#  Long-time existence for systems of quasilinear wave equations

Jason Metcalfe  Department of Mathematics, University of North Carolina,
Chapel Hill, NC 27599-3250<EMAIL_ADDRESS>and  Taylor Rhoads
Department of Mathematics, Vanderbilt University, Nashville, TN 37240
<EMAIL_ADDRESS>
###### Abstract.

We consider quasilinear wave equations in $(1+3)$-dimensions where the
nonlinearity $F(u,u^{\prime},u^{\prime\prime})$ is permitted to depend on the
solution rather than just its derivatives. For scalar equations, if
$(\partial_{u}^{2}F)(0,0,0)=0$, almost global existence was established by
Lindblad. We seek to show a related almost global existence result for coupled
systems of such equations. To do so, we will rely upon a variant of the
$r^{p}$-weighted local energy estimate of Dafermos and Rodnianski that
includes a ghost weight akin to those used by Alinhac. The decay that is
needed to close the argument comes from space-time Klainerman-Sobolev type
estimates from the work of Metcalfe, Tataru, and Tohaneanu.

The first author was supported in part by Simons Foundation Collaboration
Grant 711724 and NSF grant DMS-2054910.

##  1\. Introduction

In this article, we shall examine long-time existence for systems of
$(1+3)$-dimensional quasilinear wave equations with small initial data where
the nonlinearity is permitted to depend on the solution rather than just its
derivatives. In particular, for $\Box=\partial_{t}^{2}-\Delta$, we shall
examine

(1.1) $\begin{cases}\Box
u^{I}=F^{I}(u,u^{\prime},u^{\prime\prime}),\quad(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{3},\quad
I=1,2,\dots,M,\\\
u(0,{\,\cdot\,})=f,\quad\partial_{t}u(0,{\,\cdot\,})=g.\end{cases}$

Here $u=(u^{1},\dots,u^{M})$. We use the notation $u^{\prime}=\partial
u=(\partial_{t}u,\nabla u)$ for the space-time gradient. The smooth function
$F$ vanishes to second order at the origin, and it is linear in the
$u^{\prime\prime}$ components. Moreover, we shall assume that

(1.2)
$(\partial_{u^{J}}\partial_{u^{K}}F^{I})(0,0,0)=0,\quad\forall\,I,J,K=1,2,\dots,M,$

which has the effect of disallowing $u^{2}$-type terms at the quadratic level
for $F$. In [11], almost global existence, which shows that the lifespan grows
exponentially as the size of the data shrinks, was proved for scalar
equations. In the current article, we seek similar lower bounds on the
lifespan for systems.

In the sequel, we use Einstein’s summation convention. Repeated Greek letters
$\alpha,\beta,\gamma$ are understood to sum from $0$ to $3$ where $x_{0}=t$.
Repeated lower case Roman letters $i,j,k$ are summed from $1$ to $3$, and
repeated upper case letters $I,J,K$ will be summed from $1$ to $M$.

For simplicity, we shall truncate the nonlinearity at the quadratic level:

(1.3)
$F^{I}(u,u^{\prime},u^{\prime\prime})=a^{I,\alpha}_{JK}u^{J}\partial_{\alpha}u^{K}+b^{I,\alpha\beta}_{JK}\partial_{\alpha}u^{J}\partial_{\beta}u^{K}+A^{I,\alpha\beta}_{JK}u^{K}\partial_{\alpha}\partial_{\beta}u^{J}+B^{I,\alpha\beta\gamma}_{JK}\partial_{\gamma}u^{K}\partial_{\alpha}\partial_{\beta}u^{J}.$

In the small data regime, higher order terms are better behaved. The constants
will be assumed to satisfy the symmetry conditions

(1.4)
$A^{I,\alpha\beta}_{JK}=A^{I,\beta\alpha}_{JK}=A^{J,\alpha\beta}_{IK},\quad
B^{I,\alpha\beta\gamma}_{JK}=B^{I,\beta\alpha\gamma}_{JK}=B^{J,\alpha\beta\gamma}_{IK}.$

Our main result is the following statement of almost global existence.

######  Theorem 1.1.

Suppose that $f,g\in(C^{\infty}_{c}({\mathbb{R}}^{3}))^{M}$. Moreover, assume
that the smooth function $F$ vanishes to second order at the origin, satisfies
(1.2), and is subject to the symmetry conditions (1.4). Then, for
$N\in\mathbb{N}$ sufficiently large, there are constants $c,\varepsilon_{0}>0$
so that if

(1.5) $\sum_{|\mu|\leq N+1}\|\partial^{\mu}f\|_{L^{2}}+\sum_{|\mu|\leq
N}\|\partial^{\mu}g\|_{L^{2}}\leq\varepsilon$

with $\varepsilon<\varepsilon_{0}$, then (1.1) has a unique solution with
$u\in(C^{\infty}([0,T_{\varepsilon}]\times{\mathbb{R}}^{3}))^{M}$ where

(1.6) $T_{\varepsilon}=\exp(c/\varepsilon^{\frac{1}{3}}).$

To aid the exposition, we have restricted to the case of compactly supported
initial data. Without loss of generality, we shall take

(1.7) $\text{supp}\,f^{I},\text{supp}\,g^{I}\subset\\{|x|\leq
2\\},\quad\forall\,I=1,2,\dots,M.$

Small data in a sufficiently weighted space would also suffice. In [11], the
lower bound of the lifespan was $\exp(c/\varepsilon)$. The difference in the
lifespan between [11] and Theorem 1.1 is primarily due to a logarithmic loss
that occurs as a part of an endpoint Hardy inequality. See Lemma 2.5. While
refinements of our argument to improve the power in (1.6) are likely possible,
it is not clear what the sharp power is.

Equations such as (1.1) with nonlinearity that depends on the solution rather
than just its derivatives do not mesh as simply with the energy methods that
are typically employed to prove long-time existence. In [7], without the
hypothesis (1.2), a lower bound of $\exp(c/\varepsilon)$ was established in
$(1+4)$-dimensions. For scalar equations, the additional hypothesis (1.2) was,
moreover, found to be sufficient to guarantee global existence for
sufficiently small data. The analogous results in $(1+3)$-dimensions appeared
in [11] where the lifespan was shown to exceed $c/\varepsilon^{2}$ without
(1.2) and almost global existence was provided for scalar equations.

In both [7] and [11], the restriction to scalar equations is necessitated by
the use of the chain rule to write $u\cdot\partial u=\frac{1}{2}\partial
u^{2}$ interactions in divergence form. This special form, in turn, allows for
easier estimation of the solution rather than only its derivatives. See, e.g.,
[11, Proposition 1.8]. The article [13] extended the result of [7] by
establishing small data global existence for systems (1.1) subject to (1.2) in
$(1+4)$-dimensions. We establish the $(1+3)$-dimensional analog here.

The principle source of decay in our proof is obtained from space-time
Klainerman-Sobolev estimates as were proved in [20]. This will be paired with
variants of the integrated local energy decay estimates. In [2],
$r^{p}$-weighted local energy estimates, which provide improved bounds on the
“good” components of the gradient: $\partial_{t}+\partial_{r}$ and
${\not\negmedspace\nabla}:=\nabla-\frac{x}{r}\partial_{r}$, were proved. These
will be combined with a ghost weight, which originates from [1]. This permits
a further improvement of the bounds in the vicinity of the light cone and
meshes particularly well with the space-time Klainerman-Sobolev estimates of
[20].

While the proof uses the method of invariant vector fields, it will not rely
on the Lorentz boosts. While Lorentz boosts are perfectly acceptable for
systems such as (1.1), they can limit further extensions to, e.g., multiple
speed settings, equations in exterior domains, or equations on stationary
background geometries. Our proof is readily adaptable to Dirichlet wave
equations exterior to, say, star-shaped obstacles. We do not include these
extraneous details here. When combined with [4], [3], and [16], it (largely)
completes the extension of the long-time existence results of [7] and [11] to
exterior domains. The work on systems in [13] was also completed exterior to
star-shaped obstacles. The restriction to star-shaped obstacles is likely a
convenience. It is anticipated that any geometry that permits a sufficiently
rapid decay of local energy would suffice. See, e.g., [5, 6].

The key aspects to the proof are to effectively bound the solution $u$ when
typical energy methods estimate $\partial u$ and to obtain additional decay
from the derivative that must be present in at least one factor of every
nonlinear term. In particular, it was the former that restricted the analysis
to scalar equations in preceding results. The $r^{p}$-weights and ghost
weights help with both aspects. In particular, when combined with rather
standard Hardy-type estimates, improved bounds on the local energy of the
solution without derivatives, when compared to e.g. (2.2), result. When
attempting to gain additional decay from the derivative that must appear in at
least one factor of every term of the nonlinearity, one often relies on the
scaling vector field, which near the light cone gives additional decay in
$t-|x|$. The ghost weight allows us to take advantage of this additional decay
off of the light cone.

Since derivatives can be exchanged for extra decay using the scaling vector
field and since the $r^{p}$-weighted and ghost weighted estimates allow for
much larger weights for the good derivatives, which in essence provides
additional decay that can be used on other factors, one can quickly become
convinced that the worst possible nonlinear terms are of the form
$u(\partial_{t}-\partial_{r})u$. Moreover when all of the vector fields land
on the differentiated factor, one cannot afford to lose the additional vector
field that would result from using the scaling vector field to get additional
decay. In this case, we move the $\partial_{t}-\partial_{r}$ using integration
by parts. Within the local energy estimates, it could land on the weights or
the lower order factor, which allows us to gain additional decay, in the
latter case by using the scaling vector field. Additionally it could land on
the multiplier, which has the basic form $\partial_{t}+\partial_{r}$ and
modulo better behaved terms results in $\Box$ effectively turning these
quadratic interactions into better cubic interactions.

This article is organized as follows. In the next subsection, we shall gather
some notation and preliminary results that will be used frequently throughout
the paper. In Section 2, the integrated local energy estimates will be proved.
Section 3 contains our sources of decay, which are primarily space-time
versions of the Klainerman-Sobolev inequality. Finally, the main theorem is
proved in Section 4.

###  1.1. Notation

The vector fields that we rely on are

$Z=(\partial_{t},\partial_{1},\partial_{2},\partial_{3},\Omega_{1},\Omega_{2},\Omega_{3},S)$

where

$S=t\partial_{t}+r\partial_{r},\quad\Omega=x\times\nabla$

represent the scaling vector field and generators of (spatial) rotations,
respectively. We will frequently rely on the orthogonal decomposition

$\nabla=\frac{x}{r}\partial_{r}+{\not\negmedspace\nabla},$

and we shall use
${\not\negmedspace\partial}=(\partial_{t}+\partial_{r},{\not\negmedspace\nabla})$
as an abbreviation for the “good” derivatives. We note that

${\not\negmedspace\nabla}=-\frac{x}{r^{2}}\times\Omega,$

and as such,

(1.8) $|{\not\negmedspace\nabla}u|\leq\frac{1}{r}|Zu|.$

Moreover, the following commutator will be used frequently in the proofs of
local energy estimates, as it was in the seminal work [21]:

(1.9)
$[\nabla,\partial_{r}]=[{\not\negmedspace\nabla},\partial_{r}]=\frac{1}{r}{\not\negmedspace\nabla}.$

The admissible vector fields are well-known to satisfy

(1.10) $[\Box,\partial]=[\Box,\Omega_{j}]=0,\quad[\Box,S]=2\Box.$

Moreover, we have

(1.11)
$[Z,\partial]\in\text{span}(\partial),\quad|[Z,{\not\negmedspace\partial}]u|\leq\frac{1}{r}|Zu|+|{\not\negmedspace\partial}u|.$

We shall abbreviate

$|Z^{\leq N}u|=\sum_{|\mu|\leq N}|Z^{\mu}u|,\quad|\partial^{\leq
N}u|=\sum_{|\mu|\leq N}|\partial^{\mu}u|.$

We use $L^{p}L^{q}$ as an abbreviation for
$L^{p}_{t}L^{q}_{x}([0,T]\times{\mathbb{R}}^{3})=L^{p}([0,T];L^{q}({\mathbb{R}}^{3}))$.
In several circumstances, it will be convenient to do an inhomogeneous dyadic
decomposition of ${\mathbb{R}}^{3}$, and for this purpose we denote

$A_{R}=\\{R<|x|<2R\\},\quad\tilde{A}_{R}=\Bigl{\\{}\frac{7}{8}R<|x|<\frac{17}{8}R\Bigr{\\}}\quad\text{if
}R>1,$

and $A_{1}=\\{|x|<2\\}$, $\tilde{A}_{1}=\\{|x|<17/4\\}$. For the standard
integrated local energy estimates, we shall employ the following notations
from [24], [20]:

$\|u\|_{LE}=\sup_{j\geq 0}2^{-j/2}\|u\|_{L^{2}L^{2}([0,T]\times
A_{2^{j}})},\quad\|u\|_{LE^{1}}=\|(\partial u,|x|^{-1}u)\|_{LE}.$

In the proof of the local energy estimates, we will often desire a
$C^{1}({\mathbb{R}})$, bounded, nondecreasing function and for these purposes
set

$\sigma_{U}(z)=\frac{z}{U+|z|},\quad U>0.$

In the sequel, we shall also need dyadic decompositions in $t-r$, so we set

$X_{U}=\\{(t,x)\in[0,T]\times{\mathbb{R}}^{3}\,:\,U<t-r<2U\\}\>\text{for
$U>1$},\quad X_{1}=\\{(t,x)\in[0,T]\times{\mathbb{R}}^{3}\,:\,|t-r|<2\\},$

with a similar enlargement being denoted by $\tilde{X}_{U}$.

The estimates from [20] rely on a mixed decomposition where the cone is
divided in $|x|$ away from the light cone and in $t-|x|$ near the cone. We set
$C=\\{r\leq t+2\\}$. Due to the simplifying assumption that the data are
compactly supported, it suffices to consider only this region, though we
believe that these estimates can be extended to all of
$[0,T]\times{\mathbb{R}}^{3}$ in a straightforward fashion.

We first divide into dyadic intervals in time
$C_{\tau}=\\{t\in[\tau,2\tau]\cap[0,T],r\leq t+2\\}$. Next, we shall decompose
dyadically in $r$ or $t-r$ depending on the proximity to the light cone. For
$R,U>1$, we set

$C^{R}_{\tau}=C_{\tau}\cap\\{R<r<2R\\},\quad
C_{\tau}^{U}=C_{\tau}\cap\\{U<t-r<2U\\}$

and

$C^{R=1}_{\tau}=C_{\tau}\cap\\{r<2\\},\quad
C^{U=1}_{\tau}=C_{\tau}\cap\\{|t-r|<2\\}.$

We note that

$C_{\tau}=\Bigl{(}\bigcup_{1\leq
R\leq\tau/4}C_{\tau}^{R}\Bigr{)}\cup\Bigl{(}\bigcup_{1\leq
U\leq\tau/4}C^{U}_{\tau}\Bigr{)}\cup C^{\frac{\tau}{2}}_{\tau},$

where

$C^{\frac{\tau}{2}}_{\tau}=C_{\tau}\cap\\{t-r\geq\frac{\tau}{2}\\}\cap\\{r\geq\frac{\tau}{2}\\}.$

We use $\tilde{C}^{R}_{\tau}$, $\tilde{C}_{\tau}^{U}$ to denote enlargements
of these sets on both the $R/U$ and $\tau$ scales. In the latter case, we
enlarge from $[\tau,2\tau]\cap[0,T]$ to $[(7/8)\tau,2\tau]\cap[0,T]$.
Subsequently $\tilde{\tilde{C}}^{R}_{\tau}$ and $\tilde{\tilde{C}}^{U}_{\tau}$
will indicate further enlargements. The key observation is that

$\langle r\rangle\approx R,\quad t-r\approx\tau,\quad\text{ on
}C^{R}_{\tau},\tilde{C}^{R}_{\tau},\tilde{\tilde{C}}^{R}_{\tau},\quad\text{with
}1\leq R\leq\tau/4$

and

$r\approx\tau,\quad\langle t-r\rangle\approx U,\quad\text{ on
}C^{U}_{\tau},\tilde{C}^{U}_{\tau},\tilde{\tilde{C}}^{U}_{\tau},\quad\text{with
}1\leq U\leq\tau/4.$

In this sense, the $C^{\frac{\tau}{2}}_{\tau}$ region can be thought of as
either a $C^{R}_{\tau}$ or a $C^{U}_{\tau}$ region. Here and throughout, $R,U$
are understood to run over dyadic values. Here we have set $\langle r\rangle$
to be a smooth function so that $\langle r\rangle\geq 3$ and $\langle
r\rangle\approx r$ for $r\gg 1$. For simplicity, we could simply take $\langle
r\rangle=(3+r)$ and $\langle T\rangle=(3+T)$.

In the sequel, we shall need cutoffs to localize to certain regions. We fix
$\beta\in C^{\infty}({\mathbb{R}})$ so that $\beta\equiv 1$ for $z<1$ and
$\beta\equiv 0$ for $z>2$. We then set

$\beta_{<R}(z)=\beta(z/R),\quad\beta_{>R}(z)=1-\beta_{<R}(z).$

##  2\. Local energy estimate

In this section, we shall collect some integrated local energy estimates,
which will serve as the main linear estimate for our proof of almost global
existence. The first of these is a now standard version of the original
estimates of [21]. We proceed to explore a variant of this that only bounds
the “good” derivatives but with much better weights. What results is a mixture
of the ghost weight method of [1] (see also the related estimates in [12]) and
the $r^{p}$-weighted local energy estimates of [2].

In this examination of a quasilinear problem, it will be helpful to have
estimates on small, time-dependent perturbations of the flat operator $\Box$.
To this end, we define

$(\Box_{h}u)^{I}=(\partial_{t}^{2}-\Delta)u^{I}-h^{I,\alpha\beta}_{J}(t,x)\partial_{\alpha}\partial_{\beta}u^{J}.$

Here we shall assume that

(2.1)
$h^{I,\alpha\beta}_{J}=h^{I,\beta\alpha}_{J}=h^{J,\alpha\beta}_{I},\qquad
h^{I,\alpha\beta}_{J}\in C^{1}([0,T]\times{\mathbb{R}}^{3}).$

We shall also abbreviate

$|h|=\sum_{I,J=1}^{M}\sum_{\alpha,\beta=0}^{3}|h^{I,\alpha\beta}_{J}|,\quad|\partial
h|=\sum_{I,J=1}^{M}\sum_{\alpha,\beta,\gamma=0}^{3}|\partial_{\gamma}h^{I,\alpha\beta}_{J}|.$

We begin by recalling the local energy estimate for perturbations of $\Box$
that was proved in [15]. See, also, [23], [14], [19, 18], and [17].

######  Theorem 2.1.

Suppose that $h$ satisfies (2.1), that $|h|$ is sufficiently small, that
$u\in(C^{2}([0,T]\times{\mathbb{R}}^{3}))^{M}$, and that for all $t\in[0,T]$,
$|\partial^{\leq 1}u(t,x)|\to 0$ as $|x|\to\infty$. Then,

(2.2) $\|\partial
u\|^{2}_{L^{\infty}L^{2}}+\|u\|^{2}_{LE^{1}}\lesssim\|\partial
u(0,{\,\cdot\,})\|^{2}_{L^{2}}+\int_{0}^{T}\int|\Box_{h}u|\Bigl{(}|\partial
u|+\frac{|u|}{r}\Bigr{)}\,dx\,dt\\\ +\int_{0}^{T}\int\Bigl{(}|\partial
h|+\frac{|h|}{\langle r\rangle}\Bigr{)}|\partial u|\Bigl{(}|\partial
u|+\frac{|u|}{r}\Bigr{)}\,dx\,dt.$

The proof of the theorem follows by pairing $\Box_{h}u$ with the multiplier
$C\partial_{t}u+\sigma_{2^{j}}(r)\partial_{r}u+\frac{\sigma_{2^{j}}(r)}{r}u$,
integrating over a space-time slab, and integrating by parts.

We next consider the following variant of Theorem 2.1. It represents a
combination of the ideas of [2] and [1]. The former considered multipliers
with principal part of the form $r^{p}(\partial_{t}+\partial_{r})u$. Rather
than considering associated flux terms to bound terms similar to the third
term in the left side below, we shall instead modify the multiplier using a
ghost weight, which originates in [1]. This more readily allows us to perform
necessary manipulations to prevent a loss of regularity due to the quasilinear
nature of the problem. It will also allow us to subsequently integrate by
parts to control a term using ideas akin to normal forms.

######  Theorem 2.2.

Fix $0<p<2$. Suppose that $h$ satisfies (2.1), that
$u\in(C^{2}([0,T]\times{\mathbb{R}}^{3}))^{M}$, and that for all $t\in[0,T]$,
$|r^{\frac{p+2}{2}}\partial^{\leq 1}u(t,x)|\to 0$ as $|x|\to\infty$. Then for
any $U>0$,

(2.3)
$\|r^{\frac{p-2}{2}}{\not\negmedspace\partial}(ru)\|^{2}_{L^{\infty}L^{2}}+\|r^{\frac{p-3}{2}}{\not\negmedspace\partial}(ru)\|^{2}_{L^{2}L^{2}}+\|r^{\frac{p-2}{2}}(\sigma^{\prime}_{U}(t-r))^{\frac{1}{2}}(\partial_{t}+\partial_{r})(ru)\|^{2}_{L^{2}L^{2}}\\\
\lesssim\|r^{\frac{p-2}{2}}{\not\negmedspace\partial}(ru)(0,{\,\cdot\,})\|^{2}_{L^{2}}+\sup_{t\in[0,T]}\int
r^{p}|h||\partial u|\Bigl{(}|\partial u|+\frac{|u|}{r}\Bigr{)}\,dx\\\
+\sup_{t\in[0,T]}\Bigl{|}\int_{0}^{t}\int
r^{p-1}e^{-\sigma_{U}(t-r)}\Box_{h}u\cdot\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru)\,dx\,dt\Bigr{|}+\int_{0}^{T}\int
r^{p-1}\Bigl{(}|\partial h|+\frac{|h|}{r}\Bigr{)}|\partial
u||(\partial_{t}+\partial_{r})(ru)|\,dx\,dt\\\ +\int_{0}^{T}\int
r^{p-1}|h||\partial
u|\Bigl{(}|{\not\negmedspace\nabla}u|+\frac{|u|}{r}\Bigr{)}\,dx\,dt+\int_{0}^{T}\int|h|r^{p-1}\sigma_{U}^{\prime}(t-r)|\partial
u||(\partial_{t}+\partial_{r})(ru)|\,dx\,dt\\\ +\int_{0}^{T}\int
r^{p}\Bigl{(}|(\partial_{t}+\partial_{r})h|+\frac{|h|}{r}\Bigr{)}|\partial
u|^{2}\,dx\,dt.$

The implicit constant is independent of both $T$ and $U$.

###### Proof.

We first note that

$\int_{0}^{t}\int(\Box
u)^{I}r^{p}e^{-\sigma_{U}(t-r)}\Bigl{(}\partial_{t}+\partial_{r}+\frac{1}{r}\Bigr{)}u^{I}\,dx\,dt\\\
=\int_{0}^{t}\int_{0}^{\infty}\int_{{\mathbb{S}}^{2}}\Bigl{(}\partial_{t}^{2}-\partial_{r}^{2}-{\not\negmedspace\nabla}\cdot{\not\negmedspace\nabla}\Bigr{)}(ru)^{I}\cdot
r^{p}e^{-\sigma_{U}(t-r)}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru)^{I}\,d\omega\,dr\,dt.$

Using integration by parts, we see that the right side is equal to

$\frac{1}{2}\int_{0}^{t}\int_{0}^{\infty}\int_{{\mathbb{S}}^{2}}r^{p}e^{-\sigma_{U}(t-r)}\Bigl{(}\partial_{t}-\partial_{r}\Bigr{)}\Bigl{|}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru)\Bigr{|}^{2}\,d\omega\,dr\,dt\\\
+\int_{0}^{t}\int_{0}^{\infty}\int_{{\mathbb{S}}^{2}}r^{p}e^{-\sigma_{U}(t-r)}{\not\negmedspace\nabla}(ru)^{I}\cdot{\not\negmedspace\nabla}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru)^{I}\,d\omega\,dr\,dt.$

Relying on (1.9), we further see that this is the same as

$\frac{1}{2}\int_{0}^{\infty}\int_{{\mathbb{S}}^{2}}r^{p}e^{-\sigma_{U}(t-r)}\Bigl{|}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru)\Bigr{|}^{2}\,d\omega\,dr\Bigl{|}_{t=0}^{t}+\frac{p}{2}\int_{0}^{t}\int_{0}^{\infty}\int_{{\mathbb{S}}^{2}}r^{p-1}e^{-\sigma_{U}(t-r)}\Bigl{|}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru)\Bigr{|}^{2}\,d\omega\,dr\,dt\\\
+\int_{0}^{t}\int_{0}^{\infty}\int_{{\mathbb{S}}^{2}}r^{p}\sigma_{U}^{\prime}(t-r)e^{-\sigma_{U}(t-r)}\Bigl{|}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru)\Bigr{|}^{2}\,d\omega\,dr\,dt+\int_{0}^{t}\int_{0}^{\infty}\int_{{\mathbb{S}}^{2}}r^{p-1}e^{-\sigma_{U}(t-r)}|{\not\negmedspace\nabla}(ru)|^{2}\,d\omega\,dr\,dt\\\
+\frac{1}{2}\int_{0}^{t}\int_{0}^{\infty}\int_{{\mathbb{S}}^{2}}r^{p}e^{-\sigma_{U}(t-r)}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}|{\not\negmedspace\nabla}(ru)|^{2}\,d\omega\,dr\,dt.$

A final integration by parts then gives that

(2.4) $\int_{0}^{t}\int(\Box
u)^{I}r^{p}e^{-\sigma_{U}(t-r)}\Bigl{(}\partial_{t}+\partial_{r}+\frac{1}{r}\Bigr{)}u^{I}\,dx\,dt\\\
=\frac{1}{2}\int_{0}^{\infty}\int_{{\mathbb{S}}^{2}}r^{p}e^{-\sigma_{U}(t-r)}\Bigl{(}\Bigl{|}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru)\Bigr{|}^{2}+|{\not\negmedspace\nabla}(ru)|^{2}\Bigr{)}\,d\omega\,dr\Bigl{|}_{t=0}^{t}\\\
+\frac{p}{2}\int_{0}^{t}\int_{0}^{\infty}\int_{{\mathbb{S}}^{2}}r^{p-1}e^{-\sigma_{U}(t-r)}\Bigl{|}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru)\Bigr{|}^{2}\,d\omega\,dr\,dt\\\
+\Bigl{(}1-\frac{p}{2}\Bigr{)}\int_{0}^{t}\int_{0}^{\infty}\int_{{\mathbb{S}}^{2}}r^{p-1}e^{-\sigma_{U}(t-r)}|{\not\negmedspace\nabla}(ru)|^{2}\,d\omega\,dr\,dt\\\
+\frac{1}{2}\int_{0}^{t}\int_{0}^{\infty}\int_{{\mathbb{S}}^{2}}r^{p}\sigma_{U}^{\prime}(t-r)e^{-\sigma_{U}(t-r)}\Bigl{|}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru)\Bigr{|}^{2}\,d\omega\,dr\,dt.$

This string of equalities proves the desired estimate when $h\equiv 0$.

We now consider the perturbation. Using integration by parts, we note that

$-\int_{0}^{t}\int
h^{I,\alpha\beta}_{J}\partial_{\alpha}\partial_{\beta}u^{J}\cdot
r^{p}e^{-\sigma_{U}(t-r)}\Bigl{(}\partial_{t}+\partial_{r}+\frac{1}{r}\Bigr{)}u^{I}\,dx\,dt=-\int
h^{I,0\beta}_{J}r^{p-1}e^{-\sigma_{U}(t-r)}\partial_{\beta}u^{J}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru^{I})\,dx\Bigl{|}_{t=0}^{t}\\\
+\int_{0}^{t}\int(\partial_{\alpha}h^{I,\alpha\beta}_{J})r^{p-1}e^{-\sigma_{U}(t-r)}\partial_{\beta}u^{J}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru^{I})\,dx\,dt\\\
+\int_{0}^{t}\int
h^{I,\alpha\beta}_{J}r^{-1}\partial_{\alpha}\Bigl{(}r^{p}e^{-\sigma_{U}(t-r)}\Bigr{)}\partial_{\beta}u^{J}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru^{I})\,dx\,dt\\\
+\int_{0}^{t}\int
h^{I,\alpha\beta}_{J}r^{p}e^{-\sigma_{U}(t-r)}\partial_{\beta}u^{J}\partial_{\alpha}\Bigl{(}\partial_{t}+\partial_{r}+\frac{1}{r}\Bigr{)}u^{I}\,dx\,dt.$

Commuting the $\partial_{\alpha}$ using (1.9) and using the symmetries (2.1),
we see that

$\int_{0}^{t}\int
h^{I,\alpha\beta}_{J}r^{p}e^{-\sigma_{U}(t-r)}\partial_{\beta}u^{J}\partial_{\alpha}\Bigl{(}\partial_{t}+\partial_{r}+\frac{1}{r}\Bigr{)}u^{I}\,dx\,dt=\int_{0}^{t}\int
h^{I,k\beta}_{J}r^{p-1}e^{-\sigma_{U}(t-r)}\partial_{\beta}u^{J}{\not\negmedspace\nabla}_{k}u^{I}\,dx\,dt\\\
-\int_{0}^{t}\int
h^{I,k\beta}_{J}r^{p-2}e^{-\sigma_{U}(t-r)}\frac{x_{k}}{r}u^{I}\partial_{\beta}u^{J}\,dx\,dt+\frac{1}{2}\int_{0}^{t}\int
h^{I,\alpha\beta}_{J}r^{p}e^{-\sigma_{U}(t-r)}\Bigl{(}\partial_{t}+\partial_{r}+\frac{2}{r}\Bigr{)}\Bigl{[}\partial_{\beta}u^{J}\partial_{\alpha}u^{I}\Bigr{]}\,dx\,dt.$

Combining the above two identities and integrating by parts, we obtain

(2.5) $-\int_{0}^{t}\int
h^{I,\alpha\beta}_{J}\partial_{\alpha}\partial_{\beta}u^{J}\cdot
r^{p}e^{-\sigma_{U}(t-r)}\Bigl{(}\partial_{t}+\partial_{r}+\frac{1}{r}\Bigr{)}u^{I}\,dx\,dt\\\
=-\int
h^{I,0\beta}_{J}r^{p-1}e^{-\sigma_{U}(t-r)}\partial_{\beta}u^{J}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru^{I})\,dx\Bigl{|}_{t=0}^{t}+\frac{1}{2}\int
h^{I,\alpha\beta}_{J}r^{p}e^{-\sigma_{U}(t-r)}\partial_{\beta}u^{J}\partial_{\alpha}u^{I}\,dx\Bigl{|}_{t=0}^{t}\\\
+\int_{0}^{t}\int(\partial_{\alpha}h^{I,\alpha\beta}_{J})r^{p-1}e^{-\sigma_{U}(t-r)}\partial_{\beta}u^{J}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru^{I})\,dx\,dt\\\
+p\int_{0}^{t}\int
h^{I,k\beta}_{J}\frac{x_{k}}{r}r^{p-2}e^{-\sigma_{U}(t-r)}\partial_{\beta}u^{J}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru^{I})\,dx\,dt+\int_{0}^{t}\int
h^{I,k\beta}_{J}r^{p-1}e^{-\sigma_{U}(t-r)}\partial_{\beta}u^{J}{\not\negmedspace\nabla}_{k}u^{I}\,dx\,dt\\\
-\int_{0}^{t}\int
h^{I,k\beta}_{J}r^{p-2}e^{-\sigma_{U}(t-r)}\frac{x_{k}}{r}u^{I}\partial_{\beta}u^{J}\,dx\,dt+\int_{0}^{t}\int
h^{I,\alpha\beta}_{J}\omega_{\alpha}r^{p-1}\sigma_{U}^{\prime}(t-r)e^{-\sigma_{U}(t-r)}\partial_{\beta}u^{J}(\partial_{t}+\partial_{r})(ru^{I})\,dx\,dt\\\
-\frac{1}{2}\int_{0}^{t}\int\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(h^{I,\alpha\beta}_{J})r^{p}e^{-\sigma_{U}(t-r)}\partial_{\beta}u^{J}\partial_{\alpha}u^{I}\,dx\,dt-\frac{p}{2}\int_{0}^{t}\int
h^{I,\alpha\beta}_{J}r^{p-1}e^{-\sigma_{U}(t-r)}\partial_{\beta}u^{J}\partial_{\alpha}u^{I}\,dx\,dt.$

Here $\omega=(-1,x/r)$. Our estimate (2.3) is an immediate consequence of
(2.4) and (2.5) and taking the supremum over $t\in[0,T]$. ∎

We next consider a Hardy-type inequality to obtain associated bounds on the
solution, which are analogous to the bounds on $\||x|^{-1}u\|_{LE}$ in (2.2).

######  Lemma 2.3.

Let $0<p<2$. Suppose $u\in C^{1}([0,T]\times{\mathbb{R}}^{3})$ and that for
every $t\in[0,T]$, $|r^{p/2}u(t,x)|\to 0$ as $|x|\to\infty$. Then

(2.6)
$\|r^{\frac{p-3}{2}}u\|_{L^{2}L^{2}}+\|r^{\frac{p-2}{2}}u\|_{L^{\infty}L^{2}}\lesssim\|r^{\frac{p-2}{2}}u(0,{\,\cdot\,})\|_{L^{2}}+\|r^{\frac{p-3}{2}}{\not\negmedspace\partial}(ru)\|_{L^{2}L^{2}}.$

###### Proof.

By integrating by parts, for any $t\in[0,T]$, we have

$\displaystyle\frac{1}{2-p}\int r^{p-2}u^{2}(t,x)\,dx$
$\displaystyle+\int_{0}^{t}\int r^{p-3}u^{2}(\tau,x)\,dx\,d\tau$
$\displaystyle=\frac{1}{2-p}\int
r^{p-2}u^{2}(t,x)\,dx+\frac{1}{p-2}\int_{0}^{t}\int_{{\mathbb{S}}^{2}}\int_{0}^{\infty}(\partial_{\tau}+\partial_{r})(r^{p-2})(ru)^{2}\,dr\,d\omega\,d\tau$
$\displaystyle=\frac{1}{2-p}\int
r^{p-2}u^{2}(0,x)\,dx-\frac{2}{p-2}\int_{0}^{t}\int_{{\mathbb{S}}^{2}}\int_{0}^{\infty}r^{p-2}(ru)(\partial_{t}+\partial_{r})(ru)\,dr\,d\omega\,d\tau.$

As the Schwarz inequality allows us to bound the last term by

$C\Bigl{(}\int_{0}^{t}\int
r^{p-3}u^{2}\,dx\,d\tau\Bigr{)}^{1/2}\Bigl{(}\int_{0}^{t}\int
r^{p-3}\Bigl{[}(\partial_{t}+\partial_{r})(ru)\Bigr{]}^{2}\,dx\,d\tau\Bigr{)}^{1/2}$

and as the first factor can be bootstrapped, (2.6) follows upon taking a
supremum in $t\in[0,T]$. ∎

While we will not directly use the next lemma, which indicates the form of the
lower order bound with decay in $t-r$ and $0<p<1$, we include it for the sake
of completeness.

######  Lemma 2.4.

Let $0\leq p<1$. Suppose $u\in C^{1}([0,T]\times{\mathbb{R}}^{3})$ and that
for every $t\in[0,T]$, $|u(t,x)|\to 0$ as $|x|\to\infty$. Then

(2.7) $\int_{0}^{T}\int
r^{p-2}\sigma_{U}^{\prime}(t-r)e^{-\sigma_{U}(t-r)}u^{2}\,dx\,dt+\sup_{t\in[0,T]}\int
r^{p-1}\sigma_{U}^{\prime}(t-r)e^{-\sigma_{U}(t-r)}u^{2}(t,x)\,dx\\\
\lesssim\int
r^{p-1}\sigma_{U}^{\prime}(-r)e^{-\sigma_{U}(-r)}u^{2}(0,x)\,dx+\int_{0}^{T}\int
r^{p-2}\sigma_{U}^{\prime}(t-r)e^{-\sigma_{U}(t-r)}\Bigl{[}(\partial_{t}+\partial_{r})(ru)\Bigr{]}^{2}\,dx\,dt.$

###### Proof.

We argue similarly to the preceding lemma and apply integration by parts and
the Schwarz inequality to observe that

$\displaystyle\int_{0}^{t}$ $\displaystyle\int
r^{p-2}\sigma_{U}^{\prime}(\tau-r)e^{-\sigma_{U}(\tau-r)}u^{2}\,dx\,d\tau+\frac{1}{1-p}\int
r^{p-1}\sigma_{U}^{\prime}(t-r)e^{-\sigma_{U}(t-r)}u^{2}(t,x)\,dx$
$\displaystyle=\frac{1}{p-1}\int_{0}^{t}\int_{{\mathbb{S}}^{2}}\int_{0}^{\infty}(\partial_{\tau}+\partial_{r})[r^{p-1}\sigma_{U}^{\prime}(\tau-r)e^{-\sigma_{U}(\tau-r)}](ru)^{2}\,dr\,d\omega\,d\tau$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{1}{1-p}\int
r^{p-1}\sigma_{U}^{\prime}(t-r)e^{-\sigma_{U}(t-r)}u^{2}(t,x)\,dx$
$\displaystyle=\frac{1}{1-p}\int
r^{p-1}\sigma_{U}^{\prime}(-r)e^{-\sigma_{U}(-r)}u^{2}(0,x)\,dx$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\frac{2}{p-1}\int_{0}^{t}\int_{{\mathbb{S}}^{2}}\int_{0}^{\infty}r^{p-1}\sigma_{U}^{\prime}(\tau-r)e^{-\sigma_{U}(\tau-r)}ru(\partial_{t}+\partial_{r})(ru)\,dr\,d\omega\,d\tau$
$\displaystyle\lesssim\int
r^{p-1}\sigma_{U}^{\prime}(-r)e^{-\sigma_{U}(-r)}u^{2}(0,x)\,dx$
$\displaystyle\quad+\Bigl{(}\int_{0}^{t}\int
r^{p-2}\sigma_{U}^{\prime}(\tau-r)e^{-\sigma_{U}(\tau-r)}u^{2}\,dx\,d\tau\Bigr{)}^{\frac{1}{2}}\Bigl{(}\int_{0}^{t}\int
r^{p-2}\sigma_{U}^{\prime}(\tau-r)e^{-\sigma_{U}(\tau-r)}\Bigl{[}(\partial_{t}+\partial_{r})(ru)\Bigr{]}^{2}\,dx\,d\tau\Bigr{)}^{\frac{1}{2}}.$

We may then bootstrap the first factor of the last term and take a supremum in
$t$ to complete the argument. ∎

We shall need the analog of the above when $p=1$, which comes with a
logarithmic loss. It is this logarithm that is largely responsible for the
difference between (1.6) and the $\exp(c/\varepsilon)$ lifespan of [11].

######  Lemma 2.5.

Suppose $u\in C^{1}([0,T]\times{\mathbb{R}}^{3})$ and that for every
$t\in[0,T]$, $|u(t,x)|\to 0$ as $|x|\to\infty$. Then

(2.8) $\int_{0}^{T}\int\beta_{>2}(r)\frac{1}{r(\log\langle
r\rangle)^{2}}\sigma_{U}^{\prime}(t-r)e^{-\sigma_{U}(t-r)}u^{2}\,dx\,dt\\\
+\sup_{t\in[0,T]}\int\beta_{>2}(r)\frac{1}{\log\langle
r\rangle}\sigma_{U}^{\prime}(t-r)e^{-\sigma_{U}(t-r)}u^{2}(t,x)\,dx\\\
\lesssim\int\frac{1}{\log\langle
r\rangle}\sigma_{U}^{\prime}(-r)e^{-\sigma_{U}(-r)}u^{2}(0,x)\,dx\\\
+\int_{0}^{T}\int
r^{-1}\sigma_{U}^{\prime}(t-r)e^{-\sigma_{U}(t-r)}\Bigl{[}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru)\Bigr{]}^{2}\,dx\,dt+\|u\|^{2}_{LE^{1}}.$

###### Proof.

We observe that

$\displaystyle\int_{0}^{t}\int$ $\displaystyle\beta_{>2}(r)\frac{1}{r(\log
r)^{2}}\sigma_{U}^{\prime}(\tau-r)e^{-\sigma_{U}(\tau-r)}u^{2}\,dx\,d\tau+\int\beta_{>2}(r)\frac{1}{\log
r}\sigma_{U}^{\prime}(t-r)e^{-\sigma_{U}(t-r)}u^{2}(t,x)\,dx$
$\displaystyle=\int_{0}^{t}\int_{{\mathbb{S}}^{2}}\int_{0}^{\infty}\beta_{>2}(r)\Bigl{(}\partial_{\tau}+\partial_{r}\Bigr{)}\Bigl{[}-\frac{1}{\log
r}\sigma_{U}^{\prime}(\tau-r)e^{-\sigma_{U}(\tau-r)}\Bigr{]}(ru)^{2}\,dr\,d\omega\,d\tau$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\int\beta_{>2}(r)\frac{1}{\log
r}\sigma_{U}^{\prime}(t-r)e^{-\sigma_{U}(t-r)}u^{2}(t,x)\,dx$
$\displaystyle=2\int_{0}^{t}\int_{{\mathbb{S}}^{2}}\int_{0}^{\infty}\beta_{>2}(r)\frac{1}{\log
r}\sigma_{U}^{\prime}(\tau-r)e^{-\sigma_{U}(\tau-r)}(ru)\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru)\,dr\,d\omega\,d\tau$
$\displaystyle\qquad\qquad+\int\beta_{>2}(r)\frac{1}{\log
r}\sigma_{U}^{\prime}(-r)e^{-\sigma_{U}(-r)}u^{2}(0,x)\,dx+\int_{0}^{t}\int\beta_{>2}^{\prime}(r)\frac{1}{\log
r}\sigma_{U}^{\prime}(\tau-r)e^{-\sigma_{U}(\tau-r)}u^{2}\,dx\,d\tau.$

As $\text{supp}\,\beta^{\prime}_{>2}\subset[2,4]$, the last term is bounded by
$\|u\|_{LE^{1}}^{2}$. The Schwarz inequality shows that the first term in the
right side is

$\lesssim\Bigl{(}\int_{0}^{t}\int\beta_{>2}(r)\frac{1}{r(\log
r)^{2}}\sigma_{U}^{\prime}(\tau-r)e^{-\sigma_{U}(\tau-r)}u^{2}\,dx\,d\tau\Bigr{)}^{1/2}\\\
\times\Bigl{(}\int_{0}^{t}\int\beta_{>2}(r)r^{-1}\sigma_{U}^{\prime}(\tau-r)e^{-\sigma_{U}(\tau-r)}\Big{[}\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(ru)\Bigr{]}^{2}\,dx\,d\tau\Bigr{)}^{1/2}.$

The first factor may be bootstrapped, and upon taking a supremum over
$t\in[0,T]$, the proof is complete. ∎

The following corollary combines (2.3), (2.6), and (2.8) when $p=1$ with
(1.10) and provides the primary linear estimate that our proof is based upon.

######  Corollary 2.6.

Fix $N\in\mathbb{N}$. Suppose that $h$ satisfies (2.1), that $u\in
C^{2}([0,T]\times{\mathbb{R}}^{3})$, and that for all $t\in[0,T]$,
$|r^{\frac{p+2}{2}}\partial^{\leq 1}Z^{\leq N}u(t,x)|\to 0$ as $|x|\to\infty$.
Then,

(2.9) $\|\langle r\rangle^{\frac{1}{2}}{\not\negmedspace\partial}Z^{\leq
N}u\|^{2}_{L^{\infty}L^{2}}+\|r^{-\frac{1}{2}}Z^{\leq
N}u\|_{L^{\infty}L^{2}}^{2}+\|{\not\negmedspace\partial}Z^{\leq
N}u\|^{2}_{L^{2}L^{2}}+\|r^{-1}Z^{\leq N}u\|_{L^{2}L^{2}}^{2}\\\ +\sup_{U\geq
1}\Bigl{(}\|r^{-\frac{1}{2}}\langle
t-r\rangle^{-\frac{1}{2}}(\partial_{t}+\partial_{r})(rZ^{\leq
N}u)\|^{2}_{L^{2}L^{2}(X_{U})}+\|r^{-\frac{1}{2}}(\log\langle
r\rangle)^{-1}\langle t-r\rangle^{-\frac{1}{2}}Z^{\leq
N}u\|^{2}_{L^{2}L^{2}(X_{U})}\Bigr{)}\\\
\lesssim\|r^{-\frac{1}{2}}(r{\not\negmedspace\partial})^{\leq 1}Z^{\leq
N}u(0,{\,\cdot\,})\|^{2}_{L^{2}}+\sup_{t\in[0,T]}\int r|h||\partial Z^{\leq
N}u|\Bigl{(}|\partial Z^{\leq N}u|+\frac{|Z^{\leq N}u|}{r}\Bigr{)}\,dx\\\
+\sup_{U\geq 1}\sup_{t\in[0,T]}\Bigl{|}\int_{0}^{t}\int
e^{-\sigma_{U}(t-r)}\Box_{h}Z^{\leq
N}u\cdot\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(rZ^{\leq
N}u)\,dx\,dt\Bigr{|}\\\ +\int_{0}^{T}\int\Bigl{(}|\partial
h|+\frac{|h|}{r}\Bigr{)}|\partial Z^{\leq
N}u||(\partial_{t}+\partial_{r})(rZ^{\leq
N}u)|\,dx\,dt+\int_{0}^{T}\int|h||\partial Z^{\leq
N}u|\Bigl{(}|{\not\negmedspace\nabla}Z^{\leq N}u|+\frac{|Z^{\leq
N}u|}{r}\Bigr{)}\,dx\,dt\\\ +\int_{0}^{T}\int|h|\langle
t-r\rangle^{-1}|\partial Z^{\leq N}u||(\partial_{t}+\partial_{r})(rZ^{\leq
N}u)|\,dx\,dt+\int_{0}^{T}\int
r\Bigl{(}|(\partial_{t}+\partial_{r})h|+\frac{|h|}{r}\Bigr{)}|\partial Z^{\leq
N}u|^{2}\,dx\,dt\\\ +\|\partial Z^{\leq N}u\|^{2}_{L^{\infty}L^{2}}+\|Z^{\leq
N}u\|^{2}_{LE^{1}}.$

###### Proof.

Using (2.6) and (2.8), we may adapt (2.3) with $p=1$ to the bound

(2.10) $\|\langle r\rangle^{\frac{1}{2}}{\not\negmedspace\partial}Z^{\leq
N}u\|^{2}_{L^{\infty}L^{2}}+\|r^{-\frac{1}{2}}Z^{\leq
N}u\|^{2}_{L^{\infty}L^{2}}+\|{\not\negmedspace\partial}Z^{\leq
N}u\|^{2}_{L^{2}L^{2}}+\|r^{-1}Z^{\leq N}u\|^{2}_{L^{2}L^{2}}\\\
+\|r^{-\frac{1}{2}}(\sigma^{\prime}_{U}(t-r))^{\frac{1}{2}}(\partial_{t}+\partial_{r})(rZ^{\leq
N}u)\|^{2}_{L^{2}L^{2}}+\|r^{-\frac{1}{2}}(\log(\langle
r\rangle)^{-1}(\sigma_{U}^{\prime}(t-r))^{\frac{1}{2}}Z^{\leq
N}u\|^{2}_{L^{2}L^{2}}\\\
\lesssim\|r^{-\frac{1}{2}}(r{\not\negmedspace\partial})^{\leq 1}Z^{\leq
N}u(0,{\,\cdot\,})\|^{2}_{L^{2}}+\sup_{t\in[0,T]}\int r|h||\partial Z^{\leq
N}u|\Bigl{(}|\partial Z^{\leq N}u|+\frac{|Z^{\leq N}u|}{r}\Bigr{)}\,dx\\\
+\sup_{t\in[0,T]}\Bigl{|}\int_{0}^{t}\int e^{-\sigma_{U}(t-r)}\Box_{h}Z^{\leq
N}u\cdot\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(rZ^{\leq
N}u)\,dx\,dt\Bigr{|}\\\ +\int_{0}^{T}\int\Bigl{(}|\partial
h|+\frac{|h|}{r}\Bigr{)}|\partial Z^{\leq
N}u||(\partial_{t}+\partial_{r})(rZ^{\leq
N}u)|\,dx\,dt+\int_{0}^{T}\int|h||\partial Z^{\leq
N}u|\Bigl{(}|{\not\negmedspace\nabla}Z^{\leq N}u|+\frac{|Z^{\leq
N}u|}{r}\Bigr{)}\,dx\,dt\\\
+\int_{0}^{T}\int|h|\sigma_{U}^{\prime}(t-r)|\partial Z^{\leq
N}u||(\partial_{t}+\partial_{r})(rZ^{\leq N}u)|\,dx\,dt+\int_{0}^{T}\int
r\Bigl{(}|(\partial_{t}+\partial_{r})h|+\frac{|h|}{r}\Bigr{)}|\partial Z^{\leq
N}u|^{2}\,dx\,dt\\\ +\|\partial Z^{\leq N}u\|^{2}_{L^{\infty}L^{2}}+\|Z^{\leq
N}u\|^{2}_{LE^{1}}.$

We note that

$\sigma^{\prime}_{U}(t-r)\gtrsim\frac{1}{\langle t-r\rangle}\text{ on
}X_{U},\quad\text{ and }\quad\sigma^{\prime}_{U}(t-r)\lesssim\frac{1}{\langle
t-r\rangle}\text{ provided }U\geq 1.$

Using these facts in (2.10) and subsequently taking a supremum in $U$ yields
(2.9). ∎

This last Hardy-type inequality is not strictly necessary. When we set up an
iteration to solve (1.1), this will be a convenience when showing that the
sequence converges. In particular, it will allow us to focus only on energy
and integrated local energy spaces for this portion of the argument. A closely
related calculation appears in [11].

######  Lemma 2.7.

Suppose that $u\in C^{1}([0,T]\times{\mathbb{R}}^{3})$ is supported where
$\\{r\leq t+2\\}$. Then

(2.11)
$\int\frac{1}{(1+r)(t-r+3)^{2}}u^{2}\,dx\lesssim\int\frac{1}{(1+r)r^{2}}u^{2}\,dx+\int\frac{1}{(1+r)}(\partial_{r}u)^{2}\,dx.$

###### Proof.

For $t\in[0,T]$ fixed, we integrate by parts and apply the Schwarz inequality
to obtain

$\displaystyle\int\frac{1}{(1+r)(t-r+3)^{2}}$ $\displaystyle
u^{2}\,dx=\int_{{\mathbb{S}}^{2}}\int_{0}^{\infty}\partial_{r}[(t-r+3)^{-1}]\frac{r^{2}}{(1+r)}u^{2}\,dr\,d\omega$
$\displaystyle=-\int\frac{1}{t-r+3}\cdot\frac{2+r}{r(1+r)^{2}}u^{2}\,dx-2\int\frac{1}{(1+r)(t-r+3)}u\,\partial_{r}u\,dx$
$\displaystyle\lesssim\Bigl{(}\int\frac{1}{(1+r)(t-r+3)^{2}}u^{2}\,dx\Bigl{)}^{\frac{1}{2}}\Bigl{[}\Bigl{(}\int\frac{1}{r^{2}(1+r)}u^{2}\,dx\Bigr{)}^{\frac{1}{2}}+\Bigl{(}\int\frac{1}{(1+r)}(\partial_{r}u)^{2}\,dx\Bigr{)}^{\frac{1}{2}}\Bigr{]}.$

Dividing both sides by the first factor in the right completes the proof. ∎

##  3\. Sobolev estimates

The main decay estimate that we shall rely upon is a space-time variant of the
Klainerman-Sobolev estimate [9] that was established in [20] and is
particularly well-adapted to integrated local energy estimates.

As is described in Section 1.1, we will break space-time up into
$C^{R}_{\tau}$ and $C^{U}_{\tau}$ regions where $\tau\in[0,T]$ and $1\leq
R,U\leq\tau/4$. On these regions, we have the following weighted Sobolev
estimates, which will serve as our source of decay.

######  Lemma 3.1.

For any $\tau\geq 1$ and $1\leq R,U\leq\tau/4$, we have

(3.1) $\displaystyle\|w\|_{L^{\infty}L^{\infty}(C^{R}_{\tau})}$
$\displaystyle\lesssim\frac{1}{\tau^{\frac{1}{2}}R^{\frac{3}{2}}}\|Z^{\leq
4}w\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+\frac{1}{\tau^{\frac{1}{2}}R^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})Z^{\leq
3}w\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})},$ (3.2)
$\displaystyle\|w\|_{L^{\infty}L^{\infty}(C_{\tau}^{U})}$
$\displaystyle\lesssim\frac{1}{\tau^{\frac{3}{2}}U^{\frac{1}{2}}}\|Z^{\leq
4}w\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}+\frac{1}{\tau^{\frac{3}{2}}U^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
3}w)\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})},$ (3.3)
$\displaystyle\|w\|_{L^{\infty}L^{\infty}(C^{\frac{\tau}{2}}_{\tau})}$
$\displaystyle\lesssim\frac{1}{\tau^{2}}\|Z^{\leq
4}w\|_{L^{2}L^{2}(\tilde{C}^{\frac{\tau}{2}}_{\tau})}+\frac{1}{\tau}\|(\partial_{t}+\partial_{r})Z^{\leq
3}w\|_{L^{2}L^{2}(\tilde{C}^{\frac{\tau}{2}}_{\tau})}$

See [20, Lemma 3.8]. The proof of (3.1) follows by changing coordinates to
$t=e^{s}$, $r=e^{s+\rho}$ and applying Sobolev embeddings in $\omega$ and the
Fundamental Theorem of Calculus in $s$ and $\rho$. In fact, this yields

(3.4)
$\|w\|_{L^{\infty}L^{\infty}(C^{R}_{\tau})}\lesssim\frac{1}{\tau^{\frac{1}{2}}R^{\frac{3}{2}}}\|Z^{\leq
3}w\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+\frac{1}{\tau^{\frac{1}{2}}R}\|Z^{\leq
3}w\|^{1/2}_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}\|\partial_{r}Z^{\leq
3}w\|^{1/2}_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}$

for any $R>1$. In order to get additional decay out of differentiated terms,
such as those appearing in the last term of (3.4), the preceding work [13] in
$(1+4)$-dimensions relied upon [20, Lemma 3.11]. As (2.9) provides better
control on the good derivatives, we can argue more simply and instead use

(3.5)
$(\partial_{t}-\partial_{r})=\frac{2}{t-r}S-\frac{t+r}{t-r}(\partial_{t}+\partial_{r}).$

As $(t+r)/(t-r)={\mathcal{O}}(1)$ on $\tilde{C}^{R}_{\tau}$ with
$R\leq\tau/4$, (3.1) follows immediately. Replacing $w$ by
$\beta_{>\tau/2}(t-r)w$ in (3.4) and using that
$S(\beta_{>\tau/2}(t-r))={\mathcal{O}}(1)$, (3.3) is obtained similarly.

When $U=1$, the other estimate (3.2) is an immediate corollary of (3.4) as we
need only consider $\partial_{r}$ as a vector field. When $U>1$,

(3.6)
$\|w\|_{L^{\infty}L^{\infty}(C^{U}_{\tau})}\lesssim\frac{1}{\tau^{\frac{3}{2}}U^{\frac{1}{2}}}\|Z^{\leq
3}w\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}+\frac{U^{\frac{1}{2}}}{\tau^{\frac{3}{2}}}\|\partial
Z^{\leq 3}w\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}$

follows from arguments similar to the above in coordinates $t+r=e^{s}$,
$t-r=e^{s+\rho}$. Subsequently applying (3.5) yields (3.2).

It will also be helpful to have the following common weighted Sobolev estimate
of [8].

######  Lemma 3.2.

Provided that $h\in C^{\infty}({\mathbb{R}}^{3})$,

(3.7) $\|h\|_{L^{\infty}(A_{R})}\lesssim R^{-1}\|Z^{\leq
2}h\|_{L^{2}(\tilde{A}_{R})}.$

For $R=1$, standard Sobolev embeddings yield the result. And for $R>1$, after
localizing, one only needs to apply Sobolev embeddings in $(r,\omega)$. The
decay is then a consequence of converting the volume element $dr\,d\omega$ to
$dx=r^{2}\,dr\,d\omega$.

##  4\. Proof of Theorem 1.1

We shall solve (1.1) via iteration. We set $u_{0}\equiv 0$, and for $k\geq 1$,
let $u_{k}$ solve

(4.1) $\begin{cases}\Box
u_{k}^{I}=a^{I,\alpha}_{JK}u_{k-1}^{J}\partial_{\alpha}u_{k-1}^{K}+b^{I,\alpha\beta}_{JK}\partial_{\alpha}u_{k-1}^{J}\partial_{\beta}u_{k-1}^{K}+A^{I,\alpha\beta}_{JK}u_{k-1}^{K}\partial_{\alpha}\partial_{\beta}u_{k}^{J}+B^{I,\alpha\beta\gamma}_{JK}\partial_{\gamma}u_{k-1}^{K}\partial_{\alpha}\partial_{\beta}u_{k}^{J},\\\
u_{k}(0,{\,\cdot\,})=f,\quad\partial_{t}u_{k}(0,{\,\cdot\,})=g.\end{cases}$

Note that the right side of the equation is
$F^{I}(u_{k-1},u^{\prime}_{k-1},u^{\prime\prime}_{k})$. We will show that the
sequence $(u_{k})$ converges. The limit of this sequence is the desired
solution.

We shall work with $N=60$, though this is far from sharp. In the
$r^{p}$-weighted estimates, we use $p=1$ throughout.

To show that the sequence is convergent, we first show a certain boundedness.
Relying on that, we next demonstrate that the sequence is Cauchy, which due to
completeness of the spaces we are working in, finishes the proof.

###  4.1. Boundedness

For any fixed $T\leq T_{\varepsilon}$, we set

(4.2) $M_{k}=\|\partial Z^{\leq 60}u_{k}\|_{L^{\infty}L^{2}}+\|Z^{\leq
60}u_{k}\|_{LE^{1}}+\|{\not\negmedspace\partial}(Z^{\leq
60}u_{k})\|_{L^{2}L^{2}}\\\ +\|r^{-1}Z^{\leq 60}u_{k}\|_{L^{2}L^{2}}+\|\langle
r\rangle^{\frac{1}{2}}{\not\negmedspace\partial}Z^{\leq
60}u_{k}\|_{L^{\infty}L^{2}}+\|r^{-\frac{1}{2}}Z^{\leq
60}u_{k}\|_{L^{\infty}L^{2}}\\\ +\sup_{U}\|r^{-\frac{1}{2}}\langle
t-r\rangle^{-\frac{1}{2}}(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})\|_{L^{2}L^{2}(X_{U})}+\sup_{U}\|r^{-\frac{1}{2}}(\log\langle
r\rangle)^{-1}\langle t-r\rangle^{-\frac{1}{2}}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(X_{U})}\\\ +\Bigl{(}\sum_{\tau\leq
T}\sum_{R\leq\tau/2}\|\partial Z^{\leq
50}u_{k}\|^{2}_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}\Bigr{)}^{\frac{1}{2}}+\Bigl{[}\sum_{\tau\leq
T}\sum_{R\leq\tau/2}\Bigl{(}R\|{\not\negmedspace\partial}\partial Z^{\leq
40}u_{k}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}\Bigr{)}^{2}\Bigr{]}^{\frac{1}{2}}\\\
+\sup_{U}\Bigl{[}\sum_{\tau\geq
4U}\Bigl{(}\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}\log\langle\tau\rangle}\|\partial
Z^{\leq
50}u_{k}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}\Bigr{)}^{2}\Bigr{]}^{\frac{1}{2}}+\sup_{U}\Bigl{[}\sum_{\tau\geq
4U}\Bigl{(}\frac{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}{\log\langle\tau\rangle}\|{\not\negmedspace\partial}\partial
Z^{\leq
40}u_{k}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}\Bigr{)}^{2}\Bigr{]}^{\frac{1}{2}}.$

We call these terms $I_{k}$, $II_{k}$, …, $XI_{k}$, $XII_{k}$ respectively. We
shall argue inductively to show that

(4.3) $M_{k}\leq 2C_{0}\varepsilon$

for a uniform constant $C_{0}$ provided that $T\leq T_{\varepsilon}$. Indeed,
for a universal constant $C_{0}$, we shall show that

(4.4) $M_{k}^{2}\leq(C_{0}\varepsilon)^{2}+C(\log\langle
T\rangle)^{3}M_{k-1}^{2}M_{k}+C(\log\langle
T\rangle)^{3}M_{k-1}M_{k}^{2}+C(\log\langle T\rangle)^{2}M_{k-1}^{4}\\\
+C(\log\langle T\rangle)^{2}M_{k-1}^{2}M_{k}^{2}+C(\log\langle
T\rangle)^{5}M_{k-1}^{4}+C(\log\langle T\rangle)^{5}M_{k-1}^{3}M_{k}.$

From this, it follows that $M_{1}\leq C_{0}\varepsilon$. Then by the inductive
hypothesis and (1.6), provided that $c$ and $\varepsilon$ are sufficiently
small (compared to $C_{0}$), we obtain (4.3).

We briefly summarize the proof of (4.4) that is to follow. Note that terms
$I_{k}$ and $II_{k}$ are bounded using the energy and integrated local energy
estimate (2.2), while terms $III_{k},\dots,VIII_{k}$ represent the left side
of (2.9). These eight terms are the principal portions. To prove (4.4), upon
applying (2.2) and (2.9), the product rule will guarantee that one factor from
each nonlinear term will be lower order (in terms of the number of vector
fields). As this factor can afford additional vector fields, we may apply our
decay estimates (3.1), (3.2), (3.3), or (3.7) to it.

Closing the argument requires that we obtain additional decay from the
derivative that must be present on at least one factor of every nonlinear
term. When this derivative is a “good” derivative
${\not\negmedspace\partial}$, this is relatively simple as the $r^{p}$-weight
allows it to be bounded with a larger weight, which effectively provides
additional decay to be used for the other factors. For the
$\partial_{t}-\partial_{r}$ directions, provided that the factor can admit an
additional vector field (3.5) yields additional decay. Here the decay is in
$t-r$, and the use of the ghost weight allows our estimates to take advantage
of this. Terms $IX_{k},\dots,XII_{k}$ of (4.2) are commonly occurring factors
where such a procedure is utilized.

The resulting worst nonlinear term is when
$u_{k-1}(\partial_{t}-\partial_{r})Z^{\leq 60}u_{k-1}$ occurs within the right
side of (2.9). Here one integrates $(\partial_{t}-\partial_{r})$ by parts.
When it lands on the lower order factor, the procedure based in (3.5)
described above can be used. When it instead lands on the multiplier term, up
to better behaved terms, $\Box u_{k}$ is reproduced. This term is replaced
using the nonlinear equation, and quartic interactions result. The majority of
the terms can be handled as above and the worst case is again the
$u_{k-1}(\partial_{t}-\partial_{r})Z^{\leq 60}u_{k-1}$ interactions. At this
point, however, the two high order factors can be combined using the chain
rule $w\partial w=\frac{1}{2}\partial w^{2}$, and integration by parts can be
used to move the derivative to the lower order factors where (3.5) can once
again be applied.

We note that the extra logarithmic factor in term $VIII_{k}$ is largely
responsible for our lifespan being $\exp(c/\varepsilon^{\frac{1}{3}})$ rather
than $\exp(c/\varepsilon)$ as is known for scalar equations.

In our applications of (2.2) and (2.9), we set

(4.5)
$h^{I,\alpha\beta}_{J}=A^{I,\alpha\beta}_{JK}u_{k-1}^{K}+B^{I,\alpha\beta\gamma}_{JK}\partial_{\gamma}u_{k-1}^{K}.$

We proceed with establishing the necessary bound for each of
$I_{k},\dots,XII_{k}$.

$\mathbf{[I_{k}+II_{k}]}:$ We begin by showing

(4.6) $I_{k}^{2}+II_{k}^{2}\leq(C_{0}\varepsilon)^{2}+(\log\langle
T\rangle)^{\frac{5}{2}}M_{k-1}M^{2}_{k}+(\log\langle
T\rangle)^{\frac{5}{2}}M^{2}_{k-1}M_{k}.$

Using (2.2) and (1.5), we have that

(4.7) $I_{k}^{2}+II_{k}^{2}\leq
C_{0}^{2}\varepsilon^{2}+C\int_{0}^{T}\int|\Box_{h}Z^{\leq
60}u_{k}|\Bigl{(}|\partial Z^{\leq 60}u_{k}|+\frac{|Z^{\leq
60}u_{k}|}{r}\Bigr{)}\,dx\,dt\\\
+C\int_{0}^{T}\int\Bigl{(}|\partial\partial^{\leq
1}u_{k-1}|+\frac{|\partial^{\leq 1}u_{k-1}|}{\langle
r\rangle}\Bigr{)}|\partial Z^{\leq 60}u_{k}|\Bigl{(}|\partial Z^{\leq
60}u_{k}|+\frac{|Z^{\leq 60}u_{k}|}{r}\Bigr{)}\,dx\,dt.$

We first note that we may apply (3.7) and the finite speed of propagation to
see that the last term in the right of (4.7) is bounded by

$\sum_{0\leq j\lesssim\log\langle T\rangle}2^{-j}\|r^{-1}(r\partial)^{\leq
1}Z^{\leq 3}u_{k-1}\|_{L^{\infty}L^{2}}\int_{0}^{T}\int_{A_{2^{j}}}|\partial
Z^{\leq 60}u_{k}|\Bigl{(}|\partial Z^{\leq 60}u_{k}|+\frac{|Z^{\leq
60}u_{k}|}{r}\Bigr{)}\,dx\,dt.$

The Schwarz inequality and a Hardy inequality then give that

(4.8) $\begin{split}\int_{0}^{T}\int\Bigl{(}|\partial\partial^{\leq
1}u_{k-1}|+&\frac{|\partial^{\leq 1}u_{k-1}|}{\langle
r\rangle}\Bigr{)}|\partial Z^{\leq 60}u_{k}|\Bigl{(}|\partial Z^{\leq
60}u_{k}|+\frac{|Z^{\leq 60}u_{k}|}{r}\Bigr{)}\,dx\,dt\\\ &\lesssim\|\partial
Z^{\leq 3}u_{k-1}\|_{L^{\infty}L^{2}}\Bigl{(}\log\langle T\rangle\|Z^{\leq
60}u_{k}\|^{2}_{LE^{1}}\Bigr{)}\\\ &\lesssim I_{k-1}\Bigl{(}\log\langle
T\rangle II_{k}^{2}\Bigr{)},\end{split}$

which is controlled by the right side of (4.6).

To address the second term in the right side of (4.7), we note that

(4.9) $|\Box_{h}Z^{\leq 60}u_{k}|\lesssim\Bigl{(}|\partial Z^{\leq
30}u_{k-1}|+|\partial Z^{\leq 31}u_{k}|\Bigr{)}|\partial^{\leq 1}Z^{\leq
60}u_{k-1}|+|Z^{\leq 31}u_{k-1}|\Bigl{(}|\partial Z^{\leq
60}u_{k-1}|+|\partial Z^{\leq 60}u_{k}|\Bigr{)}.$

We first write

(4.10) $\int_{0}^{T}\int|\Box_{h}Z^{\leq 60}u_{k}|\Bigl{(}|\partial Z^{\leq
60}u_{k}|+\frac{|Z^{\leq 60}u_{k}|}{r}\Bigr{)}\,dx\,dt\\\
\lesssim\sum_{\tau\leq
T}\Bigl{(}\sum_{R\leq\tau/2}\int\int_{C^{R}_{\tau}}|\Box_{h}Z^{\leq
60}u_{k}|\Bigl{(}|\partial Z^{\leq 60}u_{k}|+\frac{|Z^{\leq
60}u_{k}|}{r}\Bigr{)}\,dx\,dt\\\
+\sum_{U\leq\tau/4}\int\int_{C^{U}_{\tau}}|\Box_{h}Z^{\leq
60}u_{k}|\Bigl{(}|\partial Z^{\leq 60}u_{k}|+\frac{|Z^{\leq
60}u_{k}|}{r}\Bigr{)}\,dx\,dt\Bigr{)}.$

By (3.1) and (3.3), we have

$\int\int_{C^{R}_{\tau}}\Bigl{(}|\partial Z^{\leq 30}u_{k-1}|+|\partial
Z^{\leq 31}u_{k}|\Bigr{)}|\partial^{\leq 1}Z^{\leq
60}u_{k-1}|\Bigl{(}|\partial Z^{\leq 60}u_{k}|+\frac{|Z^{\leq
60}u_{k}|}{r}\Bigr{)}\,dx\,dt\\\ \lesssim\Bigl{(}\|\partial Z^{\leq
34}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+\|\partial Z^{\leq
35}u_{k}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+R\|{\not\negmedspace\partial}\partial
Z^{\leq
33}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+R\|{\not\negmedspace\partial}\partial
Z^{\leq 34}u_{k}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}\Bigr{)}\\\
\times\|\langle r\rangle^{-1}\partial^{\leq 1}Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{R}_{\tau})}\|\langle r\rangle^{-1}(\partial
Z^{\leq 60}u_{k},r^{-1}Z^{\leq 60}u_{k})\|_{L^{2}L^{2}(C^{R}_{\tau})}.$

Noting, for example, that

$\|\langle r\rangle^{-1}\partial Z^{\leq
60}u\|^{2}_{L^{2}L^{2}}=\sum_{j}2^{-j}\Bigl{(}2^{-j}\|\partial Z^{\leq
60}u\|^{2}_{L^{2}L^{2}([0,T]\times A_{2^{j}})}\Bigr{)}\lesssim\|Z^{\leq
60}u\|^{2}_{LE^{1}},$

we thus have

(4.11) $\sum_{\tau\leq
T}\sum_{R\leq\tau/2}\int\int_{C^{R}_{\tau}}\Bigl{(}|\partial Z^{\leq
30}u_{k-1}|+|\partial Z^{\leq 31}u_{k}|\Bigr{)}|\partial^{\leq 1}Z^{\leq
60}u_{k-1}|\Bigl{(}|\partial Z^{\leq 60}u_{k}|+\frac{|Z^{\leq
60}u_{k}|}{r}\Bigr{)}\,dx\,dt\\\
\lesssim\Bigl{(}IX_{k-1}+IX_{k}+X_{k-1}+X_{k}\Bigr{)}\Bigl{(}IV_{k-1}+II_{k-1}\Bigr{)}II_{k}.$

Similarly,

$\int\int_{C^{R}_{\tau}}|Z^{\leq 31}u_{k-1}|\Bigl{(}|\partial Z^{\leq
60}u_{k-1}|+|\partial Z^{\leq 60}u_{k}|\Bigr{)}\Bigl{(}|\partial Z^{\leq
60}u_{k}|+\frac{|Z^{\leq 60}u_{k}|}{r}\Bigr{)}\,dx\,dt\\\
\lesssim\Bigl{(}\|r^{-1}Z^{\leq
35}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
34}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}\Bigr{)}\|\langle
r\rangle^{-\frac{1}{2}}(\partial Z^{\leq 60}u_{k},r^{-1}Z^{\leq
60}u_{k})\|_{L^{2}L^{2}(C^{R}_{\tau})}\\\ \times\Bigl{(}\|\langle
r\rangle^{-\frac{1}{2}}\partial Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{R}_{\tau})}+\|\langle
r\rangle^{-\frac{1}{2}}\partial Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}\Bigr{)}.$

Since the initial data are supported in $\\{|x|\leq 2\\}$, we get

(4.12) $\|\langle r\rangle^{-\frac{1}{2}}\partial Z^{\leq
60}u\|^{2}_{L^{2}L^{2}}\leq\sum_{j\lesssim\log\langle
T\rangle}2^{-j}\|\partial Z^{\leq 60}u\|^{2}_{L^{2}L^{2}([0,T]\times
A_{2^{j}})}\lesssim\log\langle T\rangle\|Z^{\leq 60}u\|_{LE^{1}}^{2}.$

It follows that

(4.13) $\sum_{\tau\leq T}\sum_{R\leq\tau/2}\int\int_{C^{R}_{\tau}}|Z^{\leq
31}u_{k-1}|\Bigl{(}|\partial Z^{\leq 60}u_{k-1}|+|\partial Z^{\leq
60}u_{k}|\Bigr{)}\Bigl{(}|\partial Z^{\leq 60}u_{k}|+\frac{|Z^{\leq
60}u_{k}|}{r}\Bigr{)}\,dx\,dt\\\
\lesssim\Bigl{(}IV_{k-1}+III_{k-1}\Bigr{)}\cdot II_{k}\cdot(\log\langle
T\rangle)^{\frac{1}{2}}\Bigl{(}II_{k-1}+II_{k}\Bigr{)}.$

By (3.2),

$\int\int_{C^{U}_{\tau}}\Bigl{(}|\partial Z^{\leq 30}u_{k-1}|+|\partial
Z^{\leq 31}u_{k}|\Bigr{)}|\partial^{\leq 1}Z^{\leq
60}u_{k-1}|\Bigl{(}|\partial Z^{\leq 60}u_{k}|+\frac{|Z^{\leq
60}u_{k}|}{r}\Bigr{)}\,dx\,dt\\\
\lesssim\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}}\Bigl{(}\|\partial Z^{\leq
34}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}+\|\partial Z^{\leq
35}u_{k}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}+\tau\|{\not\negmedspace\partial}\partial
Z^{\leq
33}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}+\tau\|{\not\negmedspace\partial}\partial
Z^{\leq 34}u_{k}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}\Bigr{)}\\\ \times
U^{-1}\|\langle r\rangle^{-\frac{1}{2}}\partial^{\leq 1}Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}\|\langle
r\rangle^{-\frac{1}{2}}(\partial Z^{\leq 60}u_{k},r^{-1}Z^{\leq
60}u_{k})\|_{L^{2}L^{2}(C^{U}_{\tau})}.$

Thus, using (4.12),

(4.14) $\sum_{\tau\leq
T}\sum_{U\leq\tau/4}\int\int_{C^{U}_{\tau}}\Bigl{(}|\partial Z^{\leq
30}u_{k-1}|+|\partial Z^{\leq 31}u_{k}|\Bigr{)}|\partial^{\leq 1}Z^{\leq
60}u_{k-1}|\Bigl{(}|\partial Z^{\leq 60}u_{k}|+\frac{|Z^{\leq
60}u_{k}|}{r}\Bigr{)}\,dx\,dt\\\ \lesssim\log\langle
T\rangle\Bigl{(}XI_{k-1}+XI_{k}+XII_{k-1}+XII_{k}\Bigr{)}\Bigl{(}\log\langle
T\rangle VIII_{k-1}+(\log\langle
T\rangle)^{\frac{1}{2}}II_{k-1}\Bigr{)}\Bigl{(}(\log\langle
T\rangle)^{\frac{1}{2}}II_{k}\Bigr{)}.$

Relying on (3.2) again, we have

$\int\int_{C^{U}_{\tau}}|Z^{\leq 31}u_{k-1}|\Bigl{(}|\partial Z^{\leq
60}u_{k-1}|+|\partial Z^{\leq 60}u_{k}|\Bigr{)}\Bigl{(}|\partial Z^{\leq
60}u_{k}|+\frac{|Z^{\leq 60}u_{k}|}{r}\Bigr{)}\,dx\,dt\\\
\lesssim\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\Bigl{(}\|Z^{\leq
35}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}+\|(\partial_{t}+\partial_{r})(rZ^{\leq
34}u_{k-1})\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}\Bigr{)}\\\
\times\Bigl{(}\|\langle r\rangle^{-\frac{1}{2}}\partial Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}+\|\langle
r\rangle^{-\frac{1}{2}}\partial Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{U}_{\tau})}\Bigr{)}\|\langle
r\rangle^{-\frac{1}{2}}(\partial Z^{\leq 60}u_{k},r^{-1}Z^{\leq
60}u_{k})\|_{L^{2}L^{2}(C^{U}_{\tau})}.$

Upon using (4.12), it follows that

(4.15) $\sum_{\tau\leq T}\sum_{U\leq\tau/4}\int\int_{C^{U}_{\tau}}|Z^{\leq
31}u_{k-1}|\Bigl{(}|\partial Z^{\leq 60}u_{k-1}|+|\partial Z^{\leq
60}u_{k}|\Bigr{)}\Bigl{(}|\partial Z^{\leq 60}u_{k}|+\frac{|Z^{\leq
60}u_{k}|}{|x|}\Bigr{)}\,dx\,dt\\\ \lesssim\Bigl{(}\log\langle T\rangle
VIII_{k-1}+VII_{k-1}\Bigr{)}\Bigl{(}(\log\langle
T\rangle)^{\frac{1}{2}}II_{k-1}+(\log\langle
T\rangle)^{\frac{1}{2}}II_{k}\Bigr{)}\Bigl{(}(\log\langle
T\rangle)^{\frac{1}{2}}II_{k}\Bigr{)}.$

By plugging bounds (4.8), (4.11), (4.13), (4.14), (4.15) into (4.7) and(4.10)
we obtain the desired bound (4.6).

$\mathbf{[III_{k}+IV_{k}+V_{k}+VI_{k}+VII_{k}+VIII_{k}]}$: Here, relying on
(2.9), we show that

(4.16)
$III^{2}_{k}+IV^{2}_{k}+V^{2}_{k}+VI^{2}_{k}+VII^{2}_{k}+VIII^{2}_{k}\leq(C_{0}\varepsilon)^{2}+C(\log\langle
T\rangle)^{3}M_{k-1}M^{2}_{k}+C(\log\langle T\rangle)^{3}M^{2}_{k-1}M_{k}\\\
+C(\log\langle T\rangle)^{5}M_{k-1}^{4}+C(\log\langle
T\rangle)^{5}M_{k-1}^{3}M_{k}.$

The first term in the right of (2.9) is bounded by $C_{0}^{2}\varepsilon^{2}$
due to (1.5). We will proceed, in order, to showing that each of the terms,
other than the $\Box_{h}Z^{\leq 60}u_{k}$ term, in the right side of (2.9) are
bounded by

(4.17) $C\log\langle
T\rangle^{3}\Bigl{(}M_{k-1}^{2}M_{k}+M_{k-1}M_{k}^{2}\Bigr{)}.$

We will argue separately that the $\Box_{h}Z^{\leq 60}u_{k}$ term is bounded
by the right side of (4.16), which will establish the desired bound.

To control the second term in the right side of (2.9), we will consider the
integral at an arbitrary $t\in[0,T]$, and we fix a dyadic value $\tau$ so that
$t\in[\tau,2\tau]$. For $1\leq R\leq\tau/2$, we can apply (3.1) and (3.3) and
a Hardy inequality (after expanding the range of integration of the norm of
$r^{-1}|Z^{\leq 60}u_{k}|$ from $A_{R}$ to ${\mathbb{R}}^{3}$) to see

$\int_{A_{R}}r|\partial^{\leq 1}u_{k-1}||\partial Z^{\leq
60}u_{k}|\Bigl{(}|\partial Z^{\leq 60}u_{k}|+\frac{|Z^{\leq
60}u_{k}|}{r}\Bigr{)}\,dx\\\ \lesssim\Bigl{(}R^{-1}\|Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}\Bigr{)}\|\partial Z^{\leq
60}u_{k}(t,{\,\cdot\,})\|_{L^{2}(A_{R})}\|\partial Z^{\leq
60}u_{k}\|_{L^{\infty}L^{2}}.$

And hence, using the Schwarz inequality,

$\sup_{t\in[0,T]}\sum_{R\leq\tau/2}\int_{A_{R}}r|\partial^{\leq
1}u_{k-1}||\partial Z^{\leq 60}u_{k}|\Bigl{(}|\partial Z^{\leq
60}u_{k}|+\frac{|Z^{\leq
60}u_{k}|}{r}\Bigr{)}\,dx\lesssim\Bigl{(}IV_{k-1}+III_{k-1}\Bigr{)}I_{k}^{2},$

which is dominated by (4.17). For the remainder of this term, (3.2) and a
Hardy inequality show that

$\sum_{1\leq U\leq\tau/4}\int_{\langle t-r\rangle\approx U}r|\partial^{\leq
1}u_{k-1}||\partial Z^{\leq 60}u_{k}|\Bigl{(}|\partial Z^{\leq
60}u_{k}|+\frac{|Z^{\leq 60}u_{k}|}{r}\Bigr{)}\,dx\\\
\lesssim\sup_{U}\Bigl{(}\frac{1}{\tau^{\frac{1}{2}}U^{\frac{1}{2}}}\|Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}+\frac{1}{\tau^{\frac{1}{2}}U^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
3}u_{k-1})\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}\Bigr{)}\|\partial Z^{\leq
60}u_{k}(t,{\,\cdot\,})\|^{2}_{L^{2}}.$

The supremum of this is bounded by

$\Bigl{(}\log\langle T\rangle VIII_{k-1}+VII_{k-1}\Bigr{)}I_{k}^{2}.$

Thus,

(4.18) $\sup_{t\in[0,T]}\int r|\partial^{\leq 1}u_{k-1}||\partial Z^{\leq
60}u_{k}|\Bigl{(}|\partial Z^{\leq 60}u_{k}|+\frac{|Z^{\leq
60}u_{k}|}{r}\Bigr{)}\,dx\\\ \lesssim\Bigl{(}IV_{k-1}+III_{k-1}+\log\langle
T\rangle VIII_{k-1}+VII_{k-1}\Bigr{)}I_{k}^{2}.$

We proceed to showing that

(4.19) $\sup_{U\geq 1}\sup_{t\in[0,T]}\Bigl{|}\int_{0}^{t}\int
e^{-\sigma_{U}(t-r)}\Box_{h}Z^{\leq
60}u_{k}\cdot\Bigl{(}\partial_{t}+\partial_{r}\Bigr{)}(rZ^{\leq
60}u_{k})\,dx\,dt\Bigr{|}\\\ \lesssim(\log\langle
T\rangle)^{3}\Bigl{(}M_{k-1}^{2}M_{k}+M_{k-1}M_{k}^{2}\Bigr{)}+(\log\langle
T\rangle)^{5}M_{k-1}^{4}+(\log\langle T\rangle)^{5}M_{k-1}^{3}M_{k}.$

###### Proof of (4.19).

The most delicate terms in this analysis are those of the form
$u_{k-1}(\partial_{t}-\partial_{r})Z^{\leq 60}u_{k-1}$. Here we have a bad
derivative occurring at the highest regularity, and thus there is not room to
apply, for example, (3.5) directly in order to get additional decay.

We begin by examining the other terms. To this end, we set $\omega=(1,-x/r)$
and note that

(4.20) $|\Box_{h}Z^{\leq
60}u_{k}^{I}-a^{I,\alpha}_{JK}\omega_{\alpha}u_{k-1}^{J}(\partial_{t}-\partial_{r})Z^{\leq
60}u_{k-1}^{K}|\lesssim|Z^{\leq 30}u_{k-1}||{\not\negmedspace\partial}Z^{\leq
60}u_{k-1}|+|\partial Z^{\leq 30}u_{k-1}||\partial^{\leq 1}Z^{\leq
60}u_{k-1}|\\\ +|Z^{\leq 30}u_{k-1}||\partial Z^{\leq 59}u_{k-1}|+|\partial
Z^{\leq 31}u_{k}||\partial^{\leq 1}Z^{\leq 60}u_{k-1}|+|\partial^{\leq
1}Z^{\leq 30}u_{k-1}||\partial^{2}Z^{\leq 59}u_{k}|.$

Using (3.1), (3.2), and (3.3) gives us that

$\int\int_{C^{R}_{\tau}}r|Z^{\leq
30}u_{k-1}||{\not\negmedspace\partial}Z^{\leq
60}u_{k-1}|\Bigl{(}|{\not\negmedspace\partial}Z^{\leq 60}u_{k}|+r^{-1}|Z^{\leq
60}u_{k}|\Bigr{)}\,dx\,dt\\\ \lesssim\Bigl{(}R^{-1}\|Z^{\leq
34}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
33}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}\Bigr{)}\|{\not\negmedspace\partial}Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{R}_{\tau})}\\\
\times\Bigl{(}\|{\not\negmedspace\partial}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}+\|r^{-1}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}\Bigr{)},$

and respectively

$\int\int_{C^{U}_{\tau}}r|Z^{\leq
30}u_{k-1}||{\not\negmedspace\partial}Z^{\leq
60}u_{k-1}|\Bigl{(}|{\not\negmedspace\partial}Z^{\leq 60}u_{k}|+r^{-1}|Z^{\leq
60}u_{k}|\Bigr{)}\,dx\,dt\\\
\lesssim\Bigl{(}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
34}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}+\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
33}u_{k-1})\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}\Bigr{)}\|{\not\negmedspace\partial}Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}\\\
\times\Bigl{(}\|{\not\negmedspace\partial}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{U}_{\tau})}+\|r^{-1}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{U}_{\tau})}\Bigr{)}.$

Upon summing over $R\leq\tau/2$, $U\leq\tau/4$ and $\tau\leq T$, we get

(4.21) $\int_{0}^{T}\int r|Z^{\leq
30}u_{k-1}||{\not\negmedspace\partial}Z^{\leq
60}u_{k-1}|\Bigl{(}|{\not\negmedspace\partial}Z^{\leq 60}u_{k}|+r^{-1}|Z^{\leq
60}u_{k}|\Bigr{)}\,dx\,dt\\\ \lesssim(IV_{k-1}+III_{k-1}+\log\langle T\rangle
VIII_{k-1}+VII_{k-1})III_{k-1}(III_{k}+IV_{k}).$

Another application of (3.1) and (3.3) gives

$\int\int_{C^{R}_{\tau}}r|\partial Z^{\leq 30}u_{k-1}||\partial^{\leq
1}Z^{\leq 60}u_{k-1}|\Bigl{(}|{\not\negmedspace\partial}Z^{\leq
60}u_{k}|+r^{-1}|Z^{\leq 60}u_{k}|\Bigr{)}\,dx\,dt\\\
\lesssim\Bigl{(}\|\partial Z^{\leq
34}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+R\|{\not\negmedspace\partial}\partial
Z^{\leq
33}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}\Bigr{)}R^{-1}\|\partial^{\leq
1}Z^{\leq 60}u_{k-1}\|_{L^{2}L^{2}(C^{R}_{\tau})}\\\
\times\Bigl{(}\|{\not\negmedspace\partial}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}+\|r^{-1}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}\Bigr{)}.$

Similarly, using (3.2), we get

$\int\int_{C^{U}_{\tau}}|\partial Z^{\leq 30}u_{k-1}||\partial^{\leq 1}Z^{\leq
60}u_{k-1}|\Bigl{(}|(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})|\Bigr{)}\,dx\,dt\\\
\lesssim\Bigl{(}\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}}\|\partial Z^{\leq
34}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}+U^{\frac{1}{2}}\tau^{\frac{1}{2}}\|(\partial_{t}+\partial_{r})\partial
Z^{\leq
33}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}\Bigr{)}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|\partial^{\leq
1}Z^{\leq 60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}\\\
\times\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})\|_{L^{2}L^{2}(C^{U}_{\tau})}.$

Upon summing, these give

(4.22) $\int_{0}^{T}\int|\partial Z^{\leq 30}u_{k-1}||\partial^{\leq 1}Z^{\leq
60}u_{k-1}||(\partial_{t}+\partial_{r})(rZ^{\leq 60}u_{k})|\,dx\,dt\\\
\lesssim(IX_{k-1}+X_{k-1})(II_{k-1}+IV_{k-1})(III_{k}+IV_{k})\\\ +(\log\langle
T\rangle)^{2}\Bigl{(}XI_{k-1}+XII_{k-1}\Bigr{)}\Bigl{(}(\log\langle
T\rangle)^{\frac{1}{2}}II_{k-1}+\log\langle T\rangle
VIII_{k-1}\Bigr{)}VII_{k}.$

Following the same argument, we also obtain

(4.23) $\int_{0}^{T}\int|\partial Z^{\leq 31}u_{k}||\partial^{\leq 1}Z^{\leq
60}u_{k-1}||(\partial_{t}+\partial_{r})(rZ^{\leq 60}u_{k})|\,dx\,dt\\\
\lesssim(IX_{k}+X_{k})(II_{k-1}+IV_{k-1})(III_{k}+IV_{k})\\\ +(\log\langle
T\rangle)^{2}\Bigl{(}XI_{k}+XII_{k}\Bigr{)}\Bigl{(}(\log\langle
T\rangle)^{\frac{1}{2}}II_{k-1}+\log\langle T\rangle
VIII_{k-1}\Bigr{)}VII_{k}.$

We now apply both (3.5) and (3.1) (and (3.3)) to see that

$\int\int_{C^{R}_{\tau}}|Z^{\leq 30}u_{k-1}||\partial Z^{\leq
59}u_{k-1}||(\partial_{t}+\partial_{r})(rZ^{\leq 60}u_{k})|\,dx\,dt\\\
\lesssim\Bigl{(}\frac{1}{R}\|Z^{\leq
34}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
33}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}\Bigr{)}\Bigl{(}\frac{1}{R}\|Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
59}u_{k-1}\|_{L^{2}L^{2}(C^{R}_{\tau})}\Bigr{)}\\\
\times\Bigl{(}\|{\not\negmedspace\partial}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}+\frac{1}{R}\|Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}\Bigr{)}.$

While (3.5) and (3.2) give

$\int\int_{C^{U}_{\tau}}|Z^{\leq 30}u_{k-1}||\partial Z^{\leq
59}u_{k-1}||(\partial_{t}+\partial_{r})(rZ^{\leq 60}u_{k})|\,dx\,dt\\\
\lesssim\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\Bigl{(}\|Z^{\leq
34}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}+\|(\partial_{t}+\partial_{r})(rZ^{\leq
33}u_{k-1})\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}\Bigr{)}\\\
\times\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\Bigl{(}\|Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}+\|(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k-1})\|_{L^{2}L^{2}(C^{U}_{\tau})}\Bigr{)}\\\
\times\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})\|_{L^{2}L^{2}(C^{U}_{\tau})}.$

Upon summation, this gives

(4.24) $\int_{0}^{T}\int|Z^{\leq 30}u_{k-1}||\partial Z^{\leq
59}u_{k-1}||(\partial_{t}+\partial_{r})(rZ^{\leq 60}u_{k})|\,dx\,dt\\\
\lesssim(IV_{k-1}+III_{k-1})^{2}(III_{k}+IV_{k})\\\ +\log\langle
T\rangle\Bigl{(}\log\langle T\rangle VIII_{k-1}+VII_{k-1}\Bigr{)}^{2}VII_{k}.$

Very similar arguments give

$\int\int_{C^{R}_{\tau}}|\partial^{\leq 1}Z^{\leq
30}u_{k-1}||\partial^{2}Z^{\leq 59}u_{k}||(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})|\,dx\,dt\\\ \lesssim\Bigl{(}\frac{1}{R}\|Z^{\leq
35}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
34}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}\Bigr{)}\Bigl{(}\frac{1}{R}\|\partial
Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}\Bigr{)}\\\
\times\Bigl{(}\|{\not\negmedspace\partial}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}+\frac{1}{R}\|Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}\Bigr{)}$

and

$\int\int_{C^{U}_{\tau}}|\partial^{\leq 1}Z^{\leq
30}u_{k-1}||\partial^{2}Z^{\leq 59}u_{k}||(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})|\,dx\,dt\\\
\lesssim\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\Bigl{(}\|Z^{\leq
35}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}+\|(\partial_{t}+\partial_{r})(rZ^{\leq
34}u_{k-1})\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}\Bigr{)}\\\
\times\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\Bigl{(}\|\partial Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{U}_{\tau})}+\|(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})\|_{L^{2}L^{2}(C^{U}_{\tau})}\Bigr{)}\\\
\times\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})\|_{L^{2}L^{2}(C^{U}_{\tau})},$

which yield

(4.25) $\int_{0}^{T}\int|\partial^{\leq 1}Z^{\leq
30}u_{k-1}||\partial^{2}Z^{\leq 59}u_{k}||(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})|\,dx\,dt\\\
\lesssim(IV_{k-1}+III_{k-1})(II_{k}+III_{k})(III_{k}+IV_{k})\\\
+\Bigl{(}\log\langle T\rangle
VIII_{k-1}+VII_{k-1}\Bigr{)}\Bigl{(}II_{k}+\log\langle T\rangle
VII_{k}\Bigr{)}VII_{k}.$

As the right sides of (4.21), (4.22), (4.23), (4.24), and (4.25) are bounded
by (4.17), to complete the bound (4.19), we need only examine

$\sup_{U\geq 1}\sup_{t\in[0,T]}\Bigl{|}\int_{0}^{t}\int
e^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u_{k-1}^{J}(\partial_{t}-\partial_{r})Z^{\leq
60}u_{k-1}^{K}(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k}^{I})\,dx\,dt\Bigr{|}.$

The argument that we shall use here is reminiscent of normal forms.

We first integrate by parts to see that

(4.26) $\int_{0}^{t}\int
e^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u_{k-1}^{J}(\partial_{t}-\partial_{r})Z^{\leq
60}u_{k-1}^{K}(\partial_{t}+\partial_{r})(rZ^{\leq 60}u_{k}^{I})\,dx\,dt\\\
=\int e^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u^{J}_{k-1}Z^{\leq
60}u^{K}_{k-1}(\partial_{t}+\partial_{r})(rZ^{\leq
60}u^{I}_{k})\,dx\Bigl{|}_{0}^{t}\\\ +2\int_{0}^{t}\int
r^{-1}e^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u^{J}_{k-1}Z^{\leq
60}u^{K}_{k-1}(\partial_{t}+\partial_{r})(rZ^{\leq 60}u^{I}_{k})\,dx\,dt\\\
+2\int_{0}^{t}\int\sigma^{\prime}_{U}(t-r)e^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u^{J}_{k-1}Z^{\leq
60}u^{K}_{k-1}(\partial_{t}+\partial_{r})(rZ^{\leq 60}u_{k}^{I})\,dx\,dt\\\
-\int_{0}^{t}\int
e^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}(\partial_{t}-\partial_{r})u_{k-1}^{J}Z^{\leq
60}u^{K}_{k-1}(\partial_{t}+\partial_{r})(rZ^{\leq 60}u_{k}^{I})\,dx\,dt\\\
-\int_{0}^{t}\int
e^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u^{J}_{k-1}Z^{\leq
60}u_{k-1}^{K}(\partial_{t}^{2}-\partial_{r}^{2})(rZ^{\leq
60}u_{k}^{I})\,dx\,dt.$

We shall proceed through arguments that will bound each term in (4.26).

For $t$ fixed and $\tau\approx t$, we may apply (3.1), (3.3), and the Schwarz
inequality to see that

$\sum_{R\leq\tau/2}\int_{A_{R}}r|u_{k-1}||Z^{\leq
60}u_{k-1}|\Bigl{(}|{\not\negmedspace\partial}Z^{\leq 60}u_{k}|+r^{-1}|Z^{\leq
60}u_{k}|\Bigr{)}\,dx\\\ \lesssim\Bigl{(}\|r^{-1}Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}}+\|{\not\negmedspace\partial}Z^{\leq
3}u_{k-1}\|_{L^{2}L^{2}}\Bigr{)}\|r^{-\frac{1}{2}}Z^{\leq
60}u_{k-1}(t,{\,\cdot\,})\|_{L^{2}}\\\ \times\Bigl{(}\|\langle
r\rangle^{\frac{1}{2}}{\not\negmedspace\partial}Z^{\leq
60}u_{k}(t,{\,\cdot\,})\|_{L^{2}}+\|r^{-\frac{1}{2}}Z^{\leq
60}u_{k}(t,{\,\cdot\,})\|_{L^{2}}\Bigr{)}.$

Using (3.2), we instead get

$\sum_{U\leq\tau/4}\int_{\langle t-r\rangle\approx U}r|u_{k-1}||Z^{\leq
60}u_{k-1}|\Bigl{(}|{\not\negmedspace\partial}Z^{\leq 60}u_{k}|+r^{-1}|Z^{\leq
60}u_{k}|\Bigr{)}\,dx\\\
\lesssim\sup_{U}\Bigl{[}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\Bigl{(}\|Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\|(\partial_{t}+\partial_{r})(rZ^{\leq
3}u_{k-1})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}\Bigr{]}\|r^{-\frac{1}{2}}Z^{\leq
60}u_{k-1}(t,{\,\cdot\,})\|_{L^{2}}\\\ \times\Bigl{(}\|\langle
r\rangle^{\frac{1}{2}}{\not\negmedspace\partial}Z^{\leq
60}u_{k}(t,{\,\cdot\,})\|_{L^{2}}+\|r^{-\frac{1}{2}}Z^{\leq
60}u_{k}(t,{\,\cdot\,})\|_{L^{2}}\Bigr{)}.$

As such,

(4.27) $\sup_{U}\sup_{t\in[0,T]}\Bigl{|}\int
e^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u^{J}_{k-1}Z^{\leq
60}u^{K}_{k-1}(\partial_{t}+\partial_{r})(rZ^{\leq
60}u^{I}_{k})\,dx\Bigl{|}_{0}^{t}\Bigr{|}\\\
\lesssim\Bigl{(}IV_{k-1}+III_{k-1}+\log\langle T\rangle
VIII_{k-1}+VII_{k-1}\Bigr{)}VI_{k-1}(V_{k}+VI_{k}).$

For the second term in the right of (4.26), provided $R\leq\tau/2$, we may
apply (3.1) or (3.3) to see that

(4.28) $\int\int_{C^{R}_{\tau}}|u_{k-1}||Z^{\leq
60}u_{k-1}|\Bigl{(}|{\not\negmedspace\partial}Z^{\leq 60}u_{k}|+r^{-1}|Z^{\leq
60}u_{k}|\Bigr{)}\,dx\,dt\\\ \lesssim\Bigl{(}\|r^{-1}Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
3}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\Bigr{)}\|r^{-1}Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{R}_{\tau})}\\\
\times\Bigl{(}\|{\not\negmedspace\partial}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}+\|r^{-1}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}\Bigr{)},$

and for $U\leq\tau/4$, (3.2) gives

(4.29) $\int\int_{C^{U}_{\tau}}r^{-1}|u_{k-1}||Z^{\leq
60}u_{k-1}||(\partial_{t}+\partial_{r})(rZ^{\leq 60}u_{k})|\,dx\,dt\\\
\lesssim\Bigl{(}\frac{1}{\tau}\|Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\frac{1}{\tau}\|(\partial_{t}+\partial_{r})(rZ^{\leq
3}u_{k-1})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}\frac{1}{\tau}\|Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}\\\
\times\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})\|_{L^{2}L^{2}(C^{U}_{\tau})}.$

Upon summing, this results in

(4.30) $\int_{0}^{T}\int r^{-1}|u_{k-1}||Z^{\leq
60}u_{k-1}||(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})|\,dx\,dt\lesssim(IV_{k-1}+III_{k-1})IV_{k-1}(III_{k}+IV_{k}+VII_{k}),$

which suffices for the bound in the supremum (in both $t$ and $U$) of the
second term in the right side of (4.26).

Similar to (4.29), we estimate

$\int\int_{C^{U}_{\tau}}\frac{1}{U}|u_{k-1}||Z^{\leq
60}u_{k-1}||(\partial_{t}+\partial_{r})(rZ^{\leq 60}u_{k})|\,dx\,dt\\\
\lesssim\Bigl{(}\frac{1}{\tau^{\frac{1}{2}}U^{\frac{1}{2}}}\|Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\frac{1}{\tau^{\frac{1}{2}}U^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
3}u_{k-1})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}\frac{1}{\tau^{\frac{1}{2}}U^{\frac{1}{2}}}\|Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}\\\
\times\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})\|_{L^{2}L^{2}(C^{U}_{\tau})}.$

Since

$\sigma_{U}^{\prime}(t-r)\lesssim\frac{1}{\tau}\,\text{ on
}C^{R}_{\tau},\quad\sigma^{\prime}_{U}(t-r)\lesssim\frac{1}{U}\,\text{ on
}C^{U}_{\tau},$

we may combine this with (4.28) to see that

(4.31) $\int_{0}^{T}\int\sigma^{\prime}_{U}(t-r)|u_{k-1}||Z^{\leq
60}u_{k-1}||(\partial_{t}+\partial_{r})(rZ^{\leq 60}u_{k})|\,dx\,dt\\\
\lesssim(IV_{k-1}+III_{k-1})IV_{k-1}(III_{k}+IV_{k})+(\log\langle
T\rangle)^{2}\Bigl{(}\log\langle T\rangle
VIII_{k-1}+VII_{k-1}\Bigr{)}VIII_{k-1}VII_{k},$

which provides the appropriate bound for the supremums of the third term in
(4.26).

As (4.22) suffices to bound the fourth term in the right side of (4.26), it
remains to consider

(4.32) $\int_{0}^{t}\int
e^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u_{k-1}^{J}Z^{\leq
60}u_{k-1}^{K}(\partial_{t}^{2}-\partial_{r}^{2})(rZ^{\leq
60}u_{k}^{I})\,dx\,dt\\\ =\int_{0}^{t}\int
e^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u_{k-1}^{J}Z^{\leq
60}u_{k-1}^{K}{\not\negmedspace\nabla}\cdot{\not\negmedspace\nabla}(rZ^{\leq
60}u_{k}^{I})\,dx\,dt\\\ +\int_{0}^{t}\int
re^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u_{k-1}^{J}Z^{\leq
60}u_{k-1}^{K}h^{I,\beta\gamma}_{\tilde{J}}\partial_{\beta}\partial_{\gamma}Z^{\leq
60}u_{k}^{\tilde{J}}\,dx\,dt\\\ +\int_{0}^{t}\int
re^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u_{k-1}^{J}Z^{\leq
60}u_{k-1}^{K}\Box_{h}Z^{\leq 60}u_{k}^{I}\,dx\,dt.$

For the first term in the right, we integrate by parts and use (1.8) to see
that

$\sup_{U}\sup_{t\in[0,T]}\Bigl{|}\int_{0}^{t}\int
e^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u_{k-1}^{J}Z^{\leq
60}u_{k-1}^{K}{\not\negmedspace\nabla}\cdot{\not\negmedspace\nabla}(rZ^{\leq
60}u_{k}^{I})\,dx\,dt\Bigr{|}\\\ \lesssim\int_{0}^{T}\int|Z^{\leq
1}u_{k-1}||Z^{\leq 60}u_{k-1}||{\not\negmedspace\nabla}Z^{\leq
60}u_{k}|\,dx\,dt+\int_{0}^{T}\int r|u_{k-1}||{\not\negmedspace\nabla}Z^{\leq
60}u_{k-1}||{\not\negmedspace\nabla}Z^{\leq 60}u_{k}|\,dx\,dt.$

The preceding bound (4.21) shows that the latter term is controlled by (4.17).
And (3.1) (and (3.3)) and (3.2), respectively, give

$\int\int_{C^{R}_{\tau}}|Z^{\leq 1}u_{k-1}||Z^{\leq
60}u_{k-1}||{\not\negmedspace\nabla}Z^{\leq 60}u_{k}|\,dx\,dt\\\
\lesssim\Bigl{(}\|r^{-1}Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\Bigr{)}\|r^{-1}Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{R}_{\tau})}\|{\not\negmedspace\partial}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}$

and

$\int\int_{C^{U}_{\tau}}|Z^{\leq 1}u_{k-1}||Z^{\leq
60}u_{k-1}||{\not\negmedspace\nabla}Z^{\leq 60}u_{k}|\,dx\,dt\\\
\lesssim\Bigl{(}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
4}u_{k-1})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}\\\
\times\|r^{-1}Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}\|{\not\negmedspace\partial}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{U}_{\tau})}.$

Hence,

$\int_{0}^{T}\int|Z^{\leq 1}u_{k-1}||Z^{\leq
60}u_{k-1}||{\not\negmedspace\nabla}Z^{\leq
60}u_{k}|\,dx\,dt\lesssim(IV_{k-1}+III_{k-1}+\log\langle T\rangle
VIII_{k-1}+VII_{k-1})IV_{k-1}III_{k},$

which shows that

(4.33) $\sup_{U}\sup_{t\in[0,T]}\Bigl{|}\int_{0}^{t}\int
e^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u_{k-1}^{J}Z^{\leq
60}u_{k-1}^{K}{\not\negmedspace\nabla}\cdot{\not\negmedspace\nabla}(rZ^{\leq
60}u_{k}^{I})\,dx\,dt\Bigr{|}\lesssim\log\langle T\rangle M_{k-1}^{2}M_{k}.$

Another integration by parts gives

(4.34) $\sup_{U}\sup_{t\in[0,T]}\Bigl{|}\int_{0}^{t}\int
re^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u_{k-1}^{J}Z^{\leq
60}u_{k-1}^{K}h^{I,\beta\gamma}_{\tilde{J}}\partial_{\beta}\partial_{\gamma}Z^{\leq
60}u_{k}^{\tilde{J}}\,dx\,dt\Bigr{|}\\\ \lesssim\sup_{t\in[0,T]}\int
r|\partial^{\leq 1}u_{k-1}|^{2}|Z^{\leq 60}u_{k-1}||\partial Z^{\leq
60}u_{k}|\,dx+\int_{0}^{T}\int|\partial^{\leq 1}u_{k-1}|^{2}|Z^{\leq
60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\,dt\\\
+\int_{0}^{T}\int\frac{r}{\langle t-r\rangle}|\partial^{\leq
1}u_{k-1}|^{2}|Z^{\leq 60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\,dt\\\
+\int_{0}^{T}\int r|\partial^{\leq 1}u_{k-1}||\partial\partial^{\leq
1}u_{k-1}||Z^{\leq 60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\,dt\\\
+\int_{0}^{T}\int r|\partial^{\leq 1}u_{k-1}|^{2}|\partial Z^{\leq
60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\,dt.$

If we argue precisely as in (4.27), we see that

$\sup_{t\in[0,T]}\int r|\partial^{\leq 1}u_{k-1}|^{2}|Z^{\leq
60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\\\
\lesssim\Bigl{(}IV_{k-1}+III_{k-1}+\log\langle T\rangle
VIII_{k-1}+VII_{k-1}\Bigr{)}VI_{k-1}\|\langle
r\rangle^{\frac{1}{2}}|\partial^{\leq 1}u_{k-1}||\partial Z^{\leq
60}u_{k}|\|_{L^{\infty}L^{2}}.$

Subsequently applying (3.7) gives that

$\|\langle r\rangle^{\frac{1}{2}}|\partial^{\leq 1}u_{k-1}||\partial Z^{\leq
60}u_{k}|\|_{L^{\infty}L^{2}}\lesssim\|r^{-\frac{1}{2}}Z^{\leq
3}u_{k-1}\|_{L^{\infty}L^{2}}\|\partial Z^{\leq 60}u_{k}\|_{L^{\infty}L^{2}}.$

And hence,

(4.35) $\sup_{t\in[0,T]}\int r|\partial^{\leq 1}u_{k-1}|^{2}|Z^{\leq
60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\\\
\lesssim\Bigl{(}IV_{k-1}+III_{k-1}+\log\langle T\rangle
VIII_{k-1}+VII_{k-1}\Bigr{)}VI_{k-1}VI_{k-1}I_{k}.$

For the remaining terms in (4.34), we continue to apply (3.1), (3.2), and
(3.3) repeatedly. These give

$\int\int_{C^{R}_{\tau}}|\partial^{\leq 1}u_{k-1}|^{2}|Z^{\leq
60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\,dt\\\
\lesssim\Bigl{(}\|r^{-1}Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\Bigr{)}^{2}\|r^{-1}Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{R}_{\tau})}\frac{1}{R^{\frac{1}{2}}}\|r^{-\frac{1}{2}}\partial
Z^{\leq 60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}$

and

$\int\int_{C^{U}_{\tau}}|\partial^{\leq 1}u_{k-1}|^{2}|Z^{\leq
60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\,dt\\\
\lesssim\Bigl{(}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
4}u_{k-1})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}^{2}\\\
\times\|r^{-1}Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}\frac{1}{\tau^{\frac{1}{2}}}\|r^{-\frac{1}{2}}\partial
Z^{\leq 60}u_{k}\|_{L^{2}L^{2}(C^{U}_{\tau})}.$

Similarly,

$\int\int_{C^{U}_{\tau}}\frac{r}{\langle t-r\rangle}|\partial^{\leq
1}u_{k-1}|^{2}|Z^{\leq 60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\,dt\\\
\lesssim\Bigl{(}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
4}u_{k-1})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}^{2}\\\ \times
U^{-1}\|r^{-\frac{1}{2}}Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}\|r^{-\frac{1}{2}}\partial Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{U}_{\tau})}.$

These combine to give

(4.36) $\int_{0}^{T}\int|\partial^{\leq 1}u_{k-1}|^{2}|Z^{\leq
60}u_{k-1}||\partial Z^{\leq
60}u_{k}|\,dx\,dt+\int\int_{C^{U}_{\tau}}\frac{r}{\langle
t-r\rangle}|\partial^{\leq 1}u_{k-1}|^{2}|Z^{\leq 60}u_{k-1}||\partial Z^{\leq
60}u_{k}|\,dx\,dt\\\
\lesssim(IV_{k-1}+III_{k-1})^{2}IV_{k-1}II_{k}+(\log\langle T\rangle
VIII_{k-1}+VII_{k-1})^{2}\Bigl{(}IV_{k-1}II_{k}+\log\langle T\rangle
VIII_{k-1}II_{k}\Bigr{)}.$

By related arguments,

$\int\int_{C^{R}_{\tau}}r|\partial^{\leq 1}u_{k-1}||\partial\partial^{\leq
1}u_{k-1}||Z^{\leq 60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\,dt\\\
\lesssim\Bigl{(}\|r^{-1}Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\Bigr{)}\|r^{-1}Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{R}_{\tau})}\\\ \times\Bigl{(}\|\partial Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}+R\|{\not\negmedspace\partial}\partial
Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\Bigr{)}R^{-\frac{1}{2}}\|\langle
r\rangle^{-\frac{1}{2}}\partial Z^{\leq 60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}$

and

$\int\int_{C^{U}_{\tau}}r|\partial^{\leq 1}u_{k-1}||\partial\partial^{\leq
1}u_{k-1}||Z^{\leq 60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\,dt\\\
\lesssim\frac{1}{U^{\frac{1}{2}}}\Bigl{(}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
4}u_{k-1})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}\\\
\times\Bigl{(}\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}}\|\partial Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+U^{\frac{1}{2}}\tau^{\frac{1}{2}}\|{\not\negmedspace\partial}\partial
Z^{\leq 4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}\\\
\times\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}\frac{1}{\tau^{\frac{1}{2}}}\|\partial
Z^{\leq 60}u_{k}\|_{L^{2}L^{2}(C^{U}_{\tau})},$

which gives

(4.37) $\int_{0}^{T}\int r|\partial^{\leq 1}u_{k-1}||\partial\partial^{\leq
1}u_{k-1}||Z^{\leq 60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\,dt\\\
\lesssim(IV_{k-1}+III_{k-1})(IX_{k-1}+X_{k-1})IV_{k-1}II_{k}\\\ +(\log\langle
T\rangle)^{2}(\log\langle T\rangle
VIII_{k-1}+VII_{k-1})(XI_{k-1}+XII_{k-1})VIII_{k-1}II_{k}.$

For the last term in (4.34), we get

$\int\int_{C^{R}_{\tau}}r|\partial^{\leq 1}u_{k-1}|^{2}|\partial Z^{\leq
60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\,dt\lesssim\Bigl{(}\|r^{-1}Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\Bigr{)}^{2}\\\
\times\|r^{-\frac{1}{2}}\partial Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{R}_{\tau})}\|r^{-\frac{1}{2}}\partial Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}$

and

$\int\int_{C^{U}_{\tau}}r|\partial^{\leq 1}u_{k-1}|^{2}|\partial Z^{\leq
60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\,dt\\\
\lesssim\Bigl{(}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
4}u_{k-1})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}^{2}\\\
\times\|r^{-\frac{1}{2}}\partial Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}\|r^{-\frac{1}{2}}\partial Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{U}_{\tau})},$

yielding

(4.38) $\int_{0}^{T}\int r|\partial^{\leq 1}u_{k-1}|^{2}|\partial Z^{\leq
60}u_{k-1}||\partial Z^{\leq 60}u_{k}|\,dx\,dt\\\
\lesssim(IV_{k-1}+III_{k-1})^{2}II_{k-1}II_{k}+\log\langle
T\rangle(\log\langle T\rangle VIII_{k-1}+VII_{k-1})^{2}II_{k-1}II_{k}.$

The combination of (4.35), (4.36), (4.37), and (4.38) then establish that

(4.39) $\sup_{U}\sup_{t\in[0,T]}\Bigl{|}\int_{0}^{t}\int
re^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u_{k-1}^{J}Z^{\leq
60}u_{k-1}^{K}h^{I,\alpha\beta}_{\tilde{J}}\partial_{\alpha}\partial_{\beta}Z^{\leq
60}u_{k}^{\tilde{J}}\,dx\,dt\Bigr{|}\lesssim(\log\langle
T\rangle)^{3}M_{k-1}^{3}M_{k}.$

We now turn our attention to

$\sup_{U}\sup_{t\in[0,T]}\Bigl{|}\int_{0}^{t}\int
re^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u^{J}_{k-1}Z^{\leq
60}u^{K}_{k-1}\Box_{h}Z^{\leq 60}u_{k}^{I}\,dx\,dt\Bigr{|}.$

We shall first show

(4.40) $\sup_{U}\sup_{t\in[0,T]}\Bigl{|}\int_{0}^{t}\int
re^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u^{J}_{k-1}Z^{\leq
60}u^{K}_{k-1}\Bigl{(}\Box_{h}Z^{\leq
60}u_{k}^{I}-a^{I,\beta}_{{\tilde{J}}{\tilde{K}}}\omega_{\beta}u^{\tilde{J}}_{k-1}(\partial_{t}-\partial_{r})Z^{\leq
60}u^{\tilde{K}}_{k-1}\Bigr{)}\,dx\,dt\Bigr{|}\\\ \lesssim(\log\langle
T\rangle)^{5}M_{k-1}^{4}+(\log\langle T\rangle)^{5}M_{k-1}^{3}M_{k}.$

By (4.20), we have

(4.41) $\sup_{U}\sup_{t\in[0,T]}\Bigl{|}\int_{0}^{t}\int
re^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u^{J}_{k-1}Z^{\leq
60}u^{K}_{k-1}\Bigl{(}\Box_{h}Z^{\leq
60}u_{k}^{I}-a^{I,\beta}_{{\tilde{J}}{\tilde{K}}}\omega_{\beta}u^{\tilde{J}}_{k-1}(\partial_{t}-\partial_{r})Z^{\leq
60}u^{\tilde{K}}_{k-1}\Bigr{)}\,dx\,dt\Bigr{|}\\\ \lesssim\int_{0}^{T}\int
r|u_{k-1}||Z^{\leq 60}u_{k-1}||Z^{\leq
30}u_{k-1}||{\not\negmedspace\partial}Z^{\leq 60}u_{k-1}|\,dx\,dt\\\
+\int_{0}^{T}\int r|u_{k-1}||Z^{\leq 60}u_{k-1}||\partial Z^{\leq
30}u_{k-1}||\partial^{\leq 1}Z^{\leq 60}u_{k-1}|\,dx\,dt\\\ +\int_{0}^{T}\int
r|u_{k-1}||Z^{\leq 60}u_{k-1}||\partial Z^{\leq 31}u_{k}||\partial^{\leq
1}Z^{\leq 60}u_{k-1}|\,dx\,dt\\\ \int_{0}^{T}\int r|u_{k-1}||Z^{\leq
60}u_{k-1}||\partial^{\leq 1}Z^{\leq 30}u_{k-1}||\partial^{2}Z^{\leq
59}u_{k}|\,dx\,dt.$

We note that (3.1) and (3.3) give

(4.42) $\||u_{k-1}||Z^{\leq
60}u_{k-1}|\|_{L^{2}L^{2}(C^{R}_{\tau})}\lesssim\Bigl{(}\|r^{-1}Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
3}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\Bigr{)}\|r^{-1}Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{R}_{\tau})}$

and (3.2) gives

(4.43) $\frac{\tau^{\frac{1}{2}}}{U^{\frac{1}{2}}}\||u_{k-1}||Z^{\leq
60}u_{k-1}|\|_{L^{2}L^{2}(C^{U}_{\tau})}\\\
\lesssim\frac{1}{\tau^{\frac{1}{2}}U^{\frac{1}{2}}}\Bigl{(}\|Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\|(\partial_{t}+\partial_{r})(rZ^{\leq
3}u_{k-1})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}.$

Arguing as in (4.21), (4.22), (4.23), and (4.25) where

$\|{\not\negmedspace\partial}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})},\quad\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})\|_{L^{2}L^{2}(C^{U}_{\tau})}$

are replaced by (4.42) and (4.43), respectively, we immediately get (4.40).

To complete the proof of (4.19), we now consider

$\int_{0}^{t}\int
re^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u^{J}_{k-1}Z^{\leq
60}u^{K}_{k-1}\Bigl{(}a^{I,\beta}_{{\tilde{J}}{\tilde{K}}}\omega_{\beta}u^{\tilde{J}}_{k-1}(\partial_{t}-\partial_{r})Z^{\leq
60}u^{\tilde{K}}_{k-1}\Bigr{)}\,dx\,dt\\\ =\frac{1}{2}\int_{0}^{t}\int
re^{-\sigma_{U}(t-r)}u^{J}_{k-1}u^{\tilde{J}}_{k-1}(\partial_{t}-\partial_{r})\Bigl{[}a^{I,\alpha}_{JK}\omega_{\alpha}a^{I,\beta}_{{\tilde{J}}{\tilde{K}}}\omega_{\beta}Z^{\leq
60}u^{K}_{k-1}Z^{\leq 60}u^{\tilde{K}}_{k-1}\Bigr{]}\,dx\,dt.$

Integration by parts shows that

(4.44) $\sup_{U}\sup_{t\in[0,T]}\Bigl{|}\int_{0}^{t}\int
re^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u^{J}_{k-1}Z^{\leq
60}u^{K}_{k-1}\Bigl{(}a^{I,\beta}_{{\tilde{J}}{\tilde{K}}}\omega_{\beta}u^{\tilde{J}}_{k-1}(\partial_{t}-\partial_{r})Z^{\leq
60}u^{\tilde{K}}_{k-1}\Bigr{)}\,dx\,dt\Bigr{|}\\\ \lesssim\sup_{t\in[0,T]}\int
r|u_{k-1}|^{2}|Z^{\leq 60}u_{k-1}|^{2}\,dx+\int_{0}^{T}\int\frac{r}{\langle
t-r\rangle}|u_{k-1}|^{2}|Z^{\leq 60}u_{k-1}|^{2}\,dx\,dt\\\
+\int_{0}^{T}\int|u_{k-1}|^{2}|Z^{\leq
60}u_{k-1}|^{2}\,dx\,dt+\int_{0}^{T}\int r|u_{k-1}||\partial u_{k-1}||Z^{\leq
60}u_{k-1}|^{2}\,dx\,dt.$

For the first term, we first consider a fixed $t$ and set $\tau\approx t$.
Then by (3.1) (and (3.3)),

$\int_{A_{R}}r|u_{k-1}|^{2}|Z^{\leq
60}u_{k-1}|^{2}\,dx\lesssim\Bigl{(}\|r^{-1}Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
3}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\Bigr{)}^{2}\|r^{-\frac{1}{2}}Z^{\leq
60}u_{k-1}(t,{\,\cdot\,})\|_{L^{2}}^{2},$

and by (3.2)

$\int_{\langle t-r\rangle\approx U}r|u_{k-1}|^{2}|Z^{\leq
60}u_{k-1}|^{2}\,dx\lesssim\Bigl{(}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})rZ^{\leq
3}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}^{2}\\\
\times\|r^{-\frac{1}{2}}Z^{\leq
60}u_{k-1}(t,{\,\cdot\,})\|^{2}_{L^{2}(\\{\langle t-r\rangle\approx U\\})}.$

Upon summing over $R\leq\tau/2$, $U\leq\tau/4$ and then taking a supremum in
$t$, we obtain

(4.45) $\sup_{t\in[0,T]}\int r|u_{k-1}|^{2}|Z^{\leq
60}u_{k-1}|^{2}\,dx\lesssim(IV_{k-1}+III_{k-1})^{2}VI_{k-1}^{2}+(\log\langle
T\rangle VIII_{k-1}+VII_{k-1})^{2}VI_{k-1}^{2}.$

For the second term in the right of (4.44), subsequently applying (3.1) (and
(3.3)) and (3.2) when $R\leq\tau/2$, $U\leq\tau/4$ gives

(4.46) $\int\int_{C^{R}_{\tau}}\frac{r}{\langle
t-r\rangle}|u_{k-1}|^{2}|Z^{\leq 60}u_{k-1}|^{2}\,dx\,dt\\\
\lesssim\Bigl{(}\|r^{-1}Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
3}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\Bigr{)}^{2}\|r^{-1}Z^{\leq
60}u_{k-1}\|^{2}_{L^{2}L^{2}(C^{R}_{\tau})}$

and

$\int\int_{C^{U}_{\tau}}\frac{r}{\langle t-r\rangle}|u_{k-1}|^{2}|Z^{\leq
60}u_{k-1}|^{2}\,dx\,dt\\\
\lesssim\Bigl{(}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
3}u_{k-1})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}^{2}\\\
\times\Bigl{(}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}\Bigr{)}^{2}.$

Upon summing, we obtain

(4.47) $\int_{0}^{T}\int\frac{r}{\langle t-r\rangle}|u_{k-1}|^{2}|Z^{\leq
60}u_{k-1}|^{2}\,dx\,dt\lesssim(IV_{k-1}+III_{k-1})^{2}IV_{k-1}^{2}\\\
+(\log\langle T\rangle)^{3}\Bigl{(}\log\langle T\rangle
VIII_{k-1}+VII_{k-1}\Bigr{)}^{2}VIII_{k-1}^{2}.$

Moreover,

$\int\int_{C^{U}_{\tau}}|u_{k-1}|^{2}|Z^{\leq 60}u_{k-1}|^{2}\,dx\,dt\\\
\lesssim\Bigl{(}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
3}u_{k-1})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}^{2}\|r^{-1}Z^{\leq
60}u_{k-1}\|_{L^{2}L^{2}(C^{U}_{\tau})}^{2}.$

Hence when combined with (4.46), this gives

(4.48) $\int_{0}^{T}\int|u_{k-1}|^{2}|Z^{\leq
60}u_{k-1}|^{2}\,dx\,dt\lesssim(IV_{k-1}+III_{k-1})^{2}IV_{k-1}^{2}+\Bigl{(}\log\langle
T\rangle VIII_{k-1}+VII_{k-1}\Bigr{)}^{2}IV_{k-1}^{2}.$

We may use the estimate on the second term in the right side of (4.41) to
bound the last term in (4.44). Combining this with (4.45), (4.47), and (4.48)
then yields

$\sup_{U}\sup_{t\in[0,T]}\Bigl{|}\int_{0}^{t}\int
re^{-\sigma_{U}(t-r)}a^{I,\alpha}_{JK}\omega_{\alpha}u^{J}_{k-1}Z^{\leq
60}u^{K}_{k-1}\Bigl{(}a^{I,\beta}_{{\tilde{J}}{\tilde{K}}}\omega_{\beta}u^{\tilde{J}}_{k-1}(\partial_{t}-\partial_{r})Z^{\leq
60}u^{\tilde{K}}_{k-1}\Bigr{)}\,dx\,dt\Bigr{|}\lesssim(\log\langle
T\rangle)^{5}M_{k-1}^{4},$

which completes the proof of (4.19). ∎

To finish the proof of (4.16), we consider the remainder of the terms in the
right side of (2.9) and show that they are each controlled by (4.17).

By (3.1) and (3.3),

$\int\int_{C^{R}_{\tau}}\Bigl{(}|\partial Z^{\leq 1}u_{k-1}|+\frac{|Z^{\leq
1}u_{k-1}|}{r}\Bigr{)}|\partial Z^{\leq
60}u_{k}||(\partial_{t}+\partial_{r})(rZ^{\leq 60}u_{k})|\,dx\,dt\\\
\lesssim\Bigl{(}\|r^{-1}Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+\|\partial Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+R\|{\not\negmedspace\partial}\partial
Z^{\leq 4}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}\Bigr{)}\\\
\times\|r^{-1}\partial Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}\Bigl{(}\|{\not\negmedspace\partial}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}+\|r^{-1}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{R}_{\tau})}\Bigr{)}.$

And by (3.2),

$\int\int_{C^{U}_{\tau}}\Bigl{(}|\partial Z^{\leq 1}u_{k-1}|+\frac{|Z^{\leq
1}u_{k-1}|}{r}\Bigr{)}|\partial Z^{\leq
60}u_{k}||(\partial_{t}+\partial_{r})(rZ^{\leq 60}u_{k})|\,dx\,dt\\\
\lesssim\Bigl{(}\|r^{-1}Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}+\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}}\|\partial
Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}+U^{\frac{1}{2}}\tau^{\frac{1}{2}}\|{\not\negmedspace\partial}\partial
Z^{\leq 4}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{U}_{\tau})}\Bigr{)}\\\
\times\frac{1}{U^{\frac{1}{2}}}\|\langle r\rangle^{-\frac{1}{2}}\partial
Z^{\leq
60}u_{k}\|_{L^{2}L^{2}(C^{U}_{\tau})}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})\|_{L^{2}L^{2}(C^{U}_{\tau})}.$

Upon summing over $R\leq\tau/2$, $U\leq\tau/4$ and $\tau\leq T$, we get

(4.49) $\int_{0}^{T}\int\Bigl{(}|\partial Z^{\leq 1}u_{k-1}|+\frac{|Z^{\leq
1}u_{k-1}|}{r}\Bigr{)}|\partial Z^{\leq
60}u_{k}||(\partial_{t}+\partial_{r})(rZ^{\leq 60}u_{k})|\,dx\,dt\\\
\lesssim(IV_{k-1}+IX_{k-1}+X_{k-1})II_{k}(III_{k}+IV_{k})\\\
+\Bigl{(}IV_{k-1}+\log\langle T\rangle XI_{k-1}+\log\langle T\rangle
XII_{k-1}\Bigr{)}II_{k}VII_{k}.$

For the fifth term in the right side of (2.9), applying (3.7) and the Schwarz
inequality, we have

(4.50) $\begin{split}\int_{0}^{T}\int|\partial^{\leq 1}u_{k-1}||\partial
Z^{\leq 60}&u_{k}|\Bigl{(}|{\not\negmedspace\nabla}Z^{\leq
60}u_{k}|+\frac{|Z^{\leq 60}u_{k}|}{r}\Bigr{)}\,dx\,dt\\\ &\lesssim\|\langle
r\rangle^{-1}Z^{\leq 3}u_{k-1}\|_{L^{2}L^{2}}\|\partial Z^{\leq
60}u_{k}\|_{L^{\infty}L^{2}}\Bigl{(}\|{\not\negmedspace\nabla}Z^{\leq
60}u_{k}\|_{L^{2}L^{2}}+\|r^{-1}Z^{\leq 60}u_{k}\|_{L^{2}L^{2}}\Bigr{)}\\\
&\lesssim IV_{k-1}I_{k}(III_{k}+IV_{k}).\end{split}$

And for the sixth term, using (3.7),

(4.51) $\begin{split}\int_{0}^{T}\int\langle t-r\rangle^{-1}|&\partial^{\leq
1}u_{k-1}||\partial Z^{\leq 60}u_{k}||(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})|\,dx\,dt\\\ &\lesssim\log\langle T\rangle\Bigl{(}\sup_{U}\|\langle
r\rangle^{-\frac{1}{2}}\langle t-r\rangle^{-\frac{1}{2}}Z^{\leq
3}u_{k-1}\|_{L^{2}L^{2}(X_{U})}\Bigr{)}\|\partial Z^{\leq
60}u_{k}\|_{L^{\infty}L^{2}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\times\sup_{U}\Bigl{(}\|\langle
r\rangle^{-\frac{1}{2}}\langle
t-r\rangle^{-\frac{1}{2}}(\partial_{t}+\partial_{r})(rZ^{\leq
60}u_{k})\|_{L^{2}L^{2}(X_{U})}\Bigr{)}\\\ &\lesssim(\log\langle
T\rangle)^{2}VIII_{k-1}\cdot I_{k}\cdot VII_{k}.\end{split}$

For the seventh term in (2.9), we apply (3.1) (and (3.3)), which yields

$\int\int_{C^{R}_{\tau}}r\Bigl{(}|(\partial_{t}+\partial_{r})\partial^{\leq
1}u_{k-1}|+\frac{|\partial^{\leq 1}u_{k-1}|}{r}\Bigr{)}|\partial Z^{\leq
60}u_{k}|^{2}\,dx\,dt\\\ \lesssim\Bigl{(}\|{\not\negmedspace\partial}Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+R\|{\not\negmedspace\partial}\partial
Z^{\leq 4}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+\|r^{-1}Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}\Bigr{)}\\\ \times\|\langle
r\rangle^{-\frac{1}{2}}\partial Z^{\leq
60}u_{k}\|^{2}_{L^{2}L^{2}(C^{R}_{\tau})}.$

On the $C^{U}_{\tau}$ regions, in addition to (3.2), we will also apply (3.6)
to the term that already contains a good derivative, which gives

$\int\int_{C^{U}_{\tau}}r\Bigl{(}|(\partial_{t}+\partial_{r})\partial^{\leq
1}u_{k-1}|+\frac{|\partial^{\leq 1}u_{k-1}|}{r}\Bigr{)}|\partial Z^{\leq
60}u_{k}|^{2}\,dx\,dt\\\
\lesssim\Bigl{(}\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
5}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|(\partial_{t}+\partial_{r})(rZ^{\leq
4}u_{k-1})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}}\|\partial(\partial_{t}+\partial_{r})(rZ^{\leq
4}u_{k-1})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}\\\ \times\|\langle
r\rangle^{-\frac{1}{2}}\partial Z^{\leq
60}u_{k}\|^{2}_{L^{2}L^{2}(C^{U}_{\tau})}.$

Using (1.9), note that

$\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}}\|\partial(\partial_{t}+\partial_{r})(rZ^{\leq
4}u_{k-1})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\lesssim\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}}\|\partial
Z^{\leq
4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+U^{\frac{1}{2}}\tau^{\frac{1}{2}}\|{\not\negmedspace\partial}\partial
Z^{\leq 4}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}.$

Thus, upon summing and using (4.12), we obtain

(4.52) $\int_{0}^{T}\int r\Bigl{(}|(\partial_{t}+\partial_{r})\partial^{\leq
1}u_{k-1}|+\frac{|\partial^{\leq 1}u_{k-1}|}{r}\Bigr{)}|\partial Z^{\leq
60}u_{k}|^{2}\,dx\,dt\\\ \lesssim(III_{k-1}+X_{k-1}+IV_{k-1})(\log\langle
T\rangle)^{\frac{1}{2}}II_{k}^{2}\\\ +\Bigl{[}VII_{k-1}+(\log\langle
T\rangle)\Bigl{(}VIII_{k-1}+XI_{k-1}+XII_{k-1}\Bigr{)}\Bigr{]}\log\langle
T\rangle II_{k}^{2}.$

The desired bound (4.16) now follows from (2.9) and the application of (1.5),
(4.18), (4.19), (4.49), (4.50), (4.51), (4.52), and (4.6).

$\mathbf{[IX_{k}]}$: As $R<\tau/2$ on each $C^{R}_{\tau}$, (3.5) allows us to
see that

(4.53) $IX_{k}^{2}\lesssim\sum_{\tau\leq
T}\sum_{R\leq\tau/4}\Bigl{(}\frac{1}{R^{2}}\|Z^{\leq
51}u_{k}\|^{2}_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
50}u_{k}\|^{2}_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}\Bigr{)}\lesssim
III_{k}^{2}+IV_{k}^{2}.$

And, thus, the appropriate bound is a consequence of (4.16).

$\mathbf{[X_{k}]}$: Using (1.8) and (3.5), when $R<\tau/2$, we see that

(4.54) $\begin{split}R\|{\not\negmedspace\partial}\partial Z^{\leq
40}u_{k}\|_{L^{2}L^{2}(\tilde{C}^{R}_{\tau})}&\lesssim\|\partial Z^{\leq
41}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}+R\|(\partial_{t}+\partial_{r})(\partial_{t}-\partial_{r})Z^{\leq
40}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\\\ &\lesssim\|\partial Z^{\leq
41}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}+R\|\Box Z^{\leq
40}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}.\end{split}$

On ${\tilde{C}}^{R}_{\tau}$, (4.54) is akin to the bounds of [10] and [22]. We
note that

(4.55) $|\Box Z^{\leq 40}u_{k}|\lesssim\Bigl{(}|\partial Z^{\leq
20}u_{k-1}|+|\partial Z^{\leq 21}u_{k}|\Bigr{)}|Z^{\leq 41}u_{k-1}|+|Z^{\leq
21}u_{k-1}|\Bigl{(}|\partial Z^{\leq 40}u_{k-1}|+|\partial Z^{\leq
41}u_{k}|\Bigr{)}.$

Applying (3.1), we obtain

$R\|Z^{\leq 40}\Box u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\lesssim
R^{-1}\|Z^{\leq
41}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\Bigl{(}\|\partial Z^{\leq
24}u_{k-1}\|_{L^{2}L^{2}({\tilde{\tilde{C}}}^{R}_{\tau})}+R\|{\not\negmedspace\partial}\partial
Z^{\leq 23}u_{k-1}\|_{L^{2}L^{2}({\tilde{\tilde{C}}}^{R}_{\tau})}\Bigr{)}\\\
+R^{-1}\|Z^{\leq
41}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\Bigl{(}\|\partial Z^{\leq
25}u_{k}\|_{L^{2}L^{2}({\tilde{\tilde{C}}}^{R}_{\tau})}+R\|{\not\negmedspace\partial}\partial
Z^{\leq 24}u_{k}\|_{L^{2}L^{2}({\tilde{\tilde{C}}}^{R}_{\tau})}\Bigr{)}\\\
+\Bigl{(}R^{-1}\|Z^{\leq
25}u_{k-1}\|_{L^{2}L^{2}({\tilde{\tilde{C}}}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
24}u_{k-1}\|_{L^{2}L^{2}({\tilde{\tilde{C}}}^{R}_{\tau})}\Bigr{)}\Bigl{(}\|\partial
Z^{\leq 40}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}+\|\partial Z^{\leq
41}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\Bigr{)}.$

Combining this with (4.54), we see that

(4.56) $X_{k}\lesssim IX_{k}+IV_{k-1}\Bigl{(}IX_{k-1}+X_{k-1}+\log\langle
T\rangle(XI_{k-1}+XII_{k-1})\Bigr{)}\\\
+IV_{k-1}\Bigl{(}IX_{k}+X_{k}+\log\langle
T\rangle(XI_{k}+XII_{k})\Bigr{)}+\Bigl{(}IV_{k-1}+III_{k-1}\Bigr{)}\Bigl{(}IX_{k-1}+IX_{k}\Bigr{)}.$

We note that the occurrences of terms $XI$ and $XII$ are due to the
enlargement of ${\tilde{C}}^{R}_{\tau}$ when $R=\tau/4$ and $R=\tau/2$. The
tails here can be bounded using the $C^{U}_{\tau}$ terms when $U=\tau/4$. By
(4.16), these terms are also bounded by the right side of (4.3).

$\mathbf{[XI_{k}]}$: For $U=1$, we have an immediate bound

$\sum_{\tau}\frac{1}{\tau(\log\langle\tau\rangle)^{2}}\|\partial Z^{\leq
50}u_{k}\|^{2}_{L^{2}L^{2}({\tilde{C}}^{U=1}_{\tau})}\lesssim\sum_{\tau}\frac{1}{(\log\langle\tau\rangle)^{2}}\|\partial
Z^{\leq 50}u_{k}\|^{2}_{L^{\infty}_{t}L^{2}_{x}}\leq I^{2}_{k}.$

For $U>1$, we can refer to (1.8) and (3.5) to see that

$\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}\log\langle\tau\rangle}\|\partial
Z^{\leq
50}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\leq\|r^{-\frac{1}{2}}\langle
t-r\rangle^{-\frac{1}{2}}(\log\langle r\rangle)^{-1}Z^{\leq
51}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\\\ +\|r^{-\frac{1}{2}}\langle
t-r\rangle^{-\frac{1}{2}}(\partial_{t}+\partial_{r})(rZ^{\leq
50}u_{k})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}.$

And thus,

(4.57) $XI_{k}\lesssim I_{k}+VIII_{k}+VII_{k}.$

Then (4.6) and (4.16) give the boundedness in terms of the right side of
(4.3).

$\mathbf{[XII_{k}]}$: Using (1.8), we first note that

$\displaystyle\sum_{\tau}\frac{\tau}{(\log\langle\tau\rangle)^{2}}\|{\not\negmedspace\partial}\partial
Z^{\leq 40}$ $\displaystyle
u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{U=1}_{\tau})}^{2}$
$\displaystyle\lesssim\sum_{\tau}\frac{1}{\tau(\log\langle\tau\rangle)^{2}}\Bigl{(}\|Z^{\leq
42}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{U=1}_{\tau})}^{2}+\|(\partial_{t}+\partial_{r})(rZ^{\leq
41}u_{k})\|^{2}_{L^{2}L^{2}({\tilde{C}}^{U=1}_{\tau})}\Bigr{)}$
$\displaystyle\lesssim VIII_{k}^{2}+VII_{k}^{2}.$

And for $1<U\leq\tau/4$, (1.8) and (3.5) give

$\displaystyle
U^{\frac{1}{2}}\tau^{\frac{1}{2}}\|{\not\negmedspace\partial}\partial Z^{\leq
40}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}$
$\displaystyle\lesssim\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}}\|\partial
Z^{\leq
41}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+U^{\frac{1}{2}}\tau^{\frac{1}{2}}\|(\partial_{t}+\partial_{r})(\partial_{t}-\partial_{r})Z^{\leq
40}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}$
$\displaystyle\lesssim\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}}\|\partial
Z^{\leq
41}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+U^{\frac{1}{2}}\tau^{\frac{1}{2}}\|\Box
Z^{\leq 40}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}.$

Applying (3.2) to the lower order terms of (4.55) gives

$U^{\frac{1}{2}}\tau^{\frac{1}{2}}\|\Box Z^{\leq
40}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\\\
\lesssim\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
41}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigl{(}\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}}\|\partial
Z^{\leq
24}u_{k-1}\|_{L^{2}L^{2}({\tilde{\tilde{C}}}^{U}_{\tau})}+U^{\frac{1}{2}}\tau^{\frac{1}{2}}\|{\not\negmedspace\partial}\partial
Z^{\leq 23}u_{k-1}\|_{L^{2}L^{2}({\tilde{\tilde{C}}}^{U}_{\tau})}\Bigr{)}\\\
+\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\|Z^{\leq
41}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigl{(}\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}}\|\partial
Z^{\leq
25}u_{k}\|_{L^{2}L^{2}({\tilde{\tilde{C}}}^{U}_{\tau})}+U^{\frac{1}{2}}\tau^{\frac{1}{2}}\|{\not\negmedspace\partial}\partial
Z^{\leq 24}u_{k}\|_{L^{2}L^{2}({\tilde{\tilde{C}}}^{U}_{\tau})}\Bigr{)}\\\
+\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\Bigl{(}\|Z^{\leq
25}u_{k-1}\|_{L^{2}L^{2}({\tilde{\tilde{C}}}^{U}_{\tau})}+\|(\partial_{t}+\partial_{r})(rZ^{\leq
24}u_{k-1})\|_{L^{2}L^{2}({\tilde{\tilde{C}}}^{U}_{\tau})}\Bigr{)}\\\
\times\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}}\Bigl{(}\|\partial Z^{\leq
40}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\|\partial Z^{\leq
41}u_{k}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}.$

From this, it follows that

(4.58) $XII_{k}\lesssim
VIII_{k}+VII_{k}+XI_{k}+VIII_{k-1}\Bigl{(}IX_{k-1}+X_{k-1}+\log\langle
T\rangle\Bigl{(}XI_{k-1}+XII_{k-1}\Bigr{)}\Bigr{)}\\\
+VIII_{k-1}\Bigl{(}IX_{k}+X_{k}+\log\langle
T\rangle\Bigl{(}XI_{k}+XII_{k}\Bigr{)}\Bigr{)}+\Bigl{(}\log\langle T\rangle
VIII_{k-1}+VII_{k-1}\Bigr{)}\Bigl{(}XI_{k-1}+XI_{k}\Bigr{)}.$

The proof of (4.3) is now complete as we can apply (4.16) and (4.57).

###  4.2. Convergence

We now conclude the proof by showing that the sequence is Cauchy. This is
somewhat simplified by the extremely high regularity that was used in the
preceding subsection. We set

(4.59) $A_{k}=\|\partial Z^{\leq
20}(u_{k}-u_{k-1})\|_{L^{\infty}L^{2}}+\|Z^{\leq
20}(u_{k}-u_{k-1})\|_{LE^{1}},$

and we shall show that for each $k$,

(4.60) $A_{k}\leq\frac{1}{2}A_{k-1},$

which suffices to complete the proof.

We use that

(4.61) $\Box u_{k}-\Box
u_{k-1}=a^{I,\alpha}_{JK}(u_{k-1}^{J}-u_{k-2}^{J})\partial_{\alpha}u_{k-1}^{K}+a^{I,\alpha}_{JK}u_{k-2}^{J}\partial_{\alpha}(u_{k-1}^{K}-u_{k-2}^{K})\\\
+b^{I,\alpha\beta}_{JK}\partial_{\alpha}(u_{k-1}^{J}-u_{k-2}^{J})\partial_{\beta}u_{k-1}^{K}+b^{I,\alpha\beta}_{JK}\partial_{\alpha}u_{k-2}^{J}\partial_{\beta}(u_{k-1}^{K}-u_{k-2}^{K})\\\
+A^{I,\alpha\beta}_{JK}u_{k-1}^{K}\partial_{\alpha}\partial_{\beta}(u_{k}^{J}-u_{k-1}^{J})+A^{I,\alpha\beta}_{JK}(u_{k-1}^{K}-u_{k-2}^{K})\partial_{\alpha}\partial_{\beta}u_{k-1}^{J}\\\
+B^{I,\alpha\beta\gamma}_{JK}\partial_{\gamma}u_{k-1}^{K}\partial_{\alpha}\partial_{\beta}(u^{J}_{k}-u^{J}_{k-1})+B^{I,\alpha\beta\gamma}_{JK}\partial_{\gamma}(u_{k-1}^{K}-u_{k-2}^{K})\partial_{\alpha}\partial_{\beta}u_{k-1}^{J}.$

With $h$ as in (4.5), we apply (2.2). Arguing as in (4.9), though we need not
take as much care to distinguish the lower order terms from the higher order
terms, this gives that

(4.62) $A_{k}^{2}\lesssim\int_{0}^{T}\int|\partial^{\leq 1}Z^{\leq
20}(u_{k-1}-u_{k-2})||\partial Z^{\leq 21}u_{k-1}|\Bigl{(}|\partial Z^{\leq
20}(u_{k}-u_{k-1})|+\frac{|Z^{\leq 20}(u_{k}-u_{k-1})|}{r}\Bigr{)}\,dx\,dt\\\
+\int_{0}^{T}\int|\partial^{\leq 1}Z^{\leq 20}u_{k-2}||\partial Z^{\leq
20}(u_{k-1}-u_{k-2})|\Bigl{(}|\partial Z^{\leq
20}(u_{k}-u_{k-1})|+\frac{|Z^{\leq 20}(u_{k}-u_{k-1})|}{r}\Bigr{)}\,dx\,dt\\\
+\int_{0}^{T}\int|\partial^{\leq 1}Z^{\leq 20}u_{k-1}||\partial Z^{\leq
20}(u_{k}-u_{k-1})|\Bigl{(}|\partial Z^{\leq
20}(u_{k}-u_{k-1})|+\frac{|Z^{\leq 20}(u_{k}-u_{k-1})|}{r}\Bigr{)}\,dx\,dt.$

Applying (3.1) and (3.3), we obtain

$\int\int_{C^{R}_{\tau}}|\partial^{\leq 1}Z^{\leq
20}(u_{k-1}-u_{k-2})||\partial Z^{\leq 21}u_{k-1}|\Bigl{(}|\partial Z^{\leq
20}(u_{k}-u_{k-1})|+\frac{|Z^{\leq 20}(u_{k}-u_{k-1})|}{r}\Bigr{)}\,dx\,dt\\\
\lesssim\Bigl{(}\|\partial Z^{\leq
25}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}+R\|{\not\negmedspace\partial}\partial
Z^{\leq 24}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\Bigr{)}\|\langle
r\rangle^{-\frac{3}{2}}\partial^{\leq 1}Z^{\leq
20}(u_{k-1}-u_{k-2})\|_{L^{2}L^{2}(C^{R}_{\tau})}\\\ \times\Bigl{(}\|\langle
r\rangle^{-\frac{1}{2}}\partial Z^{\leq
20}(u_{k}-u_{k-1})\|_{L^{2}L^{2}(C^{R}_{\tau})}+\|\langle
r\rangle^{-\frac{1}{2}}r^{-1}Z^{\leq
20}(u_{k}-u_{k-1})\|_{L^{2}L^{2}(C^{R}_{\tau})}\Bigr{)}.$

And (3.2) gives

$\int\int_{C^{U}_{\tau}}|\partial^{\leq 1}Z^{\leq
20}(u_{k-1}-u_{k-2})||\partial Z^{\leq 21}u_{k-1}|\Bigl{(}|\partial Z^{\leq
20}(u_{k}-u_{k-1})|+\frac{|Z^{\leq 20}(u_{k}-u_{k-1})|}{r}\Bigr{)}\,dx\,dt\\\
\lesssim\frac{U^{\frac{1}{2}}}{\tau^{\frac{1}{2}}}\Bigl{(}\|\partial Z^{\leq
25}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\tau\|{\not\negmedspace\partial}\partial
Z^{\leq 24}u_{k-1}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}\|\langle
t-r\rangle^{-1}\langle r\rangle^{-\frac{1}{2}}\partial^{\leq 1}Z^{\leq
20}(u_{k-1}-u_{k-2})\|_{L^{2}L^{2}(C^{U}_{\tau})}\\\ \times\Bigl{(}\|\langle
r\rangle^{-\frac{1}{2}}\partial Z^{\leq
20}(u_{k}-u_{k-1})\|_{L^{2}L^{2}(C^{U}_{\tau})}+\|\langle
r\rangle^{-\frac{1}{2}}r^{-1}Z^{\leq
20}(u_{k}-u_{k-1})\|_{L^{2}L^{2}(C^{U}_{\tau})}\Bigr{)}.$

We may use (2.11) and (4.12) to see that

$\|\langle t-r\rangle^{-1}\langle r\rangle^{-\frac{1}{2}}\partial^{\leq
1}Z^{\leq 20}(u_{k-1}-u_{k-2})\|_{L^{2}L^{2}}\lesssim(\log\langle
T\rangle)^{\frac{1}{2}}\|Z^{\leq 20}(u_{k-1}-u_{k-2})\|_{LE^{1}}.$

Thus, upon summing over $R\leq\tau/2$, $U\leq\tau/4$ and $\tau\leq T$, we see
that

(4.63) $\int_{0}^{T}\int|\partial^{\leq 1}Z^{\leq
20}(u_{k-1}-u_{k-2})||\partial Z^{\leq 21}u_{k-1}|\Bigl{(}|\partial Z^{\leq
20}(u_{k}-u_{k-1})|+\frac{|Z^{\leq 20}(u_{k}-u_{k-1})|}{r}\Bigr{)}\,dx\,dt\\\
\lesssim(\log\langle T\rangle)^{2}M_{k-1}A_{k-1}A_{k}.$

Applying (3.1) (and (3.3)) and (3.2), respectively, give

$\int\int_{C^{R}_{\tau}}|\partial^{\leq 1}Z^{\leq 20}u_{k-2}||\partial Z^{\leq
20}(u_{k-1}-u_{k-2})|\Bigl{(}|\partial Z^{\leq
20}(u_{k}-u_{k-1})|+\frac{|Z^{\leq 20}(u_{k}-u_{k-1})|}{r}\Bigr{)}\,dx\,dt\\\
\lesssim\Bigl{(}\|r^{-1}Z^{\leq
25}u_{k-2}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}+\|{\not\negmedspace\partial}Z^{\leq
24}u_{k-2}\|_{L^{2}L^{2}({\tilde{C}}^{R}_{\tau})}\Bigr{)}\|\langle
r\rangle^{-\frac{1}{2}}\partial Z^{\leq
20}(u_{k-1}-u_{k-2})\|_{L^{2}L^{2}(C^{R}_{\tau})}\\\ \times\Bigl{(}\|\langle
r\rangle^{-\frac{1}{2}}\partial Z^{\leq
20}(u_{k}-u_{k-1})\|_{L^{2}L^{2}(C^{R}_{\tau})}+\|\langle
r\rangle^{-\frac{1}{2}}r^{-1}Z^{\leq
20}(u_{k}-u_{k-1})\|_{L^{2}L^{2}(C^{R}_{\tau})}\Bigr{)}$

and

$\int\int_{C^{U}_{\tau}}|\partial^{\leq 1}Z^{\leq 20}u_{k-2}||\partial Z^{\leq
20}(u_{k-1}-u_{k-2})|\Bigl{(}|\partial Z^{\leq
20}(u_{k}-u_{k-1})|+\frac{|Z^{\leq 20}(u_{k}-u_{k-1})|}{r}\Bigr{)}\,dx\,dt\\\
\lesssim\frac{1}{U^{\frac{1}{2}}\tau^{\frac{1}{2}}}\Bigl{(}\|Z^{\leq
25}u_{k-2}\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}+\|(\partial_{t}+\partial_{r})(rZ^{\leq
24}u_{k-2})\|_{L^{2}L^{2}({\tilde{C}}^{U}_{\tau})}\Bigr{)}\|\langle
r\rangle^{-\frac{1}{2}}\partial Z^{\leq
20}(u_{k-1}-u_{k-2})\|_{L^{2}L^{2}(C^{U}_{\tau})}\\\ \times\Bigl{(}\|\langle
r\rangle^{-\frac{1}{2}}\partial Z^{\leq
20}(u_{k}-u_{k-1})\|_{L^{2}L^{2}(C^{U}_{\tau})}+\|\langle
r\rangle^{-\frac{1}{2}}r^{-1}Z^{\leq
20}(u_{k}-u_{k-1})\|_{L^{2}L^{2}(C^{U}_{\tau})}\Bigr{)},$

which, using (4.12), imply

(4.64) $\int_{0}^{T}\int|\partial^{\leq 1}Z^{\leq 20}u_{k-2}||\partial Z^{\leq
20}(u_{k-1}-u_{k-2})|\Bigl{(}|\partial Z^{\leq
20}(u_{k}-u_{k-1})|+\frac{|Z^{\leq 20}(u_{k}-u_{k-1})|}{r}\Bigr{)}\,dx\,dt\\\
\lesssim(\log\langle T\rangle)^{2}M_{k-2}A_{k-1}A_{k}.$

The third term in (4.62) is very much of the same from as the preceding term,
and the exact same arguments yield

(4.65) $\int_{0}^{T}\int|\partial^{\leq 1}Z^{\leq 20}u_{k-1}||\partial Z^{\leq
20}(u_{k}-u_{k-1})|\Bigl{(}|\partial Z^{\leq
20}(u_{k}-u_{k-1})|+\frac{|Z^{\leq 20}(u_{k}-u_{k-1})|}{r}\Bigr{)}\,dx\,dt\\\
\lesssim(\log\langle T\rangle)^{2}M_{k-1}A_{k}^{2}.$

It now follows from (4.62), (4.63), (4.64), and (4.65) that

$A_{k}^{2}\lesssim(\log\langle
T\rangle)^{2}(M_{k-1}+M_{k-2})A_{k-1}A_{k}+(\log\langle
T\rangle)^{2}M_{k-1}A_{k}^{2}.$

Using (4.3) and (1.6), provided $c\ll 1$, we may bootstrap and obtain

$A_{k}^{2}\lesssim c^{4}\varepsilon^{\frac{2}{3}}A^{2}_{k-1}.$

Thus, for $\varepsilon$ sufficiently small, we recover (4.60), which implies
that the sequence is Cauchy and thus convergent. This completes the proof.

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