# Nonlinear Landau damping and wave operators in sharp Gevrey spaces

Alexandru D. Ionescu  Princeton University<EMAIL_ADDRESS>,
Benoit Pausader  Brown University<EMAIL_ADDRESS>,  Xuecheng Wang
YMSC, Tsinghua University & BIMSA<EMAIL_ADDRESS>and  Klaus
Widmayer  University of Zurich and University of Vienna
<EMAIL_ADDRESS>
###### Abstract.

We prove nonlinear Landau damping in optimal weighted Gevrey-3 spaces for
solutions of the confined Vlasov-Poisson system on
$\mathbb{T}^{d}\times\mathbb{R}^{d}$ which are small perturbations of
homogeneous Penrose-stable equilibria.

We also prove the existence of nonlinear scattering operators associated to
the confined Vlasov-Poisson evolution, as well as suitable injectivity
properties and Lipschitz estimates (also in weighted Gevrey-3 spaces) on these
operators.

Our results give definitive answers to two well-known open problems in the
field, both of them stated in the recent review of Bedrossian [4, Section 6].

A.I. was supported in part by NSF grant DMS-2007008; B.P. was supported in
part by NSF grant DMS-2154162; X.W. was supported in part by NSFC-12141102,
and MOST-2020YFA0713003. K.W. gratefully acknowledges support of the SNSF
through grant PCEFP2_203059.

###### Contents

  1. 1 Introduction
  2. 2 The main equations and the bootstrap proposition
  3. 3 Proof of Proposition 2.2
  4. 4 Proof of Proposition 2.3
  5. 5 Proofs of the main theorems
  6. 6 Proof of Lemma 2.1

##  1\. Introduction

In this article we investigate asymptotic properties of solutions of the
classical Vlasov-Poisson system. In the confined case the electron
distribution is modeled by a function
$F=F(t,x,v):I\times\mathbb{T}^{d}_{x}\times\mathbb{R}^{d}_{v}\to\mathbb{R}$,
where $I\subseteq\mathbb{R}$ is an interval,
$\mathbb{T}:=\mathbb{R}/(2\pi\mathbb{Z})$, and $d\geq 1$, evolving according
to the nondimensionalized Vlasov-Poisson system

$\begin{split}\left(\partial_{t}+v\cdot\nabla_{x}\right)F+\nabla_{x}\phi\cdot\nabla_{v}F=0,\qquad\Delta_{x}\phi=\rho=\int_{\mathbb{R}^{d}}Fdv-1.\end{split}$
(1.1)

Our goal is to investigate the global nonlinear stability properties (Landau
damping) of solutions around a spatially homogeneous equilibrium
$M_{0}=M_{0}(v)$ normalized such that

$\int_{\mathbb{R}^{d}}M_{0}(v)\,dv=1.$ (1.2)

Looking for solutions of (1.1) of the form $F=M_{0}+f$, we obtain the system

$\begin{split}&\left(\partial_{t}+v\cdot\nabla_{x}\right)f+E\cdot\nabla_{v}f+E\cdot\nabla_{v}M_{0}=0,\\\
&E:=\nabla_{x}\Delta_{x}^{-1}\rho,\qquad\rho(t,x):=\int_{\mathbb{R}^{d}}f(t,x,v)\,dv.\end{split}$
(1.3)

###  1.1. The main theorems

The free transport equation $\partial_{t}f+v\cdot\nabla_{x}f=0$ exhibits phase
mixing, which leads to time decay of the spatial density $\rho$. It was a
fundamental observation of Landau [36] (see also [37, Chapter 3]) that an
interesting mechanism of decay exists also in the linearized Vlasov-Poisson
equations near homogeneous equilibria satisfying certain conditions (nowadays
called ”Penrose criterion” [45], see (1.7) below).

The classical mechanism for nonlinear stability involves trading regularity
for decay. In order to state our main theorems we need some definitions. For
any $\lambda,s\in(0,1]$ we define the Gevrey spaces
$\mathcal{G}^{\lambda,s}(\mathbb{T}^{d}\times\mathbb{R}^{d})$ induced by the
norms

$\|g\|_{\mathcal{G}^{\lambda,s}(\mathbb{T}^{d}\times\mathbb{R}^{d})}:=\big{\|}\widehat{g}(k,\xi)e^{\lambda\langle
k,\xi\rangle^{s}}\big{\|}_{L^{2}_{k,\xi}},$ (1.4)

where $\langle k,\xi\rangle:=\sqrt{1+|k|^{2}+|\xi|^{2}}$ and
$\widehat{g}:\mathbb{Z}^{d}\times\mathbb{R}^{d}\to\mathbb{C}$ denotes the
Fourier transform of the function $g$,

$\widehat{g}(k,\xi):=\int_{\mathbb{T}^{d}\times\mathbb{R}^{d}}g(x,v)e^{-ik\cdot
x}e^{-i\xi\cdot v}\,dxdv.$

For $n\in\mathbb{Z}_{+}$ we define also the weighted Gevrey spaces
$\mathcal{G}^{\lambda,s}_{n}(\mathbb{T}^{d}\times\mathbb{R}^{d})$ induced by
the norms

$\|g(x,v)\|_{\mathcal{G}^{\lambda,s}_{n}(\mathbb{T}^{d}\times\mathbb{R}^{d})}:=\sum_{|a|\leq
n}\|D^{a}_{\xi}\widehat{g}(k,\xi)e^{\lambda\langle
k,\xi\rangle^{s}}\|_{\mathcal{G}^{\lambda,s}(\mathbb{T}^{d}\times\mathbb{R}^{d})}.$
(1.5)

We also define the spaces $\mathcal{G}^{\lambda,s}(\mathbb{T}^{d})$,
$\mathcal{G}^{\lambda,s}(\mathbb{R}^{d})$, and
$\mathcal{G}^{\lambda,s}_{n}(\mathbb{R}^{d})$ in a similar fashion. For
simplicity of notation, we sometimes let
$\mathcal{G}^{\lambda,s}_{n}:=\mathcal{G}^{\lambda,s}_{n}(\mathbb{T}^{d}\times\mathbb{R}^{d})$.

We assume that there are constants $\lambda_{0},\vartheta\in(0,1]$ such that
the homogenous equilibrium $M_{0}$ satisfies the smoothness and decay bounds

$\sup_{\xi\in\mathbb{R}^{d}}e^{\lambda_{0}\langle\xi\rangle^{1/3}}\big{\\{}\big{|}\widehat{M_{0}}(\xi)\big{|}+|\xi|\,\big{|}\nabla_{\xi}\widehat{M}_{0}(\xi)\big{|}\big{\\}}+\|\nabla_{v}M_{0}\|_{\mathcal{G}^{\lambda_{0},1/3}_{d^{\prime}}(\mathbb{R}^{d})}\leq\vartheta^{-1},$
(1.6)

where $d^{\prime}:=\lfloor d/2+1\rfloor$, and the Penrose criterion

$\inf_{k\in\mathbb{Z}^{d}\setminus\\{0\\},\,\tau\in\mathbb{C},\,\Im(\tau)\leq
0}\Big{|}1+\int_{0}^{\infty}e^{-i\tau
s}s\widehat{M_{0}}(sk)\,ds\Big{|}\geq\vartheta>0.$ (1.7)

To state our main results it is convenient to define the kinetic profile

$g(t,x,v):=f(t,x+tv,v).$ (1.8)

In terms of the profile $g$, the Vlasov-Poisson system (1.3) becomes

$\begin{split}&\partial_{t}g(t,x,v)+E(t,x+tv)\cdot\nabla_{v}M_{0}(v)+E(t,x+tv)\cdot(\nabla_{v}-t\nabla_{x})g(t,x,v)=0,\\\
&E:=\nabla_{x}\Delta_{x}^{-1}\rho,\qquad\rho(t,x):=\int_{\mathbb{R}^{d}}g(t,x-tv,v)\,dv.\end{split}$
(1.9)

Our first main theorem concerns the global nonlinear stability of solutions of
the Vlasov-Poisson system (1.3).

######  Theorem 1.1.

(Nonlinear Landau damping) Assume $d\geq 1$, $\lambda_{0},\vartheta\in(0,1]$,
and $M_{0}$ is a homogeneous equilibrium satisfying the bounds (1.6)–(1.7).

(i) There is a constant
$\overline{\kappa}=\overline{\kappa}(d,\lambda_{0},\vartheta)>0$ with the
property that if $f_{0}$ is a Gevrey-smooth, small, and neutral perturbation
satisfying

$\|f_{0}\|_{\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3}(\mathbb{T}^{d}\times\mathbb{R}^{d})}\leq\kappa_{0}\leq\overline{\kappa},\qquad\int_{\mathbb{T}^{d}\times\mathbb{R}^{d}}f_{0}(x,v)\,dxdv=0$
(1.10)

then there is a unique global solution $g\in
C([0,\infty):\mathcal{G}_{d^{\prime}}^{3\lambda_{0}/4,1/3})$ of the equation
(1.3) with initial data $f(0)=f_{0}$. Moreover
$\int_{\mathbb{T}^{d}\times\mathbb{R}^{d}}f(t,x,v)\,dxdv=0$ for any
$t\in[0,\infty)$, and there is a function
$g_{\infty}\in\mathcal{G}_{d^{\prime}}^{3\lambda_{0}/4,1/3}$ (the final state)
such that

$\|\rho(t)\|_{\mathcal{G}^{\lambda_{0}/2,1/3}(\mathbb{T}^{d})}+\|g(t)-g_{\infty}\|_{\mathcal{G}^{\lambda_{0}/2,1/3}_{d^{\prime}}(\mathbb{T}^{d}\times\mathbb{R}^{d})}\leq
A\kappa_{0}e^{-\lambda_{0}\langle t\rangle^{1/3}/4}$ (1.11)

for any $t\in[0,\infty)$, where
$A=A(d,\lambda_{0},\vartheta)\in[1,\overline{\kappa}^{-1}]$ is a constant,
$\rho$ denotes the macroscopic density defined in (1.3), and $g$ denotes the
profile defined in (1.8).

(ii) The solution map $f_{0}\to g$ is a continuous map from
$B_{\overline{\kappa}}(\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3})$ to
$C\big{(}[0,\infty):\mathcal{G}_{d^{\prime}}^{3\lambda_{0}/4,1/3}\big{)}$. In
fact, for any $f_{01},f_{02}\in
B_{\overline{\kappa}}(\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3})$ and
$t\in[0,\infty)$ we have the Lipschitz estimates

$A^{-1}\|f_{01}-f_{02}\|_{\mathcal{G}_{d^{\prime}}^{\lambda_{0}/2,1/3}}\leq\|(g_{1}-g_{2})(t)\|_{\mathcal{G}_{d^{\prime}}^{3\lambda_{0}/4,1/3}}\leq
A\|f_{01}-f_{02}\|_{\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3}}.$ (1.12)

In particular, the final state operator $\mathcal{S}_{+}(f_{0}):=g_{\infty}$
is a Lipschitz continuous and injective map from
$B_{\overline{\kappa}}(\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3})$ to
$\mathcal{G}_{d^{\prime}}^{3\lambda_{0}/4,1/3}$.

Our second main theorem concerns the solvability of the final state problem,
and the construction of injective wave operators satisfying Lipschitz bounds.

######  Theorem 1.2.

(The final state problem) Assume $d$, $\lambda_{0}$, $\vartheta$, and $M_{0}$
are as in Theorem 1.1. Then there is a constant
$\overline{\kappa}^{\prime}=\overline{\kappa}^{\prime}(d,\lambda_{0},\vartheta)>0$
with the property that if $g_{\infty}$ satisfies

$\|g_{\infty}\|_{\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3}}\leq\overline{\kappa}^{\prime},\qquad\int_{\mathbb{T}^{d}\times\mathbb{R}^{d}}g_{\infty}(x,v)\,dxdv=0$
(1.13)

then there is a unique solution $g\in
C([0,\infty):\mathcal{G}_{d^{\prime}}^{3\lambda_{0}/4,1/3})$ of the system
(1.9) satisfying

$\lim_{t\to\infty}\|g(t)-g_{\infty}\|_{\mathcal{G}_{d^{\prime}}^{3\lambda_{0}/4,1/3}}=0,\qquad\int_{\mathbb{T}^{d}\times\mathbb{R}^{d}}g(t,x,v)\,dxdv=0.$
(1.14)

Moreover, for any $g_{\infty 1},g_{\infty 2}\in
B_{\overline{\kappa}^{\prime}}(\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3})$
and $t\in[0,\infty)$ we have the bi-Lipschitz estimates

$(A^{\prime})^{-1}\|g_{\infty 1}-g_{\infty
2}\|_{\mathcal{G}_{d^{\prime}}^{\lambda_{0}/2,1/3}}\leq\|(g_{1}-g_{2})(t)\|_{\mathcal{G}_{d^{\prime}}^{3\lambda_{0}/4,1/3}}\leq
A^{\prime}\|g_{\infty 1}-g_{\infty
2}\|_{\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3}},$ (1.15)

for some constant
$A^{\prime}=A^{\prime}(d,\lambda_{0},\vartheta)\in[1,1/\overline{\kappa}^{\prime}]$.
In particular, the wave operator $\mathcal{W}_{+}(g_{\infty}):=g(0)$ is a
Lipschitz continuous and injective map from
$B_{\overline{\kappa}^{\prime}}(\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3})$
to $\mathcal{G}_{d^{\prime}}^{3\lambda_{0}/4,1/3}$.

Finally, we can combine Theorems 1.1 and 1.2 to prove the existence of
nonlinear scattering operators associated to the confined Vlasov-Poisson
system. More precisely:

######  Theorem 1.3.

(The scattering operator) Assume $d$, $\lambda_{0}$, $\vartheta$, and $M_{0}$
are as in Theorem 1.1. Then there is a constant
$\overline{\kappa}^{\prime\prime}=\overline{\kappa}^{\prime\prime}(d,\lambda_{0},\vartheta)>0$
with the property that if $g_{-\infty}$ satisfies

$\|g_{-\infty}\|_{\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3}}\leq\overline{\kappa}^{\prime\prime},\qquad\int_{\mathbb{T}^{d}\times\mathbb{R}^{d}}g_{-\infty}(x,v)\,dxdv=0$
(1.16)

then there is a unique neutral solution $g\in
C(\mathbb{R}:\mathcal{G}_{d^{\prime}}^{\lambda_{0}/2,1/3})$ of the system
(1.9) satisfying

$\lim_{t\to-\infty}g(t)=g_{-\infty}\text{ in
}\mathcal{G}_{d^{\prime}}^{\lambda_{0}/2,1/3}\qquad\text{ and
}\qquad\lim_{t\to\infty}g(t)=g_{\infty}\text{ in
}\mathcal{G}_{d^{\prime}}^{\lambda_{0}/2,1/3}.$ (1.17)

Moreover the scattering operator $\mathcal{S}(g_{-\infty}):=g_{\infty}$ is a
Lipschitz continuous and injective map from
$B_{\overline{\kappa}^{\prime\prime}}(\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3})$
to $\mathcal{G}_{d^{\prime}}^{\lambda_{0}/2,1/3}$. In fact, for any
$g_{-\infty 1},g_{-\infty 2}\in
B_{\overline{\kappa}^{\prime\prime}}(\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3})$
we have

$(A^{\prime\prime})^{-1}\|g_{-\infty 1}-g_{-\infty
2}\|_{\mathcal{G}_{d^{\prime}}^{\lambda_{0}/4,1/3}}\leq\|g_{\infty
1}-g_{\infty 2}\|_{\mathcal{G}_{d^{\prime}}^{\lambda_{0}/2,1/3}}\leq
A^{\prime\prime}\|g_{-\infty 1}-g_{-\infty
2}\|_{\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3}},$ (1.18)

for some constant
$A^{\prime\prime}=A^{\prime\prime}(d,\lambda_{0},\vartheta)\in[1,1/\overline{\kappa}^{\prime\prime}]$.

We conclude this subsection with some remarks:

######  Remark 1.4.

Our main theorems provide definitive answers to two of the open problems
stated in the recent review article of Bedrossian [4, Section 6]:

(a) nonlinear Landau damping for initial perturbations in sharp weighted
Gevrey-3 spaces, which was already conjectured by Mouhot-Villani [41, Section
13];

(b) injectivity and Lipschitz bounds on the scattering operator
$g_{-\infty}\mapsto g_{\infty}$ (and similar properties for the final state
operator $f_{0}\mapsto g_{\infty}$ and the wave operator $g_{\infty}\mapsto
g(0)$).

######  Remark 1.5.

We also allow a larger class of equilibria $M_{0}$ than in the earlier work,
in the same weighted Gevrey-3 space as the initial perturbation. For
comparison, all the earlier proofs of nonlinear Landau damping in the confined
case (in [41, 5, 22]) require analytic equilibria. Our assumptions (1.6) also
allow certain slowly decreasing equilibria, like the Poisson equilibrium
corresponding to $\widehat{M_{0}}(\xi)=e^{-|\xi|}$ in all dimensions $d\geq
1$, which were not included in the earlier nonlinear theorems. The main point
is that we can still prove sufficiently strong bounds on the associated
Green’s function (as stated in (2.11) below) under the weaker and more natural
assumptions (1.6) on $M_{0}$. We provide all the details in section 6.

We notice also that we could allow non-neutral perturbations in all the
theorems, by modifying suitably the homogeneous equilibrium $M_{0}$ around
which we perturb.

######  Remark 1.6.

We note that there is exponential loss of derivative during the nonlinear
flow. For example, in Theorem 1.1 the initial data is assumed to be in the
stronger Gevrey space $\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3}$, while the
bounds on the solution are stated in the weaker Gevrey space
$\mathcal{G}_{d^{\prime}}^{\lambda_{0}/2,1/3}$. The precise amount of
derivative loss can be reduced (for example from
$\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3}$ to
$\mathcal{G}_{d^{\prime}}^{0.99\lambda_{0},1/3}$, with small adjustments in
the proof), but some loss is necessary to get time decay of macroscopic
quantities. This is an important conceptual feature of Landau damping.

######  Remark 1.7.

The global well-posedness theory of the final state problem and the
construction of scattering operators (Theorems 1.2 and 1.3) are new in the
context of asymptotic stability of the Vlasov-Poisson system near homogeneous
equilibria. To summarize, we prove that 3 nonlinear operators, the final state
operator $f_{0}\mapsto g_{\infty}$, the wave operator $g_{\infty}\mapsto
g(0)$, and the scattering operator $g_{-\infty}\mapsto g_{\infty}$ are well
defined, Lipschitz continuous (between suitable spaces, with derivative loss,
see (1.12), (1.15), (1.18)) and injective (in the quantitative sense given by
the lower bounds in these inequalities).

###  1.2. Linear and nonlinear Landau damping

Landau damping is one of the most fundamental phenomena in Plasma Physics, and
goes back to foundational work of Landau [36] in 1946 on the linearized
Vlasov-Poisson equations on $\mathbb{R}^{3}\times\mathbb{R}^{3}$ near
Maxwellian equilibria. Nowadays, Landau damping is a term that is broadly used
to denote a mechanism of decay of certain averaged quantities, as well as
asymptotic stability. It took more than 60 years to develop a first
understanding of Landau damping at the nonlinear level, for the full Vlasov-
Poisson system (1.3). This concerns the confined periodic setting
$x\in\mathbb{T}^{d}$, where the decay of macroscopic quantities (such as the
density $\rho$ and the electric field $E$) occurs due to phase mixing and
relies mainly on the high regularity of the initial data (see also [39]).
After some preliminary works [11, 30], nonlinear Landau damping was proved in
pioneering work by Mouhot–Villani [41], who showed that small and analytic
perturbations of homogeneous equilibria $M_{0}$ that satisfy the Penrose
criterion (1.7) lead to global solutions which scatter linearly, with density
functions that decay exponentially in time. The theory has been refined
significantly by Bedrossian-Masmoudi-Mouhot [5] and then Grenier-Nguyen-
Rodnianski [22], who expanded the class of acceptable perturbations to the
more flexible class of Gevrey spaces
$\mathcal{G}_{d^{\prime}}^{\lambda_{0},s}$, in the almost optimal range
$s>1/3$, and simplified the argument substantially by using energy estimates
with suitably defined weights instead of a global Newton scheme.

Our Theorem 1.1 above represents a sharp endpoint result for this classical
line of research on Landau damping in the confined case. Stability in Gevrey-3
spaces was already conjectured in [41, Section 13], and it is known by now
that these spaces are best possible, due to the recent results in [3, 4, 52].
The theorem also applies to the vacuum case $M_{0}\equiv 0$; the normalization
(1.2) is only needed to pass from (1.1) to (1.3).

A second contribution of this paper is the development of a new and complete
scattering theory for the Vlasov-Poisson system in the confined case around
homogeneous equilibria, in Theorems 1.2 and 1.3, also in optimal weighted
Gevrey-3 spaces. The existence of the scattering operator (or even just the
injectivity of the final state operator $f_{0}\mapsto g_{\infty}$ proved in
Theorem 1.1) has a natural physical interpretation: despite its name, “Landau
damping” is different from other types of damping (for example damping of
solutions of parabolic equations) in the sense that there is no loss of
information during the nonlinear evolution in passing from the initial data
$f_{0}$ to the final state $g_{\infty}$. The “damping” takes place only at the
level of macroscopic quantities, like the density, consistent with the time-
reversibility of the system.

In view of these results Landau damping is well understood by now in the
confined case. We remark, however, that nonlinear Landau damping around
homogeneous equilibria remains a major open problem in the _unconfined case_ ,
particularly in 3 dimensions $x\in\mathbb{R}^{3}$, both for the Vlasov-Poisson
system and for the related relativistic Vlasov-Maxwell system. A key
difference is that the Penrose condition (1.7) cannot hold uniformly as the
frequency $\tau$ goes to $0$; as a result the Green’s function cannot have
fast decay and macroscopic quantities like the density and the electric field
can only decay slowly, at dimension-dependent polynomial rates in $\langle
t\rangle^{-1}$, which are not sufficient in dimension $d=3$. Linear stability
results are known in these cases (see, for example, [26, 7, 34, 27] for recent
work in this direction), indicating the possibility of nonlinear stability,
but the only nonlinear result so far was proved recently by the authors in
[33] for the Poisson equilibrium.

We conclude this subsection with a short and (far from exhaustive) discussion
on related topics, and some additional references:

$\bullet\,\,$ Global existence of solutions of the Vlasov-Poisson system is
classical in low dimensions, see [1, 47, 40]. We refer to [4] for a survey of
recent results. The global dynamics of solutions, including the presence of
logarithmically modified characteristics and modified scattering, is also well
understood in the case of small initial data in $\mathbb{R}^{3}$ (which
corresponds to the nonlinear stability of vacuum) using various methods, see
[13, 50, 51, 32].

$\bullet\,\,$ The construction of nonlinear wave operators and the scattering
theory is a natural problem that has been extensively investigated for many
dispersive and hyperbolic equations. In the context of collisionless kinetic
equations, it has only been developed recently starting with the construction
of the wave operator and the scattering map for the Vlasov-Poisson system near
vacuum on $\mathbb{R}^{3}\times\mathbb{R}^{3}$ in [16]. See also the more
recent results in [8, 9, 10, 42, 49].

$\bullet\,\,$ In some variations of the Vlasov-Poisson system associated to
the ion dynamics, one considers screened interactions. In this case, the
dynamics of low-frequencies is suppressed, which leads to favorable estimates
and asymptotic stability [18], also in the unconfined setting in 3D (see [6,
25]) and even in 2D (see [28]), See also [2, 12, 14, 15, 29, 38, 39, 43, 44]
for other stability results in related kinetic models.

$\bullet\,\,$ Other classical plasma models include the hydrodynamic models of
Euler-Poisson and Euler-Maxwell type, where some stability results are known
in the unconfined Euclidean case (see [23, 24]), but no global stability
results are known in the confined case.

###  1.3. Main ideas

Our proofs are based on what we call the $Z$-norm method. This is a general
method that has been used extensively by the authors to prove global stability
of various quasi-linear dispersive and hyperbolic equations, such as water
waves, plasma models, and the Einstein equations of General Relativity. At the
abstract level, it consists of proving simultaneous bootstrap control of two
types of quantities:

(1) Energy functionals at the highest order of derivatives, sometimes
including vector-fields that commute with the linearized equation;

(2) A special $Z$-norm (usually not based on $L^{2}$ spaces) which provides
stronger control of the solutions, including time decay, but at a lower level
of derivatives.

In our specific case, to prove Theorem 1.1 we use energy functionals defined
in the Fourier space, of the form

$\mathcal{E}^{0}_{g}(t):=\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}A(t,k,\xi)^{2}\Big{\\{}\sum_{|a|\leq
d^{\prime}}\big{|}D^{a}_{\xi}\widehat{g}(t,k,\xi)\big{|}^{2}\Big{\\}}\,d\xi,$
(1.19)

where $g$ is the profile of the solution and
$A:[0,\infty)\times\mathbb{Z}^{d}\times\mathbb{R}^{d}\to[1,\infty)$ is a
family of suitable weights. The precise choice of the weights
$A(t,k,\xi)=e^{\lambda(t,|k,\xi|)}$ is very important, see the discussion in
subsection 1.4 below. These weights are decreasing in time, to reflect the
”sliding regularity” basic mechanism of the proof, and increasing in frequency
at a rate consistent with Gevrey-3 spaces, $\lambda(t,r)\approx\langle
r\rangle^{1/3}$.

We then define a matching $Z$-norm, which controls the macroscopic density, by
the formula

$\mathcal{Z}^{0}_{g}(t):=\sup_{k\in\mathbb{Z}^{d}\setminus\\{0\\}}A(t,k,tk)\big{|}\widehat{g}(t,k,tk)\big{|}|k|^{-1/2}=\sup_{k\in\mathbb{Z}^{d}\setminus\\{0\\}}A(t,k,tk)\big{|}\widehat{\rho}(t,k)\big{|}|k|^{-1/2}.$
(1.20)

This type of $Z$-norms (weighted $L^{\infty}$ norms in the Fourier space) have
been used extensively by the authors, starting with the work [35] on gravity
water waves in 2D. A similar norm was also used in [22], with a different
weight $A$.

The basic idea is to prove uniform bootstrap control of the form

$\mathcal{E}^{0}_{g}(t)+\big{[}\mathcal{Z}_{g}^{0}(t)\langle
t\rangle^{6d}\big{]}^{2}\lesssim\kappa_{0}^{2},$ (1.21)

for solutions $g$ of the system (1.9), where $\kappa_{0}$ denotes the size of
the initial data. Notice the basic interplay of the proof: the energy norm is
strong, but does not decay in time, while the $Z$-norm is comparatively weaker
in terms of regularity (due to the factor of $|k|^{-1/2}$ in the definition),
but is able to capture the time decay of the macroscopic density. The precise
powers $1/2$ and $6d$ are related in the proof, but there is some flexibility
in their choice.

The proof of the bootstrap estimates (1.21) consists of two main steps. First
we use the density equation

$\widehat{\rho}(t,k)+\int_{0}^{t}\widehat{\rho}(s,k)\widehat{M_{0}}((t-s)k)(t-s)\,ds=\widehat{\mathcal{N}}(t,k),$
(1.22)

where $\mathcal{N}$ is a suitable nonlinearity, together with bounds on
Green’s function associated with the equilibrium $M_{0}$, to control the
$Z$-norm of the solution. Then we use the basic equation (1.9) for $g$, in the
Fourier space form (2.1), to prove uniform control on the functional
$\mathcal{E}^{0}_{g}$.

###  1.4. New weights

The main novelty in our construction, which allows us to reach the sharp
Gevrey-3 regularity, is the choice of the weights $A$. The standard choice
used in [41, 5, 22] involves weights having favorable product structure of the
form

$A(t,k,\xi)=e^{\lambda(t,|k,\xi|)},\qquad\lambda(t,r)=\lambda_{1}(t)\lambda_{2}(r),$
(1.23)

for suitable functions $\lambda_{1}$ (decreasing) and $\lambda_{2}$
(increasing). This product structure easily leads to bilinear estimates of the
form

$A(t,k_{1}+k_{2},\xi_{1}+\xi_{2})\lesssim
A(t,k_{1},\xi_{1})A(t,k_{2},\xi_{2})\\{(1+|k,\xi|)^{-4}+(1+|k^{\prime},\xi^{\prime}|)^{-4}\\},$
(1.24)

for any $t,k,k^{\prime},\xi,\xi^{\prime}$, which play a crucial role in
nonlinear analysis. Weights of this type have been used in [5] and [22] (in
the context of so-called ”generator functions”) to prove Landau damping in
Gevrey spaces $\mathcal{G}_{d^{\prime}}^{\lambda_{0},s}$ in the full
subcritical range $s>1/3$, but appear to be insufficient to deal with the
critical case $s=1/3$.

Our main new idea is to work with a more general class of weights, of the form

$A(t,k,\xi)=e^{\lambda(t,|k,\xi|)},\qquad\lambda(t,r)=[\lambda_{1}+\delta(1+t)^{-\delta}]\langle
r\rangle^{1/3}+\delta\big{(}1+t\langle r\rangle^{-2/3}\big{)}^{-\delta}\langle
r\rangle^{1/3},$ (1.25)

where $\lambda_{1}\in[0.6\lambda_{0},0.9\lambda_{0}]$ and
$\delta=\lambda_{0}/200$. These weights do not have the basic product
structure (1.23), due to the last term, but we can still prove favorable
bilinear estimates like (1.24) (in the stronger form given in (2.32)), as well
as suitable commutator estimates like (2.33), which are needed to prevent
derivative loss. The key point is that we can choose the last term in the
definition (1.25) in a way that improves the analysis of the density equation;
for example we are able to prove uniform estimates like

$\sum_{a\in\mathbb{Z}^{d},\,|a|\leq|k|/8}\int_{0.99t}^{t}\frac{|t-s|A(t,k,tk)}{A(s,k,tk)}e^{-\delta(|a|^{1/3}+|sa+(t-s)k|^{1/3})}\,ds\lesssim
1,$ (1.26)

for any $t\in[10,\infty)$ and $k\in\mathbb{Z}^{d}\setminus\\{0\\}$ (see
(3.15)), which ultimately allow us to close the main bootstrap argument at the
level of the sharp Gevrey-3 spaces.

###  1.5. The final state problem

At the quantitative level, the estimates we need to prove Theorem 1.2 are
similar to the estimates in the initial data problem, using a combination of
an energy functional and a $Z$-norm. However we need a new family of weights
$A^{\sharp}(t,k,\xi)=e^{\lambda^{\sharp}(t,|k,\xi|)}:[0,\infty)\times\mathbb{Z}^{d}\times\mathbb{R}^{d}\to[1,\infty)$,
which are now increasing in time to indicate that the solution comes from
infinity and loses regularity as $t$ decreases. We also need a new density
equation, of the form

$\widehat{\rho}(t,k)-\int_{t}^{\infty}\widehat{\rho}(s,k)\widehat{M_{0}}((t-s)k)(t-s)\,ds=\widehat{\mathcal{N}^{\prime}}(t,k),$
(1.27)

for a suitable nonlinearity $\mathcal{N}^{\prime}$, which replaces the
equation (1.22) and gives us a basic formula for the density $\rho$ at time
$t$ in terms of $\rho$ at later times.

To construct the solution itself we construct a family of solutions $g_{n}$ of
the system (1.9) on the time interval $[0,n]$, with data
$g_{n}(n)=g_{\infty}$. Then we prove estimates on the differences of these
solutions, and find the full solution $g$ as the limit as $n\to\infty$ of
$g_{n}$. This construction is possible as long as we allow partial loss of
regularity, as claimed in Theorem 1.2, and provided that we control very well
differences of solutions.

###  1.6. Organization

The rest of the paper is organized as follows: in section 2 we derive our main
equations for the profile $g$ and the density $\rho$, define our two families
of weights and prove some of their basic properties, and state our two main
bootstrap propositions, Propositions 2.2 and 2.3. Then we prove these
propositions in sections 3 and 4 respectively. In section 5 we show how to use
these bootstrap propositions to complete the proofs of the main Theorems 1.1,
1.2, and 1.3. Finally, in section 6 we prove Lemma 2.1, which provides
suitable bounds on the Green’s function associated to the equilibrium $M_{0}$.

##  2\. The main equations and the bootstrap proposition

Recall the main equations (1.9) for the kinetic profile $g$. We assume that
$T_{1}\leq T_{2}\in[0,\infty)$ and $g\in
C([T_{1},T_{2}]:\mathcal{G}^{\lambda_{0}/2,1/3}_{d^{\prime}}(\mathbb{T}^{d}\times\mathbb{R}^{d}))$
is a solution of the system (1.9), satisfying the neutrality condition
$\int_{\mathbb{T}^{d}\times\mathbb{R}^{d}}g(t,x,v)\,dxdv=0$. Taking the
Fourier transform in both $x$ and $v$ variables we derive the equations

$\begin{split}&\partial_{t}\widehat{g}(t,k,\xi)+\widehat{\rho}(t,k)\widehat{M_{0}}(\xi-
tk)\frac{k\cdot(\xi-
tk)}{|k|^{2}}+\frac{1}{(2\pi)^{d}}\sum_{l\in\mathbb{Z}^{d}}\widehat{\rho}(t,l)\widehat{g}(t,k-l,\xi-
tl)\frac{l\cdot(\xi-tk)}{|l|^{2}}=0,\\\
&\widehat{\rho}(t,k):=\widehat{g}(t,k,tk).\end{split}$ (2.1)

###  2.1. The initial data problem

It follows from (2.1) that

$\begin{split}\widehat{g}(t,k,\xi)-\widehat{g}(T_{1},k,\xi)&+\int_{T_{1}}^{t}\widehat{\rho}(s,k)\widehat{M_{0}}(\xi-
sk)\frac{k\cdot(\xi-sk)}{|k|^{2}}\,ds\\\
&+\frac{1}{(2\pi)^{d}}\sum_{l\in\mathbb{Z}^{d}}\int_{T_{1}}^{t}\widehat{\rho}(s,l)\widehat{g}(s,k-l,\xi-
sl)\frac{l\cdot(\xi-sk)}{|l|^{2}}\,ds=0.\end{split}$

Letting $\xi=tk$ it follows that

$\begin{split}&\widehat{\rho}(t,k)+\int_{T_{1}}^{t}\widehat{\rho}(s,k)\widehat{M_{0}}((t-s)k)(t-s)\,ds=\widehat{\mathcal{N}}(t,k),\\\
&\widehat{\mathcal{N}}(t,k):=\widehat{g}(T_{1},k,tk)-\frac{1}{(2\pi)^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{T_{1}}^{t}\widehat{\rho}(s,l)\widehat{g}(s,k-l,tk-
sl)\frac{(t-s)l\cdot k}{|l|^{2}}\,ds.\end{split}$ (2.2)

We would like to use the Volterra equation (2.2) and the main assumptions
(1.6)–(1.7) to prove an important formula for $\rho$. We extend
$\widehat{\rho}$ to a function on $\mathbb{R}\times\mathbb{Z}^{d}$ supported
in $[T_{1},T_{2}]\times\mathbb{Z}^{d}$, and extend the function
$\widehat{\mathcal{N}}$ according to the identity in the first line of (2.2)
(clearly $\widehat{\mathcal{N}}(t,k)=0$ if $t<T_{1}$). Then we define, for
$\tau\in\mathbb{R}$ and $k\in\mathbb{Z}^{d}$,

$\widetilde{\rho}(\tau,k):=\int_{\mathbb{R}}\widehat{\rho}(t,k)e^{-it\tau}\,dt,\qquad\widetilde{\mathcal{N}}(\tau,k):=\int_{\mathbb{R}}\widehat{\mathcal{N}}(t,k)e^{-it\tau}\,dt.$
(2.3)

For $\tau\in\mathbb{H}_{-}:=\\{\tau\in\mathbb{C}:\,\Im\tau\leq 0\\}$ and
$k\in\mathbb{Z}^{d}\setminus\\{0\\}$ let

$L(\tau,k):=\int_{0}^{\infty}e^{-i\tau s}s\widehat{M_{0}}(sk)\,ds.$ (2.4)

In view of (1.6), for any $k\in\mathbb{Z}^{d}\setminus\\{0\\}$ the function
$L(.,k)$ is well-defined and continuous in $\mathbb{H}_{-}$, and analytic in
the open half-plane
$\mathbb{H}^{\circ}_{-}:=\\{\tau\in\mathbb{C}:\,\Im\tau<0\\}$.

It follows from the identity in the first line of (2.2) that

$\widetilde{\rho}(\tau,k)\big{(}1+L(\tau,k)\big{)}=\widetilde{\mathcal{N}}(\tau,k),\qquad\tau\in\mathbb{R},\,k\in\mathbb{Z}^{d}\setminus\\{0\\},$
(2.5)

where the function $L$ is defined as in (2.4). For $\tau\in\mathbb{H}_{-}$ and
$k\in\mathbb{Z}^{d}\setminus\\{0\\}$ let

$L^{\prime}(\tau,k):=\frac{L(\tau,k)}{1+L(\tau,k)}.$ (2.6)

In view of the Penrose condition (1.7), the function
$L^{\prime}:\mathbb{H}_{-}\times(\mathbb{Z}^{d}\setminus\\{0\\})\to\mathbb{C}$
is well-defined and continuous in $\mathbb{H}_{-}$, and analytic in
$\mathbb{H}^{\circ}_{-}$. Moreover, using (2.5),

$\widetilde{\rho}(\tau,k)=\widetilde{\mathcal{N}}(\tau,k)-\widetilde{\mathcal{N}}(\tau,k)L^{\prime}(\tau,k),\qquad\tau\in\mathbb{R},\,k\in\mathbb{Z}^{d}\setminus\\{0\\}.$
(2.7)

We take the inverse Fourier transform in $\tau$ to conclude that

$\begin{split}&\widehat{\rho}(t,k)=\widehat{\mathcal{N}}(t,k)-\int_{\mathbb{R}}\widehat{\mathcal{N}}(t-s,k)\widehat{G}(s,k)\,ds\quad\text{
where
}\quad\widehat{G}(s,k):=\frac{1}{2\pi}\int_{\mathbb{R}}e^{is\tau}L^{\prime}(\tau,k)\,d\tau.\end{split}$
(2.8)

Our next lemma provides estimates on the functions $L,L^{\prime}$ and the
Green’s function $\widehat{G}$, which show in particular that the formal
calculations above are justified.

######  Lemma 2.1.

(i) The functions
$L,L^{\prime}:\mathbb{H}_{-}\times(\mathbb{Z}^{d}\setminus\\{0\\})\to\mathbb{C}$
defined in (2.4) and (2.6) are well-defined and continuous in
$\mathbb{H}_{-}$, analytic in $\mathbb{H}^{\circ}_{-}$, and satisfy the bounds

$\begin{split}\big{\|}\langle\alpha\rangle
L(\alpha+i\gamma,k)\big{\|}_{L^{2}_{\alpha}}+\big{\|}\langle\alpha\rangle
L^{\prime}(\alpha+i\gamma,k)\big{\|}_{L^{2}_{\alpha}}&\leq
C_{0}^{\ast}|k|^{-1/2},\\\ \big{\|}\langle\alpha\rangle
D^{a}_{\alpha}L(\alpha+i\gamma,k)\big{\|}_{L^{\infty}_{\alpha}}+\big{\|}\langle\alpha\rangle
D^{a}_{\alpha}L^{\prime}(\alpha+i\gamma,k)\big{\|}_{L^{\infty}_{\alpha}}&\leq
C_{0}^{\ast}\frac{(3a)!(1.01)^{a}}{(|k|\lambda_{0}^{3})^{a+1}},\end{split}$
(2.9)

for any $k\in\mathbb{Z}^{d}\setminus\\{0\\}$, $\gamma\in(-\infty,0]$, and
$a\in\mathbb{Z}_{+}$. The constant $C_{0}^{\ast}$ may depend only on the
parameters $d$, $\lambda_{0}$, and $\vartheta$.

(ii) With $\widehat{\rho}$ and $\widehat{\mathcal{N}}$ as in (2.2), for any
$t\in\mathbb{R}$ and $k\in\mathbb{Z}^{d}\setminus\\{0\\}$ we have

$\widehat{\rho}(t,k)=\widehat{\mathcal{N}}(t,k)-\int_{T_{1}}^{t}\widehat{\mathcal{N}}(s,k)\widehat{G}(t-s,k)\,ds$
(2.10)

where the Green’s function
$\widehat{G}:\mathbb{R}\times(\mathbb{Z}^{d}\setminus\\{0\\})\to\mathbb{C}$
defined as in (2.8) is supported in
$[0,\infty)\times(\mathbb{Z}^{d}\setminus\\{0\\})$ and satisfies the uniform
bounds

$\big{|}\widehat{G}(s,k)\big{|}\leq
C_{0}^{\ast}e^{-0.95\lambda_{0}|sk|^{1/3}}.$ (2.11)

We provide a self-contained proof of this lemma in section 6.

####  2.1.1. Energy functionals and the first bootstrap proposition

The main idea is to estimate the increment of suitable functionals, which are
defined using special weights. For $t,r\in\mathbb{R}_{+}$,
$k,\xi\in\mathbb{R}^{d}$ we define

$\begin{split}&\lambda(t,r):=\lambda_{1}\langle
r\rangle^{1/3}+\delta(1+t)^{-\delta}\langle
r\rangle^{1/3}+\delta\big{(}1+t\langle r\rangle^{-2/3}\big{)}^{-\delta}\langle
r\rangle^{1/3},\\\ &A(t,k,\xi):=e^{\lambda(t,|k,\xi|)},\end{split}$ (2.12)

where $\lambda_{1}\in[0.5\lambda_{0},0.9\lambda_{0}]$,
$\delta:=\lambda_{0}/200$ and $\langle w\rangle:=(1+|w|^{2})^{1/2}$,
$w\in\mathbb{R}^{n}$.

The precise definition of the weight $A$ is important; some properties are
proved in section 2.3 below. Notice that the weights $\lambda$ and $A$ are
decreasing in $t$.

Our first bootstrap argument is based on controlling simultaneously an energy
functional for the profile $g$ and a decaying norm for the density $\rho$. Let
$d^{\prime}=\lfloor d/2+1\rfloor$ and define the spaces

$\begin{split}&X_{[T_{1},T_{2}]}:=\Big{\\{}h\in
C([T_{1},T_{2}]:\mathcal{G}^{\lambda_{0}/2,1/3}_{d^{\prime}}(\mathbb{T}^{d}\times\mathbb{R}^{d})):\\\
&\quad\|h\|_{X_{[T_{1},T_{2}]}}:=\sup_{t\in[T_{1},T_{2}]}\Big{\\{}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}A(t,k,\xi)^{2}\Big{[}\sum_{|a|\leq
d^{\prime}}\big{|}D^{a}_{\xi}\widehat{h}(t,k,\xi)\big{|}^{2}\Big{]}\,d\xi\Big{\\}}^{1/2}<\infty\Big{\\}},\end{split}$
(2.13)

for $T_{1}\leq T_{2}\in[0,\infty)$. For any $h\in X_{[T_{1},T_{2}]}$ and
$t\in[T_{1},T_{2}]$, $\beta\in[0,1]$, and $p\in[0,2]$ we define

$\begin{split}\mathcal{E}^{p}_{h}(t)&:=\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-2p}A(t,k,\xi)^{2}\Big{\\{}\sum_{|a|\leq
d^{\prime}}\big{|}D^{a}_{\xi}\widehat{h}(t,k,\xi)\big{|}^{2}\Big{\\}}\,d\xi,\\\
\mathcal{Z}^{p}_{h}(t)&:=\sup_{k\in\mathbb{Z}^{d}\setminus\\{0\\}}\langle
k,tk\rangle^{-p}A(t,k,tk)\big{|}\widehat{h}(t,k,tk)\big{|}|k|^{-\beta}.\end{split}$
(2.14)

There are two parameters in these definitions: the parameter $\beta$ is taken
$\beta=1/2$ in the proof, and it indicates the fact that the $Z$-norms of
solutions are measured at lower order of differentiability than the energy
norms. The parameter $p$ is taken either $p=0$ to estimate the solutions
themselves or $p=2$ to estimate the difference of solutions (due to some loss
of symmetry, the difference of solutions can only be estimated in slightly
weaker norms).

Our first main bootstrap proposition is the following:

######  Proposition 2.2.

Assume $T_{1}\leq T_{2}\in[0,\infty)$ and $g_{1},g_{2}\in X_{[T_{1},T_{2}]}$
are real-valued solutions of the system (1.9) satisfying
$\widehat{g_{i}}(t,0,0)=0$ for any $t\in[T_{1},T_{2}]$ and $i\in\\{1,2\\}$.
Assume that $\epsilon_{0}\leq\overline{\epsilon}\ll 1$,
$\lambda_{1}\in[0.5\lambda_{0},0.9\lambda_{0}]$, and

$\|g_{i}(T_{1})\|_{{\mathcal{G}_{d^{\prime}}^{\lambda_{1}+4\delta,1/3}}}\leq\overline{\epsilon},\qquad\|(g_{1}-g_{2})(T_{1})\|_{\mathcal{G}_{d^{\prime}}^{\lambda_{1}+4\delta,1/3}}\leq\epsilon_{0},$
(2.15)

for $i\in\\{1,2\\}$. Let $\beta=1/2$ and assume that for any
$t\in[T_{1},T_{2}]$ and $i\in\\{1,2\\}$

$\begin{split}&\mathcal{E}^{0}_{g_{i}}(t)+\big{[}\mathcal{Z}_{g_{i}}^{0}(t)\langle
t\rangle^{6d}\big{]}^{2}\leq 4\overline{C}^{2}\overline{\epsilon}^{2},\\\
&\mathcal{E}^{2}_{g_{1}-g_{2}}(t)+\big{[}\mathcal{Z}_{g_{1}-g_{2}}^{2}(t)\langle
t\rangle^{6d}\big{]}^{2}\leq 4\overline{C}^{2}\epsilon_{0}^{2},\end{split}$
(2.16)

where $\overline{C}\leq\overline{\epsilon}^{-1/8}$ is a sufficiently large
constant that depends only on the structural constants
$d,\lambda_{0},\vartheta$. Then for any $t\in[T_{1},T_{2}]$ and
$i\in\\{1,2\\}$ we have the improved bounds

$\begin{split}&\mathcal{E}^{0}_{g_{i}}(t)+\big{[}\mathcal{Z}_{g_{i}}^{0}(t)\langle
t\rangle^{6d}\big{]}^{2}\leq\overline{C}^{2}\overline{\epsilon}^{2},\\\
&\mathcal{E}_{g_{1}-g_{2}}^{2}(t)+\big{[}\mathcal{Z}_{g_{1}-g_{2}}^{2}(t)\langle
t\rangle^{6d}\big{]}^{2}\leq\overline{C}^{2}\epsilon_{0}^{2}.\end{split}$
(2.17)

This proposition plays a key role in the proof of Theorem 1.1 (i); we will
prove it in section 3 below. We note that the constant $\overline{C}$ will be
taken much larger than other structural constants such as the constants
$C_{0}^{\ast}$ in Lemma 2.1, $C_{0}$ in Lemma 2.5, $C_{1}$ in Lemma 3.1, and
$C_{2}$ in Proposition 3.2.

###  2.2. The final state problem

We also prove quantitative estimates going backwards in time. Assume that
$T_{1}\leq T_{2}\in[0,\infty)$ and $g\in
C([T_{1},T_{2}]:\mathcal{G}_{d^{\prime}}^{\lambda_{0}/2,1/3}(\mathbb{T}^{d}\times\mathbb{R}^{d}))$
is a solution of the system (1.9). For any $t\in[T_{1},T_{2}]$ we use (2.1)
and integrate for $s\in[t,T_{2}]$, thus

$\begin{split}-\widehat{g}(t,k,\xi)+\widehat{g}(T_{2},k,\xi)&+\int_{t}^{T_{2}}\widehat{\rho}(s,k)\widehat{M_{0}}(\xi-
sk)\frac{k\cdot(\xi-sk)}{|k|^{2}}\,ds\\\
&+\frac{1}{(2\pi)^{d}}\sum_{l\in\mathbb{Z}^{d}}\int_{t}^{T_{2}}\widehat{\rho}(s,l)\widehat{g}(s,k-l,\xi-
sl)\frac{l\cdot(\xi-sk)}{|l|^{2}}\,ds=0.\end{split}$

Letting $\xi=tk$ it follows that

$\begin{split}&\widehat{\rho}(t,k)-\int_{t}^{T_{2}}\widehat{\rho}(s,k)\widehat{M_{0}}((t-s)k)(t-s)\,ds=\widehat{\mathcal{N}^{\prime}}(t,k),\\\
&\widehat{\mathcal{N}^{\prime}}(t,k):=\widehat{g}(T_{2},k,tk)+\frac{1}{(2\pi)^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{t}^{T_{2}}\widehat{\rho}(s,l)\widehat{g}(s,k-l,tk-
sl)\frac{(t-s)l\cdot k}{|l|^{2}}\,ds.\end{split}$ (2.18)

We would like to find an integral formula for $\rho$ in terms of
$\mathcal{N}^{\prime}$. For this we define $\rho_{1}(t)=\rho(t)$ for
$t\in[T_{1},T_{2}]$ and $\rho_{1}(t)=0$ if $t\notin[T_{1},T_{2}]$, and then we
define $\mathcal{N}^{\prime}_{1}$ by

$\widehat{\mathcal{N}^{\prime}_{1}}(t,k):=\widehat{\rho_{1}}(t,k)-\int_{t}^{\infty}\widehat{\rho_{1}}(s,k)\widehat{M_{0}}((t-s)k)(t-s)\,ds.$
(2.19)

For $\tau\in\mathbb{R}$ and $k\in\mathbb{Z}^{d}\setminus\\{0\\}$ we define the
time Fourier transforms

$\widetilde{\rho_{1}}(\tau,k):=\int_{\mathbb{R}}\widehat{\rho_{1}}(t,k)e^{-it\tau}\,dt,\qquad\widetilde{\mathcal{N}^{\prime}_{1}}(\tau,k):=\int_{\mathbb{R}}\widehat{\mathcal{N}^{\prime}_{1}}(t,k)e^{-it\tau}\,dt.$

It follows from (2.19) that for any $\tau\in\mathbb{R}$ and
$k\in\mathbb{Z}^{d}\setminus\\{0\\}$ we have

$\widetilde{\mathcal{N}^{\prime}_{1}}(\tau,k)=\widetilde{\rho_{1}}(\tau,k)(1+L(-\tau,-k)),$

where $L$ is defined as in (2.4). Therefore

$\widetilde{\rho_{1}}(\tau,k)=\widetilde{\mathcal{N}^{\prime}_{1}}(\tau,k)-\widetilde{\mathcal{N}^{\prime}_{1}}(\tau,k)L^{\prime}(-\tau,-k),\qquad\tau\in\mathbb{R},\,k\in\mathbb{Z}^{d}\setminus\\{0\\},$

where $L^{\prime}$ is defined as in (2.6). We take the inverse Fourier
transform and recall the definition (2.8) of the Green’s functions
$\widehat{G}$, so

$\widehat{\rho_{1}}(t,k)=\widehat{\mathcal{N}^{\prime}_{1}}(t,k)-\int_{\mathbb{R}}\widehat{\mathcal{N}^{\prime}_{1}}(s,k)\widehat{G}(s-t,-k)\,ds.$

Recall that $\widehat{G}$ is supported in
$[0,\infty)\times(\mathbb{Z}^{d}\setminus\\{0\\})$ (see Lemma 2.1),
$\rho_{1}(t)=\rho(t)$, $\mathcal{N}^{\prime}_{1}(t)=\mathcal{N}^{\prime}(t)$
for $t\in[T_{1},T_{2}]$, and $\mathcal{N}^{\prime}_{1}(t)=0$ for $t>T_{2}$. We
can thus derive our main identity

$\widehat{\rho}(t,k)=\widehat{\mathcal{N}^{\prime}}(t,k)-\int_{t}^{T_{2}}\widehat{\mathcal{N}^{\prime}}(s,k)\widehat{G}(s-t,-k)\,ds\qquad\text{
for }t\in[T_{1},T_{2}]\text{ and }k\in\mathbb{Z}^{d}\setminus\\{0\\}.$ (2.20)

This formula serves as the main substitute of the identity (2.10) in the
analysis of the final state problem.

####  2.2.1. The second bootstrap proposition

To understand the evolution backwards in time we need to work with a new set
of weights, which are increasing in time. We define these weights as in
(2.12): for $t\in[0,\infty)$, $r\in\mathbb{R}_{+}$, $k,\xi\in\mathbb{R}^{d}$
we define

$\begin{split}&\lambda^{\sharp}(t,r):=\lambda_{1}\langle
r\rangle^{1/3}-\delta(1+t)^{-\delta}\langle
r\rangle^{1/3}-\delta\big{(}1+t\langle r\rangle^{-2/3}\big{)}^{-\delta}\langle
r\rangle^{1/3},\\\
&A^{\sharp}(t,k,\xi):=e^{\lambda^{\sharp}(t,|k,\xi|)},\end{split}$ (2.21)

where $\lambda_{1}\in[0.5\lambda_{0},0.9\lambda_{0}]$ and
$\delta=\lambda_{0}/200$. We will use these weights for certain values of
$\lambda_{1}$.

Our second bootstrap argument is also based on controlling simultaneously an
energy functional for $g$ and a decaying norm for $\rho$. As in subsection
2.1.1 we define

$\begin{split}&X^{\sharp}_{[T_{1},T_{2}]}:=\Big{\\{}h\in
C([T_{1},T_{2}]:\mathcal{G}^{\lambda_{0}/2,1/3}_{d^{\prime}}(\mathbb{T}^{d}\times\mathbb{R}^{d})):\\\
&\qquad\|h\|_{X^{\sharp}_{[T_{1},T_{2}]}}:=\sup_{t\in[T_{1},T_{2}]]}\Big{\\{}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}A^{\sharp}(t,k,\xi)^{2}\Big{[}\sum_{|a|\leq
d^{\prime}}\big{|}D^{a}_{\xi}\widehat{h}(t,k,\xi)\big{|}^{2}\Big{]}\,d\xi\Big{\\}}^{1/2}<\infty\Big{\\}}.\end{split}$
(2.22)

For any $h\in X^{\sharp}_{[T_{1},T_{2}]}$, $t\in[T_{1},T_{2}]$, and
$p\in[0,2]$ we define

$\begin{split}\mathcal{E}^{\sharp,p}_{h}(t)&:=\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-2p}A^{\sharp}(t,k,\xi)^{2}\Big{\\{}\sum_{|a|\leq
d^{\prime}}\big{|}D^{a}_{\xi}\widehat{h}(t,k,\xi)\big{|}^{2}\Big{\\}}\,d\xi,\\\
\mathcal{Z}^{\sharp,p}_{h}(t)&:=\sup_{k\in\mathbb{Z}^{d}\setminus\\{0\\}}\langle
k,tk\rangle^{-p}A^{\sharp}(t,k,tk)\big{|}\widehat{h}(t,k,tk)\big{|}|k|^{-1/2}.\end{split}$
(2.23)

Our second main bootstrap proposition is the following:

######  Proposition 2.3.

Assume $T_{1}\leq T_{2}\in[0,\infty)$ and $g_{1},g_{2}\in
X^{\sharp}_{[T_{1},T_{2}]}$ are real-valued solutions of the system (1.9)
satisfying $\widehat{g_{i}}(t,0,0)=0$ for any $t\in[T_{1},T_{2}]$ and
$i\in\\{1,2\\}$. Assume that $\theta_{0}\leq\overline{\theta}\ll 1$ and

$\|g_{i}(T_{2})\|_{{\mathcal{G}_{d^{\prime}}^{\lambda_{1}+4\delta,1/3}}}\leq\overline{\theta},\qquad\|(g_{1}-g_{2})(T_{2})\|_{\mathcal{G}_{d^{\prime}}^{\lambda_{1}+4\delta,1/3}}\leq\theta_{0}.$
(2.24)

for $i\in\\{1,2\\}$, where $\lambda_{1}\in[0.5\lambda_{0},0.9\lambda_{0}]$.
Assume that

$\begin{split}&\mathcal{E}^{\sharp,0}_{g_{i}}(t)+\big{[}\mathcal{Z}_{g_{i}}^{\sharp,0}(t)\langle
t\rangle^{6d}\big{]}^{2}\leq 4\overline{B}^{2}\overline{\theta}^{2},\\\
&\mathcal{E}^{\sharp,2}_{g_{1}-g_{2}}(t)+\big{[}\mathcal{Z}_{g_{1}-g_{2}}^{\sharp,2}(t)\langle
t\rangle^{6d}\big{]}^{2}\leq 4\overline{B}^{2}\theta_{0}^{2},\end{split}$
(2.25)

for any $t\in[T_{1},T_{2}]$ and $i\in\\{1,2\\}$, where
$\overline{B}\leq\overline{\theta}^{-1/8}$ is a sufficiently large constant.
Then for any $t\in[T_{1},T_{2}]$ and $i\in\\{1,2\\}$ we have the improved
bounds

$\begin{split}&\mathcal{E}^{\sharp,0}_{g_{i}}(t)+\big{[}\mathcal{Z}_{g_{i}}^{\sharp,0}(t)\langle
t\rangle^{6d}\big{]}^{2}\leq\overline{B}^{2}\overline{\theta}^{2},\\\
&\mathcal{E}^{\sharp,2}_{g_{1}-g_{2}}(t)+\big{[}\mathcal{Z}_{g_{1}-g_{2}}^{\sharp,2}(t)\langle
t\rangle^{6d}\big{]}^{2}\leq\overline{B}^{2}\theta_{0}^{2}.\end{split}$ (2.26)

###  2.3. Some lemmas

We prove first some properties of the main weights $A$ and $A^{\sharp}$.

######  Lemma 2.4.

(i) Assume that $\lambda_{1}\in[\lambda_{0}/2,\lambda_{0}]$. For any
$t,r\in[0,\infty)$ we have

$\lambda(t,r),\lambda^{\sharp}(t,r)\in[(\lambda_{1}-2\delta)\langle
r\rangle^{1/3},(\lambda_{1}+2\delta)\langle r\rangle^{1/3}],$ (2.27)
$-\partial_{t}\lambda(t,r)=\partial_{t}\lambda^{\sharp}(t,r)=\delta^{2}(1+t)^{-\delta-1}\langle
r\rangle^{1/3}+\delta^{2}\big{(}1+t\langle
r\rangle^{-2/3}\big{)}^{-\delta-1}\langle r\rangle^{-1/3},$ (2.28)
$\begin{split}\partial_{r}\lambda(t,r)&=\big{[}\lambda_{1}+\delta(1+t)^{-\delta}\big{]}(r/3)\langle
r\rangle^{-5/3}\\\ &+\delta\big{(}1+t\langle
r\rangle^{-2/3}\big{)}^{-\delta-1}(r/3)\langle
r\rangle^{-5/3}\big{[}1+t\langle r\rangle^{-2/3}(1+2\delta)\big{]},\\\
\partial_{r}\lambda^{\sharp}(t,r)&=\big{[}\lambda_{1}-\delta(1+t)^{-\delta}\big{]}(r/3)\langle
r\rangle^{-5/3}\\\ &-\delta\big{(}1+t\langle
r\rangle^{-2/3}\big{)}^{-\delta-1}(r/3)\langle
r\rangle^{-5/3}\big{[}1+t\langle r\rangle^{-2/3}(1+2\delta)\big{]},\\\
\end{split}$ (2.29)

In particular the functions $\lambda$ and $\lambda^{\sharp}$ are increasing in
$r$ and

$\partial_{r}\lambda(t,r),\partial_{r}\lambda^{\sharp}(t,r)\in[(\lambda_{1}/3-\delta)r\langle
r\rangle^{-5/3},(\lambda_{1}/3+\delta)r\langle r\rangle^{-5/3}]\,\,\text{ for
any }\,\,t,r\in[0,\infty).$ (2.30)

Moreover, if $a\leq b\in\mathbb{R}_{+}$ and $x,y\in\mathbb{R}_{+}$ then

$\begin{split}&\lambda(a,r)-\lambda(b,r)=\lambda^{\sharp}(b,r)-\lambda^{\sharp}(a,r)\\\
&=\delta^{2}\int_{a}^{b}\big{[}(1+u)^{-\delta-1}\langle
r\rangle^{1/3}+\big{(}1+u\langle r\rangle^{-2/3}\big{)}^{-\delta-1}\langle
r\rangle^{-1/3}\big{]}\,du.\end{split}$ (2.31)

(ii) If $k,\xi,k^{\prime},\xi^{\prime}\in\mathbb{R}^{d}$ and $t\in[0,\infty)$
then

$\begin{split}A(t,k+k^{\prime},\xi+\xi^{\prime})&\leq
A(t,k,\xi)A(t,k^{\prime},\xi^{\prime})e^{-(\lambda_{1}/4)\min(\langle
k,\xi\rangle,\langle k^{\prime},\xi^{\prime}\rangle)^{1/3}},\\\
A^{\sharp}(t,k+k^{\prime},\xi+\xi^{\prime})&\leq
A^{\sharp}(t,k,\xi)A^{\sharp}(t,k^{\prime},\xi^{\prime})e^{-(\lambda_{1}/4)\min(\langle
k,\xi\rangle,\langle k^{\prime},\xi^{\prime}\rangle)^{1/3}}.\end{split}$
(2.32)

Moreover, if $k,\xi,k^{\prime},\xi^{\prime}\in\mathbb{R}^{d}$,
$t\in[0,\infty)$, and $|k^{\prime},\xi^{\prime}|\leq|k,\xi|/8$ then

$\begin{split}\frac{|A(t,k+k^{\prime},\xi+\xi^{\prime})-A(t,k,\xi)|}{A(t,k,\xi)A(t,k^{\prime},\xi^{\prime})}&\lesssim
e^{-(\lambda_{1}/4)\langle k^{\prime},\xi^{\prime}\rangle^{1/3}}\langle
k,\xi\rangle^{-2/3},\\\
\frac{|A^{\sharp}(t,k+k^{\prime},\xi+\xi^{\prime})-A^{\sharp}(t,k,\xi)|}{A^{\sharp}(t,k,\xi)A^{\sharp}(t,k^{\prime},\xi^{\prime})}&\lesssim
e^{-(\lambda_{1}/4)\langle k^{\prime},\xi^{\prime}\rangle^{1/3}}\langle
k,\xi\rangle^{-2/3}.\end{split}$ (2.33)

###### Proof.

(i) The identities and the bounds follow directly from the definitions.

(ii) In view of (2.30), to prove the bounds in the first line of (2.32) it
suffices to show that if $t,y,b\in[0,\infty)$ and $b\leq y$ then

$\lambda(t,y+b)\leq\lambda(t,y)+\lambda(t,b)-(\lambda_{1}/4)\langle
b\rangle^{1/3}.$ (2.34)

To prove this we notice that
$\partial_{r}\lambda(t,r)\leq(\lambda_{1}/2)\langle r\rangle^{-2/3}$ for any
$r\geq 0$, due to (2.30), thus

$\lambda(t,y+b)-\lambda(t,y)\leq(\lambda_{1}/2)\langle
y\rangle^{-2/3}b\leq(\lambda_{1}/2)\langle
b\rangle^{1/3}\leq\lambda(t,b)-(\lambda_{1}/4)\langle b\rangle^{1/3}$

as desired. The bounds in the second line of (2.32) follow in a similar way.

To prove the bounds in the first line of (2.33) it suffices to prove that

$e^{\lambda(t,y+b)-\lambda(t,y)}-1\lesssim e^{(3\lambda_{1}/5)\langle
b\rangle^{1/3}}\langle y\rangle^{-2/3},$ (2.35)

provided that $t,y,b\in[0,\infty)$ and $b\leq y$. This follows easily since
$\lambda(t,y+b)-\lambda(t,y)\leq(2\lambda_{1}/5)\langle y\rangle^{-2/3}b$. The
bounds in the second line of (2.33) follow in a similar way. ∎

We prove now two pointwise estimates in the Fourier space.

######  Lemma 2.5.

Assume $T_{1}\leq T_{2}\in[0,\infty)$, $h_{1}\in X_{[T_{1},T_{2}]}$, $h_{2}\in
X^{\sharp}_{[T_{1},T_{2}]}$. For any $p\in[0,2]$, $t\in[T_{1},T_{2}]$, and
$k\in\mathbb{Z}^{d}$ let

$\begin{split}H_{p}(h_{1})(t,k)&:=\sup_{\xi\in\mathbb{R}^{d}}\big{|}\langle
k,\xi\rangle^{-p}A(t,k,\xi)\widehat{h_{1}}(t,k,\xi)\big{|},\\\
H^{\sharp}_{p}(h_{2})(t,k)&:=\sup_{\xi\in\mathbb{R}^{d}}\big{|}\langle
k,\xi\rangle^{-p}A^{\sharp}(t,k,\xi)\widehat{h_{2}}(t,k,\xi)\big{|}.\end{split}$
(2.36)

Then there is a constant $C_{0}=C_{0}(d)$ such that

$\begin{split}|H_{p}(h_{1})(t,k)|^{2}&\leq
C_{0}^{2}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-2p}A(t,k,\xi)^{2}\Big{\\{}\sum_{|a|\leq
d^{\prime}}\big{|}D^{a}_{\xi}\widehat{h_{1}}(t,k,\xi)\big{|}^{2}\Big{\\}}\,d\xi,\\\
|H_{p}^{\sharp}(h_{2})(t,k)|^{2}&\leq C_{0}^{2}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-2p}A^{\sharp}(t,k,\xi)^{2}\Big{\\{}\sum_{|a|\leq
d^{\prime}}\big{|}D^{a}_{\xi}\widehat{h_{2}}(t,k,\xi)\big{|}^{2}\Big{\\}}\,d\xi.\end{split}$
(2.37)

###### Proof.

By Sobolev embedding,

$|H_{p}(h_{1})(t,k)|^{2}\lesssim_{d}\int_{\mathbb{R}^{d}}\sum_{|a|\leq
d^{\prime}}\big{|}D^{a}_{\xi}[\langle
k,\xi\rangle^{-p}A(t,k,\xi)\widehat{h_{1}}(t,k,\xi)]\big{|}^{2}\,d\xi.$

Using the definition (2.12) it is easy to see that

$|D^{a}_{\xi}A(t,k,\xi)|\lesssim A(t,k,\xi)\qquad\text{ for any
}t\in[T_{1},T_{2}],\,k,\xi\in\mathbb{R}^{d},\,|a|\leq d^{\prime}.$

The bounds in the first line of (2.37) follow. The bounds in the second line
follow similarly. ∎

##  3\. Proof of Proposition 2.2

In this section we prove our first main bootstrap proposition.

###  3.1. Improved control on the $Z$-norm

We prove first the improved estimates (2.17) on the quantities
$\mathcal{Z}_{g_{i}}^{0}$ and $\mathcal{Z}_{g_{1}-g_{2}}^{2}$, using the
identities (2.10). We start with estimates on the nonlinearities
$\mathcal{N}_{1}$ and $\mathcal{N}_{2}$.

######  Lemma 3.1.

Assume that $g_{1},g_{2}\in X_{[T_{1},T_{2}]}$ are solutions satisfying the
hypothesis in Proposition 2.2 (recall $\beta=1/2$), let
$\widehat{\rho_{i}}(t,k):=\widehat{g_{i}}(t,k,tk)$, and define
$\mathcal{N}_{i}$ as in (2.2). Then

$\begin{split}\sup_{t\in[T_{1},T_{2}],\,k\in\mathbb{Z}^{d}\setminus\\{0\\}}A(t,k,tk)|\widehat{\mathcal{N}_{i}}(t,k)||k|^{-\beta}\langle
t\rangle^{6d}&\leq C_{1}\overline{\epsilon},\\\
\sup_{t\in[T_{1},T_{2}],\,k\in\mathbb{Z}^{d}\setminus\\{0\\}}\langle
k,tk\rangle^{-2}A(t,k,tk)|(\widehat{\mathcal{N}_{1}}-\widehat{\mathcal{N}_{2}})(t,k)||k|^{-\beta}\langle
t\rangle^{6d}&\leq C_{1}\epsilon_{0},\end{split}$ (3.1)

for $i\in\\{1,2\\}$ and a constant $C_{1}=C_{1}(d,\lambda_{0})$ sufficiently
large.

###### Proof.

The implied constants in this proof are allowed to depend only on the
parameters $d$ and $\lambda_{0}$. For functions $h_{1}\in
C([T_{1},T_{2}]:\mathcal{G}^{\alpha_{0}/2,1/3}_{d^{\prime}}(\mathbb{T}^{d}\times\mathbb{R}^{d}))$
and $\in
C([T_{1},T_{2}]:\mathcal{G}^{\alpha_{0}/2,1/3}_{d^{\prime}}(\mathbb{T}^{d}))$
we define the operators $\mathcal{A}_{a}h_{1}$, $|a|\leq d^{\prime}$, and
$\mathcal{A}h_{2}$ by multiplication in the Fourier space,

$\widehat{\mathcal{A}_{a}h_{1}}(t,k,\xi):=A(t,k,\xi)D^{a}_{\xi}\widehat{h_{1}}(t,k,\xi),\qquad\widehat{\mathcal{A}h_{2}}(t,k):=A(t,k,tk)\widehat{h_{2}}(t,k,tk).$
(3.2)

We define $\widehat{\rho_{i}}(t,k):=\widehat{g_{i}}(t,k,tk)$,
$\delta\rho:=\rho_{1}-\rho_{2}$, $\delta g:=g_{1}-g_{2}$, and then we define
$\mathcal{A}\rho_{i}$, $\mathcal{A}\delta\rho$, $\mathcal{A}_{a}g_{i}$, and
$\mathcal{A}_{a}\delta g$, $i\in\\{1,2\\}$, by the formulas above. We also
decompose $\mathcal{N}_{i}=\mathcal{N}_{i0}-\mathcal{N}_{i1}$,

$\begin{split}&\widehat{\mathcal{N}_{i0}}(t,k):=\widehat{g_{i}}(T_{1},k,tk),\\\
&\widehat{\mathcal{N}_{i1}}(t,k):=\frac{1}{(2\pi)^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{T_{1}}^{t}\widehat{\rho_{i}}(s,l)\widehat{g_{i}}(s,k-l,tk-
sl)\frac{(t-s)l\cdot k}{|l|^{2}}\,ds.\end{split}$ (3.3)

Using Sobolev embedding (as in Lemma 2.5), for any $k\in\mathbb{Z}^{d}$, and
$t\in[T_{1},T_{2}]$ we have

$\begin{split}e^{2(\lambda_{1}+4\delta)\langle
k,tk\rangle^{1/3}}|\widehat{g_{i}}(T_{1},k,tk)|^{2}&\lesssim\int_{\mathbb{R}^{d}}e^{2(\lambda_{1}+4\delta)\langle
k,\xi\rangle^{1/3}}\Big{\\{}\sum_{|a|\leq
d^{\prime}}\big{|}D^{a}_{\xi}\widehat{g_{i}}(T_{1},k,\xi)\big{|}^{2}\Big{\\}}\,d\xi,\\\
e^{2(\lambda_{1}+4\delta)\langle
k,tk\rangle^{1/3}}|(\widehat{g_{1}}-\widehat{g_{2}})(T_{1},k,tk)|^{2}&\lesssim\int_{\mathbb{R}^{d}}e^{2(\lambda_{1}+4\delta)\langle
k,\xi\rangle^{1/3}}\Big{\\{}\sum_{|a|\leq
d^{\prime}}\big{|}D^{a}_{\xi}(\widehat{g_{1}}-\widehat{g_{2}})(T_{1},k,\xi)\big{|}^{2}\Big{\\}}\,d\xi,\end{split}$

where $i\in\\{1,2\\}$. Moreover, for $k\neq 0$ we have $A(t,k,tk)^{2}\leq
e^{2(\lambda_{1}+4\delta)\langle k,tk\rangle^{1/3}}e^{-4\delta\langle
k,tk\rangle^{1/3}}$. Since $\widehat{g_{i}}(t,0,0)=0$ and using also the
assumption (2.15), we have

$\sum_{k\in\mathbb{Z}^{d}\setminus\\{0\\}}A(t,k,tk)^{2}|\widehat{\mathcal{N}_{i0}}(t,k)|^{2}\leq\sum_{k\in\mathbb{Z}^{d}}e^{2(\lambda_{1}+4\delta)\langle
k,tk\rangle^{1/3}}e^{-4\delta\langle
t\rangle^{1/3}}|\widehat{g_{i}}(T_{1},k,tk)|^{2}\lesssim\overline{\epsilon}^{2}e^{-4\delta\langle
t\rangle^{1/3}}$

and similarly

$\sum_{k\in\mathbb{Z}^{d}\setminus\\{0\\}}A(t,k,tk)^{2}|(\widehat{\mathcal{N}_{10}}-\widehat{\mathcal{N}_{20}})(t,k)|^{2}\lesssim\epsilon_{0}^{2}e^{-4\delta\langle
t\rangle^{1/3}}.$

Therefore, for any $i\in\\{1,2\\}$, $t\in[T_{1},T_{2}]$, and
$k\in\mathbb{Z}^{d}\setminus\\{0\\}$,

$\begin{split}A(t,k,tk)|\widehat{\mathcal{N}_{i0}}(t,k)||k|^{-\beta}\langle
t\rangle^{6d}&\leq C_{1}\overline{\epsilon}/2,\\\
A(t,k,tk)|(\widehat{\mathcal{N}_{10}}-\widehat{\mathcal{N}_{20}})(t,k)||k|^{-\beta}\langle
t\rangle^{6d}&\leq C_{1}\epsilon_{0}/2.\end{split}$

Since $\mathcal{N}_{i}=\mathcal{N}_{i0}-\mathcal{N}_{i1}$, for (3.1) it
remains to prove that for $i\in\\{1,2\\}$

$\begin{split}\sup_{t\in[T_{1},T_{2}],\,k\in\mathbb{Z}^{d}\setminus\\{0\\}}\langle
t\rangle^{6d}A(t,k,tk)|\widehat{\mathcal{N}_{i1}}(t,k)||k|^{-\beta}&\leq
C_{1}\overline{\epsilon}/2,\\\
\sup_{t\in[T_{1},T_{2}],\,k\in\mathbb{Z}^{d}\setminus\\{0\\}}\langle
k,tk\rangle^{-2}\langle
t\rangle^{6d}A(t,k,tk)|(\widehat{\mathcal{N}_{11}}-\widehat{\mathcal{N}_{21}})(t,k)||k|^{-\beta}&\leq
C_{1}\epsilon_{0}/2.\end{split}$ (3.4)

Using the definitions (3.2)–(3.3) we estimate, for $i\in\\{1,2\\}$,

$\begin{split}&\langle
t\rangle^{6d}A(t,k,tk)|k|^{-\beta}|\widehat{\mathcal{N}_{i1}}(t,k)|\\\
&\lesssim\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{T_{1}}^{t}\frac{\langle
t\rangle^{6d}A(t,k,tk)}{\langle s\rangle^{6d}}\frac{\langle
s\rangle^{6d}|\widehat{\mathcal{A}\rho_{i}}(s,l)|}{A(s,l,sl)|l|^{\beta}}\frac{|\widehat{\mathcal{A}_{0}g_{i}}(s,k-l,tk-
sl)|}{A(s,k-l,tk-sl)}\frac{|t-s||k|^{1-\beta}}{|l|^{1-\beta}}\,ds\\\
&\lesssim\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{T_{1}}^{t}\frac{\langle
s\rangle^{6d}|\widehat{\mathcal{A}\rho_{i}}(s,l)|}{|l|^{\beta}}G_{\ast
0i}(s,k-l)\cdot\frac{|t-s||k|^{1-\beta}\langle
t\rangle^{6d}}{|l|^{1-\beta}\langle
s\rangle^{6d}}\frac{A(t,k,tk)}{A(s,l,sl)A(s,k-l,tk-sl)}\,ds,\end{split}$

where $G_{\ast
0i}(s,n):=\sup_{\xi\in\mathbb{R}}|\widehat{\mathcal{A}_{0}g_{i}}(s,n,\xi)|$.
The assumptions (2.16) and the bounds (2.37) show that
$\big{\|}|l|^{-\beta}\langle
s\rangle^{6d}\widehat{\mathcal{A}\rho_{i}}(s,l)\big{\|}_{L^{\infty}_{s,l}}\leq
2\overline{C}\overline{\epsilon}$ and $\big{\|}G_{\ast
0i}(s,l)\big{\|}_{L^{\infty}_{s}L^{2}_{l}}\leq
2C_{0}\overline{C}\overline{\epsilon}$. Moreover, in view (2.32)

$\begin{split}&\frac{\langle k,tk\rangle^{-p}A(t,k,tk)}{\langle
l,sl\rangle^{-p}A(s,l,sl)\langle k-l,tk-sl\rangle^{-p}A(s,k-l,tk-sl)}\\\
&\qquad\leq\frac{A(s,k,tk)}{A(s,l,sl)A(s,k-l,tk-sl)}\frac{\langle
l,sl\rangle^{p}\langle k-l,tk-sl\rangle^{p}}{\langle
k,tk\rangle^{p}}\frac{A(t,k,tk)}{A(s,k,tk)}\\\ &\qquad\lesssim
e^{-8\delta\min(\langle l,sl\rangle,\langle k-l,tk-
sl\rangle)^{1/3}}\frac{A(t,k,tk)}{A(s,k,tk)},\end{split}$ (3.5)

for $p\in[0,2]$. Since $\overline{C}^{8}\leq\overline{\epsilon}^{-1}$, to
prove the bounds in the first line of (3.4) it suffices to prove that for any
$t\in[T_{1},T_{2}]$ and $k\in\mathbb{Z}^{d}\setminus\\{0\\}$ we have

$\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{T_{1}}^{t}\frac{|t-s||k|^{1-\beta}\langle
t\rangle^{6d}}{|l|^{1-\beta}\langle
s\rangle^{6d}}\frac{A(t,k,tk)}{A(s,k,tk)}\big{[}e^{-8\delta\langle
l,sl\rangle^{1/3}}+e^{-8\delta\langle k-l,tk-
sl\rangle^{1/3}}\big{]}\,ds\lesssim 1.$ (3.6)

The bounds in the second line of (3.4) also follow from (3.6), using the
bounds (3.5) with $p=2$.

We prove the bounds (3.6) in three steps.

Step 1. We show first that for any $t\in[0,\infty)$ and
$k\in\mathbb{Z}^{d}\setminus\\{0\\}$

$\begin{split}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{0}^{t}\frac{|t-s||k|^{1-\beta}\langle
t\rangle^{6d}}{|l|^{1-\beta}\langle
s\rangle^{6d}}\frac{A(t,k,tk)}{A(s,k,tk)}e^{-\delta(s^{1/3}+|l|^{1/3})}\,ds\lesssim
1.\end{split}$ (3.7)

For this we use (2.31), so if $0\leq s\leq t$

$\frac{A(t,k,tk)}{A(s,k,tk)}=e^{\lambda(t,|k,tk|)-\lambda(s,|k,tk|)}\leq
e^{-\delta^{2}|t-s|(1+t)^{-\delta-1}|k|^{1/3}(1+t^{2})^{1/6}}\leq
e^{-\delta^{2}|t-s||k|^{1/3}(1+t)^{-2/3-\delta}/2}.$ (3.8)

Therefore, if $\beta\in[1/2,1]$, $k,l\in\mathbb{Z}^{d}\setminus\\{0\\}$, and
$s\in[t/2,t]$ then

$\begin{split}\frac{|t-s||k|^{1-\beta}\langle
t\rangle^{6d}}{|l|^{1-\beta}\langle
s\rangle^{6d}}\frac{A(t,k,tk)}{A(s,k,tk)}e^{-\delta(s^{1/3}+|l|^{1/3})}&\lesssim
e^{-\delta|l|^{1/3}}e^{-\delta
t^{1/3}/2}|k|^{1-\beta}|t-s|e^{-\delta^{2}|t-s|(1+t)^{-1}|k|^{1/3}/2}\\\
&\lesssim e^{-\delta|l|^{1/3}}e^{-\delta
t^{1/3}/4}|k|^{1/2}|t-s|(1+|t-s||k|^{1/3})^{-2}\\\ &\lesssim
e^{-\delta|l|^{1/3}}e^{-\delta t^{1/3}/4}|t-s|^{-1/2}.\end{split}$

Moreover, if $\beta\in[1/2,1]$, $k,l\in\mathbb{Z}^{d}\setminus\\{0\\}$, $s\leq
t/2$, and $t\leq 1$ then

$\begin{split}\frac{|t-s||k|^{1-\beta}\langle
t\rangle^{6d}}{|l|^{1-\beta}\langle
s\rangle^{6d}}\frac{A(t,k,tk)}{A(s,k,tk)}e^{-\delta(s^{1/3}+|l|^{1/3})}&\lesssim|t-s||k|^{1-\beta}e^{-\delta^{2}|t-s||k|^{1/3}/4}e^{-\delta|l|^{1/3}}\\\
&\lesssim e^{-\delta|l|^{1/3}}|t-s|^{-1/2}.\end{split}$

Finally, if $\beta\in[1/2,1]$, $k,l\in\mathbb{Z}^{d}\setminus\\{0\\}$, $s\leq
t/2$, and $t\geq 1$ then

$\begin{split}\frac{|t-s||k|^{1-\beta}\langle
t\rangle^{6d}}{|l|^{1-\beta}\langle
s\rangle^{6d}}\frac{A(t,k,tk)}{A(s,k,tk)}e^{-\delta(s^{1/3}+|l|^{1/3})}&\lesssim\langle
t\rangle^{7d}|k|^{1-\beta}e^{-\delta^{2}t^{1/4}|k|^{1/3}/8}e^{-\delta|l|^{1/3}}\lesssim
e^{-\delta|l|^{1/3}}\langle t\rangle^{-4}.\end{split}$

Therefore, in all cases,

$\frac{|t-s||k|^{1-\beta}\langle t\rangle^{6d}}{|l|^{1-\beta}\langle
s\rangle^{6d}}\frac{A(t,k,tk)}{A(s,k,tk)}e^{-\delta(s^{1/3}+|l|^{1/3})}\lesssim
e^{-\delta|l|^{1/3}}\langle t\rangle^{-3}|t-s|^{-1/2},$ (3.9)

and the desired bounds (3.7) follow.

Step 2. We show now that for any $t\in[0,\infty)$ and
$k\in\mathbb{Z}^{d}\setminus\\{0\\}$ we have

$\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{0}^{t}\mathbf{1}_{\mathcal{R}}(l,s)\frac{|t-s||k|^{1-\beta}\langle
t\rangle^{6d}}{|l|^{1-\beta}\langle
s\rangle^{6d}}\frac{A(t,k,tk)}{A(s,k,tk)}e^{-\delta(|k-l|^{1/3}+|tk-
sl|^{1/3})}\,ds\lesssim 1,$ (3.10)

where

$\mathcal{R}:=\big{\\{}(l,s)\in(\mathbb{Z}^{d}\setminus\\{0\\})\times[0,t]:\,s\leq
0.99(t+1)\,\,\text{ or }\,\,|k-l|\geq|l|/10\big{\\}}.$ (3.11)

For this we notice that if $k,l\in\mathbb{Z}^{d}\setminus\\{0\\}$ then

$\frac{|k|^{1-\beta}}{|l|^{1-\beta}}e^{-\delta|k-l|^{1/3}}\lesssim
e^{-\delta|k-l|^{1/3}/2}.$ (3.12)

Recall also the bounds (3.8). Therefore, if
$k,l\in\mathbb{Z}^{d}\setminus\\{0\\}$ and $s\leq\min\\{t,0.99(t+1)\\}$ then

$\frac{|t-s|\langle t\rangle^{6d}}{\langle
s\rangle^{6d}}\frac{A(t,k,tk)}{A(s,k,tk)}e^{-\delta|tk-sl|^{1/3}}\lesssim
e^{-\delta^{4}(1+t)^{1/4}}.$ (3.13)

Moreover, if $k,l\in\mathbb{Z}^{d}\setminus\\{0\\}$, $s\in[0.99(t+1),t]$, and
$|k-l|\geq|l|/10$ then $|tk-sl|=|t(k-l)+(t-s)l|\geq t|k-l|/2$, and the bounds
(3.13) follow in this case as well. The desired bounds (3.10) follow from
(3.12)–(3.13).

Step 3. Finally, we show that

$\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{0}^{t}(1-\mathbf{1}_{\mathcal{R}}(l,s))\frac{|t-s||k|^{1-\beta}\langle
t\rangle^{6d}}{|l|^{1-\beta}\langle
s\rangle^{6d}}\frac{A(t,k,tk)}{A(s,k,tk)}e^{-\delta(|k-l|^{1/3}+|tk-
sl|^{1/3})}\,ds\lesssim 1,$ (3.14)

where $\mathcal{R}$ is as in (3.11). We make the change of variables $l=k-a$,
so it suffices to prove that if $k\in\mathbb{Z}^{d}\setminus\\{0\\}$ and
$t\in[10,\infty)$ then

$\begin{split}\sum_{a\in\mathbb{Z}^{d},\,|a|\leq|k|/8}\int_{0.99t}^{t}\frac{|t-s|A(t,k,tk)}{A(s,k,tk)}e^{-\delta(|a|^{1/3}+|sa+(t-s)k|^{1/3})}\,ds\lesssim
1.\end{split}$ (3.15)

The bounds (3.8) are not sufficient in this case. We examine the identity
(2.31) and use the second term in the right-hand side to see that

$\begin{split}\lambda(s,|k,tk|)-\lambda(t,|k,tk|)&\geq\delta^{2}|t-s|(1+t\langle
k,tk\rangle^{-2/3})^{-1-\delta}\langle k,tk\rangle^{-1/3}\\\
&\geq\delta^{3}|t-s|t^{-1/3}|k|^{-1/3}(1+t^{1/3}|k|^{-2/3})^{-1-\delta},\end{split}$

if $t\geq 10$, $s\in[0.99t,t]$, and $k\in\mathbb{Z}^{d}\setminus\\{0\\}$.
Therefore

$\frac{A(t,k,tk)}{A(s,k,tk)}\leq
e^{-\delta^{4}|t-s|t^{-1/3}|k|^{-1/3}(1+t|k|^{-2})^{-1/3-\delta}}.$ (3.16)

Since $|sa+(t-s)k|\geq\big{|}s|a|-(t-s)|k|\big{|}$, for (3.15) it suffices to
show that if $a\in\mathbb{Z}^{d}$, $k\in\mathbb{Z}^{d}\setminus\\{0\\}$,
$t\in[10,T]$, and $|a|\leq|k|/8$ then

$\begin{split}&\int_{0.99t}^{t}K_{a}(t,k,s)\,ds\lesssim 1,\\\
&K_{a}(t,k,s):=|t-s|e^{-\delta^{4}|t-s|t^{-1/3}|k|^{-1/3}(1+t|k|^{-2})^{-1/3-\delta}}e^{-\delta|s|a|-(t-s)|k||^{1/3}}e^{-\delta|a|^{1/3}/2}.\end{split}$
(3.17)

The bounds (3.17) are easy in the case $a=0$, since
$|K_{0}(t,k,s)|\lesssim|t-s|e^{-\delta|t-s|^{1/3}}\lesssim\langle
t-s\rangle^{-4}$. On the other hand, if $a\in\mathbb{Z}^{d}\setminus\\{0\\}$
then

$|K_{a}(t,k,s)|\lesssim|t-s|e^{-\delta|t-s|^{1/3}|k|^{1/3}/2}\lesssim\langle
t-s\rangle^{-4}$

if $(t-s)|k|\notin[t|a|/2,2t|a|]$, and

$\begin{split}|K_{a}(t,k,s)|&\lesssim
t|k|^{-1}e^{-\delta^{4}(t|k|^{-2})^{2/3}(1+t|k|^{-2})^{-1/3-\delta}/4}e^{-\delta|s|a|-(t-s)|k||^{1/3}}e^{-\delta|a|^{1/3}/4}\\\
&\lesssim|k|e^{-\delta|s|a|-(t-s)|k||^{1/3}}e^{-\delta|a|^{1/3}/4}\end{split}$

if $(t-s)|k|\in[t|a|/2,2t|a|]$. The desired bounds (3.17) follow in this case
as well. ∎

We can now prove our main estimates on the function $\rho_{1}$, $\rho_{2}$,
and $\rho_{1}-\rho_{2}$.

######  Proposition 3.2.

With the assumptions and notations of Proposition 2.2, there is a constant
$C_{2}=C_{2}(d,\lambda_{0},\vartheta)\geq 1$ such that for any
$t\in[T_{1},T_{2}]$ and $i\in\\{1,2\\}$

$\mathcal{Z}_{g_{i}}^{0}(t)\langle t\rangle^{6d}\leq
C_{2}\overline{\epsilon}\qquad\text{ and
}\qquad\mathcal{Z}_{g_{1}-g_{2}}^{2}(t)\langle t\rangle^{6d}\leq
C_{2}\epsilon_{0}.$ (3.18)

As a consequence, if the constant $\overline{C}$ is chosen sufficiently large,
$\overline{C}\geq C_{2}$, then there is a constant
$C^{\prime}_{0}=C^{\prime}_{0}(d)$ such that

$\begin{split}\big{\|}A(t,k,tk)\widehat{\rho_{i}}(t,k)|k|^{-\beta}\|_{L^{2}_{k}}&\leq
C^{\prime}_{0}(\overline{C}^{d}C_{2})^{1/(d+1)}\overline{\epsilon}\langle
t\rangle^{-3},\\\ \big{\|}\langle
k,tk\rangle^{-2}A(t,k,tk)(\widehat{\rho_{1}-\rho_{2}})(t,k)|k|^{-\beta}\|_{L^{2}_{k}}&\leq
C^{\prime}_{0}(\overline{C}^{d}C_{2})^{1/(d+1)}\epsilon_{0}\langle
t\rangle^{-3}\end{split}$ (3.19)

for any $t\in[T_{1},T_{2}]$ and $i\in\\{1,2\\}$.

###### Proof.

We remark that the $L^{2}$ bounds (3.19) are needed later, in the proof of
Lemma 3.4 below.

To prove the bounds (3.18) we use the representation formula (2.10). For any
$t\in[T_{1},T_{2}]$, $k\in\mathbb{Z}^{d}\setminus\\{0\\}$, and $i\in\\{1,2\\}$
we estimate

$\begin{split}\langle
t\rangle^{6d}A(t,k,tk)&|k|^{-\beta}\big{|}\widehat{\rho_{i}}(t,k)\big{|}\leq\langle
t\rangle^{6d}A(t,k,tk)|k|^{-\beta}\big{|}\widehat{\mathcal{N}_{i}}(t,k)\big{|}\\\
&+\int_{0}^{t}\big{[}\langle
s\rangle^{6d}A(s,k,sk)|k|^{-\beta}\big{|}\widehat{\mathcal{N}_{i}}(s,k)\big{|}\big{]}\cdot\big{|}\widehat{G}(t-s,k)\big{|}\frac{\langle
t\rangle^{6d}A(t,k,tk)}{\langle s\rangle^{6d}A(s,k,sk)}\,ds.\end{split}$

Using Lemma 3.1, the bounds (2.32), and the bounds on $\widehat{G}$ in (2.11)
we further estimate

$\begin{split}\langle
t\rangle^{6d}A(t,k,tk)|k|^{-\beta}&\big{|}\widehat{\rho_{i}}(t,k)\big{|}\leq
C_{1}\overline{\epsilon}+C_{1}\overline{\epsilon}\int_{0}^{t}\big{|}\widehat{G}(t-s,k)\big{|}\frac{\langle
t\rangle^{6d}A(s,k,tk)}{\langle s\rangle^{6d}A(s,k,sk)}\,ds\\\
&\lesssim_{d,\lambda_{0},\vartheta}\overline{\epsilon}\Big{[}1+\int_{0}^{t}e^{-0.95\lambda_{0}\langle(t-s)k\rangle^{1/3}}\langle
t-s\rangle^{6d}A(s,0,(t-s)k)\,ds\Big{]}.\end{split}$

The desired bounds on $\mathcal{Z}_{g_{i}}^{0}(t)\langle t\rangle^{6d}$ in
(3.18) follow since $A(s,0,(t-s)k)\leq
e^{(\lambda_{1}+2\delta)\langle(t-s)k\rangle^{1/3}}$. The bounds on
$\mathcal{Z}_{g_{1}-g_{2}}^{2}(t)\langle t\rangle^{6d}$ follow similarly.

To prove the bounds (3.19) we recall that $\beta=1/2$. Using Lemma 2.5 and the
bootstrap assumption (2.16), for any $t\in[T_{1},T_{2}]$ and $i\in\\{1,2\\}$
we have

$\sum_{k\in\mathbb{Z}^{d},\,|k|\geq
D}|A(t,k,tk)\widehat{\rho_{i}}(t,k)|^{2}|k|^{-1}\lesssim_{d}\overline{C}^{2}\overline{\epsilon}^{2}D^{-1},$

where $D:=\langle t\rangle^{6}(\overline{C}/C_{2})^{2/(d+1)}$ is a suitable
constant. On the other hand, using (3.18),

$\sum_{k\in\mathbb{Z}^{d}\setminus\\{0\\},\,|k|\leq
D}|A(t,k,tk)\widehat{\rho_{i}}(t,k)|^{2}|k|^{-1}\leq\sum_{k\in\mathbb{Z}^{d},\,|k|\leq
D}C_{2}^{2}\overline{\epsilon}^{2}\langle
t\rangle^{-12d}\lesssim_{d}C_{2}^{2}\overline{\epsilon}^{2}\langle
t\rangle^{-12d}D^{d}.$

The desired bounds in the first line of (3.19) follow from these two
inequalities. The bounds in the second line follow in a similar way. ∎

###  3.2. Improved control on energy functionals

In this subsection we prove the following:

######  Proposition 3.3.

With the assumptions and notations of Proposition 2.2, for any
$t\in[T_{1},T_{2}]$ and $i\in\\{1,2\\}$ we have

$\mathcal{E}^{0}_{g_{i}}(t)\leq\overline{C}^{2}\overline{\epsilon}^{2}/4\qquad\text{
and
}\qquad\mathcal{E}^{2}_{g_{1}-g_{2}}(t)\leq\overline{C}^{2}\epsilon_{0}^{2}/4.$
(3.20)

To justify formally the calculations, for any $h\in X_{[T_{1},T_{2}]}$ we
define the mollified energies

$\begin{split}&\mathcal{E}^{p,L}_{h}(t):=\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-2p}A_{L}(t,k,\xi)^{2}\Big{\\{}\sum_{|a|\leq
d^{\prime}}\big{|}D^{a}_{\xi}\widehat{h}(t,k,\xi)\big{|}^{2}\Big{\\}}\,d\xi,\\\
&A_{L}(t,k,\xi):=(1+2^{-L}\langle k,\xi\rangle)^{-4}A(t,k,\xi),\end{split}$
(3.21)

where $p\in[0,2]$ and $L\geq 4$. We calculate

$\begin{split}\frac{d}{dt}\mathcal{E}^{0,L}_{g_{i}}(t)&=2\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}A_{L}(t,k,\xi)\dot{A_{L}}(t,k,\xi)\Big{\\{}\sum_{|a|\leq
d^{\prime}}\big{|}D^{a}_{\xi}\widehat{g_{i}}(t,k,\xi)\big{|}^{2}\Big{\\}}\,d\xi\\\
&+2\Re\Big{\\{}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}A_{L}(t,k,\xi)^{2}\Big{\\{}\sum_{|a|\leq
d^{\prime}}D^{a}_{\xi}(\partial_{t}\widehat{g_{i}})(t,k,\xi)\overline{D^{a}_{\xi}\widehat{g_{i}}(t,k,\xi)}\Big{\\}}\,d\xi\Big{\\}},\end{split}$

where $\dot{A_{L}}=\partial_{t}A_{L}$. Using the formula (2.1) for
$\partial_{t}\widehat{g_{i}}$ we obtain

$\frac{d}{dt}\mathcal{E}^{0,L}_{g_{i}}(t)=I_{i}(t)+II_{i}(t)+III_{i}(t)+IV_{i}(t),$
(3.22)

where

$I_{i}(t):=2\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}A_{L}(t,k,\xi)\dot{A_{L}}(t,k,\xi)\big{|}D^{a}_{\xi}\widehat{g_{i}}(t,k,\xi)\big{|}^{2}\,d\xi,$
(3.23) $\begin{split}II_{i}(t)&:=-2\Re\Big{\\{}\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}A_{L}(t,k,\xi)^{2}\overline{D^{a}_{\xi}\widehat{g_{i}}(t,k,\xi)}\\\
&\qquad\qquad\times
D^{a}_{\xi}\Big{[}\widehat{\rho_{i}}(t,k)\widehat{M_{0}}(\xi-
tk)\frac{k\cdot(\xi-tk)}{|k|^{2}}\Big{]}\,d\xi\Big{\\}},\end{split}$ (3.24)
$\begin{split}III_{i}(t)&:=-\frac{2}{(2\pi)^{d}}\Re\Big{\\{}\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}A_{L}(t,k,\xi)^{2}\overline{D^{a}_{\xi}\widehat{g_{i}}(t,k,\xi)}\\\
&\qquad\qquad\qquad\times\Big{[}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\widehat{\rho_{i}}(t,l)D^{a}_{\xi}\widehat{g_{i}}(t,k-l,\xi-
tl)\frac{l\cdot(\xi-tk)}{|l|^{2}}\Big{]}\,d\xi\Big{\\}},\end{split}$ (3.25)
$\begin{split}IV_{i}(t)&:=-\frac{2}{(2\pi)^{d}}\Re\Big{\\{}\sum_{|a|\leq
d^{\prime}}\sum_{|a^{\prime}|=1,\,a^{\prime}\leq
a}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}A_{L}(t,k,\xi)^{2}\overline{D^{a}_{\xi}\widehat{g_{i}}(t,k,\xi)}\\\
&\qquad\qquad\qquad\times\Big{[}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\widehat{\rho_{i}}(t,l)(D^{a-a^{\prime}}_{\xi}\widehat{g_{i}})(t,k-l,\xi-
tl)\frac{D^{a^{\prime}}_{\xi}[l\cdot(\xi-
tk)]}{|l|^{2}}\Big{]}\,d\xi\Big{\\}}.\end{split}$ (3.26)

Notice that $I_{i}(t)$ are good terms, $I_{i}(t)\leq 0$, since
$\dot{A_{L}}(t,k,\xi)\leq 0$. Using (3.22) we write

$\begin{split}\mathcal{E}^{0,L}_{g_{i}}(t)&=\mathcal{E}^{0,L}_{g_{i}}(T_{1})+\int_{T_{1}}^{t}\big{[}I_{i}(s)+II_{i}(s)+III_{i}(s)+IV_{i}(s)\big{]}\,ds\\\
&\leq\mathcal{E}^{0}_{g_{i}}(T_{1})+\int_{T_{1}}^{t}|II_{i}(s)|\,ds+\int_{T_{1}}^{t}|IV_{i}(s)|\,ds+\int_{T_{1}}^{t}\big{[}|III_{i}(s)|-|I_{i}(s)|\big{]}\,ds.\end{split}$
(3.27)

The bounds on $\mathcal{E}^{0}_{g_{i}}(t)$ in (3.20) follow from Lemmas 3.4,
3.5, 3.6 below, by letting $L\to\infty$.

Similarly, we calculate

$\frac{d}{dt}\mathcal{E}^{2,L}_{g_{1}-g_{2}}(t)=\delta I(t)+\delta
II(t)+\delta III(t)+\delta IV(t),$ (3.28)

where

$\delta I(t):=2\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-4}A_{L}(t,k,\xi)\dot{A_{L}}(t,k,\xi)\big{|}D^{a}_{\xi}(\widehat{g_{1}}-\widehat{g_{2}})(t,k,\xi)\big{|}^{2}\,d\xi,$
(3.29) $\begin{split}\delta II(t)&:=-2\Re\Big{\\{}\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-4}A_{L}(t,k,\xi)^{2}\overline{D^{a}_{\xi}(\widehat{g_{1}}-\widehat{g_{2}})(t,k,\xi)}\\\
&\qquad\qquad\times
D^{a}_{\xi}\Big{[}(\widehat{\rho_{1}}-\widehat{\rho_{2}})(t,k)\widehat{M_{0}}(\xi-
tk)\frac{k\cdot(\xi-tk)}{|k|^{2}}\Big{]}\,d\xi\Big{\\}},\end{split}$ (3.30)
$\begin{split}&\delta III(t):=-\frac{2}{(2\pi)^{d}}\Re\Big{\\{}\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-4}A_{L}(t,k,\xi)^{2}\overline{D^{a}_{\xi}(\widehat{g_{1}}-\widehat{g_{2}})(t,k,\xi)}\\\
&\quad\times\Big{[}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}[\widehat{\rho_{1}}(t,l)D^{a}_{\xi}\widehat{g_{1}}(t,k-l,\xi-
tl)-\widehat{\rho_{2}}(t,l)D^{a}_{\xi}\widehat{g_{2}}(t,k-l,\xi-
tl)]\frac{l\cdot(\xi-tk)}{|l|^{2}}\Big{]}\,d\xi\Big{\\}},\end{split}$ (3.31)
$\begin{split}&\delta IV(t):=-\frac{2}{(2\pi)^{d}}\Re\Big{\\{}\sum_{|a|\leq
d^{\prime}}\sum_{|a^{\prime}|=1,\,a^{\prime}\leq
a}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-4}A_{L}(t,k,\xi)^{2}\overline{D^{a}_{\xi}(\widehat{g_{1}}-\widehat{g_{2}})(t,k,\xi)}\\\
&\quad\times\Big{[}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}[\widehat{\rho_{1}}(t,l)(D^{a-a^{\prime}}_{\xi}\widehat{g_{1}})(t,k-l,\xi-
tl)-\widehat{\rho_{2}}(t,l)(D^{a-a^{\prime}}_{\xi}\widehat{g_{2}})(t,k-l,\xi-
tl)]\\\ &\qquad\qquad\qquad\times\frac{D^{a^{\prime}}_{\xi}[l\cdot(\xi-
tk)]}{|l|^{2}}\Big{]}\,d\xi\Big{\\}}.\end{split}$ (3.32)

As before, we notice that $\delta I(t)\leq 0$, so

$\mathcal{E}^{2,L}_{g_{1}-g_{2}}(t)\leq\mathcal{E}^{2}_{g_{1}-g_{2}}(T_{1})+\int_{T_{1}}^{t}|\delta
II(s)|\,ds+\int_{T_{1}}^{t}|\delta IV(s)|\,ds+\int_{T_{1}}^{t}\big{[}|\delta
III(s)|-|\delta I(s)|\big{]}\,ds.$ (3.33)

The bounds on $\mathcal{E}^{2}_{g_{1}-g_{2}}(t)$ in (3.20) follow from Lemmas
3.4, 3.5, 3.6 below by letting $L\to\infty$.

######  Lemma 3.4.

For any $t\in[T_{1},T_{2}]$, $L\geq 4$, and $i\in\\{1,2\\}$ we have

$\int_{T_{1}}^{t}\big{|}II_{i}(s)\big{|}\,ds\leq\overline{C}^{2}\overline{\epsilon}^{2}/8\qquad\text{
and }\qquad\int_{T_{1}}^{t}\big{|}\delta
II(s)\big{|}\,ds\leq\overline{C}^{2}\epsilon_{0}^{2}/8.$ (3.34)

###### Proof.

Let

$M_{0}^{\ast}(\eta):=\sum_{|a|\leq
d^{\prime},\,j\in\\{1,\ldots,d\\}}\big{|}D^{a}_{\eta}\big{[}\widehat{M_{0}}(\eta)\eta_{j}\big{]}\big{|}.$
(3.35)

It follows from (3.24) that

$\begin{split}|II_{i}(s)|\leq 2\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}&A(s,k,\xi)|D^{a}_{\xi}\widehat{g_{i}}(s,k,\xi)|\\\
&\times\frac{A(s,k,sk)|\widehat{\rho_{i}}(s,k)|}{|k|}\frac{A(s,k,\xi)}{A(s,k,sk)}M_{0}^{\ast}(\xi-
sk)\,d\xi.\end{split}$ (3.36)

Moreover, using (2.32),

$\frac{A(s,k,\xi)}{A(s,k,sk)}M_{0}^{\ast}(\xi-sk)\leq A(s,0,\xi-
sk)M_{0}^{\ast}(\xi-sk)\leq e^{0.95\lambda_{0}\langle\xi-
sk\rangle^{1/3}}M_{0}^{\ast}(\xi-sk).$

It follows from the assumption (1.6) and the definition (3.35) that

$\big{\|}e^{0.95\lambda_{0}\langle\eta\rangle^{1/3}}M_{0}^{\ast}(\eta)\big{\|}_{L^{2}_{\eta}}\lesssim_{d,\lambda_{0},\vartheta}1,$
(3.37)

with an implied constant that depends on $d,\lambda_{0},\vartheta$. Recall the
definition (3.2) and let

$G_{\ast i}(t,k):=\sum_{|a|\leq
d^{\prime}}\big{\|}\widehat{\mathcal{A}_{a}g_{i}}(t,k,\xi)\big{\|}_{L^{2}_{\xi}},\qquad\delta
G_{\ast}(t,k):=\sum_{|a|\leq d^{\prime}}\big{\|}\langle
k,\xi\rangle^{-2}\widehat{\mathcal{A}_{a}\delta
g}(t,k,\xi)\big{\|}_{L^{2}_{\xi}},$ (3.38)

for $t\in[T_{1},T_{2}]$ and $k\in\mathbb{Z}^{d}$. It follows from (3.36) and
the Cauchy inequality in $\xi$ that

$|II_{i}(s)|\lesssim_{d,\lambda_{0},\vartheta}\sum_{k\in\mathbb{Z}^{d}\setminus\\{0\\}}G_{\ast
i}(s,k)\frac{A(s,k,sk)|\widehat{\rho_{i}}(s,k)|}{|k|},$ (3.39)

It follows from the bootstrap assumption (2.16) that $\|G_{\ast
i}\|_{L^{\infty}_{s}L^{2}_{k}}\lesssim_{d}\overline{C}\overline{\epsilon}$.
Using also (3.19) and the Cauchy inequality we have

$|II_{i}(s)|\lesssim_{d,\lambda_{0},\vartheta}\overline{C}\overline{\epsilon}\cdot(\overline{C}^{d}C_{2})^{1/(d+1)}\overline{\epsilon}\langle
s\rangle^{-3}.$

The bounds for $|II_{i}(s)|$ in (3.34) follow if the constant $\overline{C}$
is taken sufficiently large relative to $C_{2}$. The bounds for $|\delta
II(s)|$ in (3.34) follow in a similar way. ∎

######  Lemma 3.5.

For any $t\in[T_{1},T_{2}]$, $L\geq 4$, and $i\in\\{1,2\\}$ we have

$\int_{T_{1}}^{t}\big{|}IV_{i}(s)\big{|}\,ds\leq\overline{\epsilon}^{2}\qquad\text{
and }\qquad\int_{T_{1}}^{t}\big{|}\delta
IV(s)\big{|}\,ds\leq\epsilon_{0}^{2}.$ (3.40)

###### Proof.

The implied constants in this lemma may depend only on $d$ and $\lambda_{0}$.
We provide all the details only for the slightly harder bounds on the
difference of solutions. Notice that

$\begin{split}&|\delta IV(s)|\lesssim\sum_{|a|\leq
d^{\prime}}\sum_{|a^{\prime}|=1,\,a^{\prime}\leq
a}\sum_{k\in\mathbb{Z}^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-4}A(s,k,\xi)^{2}|D^{a}_{\xi}\widehat{\delta
g}(s,k,\xi)||l|^{-1}\\\
&\qquad\times\big{\\{}|\widehat{\delta\rho}(s,l)||D^{a-a^{\prime}}_{\xi}\widehat{g_{1}}(s,k-l,\xi-
sl)|+|\widehat{\rho_{2}}(s,l)||D^{a-a^{\prime}}_{\xi}\widehat{\delta
g}(s,k-l,\xi-sl)|\big{\\}}\,d\xi,\end{split}$ (3.41)

where $\delta g=g_{1}-g_{2}$ and $\delta\rho=\rho_{1}-\rho_{2}$. We define the
functions $\mathcal{A}\delta\rho$ and $\mathcal{A}_{a}\delta g$ as in (3.2),
for $s\in[T_{1},T_{2}]$, $k\in\mathbb{Z}^{d}$, $\xi\in\mathbb{R}^{d}$, and
$|a|\leq d^{\prime}$. It follows from (3.41) that

$\begin{split}&|\delta IV(s)|\lesssim\sum_{|a|,|a^{\prime}|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-2}|\widehat{\mathcal{A}_{a}\delta g}(s,k,\xi)|\frac{\langle
k,\xi\rangle^{-2}A(s,k,\xi)}{A(s,l,sl)A(s,k-l,\xi-sl)}\\\
&\qquad\times\big{\\{}|l|^{-1}|\widehat{\mathcal{A}\delta\rho}(s,l)||\widehat{\mathcal{A}_{a^{\prime}}g_{1}}(s,k-l,\xi-
sl)|+|l|^{-1}|\widehat{\mathcal{A}\rho_{2}}(s,l)||\widehat{\mathcal{A}_{a^{\prime}}\delta
g}(s,k-l,\xi-sl)|\big{\\}}\,d\xi.\end{split}$

In view of (2.32),

$\frac{\langle k,\xi\rangle^{-2}A(s,k,\xi)}{\langle
l,sl\rangle^{-2}A(s,l,sl)\langle k-l,\xi-sl\rangle^{-2}A(s,k-l,\xi-
sl)}\lesssim e^{-8\delta\langle l,sl\rangle^{1/3}}+e^{-8\delta\langle k-l,\xi-
sl\rangle^{1/3}}.$

Therefore for any $s\in[T_{1},T_{2}]$ we can estimate

$|\delta IV(s)|\lesssim\delta IV_{1}(s)+\delta IV_{2}(s),$ (3.42)

where

$\begin{split}&\delta IV_{1}(s):=\sum_{|a|,|a^{\prime}|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-2}|\widehat{\mathcal{A}_{a}\delta g}(s,k,\xi)|e^{-\delta\langle
l,sl\rangle^{1/3}}\\\
&\quad\times\frac{|\widehat{\mathcal{A}\delta\rho}(s,l)||\widehat{\mathcal{A}_{a^{\prime}}g_{1}}(s,k-l,\xi-
sl)|+|\widehat{\mathcal{A}\rho_{2}}(s,l)||\widehat{\mathcal{A}_{a^{\prime}}\delta
g}(s,k-l,\xi-sl)|}{|l|\langle l,sl\rangle^{2}\langle k-l,\xi-
sl\rangle^{2}}\,d\xi,\end{split}$ (3.43) $\begin{split}&\delta
IV_{2}(s):=\sum_{|a|,|a^{\prime}|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-2}|\widehat{\mathcal{A}_{a}\delta g}(s,k,\xi)|e^{-\delta\langle
k-l,\xi-sl\rangle^{1/3}}\\\
&\quad\times\frac{|\widehat{\mathcal{A}\delta\rho}(s,l)||\widehat{\mathcal{A}_{a^{\prime}}g_{1}}(s,k-l,\xi-
sl)|+|\widehat{\mathcal{A}\rho_{2}}(s,l)||\widehat{\mathcal{A}_{a^{\prime}}\delta
g}(s,k-l,\xi-sl)|}{|l|\langle l,sl\rangle^{2}\langle k-l,\xi-
sl\rangle^{2}}\,d\xi.\end{split}$ (3.44)

It follows from (2.16) that
$\|\widehat{\mathcal{A}_{a}g_{1}}(t,k,\xi)\|_{L^{2}_{k,\xi}}\leq
2\overline{C}\overline{\epsilon}$ and $\|\langle
k,\xi\rangle^{-2}\widehat{\mathcal{A}_{a}\delta
g}(t,k,\xi)\|_{L^{2}_{k,\xi}}\leq 2\overline{C}\epsilon_{0}$ for any
$t\in[T_{1},T_{2}]$ and multi-index $a$ with $|a|\leq d^{\prime}$. Therefore,
using the Cauchy inequality in $k,\xi$ and recalling the definition (2.14),

$\delta
IV_{1}(s)\lesssim(\overline{C}\epsilon_{0})\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\frac{\overline{C}\overline{\epsilon}|\widehat{\mathcal{A}\delta\rho}(s,l)|+\overline{C}\epsilon_{0}|\widehat{\mathcal{A}\rho_{2}}(s,l)|}{|l|\langle
l,sl\rangle^{2}}e^{-\delta\langle
l\rangle^{1/3}}\lesssim\overline{C}^{3}\epsilon_{0}^{2}\overline{\epsilon}\langle
s\rangle^{-6d}.$

Similarly, recalling the definition (3.38) and using the Cauchy inequality in
$\xi$,

$\begin{split}\delta
IV_{2}(s)&\lesssim\sum_{k\in\mathbb{Z}^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}e^{-\delta\langle
k-l\rangle^{1/3}}\delta
G_{\ast}(s,k)\frac{|\widehat{\mathcal{A}\delta\rho}(s,l)|G_{\ast
1}(s,k-l)+|\widehat{\mathcal{A}\rho_{2}}(s,l)|\delta
G_{\ast}(s,k-l)}{|l|\langle l,sl\rangle^{2}}\\\
&\lesssim\overline{C}^{3}\epsilon_{0}^{2}\overline{\epsilon}\langle
s\rangle^{-3}.\end{split}$

The last inequality holds since $\|G_{\ast
1}\|_{L^{\infty}_{s}L^{2}_{k}}\lesssim\overline{C}\overline{\epsilon}$,
$\|\delta
G_{\ast}\|_{L^{\infty}_{s}L^{2}_{k}}\lesssim\overline{C}\epsilon_{0}$, and
using also (3.19). The desired bounds follow from (3.42), since
$\overline{C}\leq\overline{\epsilon}^{-1/8}$. ∎

######  Lemma 3.6.

For any $t\in[T_{1},T_{2}]$, $L\geq 4$, and $i\in\\{1,2\\}$ we have

$\int_{T_{1}}^{t}\big{[}|III_{i}(s)|-|I_{i}(s)|/2\big{]}\,ds\leq\overline{\epsilon}^{2}\qquad\text{
and }\qquad\int_{T_{1}}^{t}\big{[}|\delta III(s)|-|\delta
I(s)|/2\big{]}\,ds\leq\epsilon_{0}^{2}.$ (3.45)

###### Proof.

As before, the implied constants in this proof are allowed to depend on $d$
and $\lambda_{0}$. We provide all the details only for the slightly harder
bounds on the difference of solutions.

We notice that there is derivative loss in the term $\delta III$, coming from
the factor $\xi-tk$, which we can fortunately eliminate by symmetrization. For
this, we notice that

$\begin{split}\Re\Big{\\{}&\sum_{k\in\mathbb{Z}^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-2}A_{L}(t,k,\xi)\cdot\langle k-l,\xi-
tl\rangle^{-2}A_{L}(t,k-l,\xi-tl)\\\
&\qquad\qquad\times\Big{[}\,\overline{D^{a}_{\xi}\widehat{\delta
g}(t,k,\xi)}\widehat{\rho_{1}}(t,l)D^{a}_{\xi}\widehat{\delta g}(t,k-l,\xi-
tl)\frac{l\cdot(\xi-tk)}{|l|^{2}}\Big{]}\,d\xi\Big{\\}}=0,\end{split}$

for any $t\in[T_{1},T_{2}]$ and any multi-index $a$ satisfying $|a|\leq
d^{\prime}$. This follows easily by expanding $\Re z=(z+\overline{z})/2$ and
observing that $\overline{\widehat{\rho_{1}}(t,l)}=\widehat{\rho_{1}}(t,-l)$
since $\rho_{1}$ is real-valued. Therefore

$|\delta III(s)|\lesssim\delta III_{1}(s)+\delta III_{2}(s)$ (3.46)

where

$\begin{split}\delta III_{1}(s):=\sum_{|a|\leq
d^{\prime}}&\sum_{k\in\mathbb{Z}^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-2}A_{L}(s,k,\xi)\Big{|}\frac{A_{L}(s,k,\xi)}{\langle
k,\xi\rangle^{2}}-\frac{A_{L}(s,k-l,\xi-sl)}{\langle k-l,\xi-
sl\rangle^{2}}\Big{|}\\\ &\times\big{|}D^{a}_{\xi}\widehat{\delta
g}(s,k,\xi)\big{|}\,|\widehat{\rho_{1}}(s,l)|\,\big{|}D^{a}_{\xi}\widehat{\delta
g}(s,k-l,\xi-sl)\big{|}\frac{|\xi-sk|}{|l|}\,d\xi,\end{split}$ (3.47)
$\begin{split}\delta III_{2}(s):=\sum_{|a|\leq
d^{\prime}}&\sum_{k\in\mathbb{Z}^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-4}A_{L}(s,k,\xi)^{2}\\\
&\times\big{|}D^{a}_{\xi}\widehat{\delta
g}(s,k,\xi)\big{|}|\widehat{\delta\rho}(s,l)|\,\big{|}D^{a}_{\xi}\widehat{g_{2}}(s,k-l,\xi-
sl)\big{|}\frac{|\xi-sk|}{|l|}\,d\xi.\end{split}$ (3.48)

Step 1. We bound first the contribution of the term $\delta III_{1}$.
Recalling the definitions (3.2) and letting $\widehat{\mathcal{A}_{a,L}\delta
g}(s,k,\xi):=A_{L}(s,k,\xi)D^{a}_{\xi}\widehat{\delta g}(s,k,\xi)$, we have

$\begin{split}\delta III_{1}(s)\lesssim&\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}\frac{|\widehat{\mathcal{A}_{a,L}\delta
g}(s,k,\xi)|}{\langle k,\xi\rangle^{2}}\frac{|\widehat{\mathcal{A}_{a,L}\delta
g}(s,k-l,\xi-sl)|}{\langle k-l,\xi-
sl\rangle^{2}}\frac{|\widehat{\mathcal{A}\rho_{1}}(s,l)|}{|l|}\\\
&\qquad\times\Big{|}\frac{A_{L}(s,k,\xi)}{\langle
k,\xi\rangle^{2}}-\frac{A_{L}(s,k-l,\xi-sl)}{\langle k-l,\xi-
sl\rangle^{2}}\Big{|}\frac{\langle k-l,\xi-sl\rangle^{2}|\xi-
sk|}{A(s,l,sl)A_{L}(s,k-l,\xi-sl)}\,d\xi.\end{split}$ (3.49)

We bound now the expression in the second line of (3.49). If
$10|l,sl|\geq|k-l,\xi-sl|$ then we use (2.32) to estimate

$\begin{split}\Big{|}\frac{A_{L}(s,k,\xi)}{\langle
k,\xi\rangle^{2}}-\frac{A_{L}(s,k-l,\xi-sl)}{\langle k-l,\xi-
sl\rangle^{2}}\Big{|}\frac{\langle k-l,\xi-sl\rangle^{2}|\xi-
sk|}{A(s,l,sl)A_{L}(s,k-l,\xi-sl)}&\lesssim|\xi-sk|e^{-2\delta|k-l,\xi-
sl|^{1/3}}\\\ &\lesssim\langle s\rangle e^{-\delta|k-l,\xi-
sl|^{1/3}}.\end{split}$

On the other hand, if $|l,sl|\leq|k-l,\xi-sl|/10$ and $|l|\geq 1$ then we use
(2.33) to estimate

$\begin{split}\Big{|}\frac{A_{L}(s,k,\xi)}{\langle
k,\xi\rangle^{2}}-\frac{A_{L}(s,k-l,\xi-sl)}{\langle k-l,\xi-
sl\rangle^{2}}\Big{|}\frac{\langle k-l,\xi-sl\rangle^{2}|\xi-
sk|}{A(s,l,sl)A_{L}(s,k-l,\xi-sl)}&\lesssim|\xi-sk|e^{-2\delta\langle
l,sl\rangle^{1/3}}\langle k,\xi\rangle^{-2/3}\\\ &\lesssim e^{-\delta\langle
l,sl\rangle^{1/3}}\langle k,\xi\rangle^{1/3}.\end{split}$

Therefore

$\begin{split}\delta III_{1}(s)&\lesssim J_{11}(s)+J_{12}(s),\\\
J_{11}(s):=\sum_{|a|\leq
d^{\prime}}&\sum_{k\in\mathbb{Z}^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}\frac{|\widehat{\mathcal{A}_{a,L}\delta
g}(s,k,\xi)|}{\langle k,\xi\rangle^{2}}\frac{|\widehat{\mathcal{A}_{a,L}\delta
g}(s,k-l,\xi-sl)|}{\langle k-l,\xi-
sl\rangle^{2}}\frac{|\widehat{\mathcal{A}\rho_{1}}(s,l)|}{|l|}\\\
&\qquad\qquad\qquad\qquad\times\langle s\rangle
e^{-\delta|k-l|^{1/3}}\,d\xi,\\\ J_{12}(s):=\sum_{|a|\leq
d^{\prime}}&\sum_{k\in\mathbb{Z}^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}\frac{|\widehat{\mathcal{A}_{a,L}\delta
g}(s,k,\xi)|}{\langle k,\xi\rangle^{2}}\frac{|\widehat{\mathcal{A}_{a,L}\delta
g}(s,k-l,\xi-sl)|}{\langle k-l,\xi-
sl\rangle^{2}}\frac{|\widehat{\mathcal{A}\rho_{1}}(s,l)|}{|l|}\\\
&\qquad\qquad\qquad\qquad\times e^{-\delta\langle l,sl\rangle^{1/3}}\langle
k,\xi\rangle^{1/6}\langle k-l,\xi-sl\rangle^{1/6}\,d\xi.\end{split}$ (3.50)

We use the Cauchy inequality (in $\xi$), the definition (3.38), and (3.19) to
estimate

$J_{11}(s)\lesssim\sum_{k\in\mathbb{Z}^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}(\delta
G_{\ast})(s,k)(\delta
G_{\ast})(s,k-l)\frac{|\widehat{\mathcal{A}\rho_{1}}(s,l)|\langle
s\rangle}{|l|}e^{-\delta|k-l|^{1/3}}\lesssim\overline{C}^{3}\epsilon_{0}^{2}\overline{\epsilon}\langle
s\rangle^{-2}.$ (3.51)

Moreover, the definition (3.29) and the identity (2.28) show that

$|\delta I(s)|\gtrsim(1+s)^{-\delta-1}\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-4}A_{L}(s,k,\xi)^{2}\langle
k,\xi\rangle^{1/3}\big{|}D^{a}_{\xi}\widehat{\delta
g}(s,k,\xi)\big{|}^{2}\,d\xi.$

Therefore $J_{12}(s)\lesssim(\overline{C}\overline{\epsilon})|\delta I(s)|$,
using (3.18) and the Cauchy inequality in $(k,\xi)$. Finally, using also the
bounds (3.50)–(3.51) we see that

$\int_{0}^{t}\big{[}|\delta III_{1}(s)|-|\delta
I(s)|/2\big{]}\,ds\leq\epsilon_{0}^{2}/2.$ (3.52)

Step 2. We bound now the contribution of $\delta III_{2}$. Recalling (3.2) and
(3.38), we have

$\begin{split}\delta III_{2}(s)\lesssim&\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}\frac{|\widehat{\mathcal{A}_{a}\delta
g}(s,k,\xi)|}{\langle
k,\xi\rangle^{2}}|\widehat{\mathcal{A}_{a}g_{2}}(s,k-l,\xi-
sl)|\frac{|\widehat{\mathcal{A}\delta\rho}(s,l)|}{|l|\langle
l,sl\rangle^{2}}\\\ &\qquad\qquad\times\frac{\langle l,sl\rangle^{2}|\xi-
sk|}{\langle k,\xi\rangle^{2}}\frac{A(s,k,\xi)}{A(s,l,sl)A(s,k-l,\xi-
sl)}\,d\xi.\end{split}$ (3.53)

It follows easily from (2.32) that

$\frac{\langle l,sl\rangle^{2}|\xi-sk|}{\langle
k,\xi\rangle^{2}}\frac{A(s,k,\xi)}{A(s,l,sl)A(s,k-l,\xi-sl)}\lesssim\langle
s\rangle\big{[}e^{-8\delta\langle l,sl\rangle^{1/3}}+e^{-8\delta\langle
k-l,\xi-sl\rangle^{1/3}}\big{]}.$

As before, we can use the bootstrap assumptions (2.16), the bounds (3.19), and
the Cauchy inequality to see that $|\delta
III_{2}(s)|\lesssim\overline{C}^{3}\overline{\epsilon}\epsilon_{0}^{2}\langle
s\rangle^{-2}$ for any $s\in[T_{1},T_{2}]$. The desired bounds on the
difference of solutions in (3.45) follow using also (3.52). ∎

##  4\. Proof of Proposition 2.3

The proof of Proposition 2.3 is similar to the proof of Proposition 2.2 in the
previous section. As before, the constant $\overline{B}$ depends only on
$d,\lambda_{0},\vartheta$, and is thought of as much larger than other
structural constants like $C_{0}^{\ast}$ in Lemma 2.1, $C_{0}$ in Lemma 2.5,
and $B_{1},B_{2}$ in Lemma 4.1 and Proposition 4.2 below.

The conclusions follow from Propositions 4.2 and 4.3 below. We prove first
bounds on the nonlinearities $\mathcal{N}^{\prime}_{i}$ defined in (2.18).

######  Lemma 4.1.

Assume that $g_{1},g_{2}\in X^{\sharp}_{[T_{1},T_{2}]}$ are as in Proposition
2.3. Then

$\begin{split}\sup_{t\in[T_{1},T_{2}],\,k\in\mathbb{Z}^{d}\setminus\\{0\\}}A^{\sharp}(t,k,tk)|\widehat{\mathcal{N}^{\prime}_{i}}(t,k)||k|^{-1/2}\langle
t\rangle^{6d}&\leq B_{1}\overline{\theta},\\\
\sup_{t\in[T_{1},T_{2}],\,k\in\mathbb{Z}^{d}\setminus\\{0\\}}\langle
k,tk\rangle^{-2}A^{\sharp}(t,k,tk)|(\widehat{\mathcal{N}^{\prime}_{1}}-\widehat{\mathcal{N}^{\prime}_{2}})(t,k)||k|^{-1/2}\langle
t\rangle^{6d}&\leq B_{1}\theta_{0},\end{split}$ (4.1)

for $i\in\\{1,2\\}$ and a constant $B_{1}=B_{1}(d,\lambda_{0})$ sufficiently
large.

###### Proof.

This is similar to the proof of Lemma 3.1, since the weights $A^{\sharp}$
satisfy similar bounds as the weights $A$. The only non-trivial difference is
that we need to replace the bounds (3.6) with different uniform bounds: for
any $t\in[0,\infty)$ and $k\in\mathbb{Z}^{d}\setminus\\{0\\}$ we have

$\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{t}^{\infty}\frac{|t-s||k|^{1/2}\langle
t\rangle^{6d}}{|l|^{1/2}\langle
s\rangle^{6d}}\frac{A^{\sharp}(t,k,tk)}{A^{\sharp}(s,k,tk)}\big{[}e^{-8\delta\langle
l,sl\rangle^{1/3}}+e^{-8\delta\langle k-l,tk-
sl\rangle^{1/3}}\big{]}\,ds\lesssim 1.$ (4.2)

As before, we prove these bounds in three steps.

Step 1. We show first that

$\begin{split}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{t}^{\infty}\frac{|t-s||k|^{1/2}\langle
t\rangle^{6d}}{|l|^{1/2}\langle
s\rangle^{6d}}\frac{A^{\sharp}(t,k,tk)}{A^{\sharp}(s,k,tk)}e^{-\delta(s^{1/3}+|l|^{1/3})}\,ds\lesssim
1.\end{split}$ (4.3)

This is similar to the proof of (3.7). The definition (2.21) shows that

$\lambda^{\sharp}(s,r)-\lambda^{\sharp}(t,r)=\delta^{2}\int_{t}^{s}\big{[}(1+u)^{-\delta-1}\langle
r\rangle^{1/3}+\big{(}1+u\langle r\rangle^{-2/3}\big{)}^{-\delta-1}\langle
r\rangle^{-1/3}\big{]}\,du,$ (4.4)

if $0\leq t\leq s$ and $r\in[0,\infty)$. Therefore

$\frac{A^{\sharp}(t,k,tk)}{A^{\sharp}(s,k,tk)}=e^{\lambda^{\ast}(t,|k,tk|)-\lambda^{\ast}(s,|k.tk|)}\leq
e^{-\delta^{2}|t-s|(1+s)^{-\delta-1}|k|^{1/3}\langle t\rangle^{1/3}}.$ (4.5)

Therefore, if $k,l\in\mathbb{Z}^{d}\setminus\\{0\\}$, and $s\geq t\geq 0$ then

$\begin{split}\frac{|t-s||k|^{1/2}\langle t\rangle^{6d}}{|l|^{1/2}\langle
s\rangle^{6d}}&\frac{A^{\sharp}(t,k,tk)}{A^{\sharp}(s,k,tk)}e^{-\delta(s^{1/3}+|l|^{1/3})}\\\
&\lesssim e^{-\delta|l|^{1/3}}e^{-\delta
s^{1/3}}|t-s||k|^{1/2}(1+|t-s||k|^{1/3}\langle s\rangle^{-1-\delta})^{-4}\\\
&\lesssim e^{-\delta|l|^{1/3}}e^{-\delta
s^{1/3}/2}|t-s||k|^{1/2}(1+|t-s||k|^{1/3})^{-4}.\end{split}$

The desired bounds (4.3) follow.

Step 2. We show now that for any $t\geq 0$ and
$k\in\mathbb{Z}^{d}\setminus\\{0\\}$ we have

$\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{t}^{\infty}\mathbf{1}_{\mathcal{R}^{\prime}}(l,s)\frac{|t-s||k|^{1/2}\langle
t\rangle^{6d}}{|l|^{1/2}\langle
s\rangle^{6d}}\frac{A^{\sharp}(t,k,tk)}{A^{\sharp}(s,k,tk)}e^{-\delta(|k-l|^{1/3}+|tk-
sl|^{1/3})}\,ds\lesssim 1,$ (4.6)

where

$\mathcal{R}^{\prime}:=\big{\\{}(l,s)\in(\mathbb{Z}^{d}\setminus\\{0\\})\times[t,\infty):\,s\geq
1.01(t-1)\,\,\text{ or }\,\,|k-l|\geq|k|/10\big{\\}}.$ (4.7)

Recall the identity (4.4). Therefore, if
$k,l\in\mathbb{Z}^{d}\setminus\\{0\\}$ and $s\geq\max\\{t,1.01(t-1)\\}$ then

$\frac{|t-s|\langle t\rangle^{6d}}{\langle
s\rangle^{6d}}\frac{A^{\sharp}(t,k,tk)}{A^{\sharp}(s,k,tk)}e^{-\delta|tk-
sl|^{1/3}}\lesssim\frac{|t-s|\langle t\rangle^{6d}}{\langle
s\rangle^{6d}}e^{-\delta^{4}(1+t)^{1/4}}.$ (4.8)

Moreover, if $k,l\in\mathbb{Z}^{d}\setminus\\{0\\}$, $t\leq s\leq 1.01(t-1)$,
and $|k-l|\geq|k|/10$ then $|tk-sl|=|s(k-l)+(t-s)k|\geq s|k-l|/2$, and the
bounds (4.8) follow in this case as well. The desired bounds (4.6) follow from
(3.12) and (4.8).

Step 3. Finally, we show that

$\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{t}^{\infty}(1-\mathbf{1}_{\mathcal{R}^{\prime}}(l,s))\frac{|t-s||k|^{1/2}\langle
t\rangle^{6d}}{|l|^{1/2}\langle
s\rangle^{6d}}\frac{A^{\sharp}(t,k,tk)}{A^{\sharp}(s,k,tk)}e^{-\delta(|k-l|^{1/3}+|tk-
sl|^{1/3})}\,ds\lesssim 1,$ (4.9)

where $\mathcal{R}^{\prime}$ is as in (4.7). We make the change of variables
$l=k-a$, so it suffices to prove that if $k\in\mathbb{Z}^{d}\setminus\\{0\\}$
and $t\in[10,\infty]$ then

$\begin{split}\sum_{a\in\mathbb{Z}^{d},\,|a|\leq|k|/8}\int_{t}^{1.01t}\frac{|t-s|A^{\sharp}(t,k,tk)}{A^{\sharp}(s,k,tk)}e^{-\delta(|a|^{1/3}+|sa+(t-s)k|^{1/3})}\,ds\lesssim
1.\end{split}$ (4.10)

We examine the identity (2.31) and use the second term in the right-hand side
to see that

$\begin{split}\lambda^{\sharp}(s,|k,tk|)-\lambda^{\sharp}(t,|k,tk|)&\geq\delta^{2}|t-s|(1+s\langle
k,tk\rangle^{-2/3})^{-1-\delta}\langle k,tk\rangle^{-1/3}\\\
&\geq\delta^{3}|t-s|t^{-1/3}|k|^{-1/3}(1+t^{1/3}|k|^{-2/3})^{-1-\delta},\end{split}$

if $t\geq 10$, $s\in[t,1.01t]$, and $k\in\mathbb{Z}^{d}\setminus\\{0\\}$.
Therefore

$\frac{A^{\sharp}(t,k,tk)}{A^{\sharp}(s,k,tk)}\leq
e^{-\delta^{4}|t-s|t^{-1/3}|k|^{-1/3}(1+t|k|^{-2})^{-1/3-\delta}}.$ (4.11)

Since $|sa+(t-s)k|\geq\big{|}s|a|-(s-t)|k|\big{|}$, for (3.15) it suffices to
show that if $a\in\mathbb{Z}^{d}$, $k\in\mathbb{Z}^{d}\setminus\\{0\\}$,
$t\in[10,T]$, and $|a|\leq|k|/8$ then

$\begin{split}&\int_{t}^{1.01t}K^{\prime}_{a}(t,k,s)\,ds\lesssim 1,\\\
&K^{\prime}_{a}(t,k,s):=|t-s|e^{-\delta^{4}|t-s|t^{-1/3}|k|^{-1/3}(1+t|k|^{-2})^{-1/3-\delta}}e^{-\delta|s|a|-(s-t)|k||^{1/3}}e^{-\delta|a|^{1/3}/2}.\end{split}$
(4.12)

The bounds (4.12) are easy in the case $a=0$, since
$|K^{\prime}_{0}(t,k,s)|\lesssim|t-s|e^{-\delta|t-s|^{1/3}}\lesssim\langle
t-s\rangle^{-4}$. On the other hand, if $a\in\mathbb{Z}^{d}\setminus\\{0\\}$
then

$|K^{\prime}_{a}(t,k,s)|\lesssim|t-s|e^{-\delta|t-s|^{1/3}|k|^{1/3}/4}\lesssim\langle
t-s\rangle^{-4}$

if $(s-t)|k|\notin[t|a|/2,2t|a|]$, and

$\begin{split}|K_{a}(t,k,s)|&\lesssim
t|k|^{-1}e^{-\delta^{4}(t|k|^{-2})^{2/3}(1+t|k|^{-2})^{-1/3-\delta}/4}e^{-\delta|s|a|-(s-t)|k||^{1/3}}e^{-\delta|a|^{1/3}/4}\\\
&\lesssim|k|e^{-\delta|s|a|-(s-t)|k||^{1/3}}e^{-\delta|a|^{1/3}/4}\end{split}$

if $(t-s)|k|\in[t|a|/2,2t|a|]$. The desired bounds (4.12) follow in this case
as well. ∎

We can now combine these bounds with the identity (2.20) and the bounds (2.11)
to prove our main estimates on the functions $\rho_{1}$, $\rho_{2}$, and
$\rho_{1}-\rho_{2}$. The argument is identical to the argument in the proof of
Proposition 3.2.

######  Proposition 4.2.

With the assumptions and notations of Proposition 2.3, there is a constant
$B_{2}=B_{2}(d,\lambda_{0},\vartheta)\geq 1$ such that for any
$t\in[T_{1},T_{2}]$ and $i\in\\{1,2\\}$

$\mathcal{Z}_{g_{i}}^{\sharp,0}(t)\langle t\rangle^{6d}\leq
B_{2}\overline{\theta}\qquad\text{ and
}\qquad\mathcal{Z}_{g_{1}-g_{2}}^{\sharp,2}(t)\langle t\rangle^{6d}\leq
B_{2}\theta_{0}.$ (4.13)

As a consequence, if $\overline{B}\geq B_{2}$ then there is a constant
$C^{\prime}_{0}=C^{\prime}_{0}(d)$ such that

$\begin{split}\big{\|}A^{\sharp}(t,k,tk)\widehat{\rho_{i}}(t,k)|k|^{-1/2}\|_{L^{2}_{k}}&\leq
C^{\prime}_{0}(\overline{B}^{d}B_{2})^{1/(d+1)}\overline{\theta}\langle
t\rangle^{-3},\\\ \big{\|}\langle
k,tk\rangle^{-2}A^{\sharp}(t,k,tk)(\widehat{\rho_{1}-\rho_{2}})(t,k)|k|^{-1/2}\|_{L^{2}_{k}}&\leq
C^{\prime}_{0}(\overline{B}^{d}B_{2})^{1/(d+1)}\theta_{0}\langle
t\rangle^{-3}\end{split}$ (4.14)

for any $t\in[T_{1},T_{2}]$ and $i\in\\{1,2\\}$.

As before, we note that the $L^{2}$ bounds (4.14) are needed later, in the
proof of the estimates (4.28) in Proposition 4.3.

Finally we prove our main improved energy estimates.

######  Proposition 4.3.

With the assumptions and notations of Proposition 2.3, for any
$t\in[T_{1},T_{2}]$ and $i\in\\{1,2\\}$ we have

$\mathcal{E}^{\sharp,0}_{g_{i}}(t)\leq\overline{B}^{2}\overline{\theta}^{2}/4\qquad\text{
and
}\qquad\mathcal{E}^{\sharp,2}_{g_{1}-g_{2}}(t)\leq\overline{B}^{2}\theta_{0}^{2}/4.$
(4.15)

###### Proof.

This is similar to the proof of Proposition 3.3. As before, to justify the
calculations we define the mollified energies

$\begin{split}&\mathcal{E}^{\sharp,p,L}_{h}(t):=\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-2p}A^{\sharp}_{L}(t,k,\xi)^{2}\Big{\\{}\sum_{|a|\leq
d^{\prime}}\big{|}D^{a}_{\xi}\widehat{h}(t,k,\xi)\big{|}^{2}\Big{\\}}\,d\xi,\\\
&A^{\sharp}_{L}(t,k,\xi):=(1+2^{-L}\langle
k,\xi\rangle)^{-4}A^{\sharp}(t,k,\xi),\end{split}$

where $p\in[0,2]$, $L\geq 4$, and $h\in X^{\sharp}_{[T_{1},T_{2}]}$. Then we
calculate

$\frac{d}{dt}\mathcal{E}^{\sharp,0,L}_{g_{i}}(t)=I^{\sharp}_{i}(t)+II^{\sharp}_{i}(t)+III^{\sharp}_{i}(t)+IV^{\sharp}_{i}(t),$
(4.16)

where, with $\dot{A_{L}^{\sharp}}=\partial_{t}A_{L}^{\sharp}$,

$I^{\sharp}_{i}(t):=2\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}A_{L}^{\sharp}(t,k,\xi)\dot{A}_{L}^{\sharp}(t,k,\xi)\big{|}D^{a}_{\xi}\widehat{g_{i}}(t,k,\xi)\big{|}^{2}\,d\xi,$
(4.17) $\begin{split}II^{\sharp}_{i}(t)&:=-2\Re\Big{\\{}\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}A_{L}^{\sharp}(t,k,\xi)^{2}\overline{D^{a}_{\xi}\widehat{g_{i}}(t,k,\xi)}\\\
&\qquad\qquad\times
D^{a}_{\xi}\Big{[}\widehat{\rho_{i}}(t,k)\widehat{M_{0}}(\xi-
tk)\frac{k\cdot(\xi-tk)}{|k|^{2}}\Big{]}\,d\xi\Big{\\}},\end{split}$ (4.18)
$\begin{split}III^{\sharp}_{i}(t)&:=-\frac{2}{(2\pi)^{d}}\Re\Big{\\{}\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}A_{L}^{\sharp}(t,k,\xi)^{2}\overline{D^{a}_{\xi}\widehat{g_{i}}(t,k,\xi)}\\\
&\qquad\qquad\qquad\times\Big{[}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\widehat{\rho_{i}}(t,l)D^{a}_{\xi}\widehat{g_{i}}(t,k-l,\xi-
tl)\frac{l\cdot(\xi-tk)}{|l|^{2}}\Big{]}\,d\xi\Big{\\}},\end{split}$ (4.19)
$\begin{split}IV^{\sharp}_{i}(t)&:=-\frac{2}{(2\pi)^{d}}\Re\Big{\\{}\sum_{|a|\leq
d^{\prime}}\sum_{|a^{\prime}|=1,\,a^{\prime}\leq
a}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}A_{L}^{\sharp}(t,k,\xi)^{2}\overline{D^{a}_{\xi}\widehat{g_{i}}(t,k,\xi)}\\\
&\qquad\qquad\qquad\times\Big{[}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\widehat{\rho_{i}}(t,l)(D^{a-a^{\prime}}_{\xi}\widehat{g_{i}})(t,k-l,\xi-
tl)\frac{D^{a^{\prime}}_{\xi}[l\cdot(\xi-
tk)]}{|l|^{2}}\Big{]}\,d\xi\Big{\\}}.\end{split}$ (4.20)

Notice that $I^{\sharp}_{i}(t)\geq 0$, since $\dot{A^{\sharp}}(t,k,\xi)\geq
0$. Using (4.16) we can therefore estimate

$\begin{split}&\mathcal{E}^{\sharp,0,L}_{g_{i}}(t)=\mathcal{E}^{\sharp,0,L}_{g_{i}}(T_{2})-\int_{t}^{T_{2}}\big{[}I^{\sharp}_{i}(s)+II^{\sharp}_{i}(s)+III^{\sharp}_{i}(s)+IV^{\sharp}_{i}(s)\big{]}\,ds\\\
&\leq\mathcal{E}^{\sharp,0}_{g_{i}}(T_{2})+\int_{t}^{T_{2}}|II^{\sharp}_{i}(s)|\,ds+\int_{t}^{T_{2}}|IV^{\sharp}_{i}(s)|\,ds+\int_{t}^{T_{2}}\big{[}|III^{\sharp}_{i}(s)|-|I^{\sharp}_{i}(s)|\big{]}\,ds.\end{split}$
(4.21)

Similarly, we calculate

$\frac{d}{dt}\mathcal{E}^{\sharp,2,L}_{g_{1}-g_{2}}(t)=\delta
I^{\sharp}(t)+\delta II^{\sharp}(t)+\delta III^{\sharp}(t)+\delta
IV^{\sharp}(t),$ (4.22)

where

$\delta I^{\sharp}(t):=2\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-4}A_{L}^{\sharp}(t,k,\xi)\dot{A_{L}^{\sharp}}(t,k,\xi)\big{|}D^{a}_{\xi}(\widehat{g_{1}}-\widehat{g_{2}})(t,k,\xi)\big{|}^{2}\,d\xi,$
(4.23) $\begin{split}\delta II^{\sharp}(t)&:=-2\Re\Big{\\{}\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-4}A_{L}^{\sharp}(t,k,\xi)^{2}\overline{D^{a}_{\xi}(\widehat{g_{1}}-\widehat{g_{2}})(t,k,\xi)}\\\
&\qquad\qquad\times
D^{a}_{\xi}\Big{[}(\widehat{\rho_{1}}-\widehat{\rho_{2}})(t,k)\widehat{M_{0}}(\xi-
tk)\frac{k\cdot(\xi-tk)}{|k|^{2}}\Big{]}\,d\xi\Big{\\}},\end{split}$ (4.24)
$\begin{split}&\delta
III^{\sharp}(t):=-\frac{2}{(2\pi)^{d}}\Re\Big{\\{}\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-4}A_{L}^{\sharp}(t,k,\xi)^{2}\overline{D^{a}_{\xi}(\widehat{g_{1}}-\widehat{g_{2}})(t,k,\xi)}\\\
&\quad\times\Big{[}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}[\widehat{\rho_{1}}(t,l)D^{a}_{\xi}\widehat{g_{1}}(t,k-l,\xi-
tl)-\widehat{\rho_{2}}(t,l)D^{a}_{\xi}\widehat{g_{2}}(t,k-l,\xi-
tl)]\frac{l\cdot(\xi-tk)}{|l|^{2}}\Big{]}\,d\xi\Big{\\}},\end{split}$ (4.25)
$\begin{split}&\delta
IV^{\sharp}(t):=-\frac{2}{(2\pi)^{d}}\Re\Big{\\{}\sum_{|a|\leq
d^{\prime}}\sum_{|a^{\prime}|=1,\,a^{\prime}\leq
a}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\langle
k,\xi\rangle^{-4}A_{L}^{\sharp}(t,k,\xi)^{2}\overline{D^{a}_{\xi}(\widehat{g_{1}}-\widehat{g_{2}})(t,k,\xi)}\\\
&\quad\times\Big{[}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}[\widehat{\rho_{1}}(t,l)(D^{a-a^{\prime}}_{\xi}\widehat{g_{1}})(t,k-l,\xi-
tl)-\widehat{\rho_{2}}(t,l)(D^{a-a^{\prime}}_{\xi}\widehat{g_{2}})(t,k-l,\xi-
tl)]\\\ &\qquad\qquad\qquad\times\frac{D^{a^{\prime}}_{\xi}[l\cdot(\xi-
tk)]}{|l|^{2}}\Big{]}\,d\xi\Big{\\}}.\end{split}$ (4.26)

As before, we notice that $\delta I^{\sharp}(t)\geq 0$, so

$\begin{split}\mathcal{E}^{\sharp,2,L}_{g_{1}-g_{2}}(t)&\leq\mathcal{E}^{\sharp,2}_{g_{1}-g_{2}}(T_{2})+\int_{t}^{T_{2}}|\delta
II^{\sharp}(s)|\,ds+\int_{t}^{T_{2}}|\delta IV^{\sharp}(s)|\,ds\\\
&+\int_{t}^{T_{2}}\big{[}|\delta III^{\sharp}(s)|-|\delta
I^{\sharp}(s)|\big{]}\,ds.\end{split}$ (4.27)

As in the proof of Lemma 3.4, we can use the bounds (3.37) and (4.14) to show
that

$\int_{t}^{T_{2}}\big{|}II^{\sharp}_{i}(s)\big{|}\,ds\leq\overline{B}^{2}\overline{\theta}^{2}/8\qquad\text{
and }\qquad\int_{t}^{T_{2}}\big{|}\delta
II^{\sharp}(s)\big{|}\,ds\leq\overline{B}^{2}\theta_{0}^{2}/8,$ (4.28)

for any $t\in[T_{1},T_{2}]$ and $i\in\\{1,2\\}$. Moreover, as in the proofs of
Lemmas 3.5 and 3.6, we can use the bounds (2.32) and (2.33) (which are the
same for the weights $A$ and $A^{\sharp}$) to show that

$\int_{t}^{T_{2}}\big{|}IV^{\sharp}_{i}(s)\big{|}\,ds\leq\overline{\theta}^{2}\qquad\text{
and }\qquad\int_{t}^{T_{2}}\big{|}\delta
IV^{\sharp}(s)\big{|}\,ds\leq\theta_{0}^{2},$
$\int_{t}^{T_{2}}\big{[}|III_{i}^{\sharp}(s)|-|I_{i}^{\sharp}(s)|\big{]}\,ds\leq\overline{\theta}^{2}\qquad\text{
and }\qquad\int_{t}^{T_{2}}\big{[}|\delta III^{\sharp}(s)|-|\delta
I^{\sharp}(s)|\big{]}\,ds\leq\theta_{0}^{2}.$

The desired bounds (4.15) follow from these last three estimates. ∎

##  5\. Proofs of the main theorems

We can now prove our three main theorems.

###  5.1. Proof of Theorem 1.1

We use the following simple continuation criterion:

######  Lemma 5.1.

Assume $T_{1}\leq T_{2}\in[0,\infty)$,
$\lambda_{1}\in[0.6\lambda_{0},0.9\lambda_{0}]$, and $g\in X_{[T_{1},T_{2}]}$
is a real-valued solution of the system (1.9) satisfying
$\widehat{g}(t,0,0)=0$ for any $t\in[T_{1},T_{2}]$. Assume that

$\sup_{t\in[T_{1},T_{2}]}\mathcal{E}^{0}_{g}(t)\leq\rho_{0}^{2},$ (5.1)

where $\rho_{0}=\rho_{0}(d,\lambda_{0},\vartheta)$ is sufficiently small. Then
$g$ can be extended to a real-valued solution $g^{\prime}\in
X_{[T_{1},T^{\prime}_{2}]}$ of the system (1.9), for
$T^{\prime}_{2}=T_{2}+c_{d,\lambda_{0},\vartheta}(1+T_{2})^{-2}$, satisfying

$\sup_{t\in[T_{1},T^{\prime}_{2}]}\mathcal{E}^{0}_{g}(t)\leq 2\rho_{0}^{2}.$
(5.2)

###### Proof.

This is a local well-posedness result and we can use a standard parabolic
regularization argument to prove it. We provide some details for the sake of
completeness.

For $\varepsilon>0$ we are looking for solutions $g^{\varepsilon}\in
X_{[T_{2},T_{2}^{\varepsilon}]}$ of the equation (in the Fourier space)

$\begin{split}&\partial_{t}\widehat{g^{\varepsilon}}(t,k,\xi)+\varepsilon(|k|^{2}+|\xi|^{2})\widehat{g^{\varepsilon}}(t,k,\xi)+\widehat{g^{\varepsilon}}(t,k,tk)\widehat{M_{0}}(\xi-
tk)\frac{k\cdot(\xi-tk)}{|k|^{2}}\\\
&\qquad+\frac{1}{(2\pi)^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\widehat{g^{\varepsilon}}(t,l,tl)\widehat{g^{\varepsilon}}(t,k-l,\xi-
tl)\frac{l\cdot(\xi-tk)}{|l|^{2}}=0,\end{split}$ (5.3)

with initial data $\widehat{g^{\varepsilon}}(T_{2})=\widehat{g}(T_{2})$
(compare with (2.1)). Such solutions can be constructed by a fixed-point
argument in the space $X_{[T_{2},T_{2}^{\varepsilon}]}$, using the identity

$\widehat{g^{\varepsilon}}(t,k,\xi)=e^{-\varepsilon(|k|^{2}+|\xi|^{2})(t-T_{2})}\widehat{g}(T_{2},k,\xi)-\sum_{i\in\\{1,2\\}}\int_{T_{2}}^{t}e^{-\varepsilon(|k|^{2}+|\xi|^{2})(t-s)}\mathcal{M}_{i}(\widehat{g^{\varepsilon}})(s,k,\xi)\,ds,$
(5.4)

where $T_{2}^{\varepsilon}-T_{2}$ is sufficiently small (depending on
$\varepsilon,T_{2},d,\lambda_{0},\vartheta$) and

$\begin{split}&\mathcal{M}_{1}(f)(t,k,\xi):=f(t,k,tk)\widehat{M_{0}}(\xi-
tk)\frac{k\cdot(\xi-tk)}{|k|^{2}},\\\
&\mathcal{M}_{2}(f)(t,k,\xi):=\frac{1}{(2\pi)^{d}}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}f(t,l,tl)f(t,k-l,\xi-
tl)\frac{l\cdot(\xi-tk)}{|l|^{2}}.\end{split}$ (5.5)

We may assume that the solutions $g^{\varepsilon}$ are real-valued and
$\mathcal{E}^{0}_{g^{\varepsilon}}(t)\leq 1.5\rho_{0}^{2}$ for any
$t\in[T_{2},T_{2}^{\varepsilon}]$. As in the proof of Proposition 3.3, we can
estimate the energy increment by

$\frac{d}{dt}\mathcal{E}_{g^{\varepsilon}}^{0}(t)\lesssim_{d,\lambda_{0},\vartheta}(1+T_{2})^{2}\mathcal{E}_{g^{\varepsilon}}^{0}(t).$

Therefore the solution $g^{\varepsilon}$ of (5.3) can be extended on the time
interval $[T_{2},T^{\prime}_{2}]$,
$T^{\prime}_{2}:=T_{2}+c_{d,\lambda_{0},\vartheta}(1+T_{2})^{-2}$, satisfying
the uniform bounds $\mathcal{E}^{0}_{g^{\varepsilon}}(t)\leq 1.5\rho_{0}^{2}$
for any $t\in[T_{2},T^{\prime}_{2}]$.

We can now let $\varepsilon\to 0$ to construct the desired extension $g$.
Indeed, let

$\mathcal{L}_{\varepsilon,\varepsilon^{\prime}}(t):=\sum_{|a|\leq
d^{\prime}}\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\big{|}D^{a}_{\xi}\widehat{(g^{\varepsilon}-g^{\varepsilon^{\prime}})}(t,k,\xi)\big{|}^{2}\,d\xi.$

As in the proof of Proposition 3.3, we can estimate

$|\partial_{t}\mathcal{L}_{\varepsilon,\varepsilon^{\prime}}(t)|\lesssim_{d,\lambda_{0},\vartheta}(\varepsilon+\varepsilon^{\prime})[\mathcal{L}_{\varepsilon,\varepsilon^{\prime}}(t)]^{1/2}+\mathcal{L}_{\varepsilon,\varepsilon^{\prime}}(t),$

for any $t\in[T_{2},T^{\prime}_{2}]$. Since
$\mathcal{L}_{\varepsilon,\varepsilon^{\prime}}(T_{2})=0$ it follows that
$\lim_{\varepsilon,\varepsilon^{\prime}\to
0}\|\mathcal{L}_{\varepsilon,\varepsilon^{\prime}}(t)\|_{L^{\infty}_{t}}=0$.
Therefore the sequence $g^{\varepsilon}$ converges to a solution $g\in
X_{[T_{1},T^{\prime}_{2}]}$ of the equation (2.1), which satisfies the uniform
bounds (5.2), as desired. ∎

###### Proof of Theorem 1.1.

First, we combine Lemma 5.1 and the bootstrap Proposition 2.2 (both with
$\lambda_{1}=0.9\lambda_{0}$) to construct a unique solution $g\in
C([0,\infty):\mathcal{G}_{d^{\prime}}^{\lambda_{0}/2,1/3}(\mathbb{T}^{d}\times\mathbb{R}^{d}))$
of the equation (1.9) with initial data $g(0)=f_{0}$, satisfying

$\sum_{|a|\leq d^{\prime}}\|D^{a}_{\xi}\widehat{g}(t,k,\xi)\cdot
A(t,k,\xi)\|_{L^{2}_{k,\xi}}\lesssim\kappa_{0},\qquad\int_{\mathbb{T}^{d}\times\mathbb{R}^{d}}g(t,x,v)\,dxdv=0,$
(5.6)

for any $t\in[0,\infty)$, where the implied constants in this proof are
allowed to depend only on $d,\lambda_{0},\vartheta$. In particular, using
Lemma 2.5, for any $t\in[0,\infty)$ we have

$\|\widehat{\rho}(t,k)\cdot A(t,k,tk)\|_{L^{2}_{k}}\lesssim\kappa_{0}.$ (5.7)

Notice that $A(t,k,tk)\geq e^{0.9\lambda_{0}\langle k,tk\rangle^{1/3}}\geq
e^{0.6\lambda_{0}\langle k\rangle^{1/3}}e^{0.3\lambda_{0}\langle
t\rangle^{1/3}}$ if $k\in\mathbb{Z}^{d}\setminus\\{0\\}$. Thus, using (5.7),

$\|\rho(t)\|_{\mathcal{G}^{0.6\lambda_{0},1/3}(\mathbb{T}^{d})}\lesssim\kappa_{0}e^{-0.3\lambda_{0}\langle
t\rangle^{1/3}}.$ (5.8)

Moreover, using the identity (2.1), for any $t\leq t^{\prime}\in[0,\infty)$ we
have

$\begin{split}\widehat{g}(t^{\prime},k,\xi)-\widehat{g}(t,k,\xi)&+\int_{t}^{t^{\prime}}\widehat{\rho}(s,k)\widehat{M_{0}}(\xi-
sk)\frac{k\cdot(\xi-sk)}{|k|^{2}}\,ds\\\
&+\frac{1}{(2\pi)^{d}}\sum_{l\in\mathbb{Z}^{d}}\int_{t}^{t^{\prime}}\widehat{\rho}(s,l)\widehat{g}(s,k-l,\xi-
sl)\frac{l\cdot(\xi-sk)}{|l|^{2}}\,ds=0.\end{split}$ (5.9)

Therefore, recalling the definitions (3.2), (3.35), and (3.38),

$\begin{split}\sum_{|a|\leq d}&e^{0.6\lambda_{0}\langle
k,\xi\rangle^{1/3}}\big{|}D^{a}_{\xi}[\widehat{g}(t^{\prime},k,\xi)-\widehat{g}(t,k,\xi)]\big{|}\lesssim\int_{t}^{t^{\prime}}|\widehat{A\rho}(s,k)|M_{0}^{\ast}(\xi-
sk)\frac{e^{0.6\lambda_{0}\langle k,\xi\rangle^{1/3}}}{A(s,k,sk)}\,ds\\\
&+\sum_{|a|\leq
d}\sum_{l\in\mathbb{Z}^{d}\setminus\\{0\\}}\int_{t}^{t^{\prime}}|\widehat{A\rho}(s,l)||\widehat{Ag_{a}}(s,k-l,\xi-
sl)|\frac{|\xi-sk|e^{0.6\lambda_{0}\langle
k,\xi\rangle^{1/3}}}{A(s,l,sl)A(s,k-l,\xi-sl)}\,ds.\end{split}$ (5.10)

Notice that

$\begin{split}\frac{e^{0.6\lambda_{0}\langle
k,\xi\rangle^{1/3}}}{A(s,k,sk)}&\lesssim\frac{e^{0.6\lambda_{0}\langle
k,sk\rangle^{1/3}}e^{0.6\lambda_{0}\langle\xi-
sk\rangle^{1/3}}}{A(s,k,sk)}\lesssim\frac{e^{0.6\lambda_{0}\langle\xi-
sk\rangle^{1/3}}}{e^{0.3\lambda_{0}\langle k,sk\rangle^{1/3}}},\\\ \frac{|\xi-
sk|e^{0.6\lambda_{0}\langle k,\xi\rangle^{1/3}}}{A(s,l,sl)A(s,k-l,\xi-
sl)}&\lesssim\frac{|\xi-sk|e^{0.6\lambda_{0}\langle
l,sl\rangle^{1/3}}e^{0.6\lambda_{0}\langle k-l,\xi-
sl\rangle^{1/3}}}{A(s,l,sl)A(s,k-l,\xi-
sl)}\lesssim\frac{1}{e^{0.29\lambda_{0}\langle
l,sl\rangle^{1/3}}}.\end{split}$ (5.11)

Notice that
$\|\widehat{Ag_{a}}(s,k,\xi)\|_{L^{2}_{k,\xi}}+\|\widehat{A\rho}(s,k)\|_{L^{2}_{k}}\lesssim\kappa_{0}$,
due to (5.6)–(5.7). Recalling also the bounds (3.37), it follows from the last
two estimates that

$\begin{split}\sum_{|a|\leq d}\big{\|}e^{0.6\lambda_{0}\langle
k,\xi\rangle^{1/3}}D^{a}_{\xi}[\widehat{g}(t^{\prime},k,\xi)-\widehat{g}(t,k,\xi)]\big{\|}_{L^{2}_{k,\xi}}\lesssim\int_{t}^{t^{\prime}}\kappa_{0}e^{-0.29\lambda_{0}\langle
s\rangle^{1/3}}\,ds\lesssim\kappa_{0}e^{-0.28\lambda_{0}\langle
t\rangle^{1/3}}.\end{split}$

Therefore, if $t\leq t^{\prime}\in[0,\infty)$,

$\|g(t^{\prime})-g(t)\|_{\mathcal{G}_{d^{\prime}}^{0.6\lambda_{0},1/3}}\lesssim\kappa_{0}e^{-0.28\lambda_{0}\langle
t\rangle^{1/3}}.$ (5.12)

In particular, we can define $g_{\infty}:=\lim_{t\to\infty}g(t)$ in
$\mathcal{G}_{d^{\prime}}^{0.6\lambda_{0},1/3}$, and the desired bounds (1.11)
follow from (5.8) and (5.12). In fact
$\|g_{\infty}\|_{\mathcal{G}_{d^{\prime}}^{0.8\lambda_{0},1/3}}\lesssim\kappa_{0}$,
using again the uniforms bounds (5.6). This completes the proof of Theorem 1.1
(i).

To prove part (ii), assume $f_{01},f_{02}\in
B_{\overline{\kappa}}(\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3})$ are initial
data and let $g_{1},g_{2}\in
C([0,\infty):\mathcal{G}_{d^{\prime}}^{\lambda_{0}/2,1/3})$ denote the
corresponding solutions of the system (1.9). We first set
$\lambda_{1}=0.9\lambda_{0}$ and notice that the function
$t\mapsto\mathcal{E}^{2}_{g_{1}-g_{2}}(t)$ is a continuous map from
$[0,\infty)$ to $[0,1]$ (since
$\mathcal{E}^{0}_{g_{i}}(t)\lesssim\overline{\kappa}^{2}$ for any
$t\in[0,\infty)$ and $i\in\\{1,2\\}$). In view of Proposition 2.2 it follows
that

$\mathcal{E}^{2}_{g_{1}-g_{2}}(t)\lesssim\|f_{01}-f_{02}\|^{2}_{\mathcal{G}_{d^{\prime}}^{\lambda_{0},1/3}}\qquad\text{
for any }t\in[0,\infty).$

This gives the second inequality in (1.12). To prove the reverse inequality we
set $\lambda_{1}=0.7\lambda_{0}$, notice that the function
$t\mapsto\mathcal{E}^{\sharp,2}_{g_{1}-g_{2}}(t)$ is a continuous map from
$[0,\infty)$ to $[0,1]$, and use Proposition 2.3 to prove that
$\mathcal{E}^{\sharp,2}_{g_{1}-g_{2}}(t^{\prime})\lesssim\|(g_{1}-g_{2})(t)\|^{2}_{\mathcal{G}_{d^{\prime}}^{3\lambda_{0}/4,1/3}}$
for any $t^{\prime}\leq t\in[0,\infty)$. This completes the proof of Theorem
1.1 (ii). ∎

###  5.2. Proof of Theorem 1.2

We need again a continuation criterion, similar to Lemma 5.1.

######  Lemma 5.2.

Assume $T_{1}\leq T_{2}\in[0,\infty)$,
$\lambda_{1}\in[0.6\lambda_{0},0.9\lambda_{0}]$, and $g\in
X^{\sharp}_{[T_{1},T_{2}]}$ is a real-valued solution of the system (1.9)
satisfying $\widehat{g}(t,0,0)=0$ for any $t\in[T_{1},T_{2}]$. Assume that

$\sup_{t\in[T_{1},T_{2}]}\mathcal{E}^{\sharp,0}_{g}(t)\leq\rho_{0}^{2},$
(5.13)

where $\rho_{0}=\rho_{0}(d,\lambda_{0},\vartheta)$ is sufficiently small. Then
$g$ can be extended to a real-valued solution $g^{\prime}\in
X^{\sharp}_{[T^{\prime}_{1},T_{2}]}$ of the system (1.9), for
$T^{\prime}_{1}=\max\\{0,T_{1}-c_{d,\lambda_{0},\vartheta}(1+T_{1})^{-2}\\}$,
satisfying

$\sup_{t\in[T^{\prime}_{1},T_{1}]}\mathcal{E}^{\sharp,0}_{g}(t)\leq
2\rho_{0}^{2}.$ (5.14)

The proof of this lemma is similar to the proof of Lemma 5.1.

To prove Theorem 1.2 we start with a final state data $g_{\infty}$ satisfying
(1.13), set $\lambda_{1}=0.9\lambda_{0}$, and construct a sequence of function
$g_{n}\in
C\big{(}[0,\infty):\mathcal{G}_{d^{\prime}}^{\lambda_{0}/2,1/3}\big{)}$,
$n\geq 1$, satisfying the following properties:

(1) For any $t\in[n,\infty)$ we have $g_{n}(t)=g_{\infty}$;

(2) The function $g_{n}\in
C\big{(}[0,n]:\mathcal{G}_{d^{\prime}}^{\lambda_{0}/2,1/3}\big{)}$ is a
solution of the system (1.9), satisfying

$\sum_{|a|\leq d^{\prime}}\|D^{a}_{\xi}\widehat{g_{n}}(t,k,\xi)\cdot
A^{\sharp}(t,k,\xi)\|_{L^{2}_{k,\xi}}\lesssim\overline{\kappa}^{\prime}\quad\text{
and }\quad\widehat{g_{n}}(t,0,0)=0\quad\text{ for any }t\in[0,n].$ (5.15)

To prove property (2) we set $g_{n}(n)=g_{\infty}$ and use Proposition 2.3 and
Lemma 5.2 to construct the solution $g_{n}\in
C\big{(}[0,n]:\mathcal{G}_{d^{\prime}}^{\lambda_{0}/2,1/3}\big{)}$ of the
system (1.9) satisfying the bounds (5.15).

We show now that $\\{g_{n}\\}_{n\geq 1}$ is a Cauchy sequence in
$C\big{(}[0,\infty):\mathcal{G}_{d^{\prime}}^{0.75\lambda_{0},1/3}\big{)}$.
Indeed assume that $n\leq n^{\prime}\in\mathbb{Z}_{+}$ are sufficiently large.
Clearly $g_{n}(t)=g_{n^{\prime}}(t)=g_{\infty}$ if $t\geq n^{\prime}$.
Moreover, if $t\in[n,n^{\prime}]$ then we use the identity (5.9), in the form

$\begin{split}\widehat{g_{n^{\prime}}}(n^{\prime},k,\xi)-\widehat{g_{n^{\prime}}}(t,k,\xi)&+\int_{t}^{n^{\prime}}\widehat{\rho_{n^{\prime}}}(s,k)\widehat{M_{0}}(\xi-
sk)\frac{k\cdot(\xi-sk)}{|k|^{2}}\,ds\\\
&+\frac{1}{(2\pi)^{d}}\sum_{l\in\mathbb{Z}^{d}}\int_{t}^{n^{\prime}}\widehat{\rho_{n^{\prime}}}(s,l)\widehat{g_{n^{\prime}}}(s,k-l,\xi-
sl)\frac{l\cdot(\xi-sk)}{|l|^{2}}\,ds=0.\end{split}$

Then we use the uniform bounds (5.15) and estimate as in (5.10)–(5.11) to show
that

$\|g_{n^{\prime}}(n^{\prime})-g_{n^{\prime}}(t)\|_{\mathcal{G}_{d^{\prime}}^{0.85\lambda_{0},1/3}}\lesssim\overline{\kappa}^{\prime}e^{-\delta\langle
t\rangle^{1/3}}.$

Since $g_{n}(t)=g_{\infty}=g_{n^{\prime}}(n^{\prime})$, this shows that
$\|g_{n}(t)-g_{n^{\prime}}(t)\|_{\mathcal{G}_{d^{\prime}}^{0.85\lambda_{0},1/3}}\lesssim\overline{\kappa}^{\prime}e^{-\delta\langle
t\rangle^{1/3}}$ if $t\in[n,n^{\prime}]$. We can then use Proposition 2.3 on
the time interval $[T_{1},T_{2}]=[0,n]$, with $\lambda_{1}=0.8\lambda_{0}$, to
show that
$\|g_{n}(t)-g_{n^{\prime}}(t)\|_{\mathcal{G}_{d^{\prime}}^{0.75\lambda_{0},1/3}}\lesssim\overline{\kappa}^{\prime}e^{-\delta\langle
n\rangle^{1/3}}$ if $t\in[0,n]$. Therefore

$\sup_{t\in[0,\infty)}\|g_{n}(t)-g_{n^{\prime}}(t)\|_{\mathcal{G}_{d^{\prime}}^{0.75\lambda_{0},1/3}}\lesssim\overline{\kappa}^{\prime}e^{-\delta\langle
n\rangle^{1/3}}\qquad\text{ for any }n\leq n^{\prime}\in\mathbb{Z}_{+}.$
(5.16)

In particular, recalling the uniform bounds (5.15), $\\{g_{n}\\}_{n\geq 1}$ is
a Cauchy sequence in
$C\big{(}[0,\infty):\mathcal{G}_{d^{\prime}}^{3\lambda_{0}/4,1/3}\big{)}$ and
its limit $g\in
C\big{(}[0,\infty):\mathcal{G}_{d^{\prime}}^{3\lambda_{0}/4,1/3}\big{)}$ is a
solution of the final state problem.

The uniqueness of this solution follows from the quantitative bounds (1.12).
The bounds (1.15) follow also as in the proof of Theorem 1.1, using
Proposition 2.2 for the first inequality and Proposition (2.3) for the second
inequality. This completes the proof of Theorem 1.2.

###  5.3. Proof of Theorem 1.3

We define $h_{\infty}(x,v):=g_{\infty}(-x,v)$ and use Theorem 1.2 to construct
the corresponding solution $h\in
C([0,\infty):\mathcal{G}_{d^{\prime}}^{3\lambda_{0}/4,1/3})$ of the system
(1.9). Then we define $g(t,x,v):=h(-t,-x,v)$, and notice that $g\in
C([-\infty,0]:\mathcal{G}_{d^{\prime}}^{3\lambda_{0}/4,1/3})$ is a solution of
the system (1.9) on the time interval $(-\infty,0]$. Finally, we extend the
solution $g$ to the time interval $[0,\infty)$, using Theorem 1.1, and define
the corresponding final state $g_{\infty}$. The desired bounds (1.18) follow
from (1.15) and (1.12).

##  6\. Proof of Lemma 2.1

In this section we provide a self-contained proof of Lemma 2.1. The implied
constants in this proof are allowed to depend only on the parameters $d$,
$\lambda_{0}$, and $\vartheta$.

(i) Using the definition (2.4) and integration by parts, we have

$i\tau L(\tau,k)=\int_{0}^{\infty}e^{-i\tau
s}\big{[}\widehat{M_{0}}(sk)+sk\cdot\nabla\widehat{M_{0}}(sk)\big{]}\,ds.$
(6.1)

Using Plancherel theorem and (1.6) it follows that

$\big{\|}\langle\alpha\rangle
L(\alpha+i\beta,k)\big{\|}_{L^{2}_{\alpha}}\lesssim|k|^{-1/2}\qquad\text{ for
any }k\in\mathbb{Z}^{d}\setminus\\{0\\},\,\beta\in(-\infty,0].$ (6.2)

The $L^{2}_{\alpha}$ estimates in the first line of (2.9) follow using the
assumption (1.7).

Moreover, for any integer $a\geq 0$ we use (1.6), (2.4), and (6.1) to estimate

$\big{|}D^{a}_{\tau}L(\tau,k)\big{|}\lesssim\int_{0}^{\infty}s^{a+1}e^{-\lambda_{0}|sk|^{1/3}}\,ds\lesssim\frac{1}{(|k|\lambda_{0}^{3})^{a+2}}\int_{0}^{\infty}t^{3a+5}e^{-t}\,ds\lesssim\frac{(3a+5)!}{(|k|\lambda_{0}^{3})^{a+2}}.$
$\big{|}D^{a}_{\tau}(\tau
L(\tau,k))\big{|}\lesssim\int_{0}^{\infty}s^{a}e^{-\lambda_{0}|sk|^{1/3}}\,ds\lesssim\frac{(3a+2)!}{(|k|\lambda_{0}^{3})^{a+1}}.$

Therefore, for any $k\in\mathbb{Z}^{d}\setminus\\{0\\}$,
$\beta\in(-\infty,0]$, and $a\in\mathbb{Z}_{+}$ we have

$\big{\|}\langle\alpha\rangle
D^{a}_{\alpha}L(\alpha+i\beta,k)\big{\|}_{L^{\infty}_{\alpha}}\lesssim\frac{(3a+5)!}{(|k|\lambda_{0}^{3})^{a+1}}.$
(6.3)

It remains to prove pointwise estimates on higher order derivatives of
$L^{\prime}$, of the form

$\big{\|}\langle\alpha\rangle
D^{a}_{\alpha}L^{\prime}(\alpha+i\beta,k)\big{\|}_{L^{\infty}_{\alpha}}\lesssim\frac{(3a)!(1.01)^{a}}{(|k|\lambda_{0}^{3})^{a+1}},$
(6.4)

for any $k\in\mathbb{Z}^{d}\setminus\\{0\\}$, $\beta\in(-\infty,0]$, and
$a\in\mathbb{Z}_{+}$. We start from the Faá di Bruno’s formula

$D^{n}(g\circ f)(x)=\sum_{\ast}\frac{n!}{m_{1}!(1!)^{m_{1}}\cdot\ldots\cdot
m_{n}!(n!)^{m_{n}}}D^{m_{1}+\ldots+m_{n}}g(f(x))\cdot\prod_{j=1}^{n}(D^{j}f(x))^{m_{j}},$
(6.5)

for any $n\geq 1$ and any sufficiently smooth functions
$f:\mathbb{R}\to\mathbb{C}$ and $g:\mathbb{C}\to\mathbb{C}$, where
$\sum\limits_{\ast}$ denotes the sum over all $n$-tuples of non-negative
integers $m_{1},\ldots,m_{n}$ satisfying

$1\cdot m_{1}+\ldots+n\cdot m_{n}=n.$ (6.6)

In our case, we fix $\beta\in(-\infty,0]$,
$k\in\mathbb{Z}^{d}\setminus\\{0\\}$, and $g(y)=y/(y+1)$. Clearly
$|D^{b}g(y)|\leq b!|y+1|^{-b-1}$ for any integer $b\geq 0$, thus

$|D^{n}_{\alpha}L^{\prime}(\alpha+i\beta,k)|\leq\sum_{\ast}\frac{n!(m_{1}+\ldots+m_{n})!}{m_{1}!\cdot\ldots\cdot
m_{n}!\vartheta^{m_{1}+\ldots+m_{n}+1}}\prod_{j=1}^{n}\Big{[}\frac{D^{j}L(\alpha+i\beta,k)}{j!}\Big{]}^{m_{j}},$

for any integer $n\geq 1$. It follows from (6.3) that

$\langle\alpha\rangle|D^{n}_{\alpha}L^{\prime}(\alpha+i\beta,k)|\leq\sum_{\ast}[C_{1}^{\ast}]^{m_{1}+\ldots+m_{n}}\frac{n!(m_{1}+\ldots+m_{n})!}{m_{1}!\cdot\ldots\cdot
m_{n}!}\prod_{j=1}^{n}\Big{[}\frac{(3j+5)!}{j!(|k|\lambda_{0}^{3})^{j+1}}\Big{]}^{m_{j}},$
(6.7)

for any $\alpha\in\mathbb{R}$ and $n\in\mathbb{Z}_{+}^{\ast}$, where
$C_{1}^{\ast}=C_{1}^{\ast}(d,\lambda_{0},\vartheta)$ is a constant. Thus

$\begin{split}&\frac{\langle\alpha\rangle|D^{n}_{\alpha}L^{\prime}(\alpha+i\beta,k)|(|k|\lambda_{0}^{3})^{n+1}}{(3n)!}\leq\sum_{\ast}\frac{[C_{2}^{\ast}]^{m_{1}+\ldots+m_{n}}}{m_{1}!\cdot\ldots\cdot
m_{n}!}\exp(H_{m_{1},\ldots,m_{n}}),\\\
&H_{m_{1},\ldots,m_{n}}:=[\ln(n!)-\ln((3n)!)]+\sum_{j=1}^{n}m_{j}[\ln((3j+5)!)-\ln(j!)]+\big{[}\ln((m_{1}+\ldots+m_{n})!)\big{]},\end{split}$
(6.8)

where $C_{2}^{\ast}=C_{2}^{\ast}(d,\lambda_{0},\vartheta)$ is a constant.

To estimate the sum in the right-hand side of (6.8) we notice that

$\Big{|}\ln 1+\ldots+\ln m-\int_{1/2}^{m+1/2}\ln x\,dx\Big{|}\leq
C\qquad\text{ for any }m\geq 1,$

where $C$ is an absolute constant. Therefore

$\big{|}\ln(m!)-(m+1/2)\ln(m/e)\big{|}\leq C^{(1)}\qquad\text{ for any }m\geq
1,$ (6.9)

where $C^{(1)}\geq 1$ is a absolute constant. Thus, for $n\geq 1$,

$\ln(n!)-\ln((3n)!)\leq-2n\ln(n/e)-3n\ln 3+2C^{(1)}$

and

$\begin{split}\sum_{j=1}^{n}m_{j}[\ln((3j+5)!)-\ln(j!)]&\leq\sum_{j=1}^{n}m_{j}[2C^{(1)}+(2j+5)\ln(j/e)+(3j+5.5)\ln((3j+5)/j)]\\\
&\leq\sum_{j=1}^{n}m_{j}[C^{(2)}+(2j+5)\ln(j/e)+3j\ln 3],\end{split}$

where $C^{(2)}\geq 2C^{(1)}$ is an absolute constant. In view of (6.6) we have
$n=\sum_{j=1}^{n}jm_{j}$, thus

$\begin{split}H_{m_{1},\ldots,m_{n}}&\leq\sum_{j=1}^{n}m_{j}[C^{\prime\prime\prime}+(2j+5)\ln(j/e)-2j\ln(n/e)+\ln(m/e)]\\\
&\leq\sum_{j=1}^{n}m_{j}[C^{(3)}+7\ln(j/e)-(2j-1)\ln(n/j)],\end{split}$ (6.10)

where $C^{(3)}\geq C^{(2)}$ is an absolute constant and
$m:=m_{1}+\ldots+m_{n}\leq n$. It is easy to see that

$\sum_{j\in[1,2n/3]\cap\mathbb{Z}}m_{j}[C^{(3)}+7\ln(j/e)-(2j-1)\ln(n/j)]\leq\sum_{j\in[1,2n/3]\cap\mathbb{Z}}C^{(4)}m_{j}$

for some constant $C^{(4)}\geq C^{(3)}$. Also

$\sum_{j\in(2n/3,n]\cap\mathbb{Z}}m_{j}[C^{(3)}+7\ln(j/e)-(2j-1)\ln(n/j)]\leq
C^{(3)}+7\ln n,$

since $\sum_{j\in(2n/3,n]\cap\mathbb{Z}}m_{j}\leq 1$ (due to the restriction
(6.6)). Therefore it follows from (6.8) and (6.10) that

$\frac{\langle\alpha\rangle|D^{n}_{\alpha}L^{\prime}(\alpha+i\beta,k)|(|k|\lambda_{0}^{3})^{n+1}}{(3n)!}\leq
n^{7}\sum_{\ast}\frac{[C_{3}^{\ast}]^{m_{1}+\ldots+m_{n}}}{m_{1}!\cdot\ldots\cdot
m_{n}!}$ (6.11)

for any $n\geq 1$, where $C_{3}^{\ast}=C_{3}^{\ast}(d,\lambda_{0},\vartheta)$
is a constant.

Let $C_{4}^{\ast}:=\sum_{m\geq 0}(C_{3}^{\ast})^{m}/(m!)$, let
$\rho\in(0,1/2]$ be such that $(C_{4}^{\ast})^{\rho}=1.005$, and let $J$
denote the largest integer $\leq\rho n$. Notice that if the $n$-tuple
$(m_{1},\ldots,m_{n})$ satisfies (6.6) then

$m_{J+1}+\ldots+m_{n}\leq 1/\rho.$

In particular, the number of possible choices of the integers
$m_{J+1},\ldots,m_{n}$ is $\lesssim n^{1/\rho}$. Thus

$\sum_{\ast}\frac{[C_{3}^{\ast}]^{m_{1}+\ldots+m_{n}}}{m_{1}!\cdot\ldots\cdot
m_{n}!}\leq\sum_{m_{1},\ldots,m_{J}\in\mathbb{Z}_{+}}\sum_{m_{J+1}+\ldots+m_{n}\leq
1/\rho}\frac{[C_{3}^{\ast}]^{m_{1}+\ldots+m_{n}}}{m_{1}!\cdot\ldots\cdot
m_{n}!}\lesssim(C_{4}^{\ast})^{J}n^{1/\rho}.$

The desired bounds (6.4) follow from (6.11).

(ii) We can complete the proof of the lemma. In view of the analyticity of
$L^{\prime}$, we can write

$\widehat{G}(s,k)=\frac{1}{2\pi}\int_{\mathbb{R}}L^{\prime}(\alpha+i\beta,k)e^{i(\alpha+i\beta)s}\,d\alpha=\frac{e^{-\beta
s}}{2\pi}\int_{\mathbb{R}}L^{\prime}(\alpha+i\beta,k)e^{i\alpha s}\,d\alpha.$

We use the uniform bounds (6.4) and let $\beta\to-\infty$ to conclude that
$\widehat{G}(s,k)=0$ if $s<0$.

To prove the main estimates (2.11) we may assume $|sk|\geq 1$, use the formula
above with $\beta=0$, and integrate by parts $a\geq 1$ times when
$|\alpha|\leq e^{s|k|}$. Therefore for any $s\geq 1/|k|$

$\begin{split}2\pi|\widehat{G}(s,k)|&\leq\int_{|\alpha|\geq
e^{s|k|}}|L^{\prime}(\alpha,k)|\,d\alpha+|s|^{-a}\Big{|}\int_{|\alpha|\leq
e^{s|k|}}e^{is\alpha}(D^{a}_{\alpha}L^{\prime})(\alpha,k)\,d\alpha\Big{|}\\\
&+\sum_{a^{\prime}=0}^{a-1}|s|^{-a^{\prime}-1}\big{[}|D^{a^{\prime}}_{\alpha}L^{\prime}(e^{s|k|},k)|+|D^{a^{\prime}}_{\alpha}L^{\prime}(-e^{s|k|},k)|\big{]}.\end{split}$

We estimate the first term using the $L^{2}$ bounds (2.9) and the other two
terms using the $L^{\infty}$ bounds (6.4). Therefore for any
$k\in\mathbb{Z}^{d}\setminus\\{0\\}$, $s\geq 1/|k|$, and $a\geq 1$ we have

$\begin{split}|\widehat{G}(s,k)|\lesssim
e^{-s|k|/2}+|sk||s|^{-a}\frac{(3a)!(1.01)^{a}}{(|k|\lambda_{0}^{3})^{a+1}}+e^{-s|k|}\sum_{a^{\prime}=0}^{a-1}|s|^{-a^{\prime}-1}\frac{(3a^{\prime})!(1.01)^{a^{\prime}}}{(|k|\lambda_{0}^{3})^{a^{\prime}+1}}.\end{split}$

Therefore for any $k\in\mathbb{Z}^{d}\setminus\\{0\\}$, $s\geq 1/|k|$, and
$a\geq 1$

$\begin{split}|\widehat{G}(s,k)|\lesssim
e^{-s|k|/2}+|sk|\frac{(3a)!(1.01)^{a}}{(s|k|\lambda_{0}^{3})^{a}}+e^{-s|k|}\sum_{a^{\prime}=0}^{a-1}\frac{(3a^{\prime})!(1.01)^{a^{\prime}}}{(s|k|\lambda_{0}^{3})^{a^{\prime}}}.\end{split}$
(6.12)

We let now $a$ denote the smallest integer $\geq\lambda_{0}(s|k|)^{1/3}/3$.
Using the bounds (6.9) below,

$\begin{split}\frac{(3a^{\prime})!(1.01)^{a^{\prime}}}{(s|k|\lambda_{0}^{3})^{a^{\prime}}}&\lesssim\exp\big{[}(3a^{\prime}+1/2)(\ln(3a^{\prime}+1)-1)-3a^{\prime}\ln(\lambda_{0}(s|k|)^{1/3})+a^{\prime}\ln(1.01)\big{]}\\\
&\lesssim\exp\big{[}-3a^{\prime}+(1/2)\ln(3a^{\prime}+1)+a^{\prime}\ln(1.01)\big{]}\end{split}$

for any integer $a^{\prime}\in[0,a]$. The desired bounds (2.11) follow from
(6.12).

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