}_{t}f=-\partial\partial_{t}^{k-1}f-\frac{\partial_{t}^{k-1}f}{r}+\sum_{i+j=k-1}\partial^{i}_{t}f\partial^{j}_{t}\mu+\partial^{k-1}_{t}g.$

Therefore we obtain $\partial^{k}_{t}f_{|t=0}$ by induction on $k$, and check
easily their initial boundedness. With that in mind, we can estimate
$\Omega^{J}Z^{I}f$:

######  Lemma B.4.

If $f$ is a smooth and global in time solution of (B.2) initially supported in
$A_{0}$, then

$|\Omega^{J}Z^{I}f|\lesssim\frac{\varepsilon}{1+r}\qquad\text{for}\quad|I|+|J|\leq
M.$ (B.5)

###### Proof.

We prove this lemma by strong induction on the value of $|I|$. The case
$|I|=0$ is a consequence of Lemma B.3. Assume now that the estimate (B.5)
holds for all value of $|I^{\prime}|\leq k$ with $k\leq M-1$ and all
$J^{\prime}$ such that $|I^{\prime}|+|J^{\prime}|\leq M$. Let $I$ be a multi-
index such that $|I|=k+1$ and let $J$ be a multi-index such that $|I|+|J|\leq
M$. Formally, the equation on $\Omega^{J}Z^{I}f$ is

$(L-\mu)(r\Omega^{J}Z^{I}f)=\sum_{\begin{subarray}{c}I_{1}+I_{2}=I\\\
J_{1}+J_{2}=J\\\ |I_{1}|\leq k\\\
|J_{1}|\leq|J|-1\end{subarray}}r\Omega^{J_{1}}Z^{I_{1}}f\Omega^{J_{2}}Z^{I_{2}}\mu+r\Omega^{J}Z^{I}g+r[\mathscr{L},\Omega^{J}Z^{I}]f.$
(B.6)

Using the induction hypothesis and (B.3), the first two terms in the RHS of
(B.6) are bounded by $\frac{\varepsilon}{(1+r)^{2}}$. It remains to estimate
the commutator. Using $[\mathscr{L},\Omega]=0$ and Proposition A.1 we have:

$\displaystyle\ |r[\mathscr{L},\Omega^{J}Z^{I}]f|$
$\displaystyle\lesssim\left|r\Omega^{J}\sum_{|I^{\prime}|\leq|I|-1}\left(\frac{a^{I,\ell}_{I^{\prime}}}{r}\overline{\partial}_{\ell}Z^{I^{\prime}}f+b^{I}_{I^{\prime}}Z^{I^{\prime}}\mathscr{L}f+\frac{c^{I}_{I^{\prime}}}{r^{2}}Z^{I^{\prime}}f\right)\right|$
$\displaystyle\lesssim\sum_{|I^{\prime}|\leq|I|-1}\left|\Omega^{J}\overline{\partial}Z^{I^{\prime}}f\right|+\sum_{|I^{\prime}|\leq|I|-1}r\left|\Omega^{J}Z^{I^{\prime}}\mathscr{L}f\right|+\sum_{|I^{\prime}|\leq|I|-1}\frac{1}{r}\left|\Omega^{J}Z^{I^{\prime}}f\right|$
(B.7)

Since $|I^{\prime}|\leq|I|-1$ we can use the induction hypothesis to obtain

$\sum_{|I^{\prime}|\leq|I|-1}\frac{1}{r}\left|\Omega^{J}Z^{I^{\prime}}f\right|\lesssim\frac{\varepsilon}{(1+r)^{2}}.$

To estimate the terms involving $\mathscr{L}f$, we simply use (B.2), the
induction hypothesis and (B.3):

$\displaystyle\sum_{|I^{\prime}|\leq|I|-1}r\left|\Omega^{J}Z^{I^{\prime}}\mathscr{L}f\right|$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}|I^{\prime}|\leq|I|-1\\\
I_{1}+I_{2}=I^{\prime}\\\
J_{1}+J_{2}=J\end{subarray}}r|\Omega^{J_{1}}Z^{I_{1}}f\Omega^{J_{2}}Z^{I_{2}}\mu|+r|\Omega^{J}Z^{I}g|\lesssim\frac{\varepsilon}{(1+r)^{2}}.$

It remains to estimate the first terms in (B.7). Using
$[\Omega,\overline{\partial}]\sim\overline{\partial}$ and
$\overline{\partial}_{\ell}=\frac{\omega^{k}}{r}\Omega_{\ell k}$ we get

$\displaystyle\sum_{|I^{\prime}|\leq|I|-1}\left|\Omega^{J}\overline{\partial}Z^{I^{\prime}}f\right|\lesssim\sum_{\begin{subarray}{c}|I^{\prime}|\leq|I|-1\\\
|J^{\prime}|\leq|J|+1\end{subarray}}\frac{1}{r}\left|\Omega^{J^{\prime}}Z^{I^{\prime}}f\right|$

In this last sum, note that $|I^{\prime}|\leq|I|-1$ and
$|I^{\prime}|+|J^{\prime}|\leq|I|-1+|J|+1\leq M$. Thus we can use our
induction hypothesis to bound this by $\frac{\varepsilon}{(1+r)^{2}}$. In
conclusion, the RHS of (B.6) is bounded by $\frac{\varepsilon}{(1+r)^{2}}$,
and we can apply Lemma B.2 and conclude the proof. ∎

In the previous lemma, the case $J=0$ gives that a global smooth solution of
(B.2) satisfies

$|Z^{I}f|\lesssim\frac{\varepsilon}{1+r}\qquad\text{for}\quad|I|\leq M.$

This a priori estimates allows us easily to prove the existence and uniqueness
of a global solution to (B.2). We summarize this discussion in the following
corollary:

######  Corollary B.1.

Let $f_{0}$ supported in $A_{0}$ and such that and let $\mu$ and $g$ satisfy
(B.3). There exists a unique global solution $f\in C^{M}$ to (B.2) with
initial data $f_{0}$. Moreover, $f$ is supported in $A^{R}$ and satisfies

$|Z^{I}f|\lesssim\frac{\varepsilon}{1+r}\qquad\text{for}\quad|I|\leq M.$

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