#  Remarks on the Large data global solutions of $3D$ RVP system and $3D$ RVM
system

Xuecheng Wang  YMSC, Tsinghua University, Beijing, China, 100084
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We show that the simplified $3D$ relativistic Vlasov-Maxwell (sRVM) system, in
which there is no magnetic field, poses a global solution for a class of
arbitrarily large cylindrically symmetric initial data. In particular, a
vanishing order condition imposed on the initial data for the relativistic
Vlasov-Poisson system (RVP) in [16] is not imposed for the sRVM system.

##  1\. Introduction

###  1.1. Motivation

The $3D$ relativistic Vlasov-Maxwell system is one of the fundamental models
in the collisionless plasma physics. It reads as follows,

$(\textup{RVM})\qquad\left\\{\begin{array}[]{c}\partial_{t}f+\hat{v}\cdot\nabla_{x}f+(E+\hat{v}\times
B)\cdot\nabla_{v}f=0,\\\ \nabla\cdot
E=4\pi\displaystyle{\int_{\mathbb{R}^{3}}f(t,x,v)dv},\qquad\nabla\cdot B=0,\\\
\partial_{t}E=\nabla\times
B-4\pi\displaystyle{\int_{\mathbb{R}^{3}}f(t,x,v)\hat{v}dv},\quad\partial_{t}B=-\nabla\times
E,\\\ f(0,x,v)=f_{0}(x,v),\quad E(0,x)=E_{0}(x),\quad
B(0,x)=B_{0}(x),\end{array}\right.$ (1.1)

where
$f:\mathbb{R}_{t}\times\mathbb{R}_{x}^{3}\times\mathbb{R}_{v}^{3}\longrightarrow\mathbb{R}_{+}$
denotes the distribution function of particles,
$E,B:\mathbb{R}_{t}\times\mathbb{R}_{x}^{3}\longrightarrow\mathbb{R}^{3}$
denote the electromagnetic field, $\hat{v}:=v/\sqrt{1+|v|^{2}}$.

In particular, if there is no magnetic field, then the electric field is curl
free. Hence, we know that $E=\nabla\phi$, where
$\phi:\mathbb{R}_{t}\times\mathbb{R}_{x}^{3}\longrightarrow\mathbb{R}$. As a
result, the RVM system is reduced to the following simplified RVM (sRVM)
system

$(\textup{sRVM})\quad\left\\{\begin{array}[]{c}\partial_{t}f+\hat{v}\cdot\nabla_{x}f+E\cdot\nabla_{v}f=0,\quad
E=\nabla\phi\\\
\Delta\phi=4\pi\displaystyle{\int_{\mathbb{R}^{3}}f(t,x,v)dv},\qquad\partial_{t}\nabla_{x}\phi=-4\pi\displaystyle{\int_{\mathbb{R}^{3}}f(t,x,v)\hat{v}dv},\quad
f(0,x,v)=f_{0}(x,v).\end{array}\right.$ (1.2)

The following conservation law holds for the above system,

$\mathcal{H}(t):=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\sqrt{1+|v|^{2}}f(t,x,v)dxdv+\int_{\mathbb{R}^{3}}|\nabla_{x}\phi(t,x)|^{2}dx=\mathcal{H}(0).$
(1.3)

We remark that, due to the divergence free condition of the magnetic field,
for any smooth localized radial initial data, the solution of the RVM system
(1.1) must have zero magnetic field.

As comparison, the following $3D$ relativistic Vlasov-Poisson system (RVP) for
the plasma physics case is also widely studied in the literature.

$(\textup{RVP})\quad\left\\{\begin{array}[]{c}\partial_{t}f+\hat{v}\cdot\nabla_{x}f+E\cdot\nabla_{v}f=0,\quad
E=\nabla\phi\\\
\Delta\phi=\rho(t),\quad\rho(t):=\displaystyle{\int_{\mathbb{R}^{3}}f(t,x,v)dv},\quad
f(0,x,v)=f_{0}(x,v).\\\ \end{array}\right.$ (1.4)

Note that the RVP is a determined system while sRVM system is over-determined
in general. However, for the radial case, these two systems are equivalent.

The properties of the solution of the RVP system are shared with the solution
of the sRVM system, however not the other way around. For example, the
solution of sRVM system (1.2) enjoys the following property over time,

$\int_{\mathbb{R}^{3}}\hat{v}\times\nabla_{x}f(t,x,v)dv=0,$

which is not necessarily true for the solution of the RVP system in general.
But, indeed, it’s true for the radial case.

In [16], we show the global existence of $3D$ RVP system (1.4) for a class of
arbitrarily large initial data with cylindrical symmetry and an extra
vanishing order condition on the planar momentum. More precisely, the
following condition is imposed on the initial data,

$\textup{(a vanishing order condition for
RVP)}\qquad\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{1}{|\not{x}\times\not{v}|^{13}}f_{0}(x,v)dxdv<+\infty.$

In [16], we exploit the benefit of the above condition via the conservation
law of planar momentum of the $3D$ RVP system for the cylindrical symmetry.
The the conservation law of planar momentum is available is mainly because of
the cylindrical symmetry assumption and the gradient structure of the electric
field. However, the gradient structure of the electric field is no longer
available for the RVM system (1.1). We don’t expect such conservation law for
the RVM system.

One of the main goals of this paper is to show that, thanks to the extra
equation in sRVM system (1.2), the difficulty of removing the vanishing order
condition for the RVP system is absent in the sRVM system as well as in the
RVM system, see [17]. In this paper, we will not use the conservation law of
the planar momentum of any kind to show large data global existence of the
sRVM system (1.2). Instead, we exploit the hyperbolic nature of the sRVM
system.

We also reveal some properties shared commonly by the RVP system and the sRVM
system, e.g., the estimate (3.23) in Lemma 3.1. We hope that it can shine some
light on the further study of the large data problem of the RVP system (1.4).

Moreover, as a toy model for the RVM system (1.1), the framework of proving
global existence used in this paper is same as the framework used in a more
complicated study of the RVM system [17]. For pedagogical purpose, we also
hope that this paper also serves as an introduction for more complicated and
more technical full study of the RVM system in [17].

###  1.2. Main result of this paper

The main result of this paper is stated as follows.

######  Theorem 1.1.

Assume that the initial data $f_{0}(x,v)\in
H^{s}(\mathbb{R}_{x}^{3}\times\mathbb{R}_{v}^{3})$, $s\in\mathbb{Z}_{+},s\geq
6$ is cylindrical symmetric in the sense that the following equality holds for
any $x,v\in\mathbb{R}^{3}$,

$f_{0}(x_{1},x_{2},x_{3},v_{1},v_{2},v_{3})=f_{0}(|\not{x}|,0,x_{3},|\not{v}|,0,v_{3}),\quad\not{x}:=(x_{1},x_{2}),\quad\not{v}:=(v_{1},v_{2}).$
(1.5)

Moreover, we assume that the initial data decays polynomially as
$(x,v)\longrightarrow\infty$ in the following sense,

$\sum_{\alpha\in\mathbb{Z}_{+}^{6},|\alpha|\leq
s}\|(1+|x|+|v|)^{N_{0}}\nabla_{x,v}^{\alpha}f_{0}(x,v)\|_{L^{2}_{x,v}}<+\infty,\quad
N_{0}:=10^{8},$ (1.6)

then the simplified relativistic Vlasov-Maxwell system (1.2) admits a global
solution $(f(t,\cdot,\cdot))\in
H^{s}(\mathbb{R}_{x}^{3}\times\mathbb{R}_{v}^{3})$.

A few remarks are in order.

######  Remark 1.1.

Although, as stated in (1.5), the initial data is cylindrical symmetric with
respect to the $x_{3}$-axis, by changing coordinate system, our main theorem
also holds for initial data with cylindrical symmetry with respect to any line
that goes through the origin.

######  Remark 1.2.

The plausible goal of optimizing the size of $N_{0},s$ is not pursued here.

###  1.3. Notation and plan of this paper

For any two numbers $A$ and $B$, we use $A\lesssim B$ and $B\gtrsim A$ to
denote $A\leq CB$, where $C$ is an absolute constant. We use the convention
that all constants which only depend on the initial data, e.g., the conserved
quantities in (1.3), will be treated as absolute constants.

We fix an even smooth function $\tilde{\psi}:\mathbb{R}\rightarrow[0,1]$,
which is supported in $[-3/2,3/2]$ and equals to “$1$” in $[-5/4,5/4]$. For
any $k,j\in\mathbb{Z},j>0$, we define the cutoff functions
$\psi_{k},\psi_{\leq k},\psi_{\geq
k}:\cup_{n\in\mathbb{Z}_{+}}\mathbb{R}^{n}\longrightarrow\mathbb{R}$ as
follows,

$\psi_{k}(x):=\tilde{\psi}(|x|/2^{k})-\tilde{\psi}(|x|/2^{k-1}),\quad\varphi_{0}(x):=\tilde{\psi}(|x|),\quad\forall
j\in(0,\infty)\cap\mathbb{Z},\quad\varphi_{j}(x):=\psi_{j}(x).$

For any $x\in\mathbb{R}^{3}/\\{0\\},u\in\mathbb{R}^{3}$, we use
$\tilde{x}:=x/|x|$ to denote the direction of the vector $x$ and use $\not{u}$
to denote the projection of $u$ onto the $x_{1}x_{2}$-plane. For an integrable
function $f(x)$, we use both $\widehat{f}(\xi)$ and $\mathcal{F}(f)(\xi)$ to
denote the Fourier transform of $f$, which is defined as follows,

$\mathcal{F}(f)(\xi)=\int e^{-ix\cdot\xi}f(x)dx.$

We use $\mathcal{F}^{-1}(g)$ to denote the inverse Fourier transform of
$g(\xi)$.

The rest of this paper is organized as follows.

  1. $\bullet$

In section 2, we introduce the set-up of problem and record some tools
developed in [16].

  2. $\bullet$

In section 3, under the assumption of the validity of Lemma 3.1 and the Lemma
3.2, we control both the projection of the velocity characteristics onto the
horizontal plane and the full velocity characteristics that start from the
$t$-majority set $R_{t}(0)$, see (2.4). With strong control of $\beta_{t}$,
see Proposition 3.1, we control the high momentum of the distribution function
over time and prove theorem 1.1.

  3. $\bullet$

In section 4, we prove Lemma 3.1.

  4. $\bullet$

In section 5, by exploiting the hyperbolic nature of the sRVM system (1.2), we
prove Lemma 3.2.

Acknowledgment The author acknowledges support from NSFC-11801299, 12141102,
and MOST-2020YFA0713003.

##  2\. Preliminary

###  2.1. The set-up

We use the classic moment method, which is propagating a high order moment of
the distribution function over time, to control directly the electric field
over time, see Lemma 2.3. We define

$M_{n}(t):=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}(1+|v|)^{n}f(t,x,v)dvdx,\quad\tilde{M}_{n}(t):=(1+t)^{n^{2}}+\sup_{s\in[0,t]}M_{n}(s),\quad
n:=N_{0}/10=10^{7}.$ (2.1)

Therefore $\tilde{M}_{n}(t)$ is an increasing function with respect to time.

Let $T\in\mathbb{R}_{+}$ be the maximal time of existence. For any fixed
$t\in[0,T]$, we let $M_{t}\in\mathbb{Z}_{+}$ to be the least integer such that
$2^{M_{t}}\geq(\tilde{M}_{n}(t))^{1/(n-1)}$. Moreover, it would be sufficient
to only consider the case that $M_{t}\gg 1$ otherwise the momentum
$\tilde{M}_{n}(t)$ is naturally bounded by an absolute constant by definition.

From (1.2), the backward characteristics associated with the sRVM system read
as follow,

$\left\\{\begin{array}[]{l}\displaystyle{\frac{d}{ds}X(x,v,s,t)=\hat{V}(x,v,s,t)},\quad\displaystyle{\frac{d}{ds}V(x,v,s,t)=E(s,X(x,v,s,t))}\\\
\\\ X(x,v,t,t)=x,\quad V(x,v,t,t)=v.\\\ \end{array}\right.$ (2.2)

Due to the transport nature of the Vlasov equation, see (1.2), for any
$s,t\in[0,T)$, we have

$f(t,x,v)=f(s,X(x,v,s,t),V(x,v,s,t)).$ (2.3)

For any $t\in[0,T)$, we define $t$-majority set of particles, which initially
localize around zero, at time $s\in[0,T)$ as follows,

$R_{t}(s):=\\{(X(x,v,s,0),V(x,v,s,0)):|V(x,v,0,0)|+|X(x,v,0,0)|\leq
2^{M_{t}/2}\\}.$ (2.4)

Moreover, we define

$\alpha_{t}(s,x,v):=\sup_{\tau\in[0,s]}\inf\\{k:k\in\mathbb{R}_{+},|\not{V}(x,v,\tau,0)|\leq
2^{kM_{t}}\\},\quad\alpha_{t}(s):=\sup_{(x,v)\in
R_{t}(0)}\alpha_{t}(s,x,v),\\\
\beta_{t}(s,x,v):=\sup_{\tau\in[0,s]}\inf\\{k:k\in\mathbb{R}_{+},|V(x,v,\tau,0)|\leq
2^{kM_{t}}\\},\quad\beta_{t}(s):=\sup_{(x,v)\in R_{t}(0)}\beta_{t}(s,x,v),\\\
\epsilon:=100/n=10^{-5},\quad\alpha_{t}:=\alpha_{t}(t),\quad\tilde{\alpha}_{t}=\min\\{\alpha_{t},1+\epsilon\\},\quad\beta_{t}:=\beta_{t}(t),\quad\tilde{\beta}_{t}=\min\\{\beta_{t},1+\epsilon\\}.$
(2.5)

From the above definition, we know that

$\forall t\in[0,T),s\in[0,t],\quad R_{s}(0)\subset
R_{t}(0),\quad\alpha_{s}M_{s}\leq\alpha_{t}(s)M_{t}\leq\alpha_{t}M_{t},\quad\beta_{s}M_{s}\leq\beta_{t}(s)M_{t}\leq\beta_{t}M_{t}.$
(2.6)

Note that

$\forall s\in[0,t],x,v\in\mathbb{R}^{3},\quad
X(X(x,v,0,s),V(x,v,0,s),s,0)=x,\quad V(X(x,v,0,s),V(x,v,0,s),s,0)=v.$

Therefore, from the above definition, for any fixed
$s\in[0,t],x\in\mathbb{R}^{3},v\in\mathbb{R}^{3}$, s.t., either $|\not{v}|\geq
2^{(\alpha_{s}+\epsilon)M_{s}}$ or $|v|\geq 2^{(\beta_{s}+\epsilon)M_{s}}$, we
have

$(X(x,v,0,s),V(x,v,0,s))\notin R_{s}(0).$ (2.7)

Thanks to the rapid polynomial decay rate of the initial data, see (1.6), to
control the high moments $\tilde{M}_{n}(t)$, it would be sufficient to control
the majority set over time. More precisely, from the assumption on the initial
data in (1.6), the following estimate holds for any
$s\in[0,t],x,v\in\mathbb{R}^{3}$, s.t., either $|\not{v}|\geq
2^{(\alpha_{s}+\epsilon)M_{s}}$ or $|v|\geq 2^{(\beta_{s}+\epsilon)M_{s}}$,

$\big{|}f(s,x,v)\big{|}=\big{|}f_{0}(X(x,v,0,s),V(x,v,0,s))\big{|}\lesssim
2^{-4nM_{s}}.$ (2.8)

To exploit the benefit of the hyperbolic natural of the sRVM system, we will
use the following Kirchhoff’s formula.

######  Lemma 2.1 (Kirchhoff’s formula).

For any $t\in\mathbb{R},x\in\mathbb{R}^{3}$, the following equality holds,

$|\nabla|^{-1}\sin(t|\nabla|)h(x)=\frac{1}{4\pi}t\int_{\mathbb{S}^{2}}h(x+t\theta)d\theta,$
(2.9)
$|\nabla|^{-1}\cos(t|\nabla|)h(x)=\frac{1}{4\pi}\int_{\mathbb{S}^{2}}|\nabla|^{-1}h(x+t\theta)d\theta+\frac{1}{4\pi}\int_{\mathbb{S}^{2}}t\theta\cdot\frac{\nabla}{|\nabla|}h(x+t\theta)d\theta.$
(2.10)

###### Proof.

Note that

$\int_{\mathbb{R}^{3}}e^{-ix\cdot\xi}\int_{\mathbb{S}^{2}}h(x+t\theta)d\theta
dx=\int_{\mathbb{S}^{2}}e^{it\xi\cdot\theta}\hat{h}(\xi)d\theta=2\pi\hat{h}(\xi)\int_{0}^{\pi}e^{it|\xi|\cos(\phi)}\sin(\phi)d\phi=\frac{4\pi\sin(t|\xi|)}{t|\xi|}\hat{h}(\xi).$
(2.11)

Hence finishing the proof of the desired formula (2.9). Our desired equality
(2.10) holds after taking derivative with respect to “$t$” for the equality
(2.9). ∎

###  2.2. Rough estimates for the electric field

Let $E_{k}(\cdot,\cdot)=P_{k}(E)(\cdot,\cdot)$ denotes the frequency localized
electric field. Moreover, we localize further the sizes of $\not{v}$ and $v$
for the frequency localized electric field as follows,

$E_{k}(t,x)=\sum_{j_{2}\in\mathbb{Z}_{+},j_{1}\in[0,j_{2}+2]\cap\mathbb{Z}_{+}}E_{k;j_{1},j_{2}}(t,x),\\\
E_{k;j_{1},j_{2}}(t,x)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}K_{k}(y)f(t,x-y,v)\varphi_{j_{1}}(\not{v})\varphi_{j_{2}}(v)dydv,\quad
K_{k}(y)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\frac{-i\xi}{|\xi|^{2}}\varphi_{k}(\xi)d\xi.$
(2.12)

After doing integration by parts in $\xi$ many times, the following estimate
holds for the kernel $K_{k}(y)$,

$\big{|}K_{k}(y)\big{|}\lesssim 2^{2k}(1+2^{k}|y|)^{-N_{0}^{3}}.$ (2.13)

The following singular weighted space-time estimate controls strongly the
distribution of particles near the $x_{3}$-axis.

######  Lemma 2.2.

Let $\epsilon^{\star}:=\epsilon/100.$ For any $t\in[0,T)$, the following
weighted space-time estimate holds,

$A(t):=\int_{0}^{t}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{|\not{v}|^{2+2\epsilon^{\star}}}{|\not{x}|^{1-2\epsilon^{\star}}\langle
v\rangle}f(s,x,v)dxdvds\lesssim 2^{5\epsilon M_{t}}.$ (2.14)

As a by-product, the following $L^{1}_{t}-L^{\infty}_{x}$-type estimate holds
for any $t_{1},t_{2}\in[0,t]$,

$\int_{t_{1}}^{t_{2}}\|E_{k;j_{1},j_{2}}(t,\cdot)\|_{L^{\infty}_{x}}dt\lesssim
2^{2\epsilon M_{t}}+2^{k-2j_{1}+j_{2}+6\epsilon M_{t}}.$ (2.15)

###### Proof.

See [16][Proposition 3.1]. ∎

We record the following rough estimates for the localized electric field
obtained in [16].

######  Lemma 2.3.

For any $s\in[0,t]\subset[0,T)$, the following rough estimate holds for the
localized electric field,

$\|E_{k;j_{1},j_{2}}(s,x)\|_{L^{\infty}_{x}}\lesssim\min\\{2^{-k+2j_{1}+j_{2}},2^{2k-j_{2}},2^{2k-nj_{2}}\tilde{M}(t)\\}.$
(2.16)

Moreover, for any $x\in\mathbb{R}^{3}$ s.t., $|\not{x}|\neq 0$, we have the
following point-wise estimate,

$|E_{k;j_{1},j_{2}}(s,x)|\lesssim 1+\min\\{\frac{2^{j_{1}+\epsilon
M_{t}}}{|\not{x}|^{1/2}},\frac{2^{k-j_{2}+\epsilon M_{t}}}{|\not{x}|}\\}.$
(2.17)

###### Proof.

See [16][Lemma 3.1].

∎

######  Lemma 2.4.

For any $s,s_{1},s_{2}\in[0,t]\subset[0,T)$, the following
$L^{1}_{t}L^{\infty}_{x}$-type and $L^{\infty}_{x}$ type rough estimates for
the electric field hold,

$\int_{s_{1}}^{s_{2}}\|E(s,\cdot)\|_{L^{\infty}_{x}}ds\lesssim
2^{(1+5\epsilon)M_{t}},$ (2.18)
$\|E(s,x)\|_{L^{\infty}_{x}}\lesssim\min\\{2^{4\tilde{\alpha}_{t}M_{t}/3+M_{t}/3+5\epsilon
M_{t}},|\not{x}|^{-1/2}2^{\tilde{\alpha}_{t}M_{t}+\epsilon M_{t}}\\}.$ (2.19)

###### Proof.

Recall the decomposition of the electric field in (2.12). From the obtained
rough estimate (2.16), we can first rule out the case
$j_{2}\geq(1+\epsilon)M_{t}$ as follows,

$\sum_{j_{2}\geq(1+\epsilon)M_{t}}\sum_{j_{1}\in[0,j_{2}+2]\cap\mathbb{Z}}\sum_{k\in\mathbb{Z}}\|E_{k;j_{1},j_{2}}(s,\cdot)\|_{L^{\infty}_{x}}\lesssim\sum_{k\in\mathbb{Z}}\sum_{j_{2}\geq(1+\epsilon)M_{t}}2^{\epsilon
j_{2}}\min\\{2^{-k+3j_{2}},2^{2k-nj_{2}+(n-1)M_{t}}\\}$
$\lesssim\sum_{k\in\mathbb{Z}}\sum_{j_{2}\geq(1+\epsilon)M_{t}}2^{-\epsilon
k}2^{(2+\epsilon)j_{2}}\big{(}2^{-nj_{2}+(n-1)M_{t}}\big{)}^{(1-\epsilon)/3}\lesssim
1.$ (2.20)

For any fixed $j_{2}\in[0,(1+\epsilon)M_{t}]\cap\mathbb{Z}$, again from the
rough estimate (2.16), we can rule out the case $k\geq 3M_{t}+5\epsilon M_{t}$
as follows,

$\sum_{k\in\mathbb{Z},k\geq 3M_{t}+5\epsilon
M_{t}}\|E_{k;j_{1},j_{2}}(s,\cdot)\|_{L^{\infty}_{x}}\lesssim\sum_{k\in\mathbb{Z},k\geq
3M_{t}+5\epsilon M_{t}}2^{-k+3(1+\epsilon)M_{t}}\lesssim 1.$ (2.21)

It remains to consider the case $k\in[0,3M_{t}+5\epsilon
M_{t}]\cap\mathbb{Z}$, $j_{2}\in[0,(1+\epsilon)M_{t}]\cap\mathbb{Z}$ and
$j_{1}\in[0,j_{2}+2]\cap\mathbb{Z}$. Since there are at most $M_{t}^{3}$
cases, which causes only logarithmic loss. It would be sufficient to let
$k,j_{1},j_{2}$ be fixed. From the estimate (2.15) and the estimate (2.16), we
have

$\int_{s_{1}}^{s_{2}}\|E_{k;j_{1},j_{2}}(s,\cdot)\|_{L^{\infty}_{x}}ds\lesssim
2^{2\epsilon M_{t}}+\min\\{2^{-k+2j_{1}+j_{2}+\epsilon
M_{t}},2^{k-2j_{1}+j_{2}+6\epsilon M_{t}}\\}\lesssim 2^{j_{2}+3.5\epsilon
M_{t}}\lesssim 2^{(1+4.5\epsilon)M_{t}}.$ (2.22)

Hence finishing the proof of our desired estimate (2.18).

It remains to prove (2.19). Note that, If $j_{1}\in[{\alpha}_{t}M_{t}+\epsilon
M_{t},j_{2}+2]\cap\mathbb{Z}$, from the estimates (2.6) and (2.8), we have

$f(t,x,v)=f_{0}(X(x,v,0,t),V(x,v,0,t))\lesssim
2^{-10M_{t}},\quad\Longrightarrow\|E_{k;j_{1},j_{2}}(s,\cdot)\|_{L^{\infty}_{x}}\lesssim
2^{-k+3j_{2}-10M_{t}}\lesssim 1.$ (2.23)

If $j_{1}\leq\min\\{{\alpha}_{t}M_{t}+\epsilon M_{t},j_{2}+2\\}$, then from
the rough estimate (2.16) in Lemma 2.3, we have

$\|E_{k;j_{1},j_{2}}(s,\cdot)\|_{L^{\infty}_{x}}\lesssim\min\\{2^{-k+2j_{1}+j_{2}},2^{2k-j_{2}}\\}\lesssim
2^{4j_{1}/3+j_{2}/3}\lesssim 2^{4\tilde{\alpha}_{t}M_{t}/3+M_{t}/3+3\epsilon
M_{t}}.$

Hence finishing the proof of the desired estimate (2.19) holds from the above
obtained estimates and the estimate (2.17) in Lemma 2.3. ∎

##  3\. Control of the $t$-majority set and the proof of main theorem

At large scale, the framework of the proof is almost same as the one we used
for the RVM system in [17]. Intuitively speaking, the main ideas of proof of
theorem 1.1 can be summarized as follows,

$\textup{smoothing effect
}\stackrel{{\scriptstyle\textup{bootstrap}}}{{\Longrightarrow}}\alpha_{t}\leq
0.71,\quad\textup{smoothing effect +hyperbolic nature}+\alpha_{t}\leq
0.71\stackrel{{\scriptstyle\textup{bootstrap}}}{{\Longrightarrow}}\beta_{t}\leq
1-2\epsilon\\\
\Longrightarrow\tilde{M}_{n}(t)\lesssim(1+t)^{n^{2}}\Longrightarrow E(t)\in
L^{\infty}([0,T)\times\mathbb{R}^{3}\overset{\textup{continuation
criteria}}{\Longrightarrow}\textup{global existence}.$ (3.1)

First of all, by assuming the validity of $\beta_{t}\leq 1-2\epsilon$, which
is stated in Proposition 3.1, we prove our main theorem. From the conservation
law (1.3), we have

$\big{|}\int_{\mathbb{R}^{3}}\int_{|v|\leq
2^{(1-\epsilon)M_{t}}}(1+|v|)^{n}f(t,x,v)dxdv\big{|}\lesssim
2^{(n-1)(1-\epsilon)M_{t}}\leq(\tilde{M}_{n}(t))^{1-\epsilon}.$ (3.2)

Moreover, from the estimate $\beta_{t}\leq(1-2\epsilon)$, we know that
$|X(x,v,0,t)|+|V(x,v,0,t)|\geq 2^{M_{t}/2}$ if $|v|\geq
2^{(1-\epsilon)M_{t}}$. From the decay assumption of the initial data in (1.6)
and the following estimate holds if $|v|\geq 2^{(1-\epsilon)M_{t}}$,

$|f(t,x,v)|=|f_{0}(X(x,v,0,t),X(x,v,0,t))|\lesssim(1+|X(x,v,0,t)|+|V(x,v,0,t)|)^{-N_{0}+10}\lesssim
2^{-5nM_{t}+5}.$ (3.3)

Recall (2.2), from the rough $L^{\infty}_{x}$ estimate of electric field in
(2.19) in Lemma 2.4, the following estimates hold if $|v|\gtrsim
2^{2M_{t}},|x|\gtrsim 2^{2\epsilon M_{t}}$,

$\big{|}v-V(x,v,0,t)|\big{|}\lesssim 2^{5M_{t}/3+10\epsilon
M_{t}},\quad\big{|}x-X(x,v,0,t)|\big{|}\lesssim 2^{3\epsilon
M_{t}/2},\quad\Longrightarrow|v|\sim|V(x,v,0,t)|,\big{|}x\big{|}\sim\big{|}X(x,v,0,t)\big{|}.$
(3.4)

To sum up, after combining the above estimate (3.3), we know that the
following estimate holds if $|v|\geq 2^{(1-\epsilon)M_{t}}$,

$|f(t,x,v)|=|f_{0}(X(x,v,0,t),X(x,v,0,t))|\lesssim(1+|x|)^{-4}(1+|v|)^{-n-4}.$
(3.5)

From the above estimate, we have

$\big{|}\int_{\mathbb{R}^{3}}\int_{|v|\geq
2^{(1-\epsilon)M_{t}}}(1+|v|)^{n}f(t,x,v)dxdv\big{|}\lesssim 1.$ (3.6)

Therefore, recall (2.1), from the estimates (3.2) and (3.6), we know that the
following estimate holds for any $t\in[0,T)$,

$M_{n}(t)\lesssim\big{(}\tilde{M}_{n}(t)\big{)}^{1-\epsilon}.$ (3.7)

From the above estimate and the fact that $\tilde{M}_{n}(t)$ is an increasing
function with respect to $t$, the following estimate holds for any
$s\in[0,t]$,

$\tilde{M}_{n}(t)=\sup_{s\in[0,t]}M_{n}(s)+(1+t)^{n^{3}}\lesssim\big{(}\tilde{M}_{n}(t)\big{)}^{1-\epsilon}+(1+t)^{n^{3}},\quad\Longrightarrow\tilde{M}_{n}(t)\lesssim(1+t)^{n^{3}}.$

Therefore, $\tilde{M}_{n}(t)$ grows at most at rate $(1+t)^{n^{3}}$ over time.
From the rough $L^{\infty}_{x}$ estimate of electric field in (2.19) in Lemma
2.4, we know that that the boundedness of $\tilde{M}_{n}(t)$ in $[0,T)$
ensures that $E\in L^{\infty}([0,T)\times\mathbb{R}_{x}^{3})$, hence the the
lifespan of the system (2.2) can be extended at any finite time, i.e., the
solution exists globally, see also the continuation criteria by Luk-Strain
[12][Theorem 5.7]. Hence finishing the proof of Theorem 1.1.

Now, it would be sufficient to prove the Proposition 3.1. For the rest of this
section, we mainly prove Proposition 3.1 by assuming the validity of Lemma 3.1
and Lemma 3.2.

######  Proposition 3.1.

For any $t\in[0,T)$, s.t., $M_{t}\gg 1$, we have

$\alpha_{t}\leq\alpha^{\star}:=7/10+\iota,\quad\beta_{t}\leq(1-2\epsilon),\quad\iota:=10^{-2},\quad\epsilon=10^{-5}.$
(3.8)

###### Proof.

The argument presented is very general, which only depends on the analysis of
the motion of the characteristics (2.2) by using bootstrap arguments.
Actually, it’s almost same as the one we used in the study of $3D$ RVM system
[17][section 3]. The key difference between RVP system (1.4), RVM system
(1.1), and the sRVM system (1.2) is that the behavior of the acceleration
force is very different in different settings. We summarize the effects of
acceleration force over time as black boxes in Lemma 3.1 and Lemma 3.2.

For the sake of readers and also for the pedagogical purpose, instead of
referring readers to [17][section 3], we still give a full proof here.

Let $t\in[0,T)$ be fixed. Recall (2.2). For convenience in notation,  we
suppress the dependence of characteristics with respect to $(x,v)\in
R_{t}(0)$. We define the first time that characteristics almost reaches the
threshold as follows,

$t^{\ast}:=\inf\\{s:s\in[0,t],\beta_{t}(s)\geq(1-3\epsilon)\\}.$ (3.9)

Due to the continuity of characteristics, and the rough estimate of the
electric field (2.19) in Lemma 2.4, we know that $t^{\ast}>0.$ Moreover,
$\forall s\in[0,t^{\ast}],$ we have $|V(s)|\leq 2^{(1-3\epsilon)M_{t}}.$ We
make the following bootstrap assumption,

$\tau:=\sup\\{\kappa:\kappa\in[0,t],\forall
s\in[t^{\ast},\kappa],|2^{\beta_{t}(s)M_{t}}|\in
2^{(1-3\epsilon)M_{t}}[99/100,101/100]\\}.$ (3.10)

We aim to improve the above bootstrap assumption and prove that $\tau=t$.

Step $\mathbf{1}$ Estimate of the projection of the velocity characteristics,
i.e., $|\not{V}(s)|$ within the time interval $[0,\tau]$.

Let $\gamma_{1}:=\alpha^{\star}-2\epsilon$. Define $\tau^{1}$ and $\tau^{2}$
be the first time and the second time that the projection of the velocity
characteristics almost reaches a threshold as follows

$\tau_{\ast}:=\sup\\{s:s\in[0,\tau],\forall\kappa\in[0,s],{\alpha_{t}(\kappa)}\leq\gamma_{1}\\},\quad\tau^{\star}:=\sup\\{s:s\in[0,\tau],\forall\kappa\in[0,s],{\alpha_{t}(\kappa)}\leq\gamma_{1}+1/M_{t}\\}.$
(3.11)

Our goal is to show that $\tau^{\star}=\tau$. If either $\tau_{\ast}$ or
$\tau^{\star}$ equals to $\tau$, then there is nothing left to be proved. We
focus on the case $\tau_{\ast}<\tau^{\star}<\tau$. Let

$\gamma_{2}:=\inf\\{k:k\in\mathbb{R}_{+},|V(x,v,\tau_{\ast},0)||\leq
2^{kM_{t}}\\}.$

We make the bootstrap assumption for the velocity characteristics $V(s)$ as
follow ,

$\tau^{\ast}:=\sup\\{s:s\in[\tau_{\ast},\tau^{\star}],\forall\kappa\in[\tau^{\ast},s],|\not{V}(s)|\in
2^{\gamma_{1}M_{t}}[99/100,101/100],|V(s)|\in
2^{\gamma_{2}M_{t}}[99/100,101/100]\\}.$ (3.12)

Now, our goal is to show that, independent of $(x,v)\in R_{t}(0)$, we can
improve the above bootstrap assumption and show that
$\tau^{\ast}=\tau^{\star}$. Hence improving the upper bound in the definition
of $\tau^{\star}$ and ruling out the case $\tau^{\star}<\tau.$

Recall (2.4) and (2.2), we have

$\forall s\in[0,t],\quad|X(s)|\leq|X(0)|+\int_{0}^{t}|\hat{V}(s)|ds\leq
2^{M_{t}/2+1}.$

Based on the possible size of the magnitude of space characteristic in the
horizontal plane, we decompose the interval $[0,M_{t}/2+1]$ into a finite
union of intervals, which overlap with at most with four other intervals. More
precisely,

$[0,2^{M_{t}/2+1}]=\cup_{i=0}^{K}I_{i},\quad I_{0}=[0,2^{{\kappa_{0}}}],\quad
I_{k}=[2^{a_{k}},2^{a_{k}+10}],\quad
a_{0}=\kappa_{0}-6,\kappa_{0}=-100M_{t},\\\ \forall
k\in\\{1,\cdots,K-2\\},a_{k+1}=a_{k}+6,\quad a_{K}=M_{t}/20+1-10,\quad
K=\lceil(M_{t}/2+1-\kappa_{0})/6\rceil.$ (3.13)

Moreover, as a result of direct computation, from (2.2), we have

$\frac{d}{ds}|\not{X}(s)|^{2}=\frac{2\not{X}(s)\cdot\not{V}(s)}{\sqrt{1+|V(s)|^{2}}},\qquad\frac{d}{ds}\frac{\not{X}(s)\cdot\not{V}(s)}{\sqrt{1+|V(s)|^{2}}}=\frac{|\not{V}(s)|^{2}}{1+|V(s)|^{2}}+\big{(}\frac{\big{(}\not{X}(s),0\big{)}}{\sqrt{1+|V(s)|^{2}}}-\frac{\not{X}(s)\cdot\not{V}(s)}{\big{(}{1+|V(s)|^{2}}\big{)}^{3/2}}V(s)\big{)}\cdot
E(s,X(s)),\\\
\frac{d}{ds}|\not{V}(s)|=|\not{V}(s)|^{-1}\big{(}\not{V}(s),0\big{)}\cdot
E(s,X(s)),\quad\frac{d}{ds}|V(s)|^{2}=2V(s)\cdot E(s,X(s)).$ (3.14)

Within the time interval $[\tau_{\ast},\tau^{\ast}]$, we show two facts about
the characteristics under the bootstrap assumption (3.12).

  1. Fact (i)

Given any $t_{1},t_{2}\in[\tau_{\ast},\tau^{\ast}]$ and any
$i\in\\{0,\cdots,K-1\\}$, s.t., $\forall s\in[t_{1},t_{2}],|\not{X}(s)|\in
I_{i}\cup I_{i+1}$, then the bounds of speed characteristics
$|\not{V}(s)|,|V(s)|$ can be improved for any $s\in[t_{1},t_{2}].$ More,
precisely, the increment of $|\not{V}(s)|$ is at most
$2^{\gamma_{1}M_{t}-\epsilon M_{t}}$. and the increment of $|V(s)|$ is at most
$2^{\gamma_{2}M_{t}-\epsilon M_{t}}$.

  2. Fact (ii)

Let $I_{-1}:=\emptyset$. For any $i\in\\{0,\cdots,K\\}$ and any
$s\in[\tau_{\ast},\tau^{\ast}]$, if $|\not{X}(s)|\in I_{i}/I_{i-1}$ with
$d|\not{X}(t)|^{2}/dt\big{|}_{t=s}\geq 0$, then for any later time
$l\in[s,\tau^{\ast}]$, $|\not{X}(l)|\in\cup_{i\leq j\leq K}I_{j}$.

Based on the possible size of $i$, we divide the proof of Fact (i) into two
cases as follows. The Fact (ii) will be obtained as a byproduct.

Case $1$: Given any $t_{1},t_{2}\in[\tau_{\ast},\tau^{\ast}]$, if $\forall
s\in[t_{1},t_{2}],|\not{X}(s)|\in I_{0}\cup I_{1}$.

Recall (3.14). From the rough estimate of the electric field (2.19) in Lemma
2.4, the following estimate holds for any $s\in[t_{1},t_{2}],$

$\frac{\not{X}(s)\cdot\not{V}(s)}{\sqrt{1+|V(s)|^{2}}}-\frac{\not{X}(t_{1})\cdot\not{V}(t_{1})}{\sqrt{1+|V(t_{1})|^{2}}}\geq\frac{4}{5}2^{2\gamma_{1}M_{t}-2\gamma_{2}M_{t}}(s-t_{1})-2^{(\kappa_{0}-\gamma_{2})M_{t}+5M_{t}/3+10\epsilon
M_{t}}(s-t_{1}),$
$\Longrightarrow\frac{\not{X}(s)\cdot\not{V}(s)}{\sqrt{1+|V(s)|^{2}}}\geq\frac{3}{4}2^{2\gamma_{1}M_{t}-2\gamma_{2}M_{t}}(s-t_{1})-\frac{11}{10}2^{\kappa_{0}M_{t}+\gamma_{1}M_{t}-\gamma_{2}M_{t}},$
$\Longrightarrow|X(s)|^{2}-|X(t_{1})|^{2}\geq\frac{3}{4}2^{2\gamma_{1}M_{t}-2\gamma_{2}M_{t}}(s-t_{1})^{2}-\frac{11}{5}2^{\kappa_{0}M_{t}+\gamma_{1}M_{t}-\gamma_{2}M_{t}}(s-t_{1}).$
(3.15)

Since $|\not{X}(t_{2})|\leq 2^{\kappa_{0}+10}$, from the above estimate, we
know that

$2^{2\gamma_{1}M_{t}-2\gamma_{2}M_{t}}(t_{2}-t_{1})^{2}\leq
2^{2\kappa_{0}+20},\quad\Longrightarrow|t_{2}-t_{1}|\leq
2^{\kappa_{0}+\gamma_{2}M_{t}-\gamma_{1}M_{t}+10}.$

From the above estimates, we know that, if the lapse of time is greater than
$2^{\kappa_{0}+\gamma_{2}M_{t}-\gamma_{1}M_{t}+10}$, then the space
characteristics will leave $I_{0}\cup I_{1}$, i.e., entering the region
$\cup_{i=2}^{K}I_{i}\cup(I_{2}/I_{1})$. Moreover, from (3.14) and the rough
estimate of the electric field (2.19) in Lemma 2.4, for any
$s\in[t_{1},t_{2}],$ we have

$\big{|}|\not{V}(s)|^{2}-|\not{V}(t_{1})|^{2}\big{|}+\big{|}|V(s)|^{2}-|V(t_{1})|^{2}\big{|}\leq
2^{\gamma_{2}M_{t}+2}\int_{t_{1}}^{t_{2}}\|E(s,x)\|_{L^{\infty}_{x}}ds\lesssim
2^{\kappa_{0}+2\gamma_{2}M_{t}}2^{(5/3+10\epsilon)M_{t}}\lesssim 1.$ (3.16)

Case $2$: Given any $t_{1},t_{2}\in[\tau_{\ast},\tau^{\ast}]$, if $\forall
s\in[t_{1},t_{2}],|\not{X}(s)|\in I_{i}\cup I_{i+1}$ .

Recall (3.14). For any $s\in[t_{1},t_{2}]$, from the estimate (3.23) in Lemma
3.1, we have

$\frac{\not{X}(s)\cdot\not{V}(s)}{\sqrt{1+|V(s)|^{2}}}-\frac{\not{X}(t_{1})\cdot\not{V}(t_{1})}{\sqrt{1+|V(t_{1})|^{2}}}\geq\frac{9}{10}2^{2\gamma_{1}M_{t}-2\gamma_{2}M_{t}}(s-t_{1})-C\big{[}\sum_{b\in\\{0,2/3,5/6,1\\}}2^{(1-b)a_{i}}2^{b(\gamma_{1}-\gamma_{2})M_{t}+(\gamma_{1}-\gamma_{2}-2\epsilon)M_{t}}(t_{2}-t_{1})$
$+2^{a_{i}+3\alpha^{\star}M_{t}/4-\gamma_{2}M_{t}+10\epsilon M_{t}}\big{]}$
$\Longrightarrow\frac{\not{X}(s)\cdot\not{V}(s)}{\sqrt{1+|V(s)|^{2}}}\geq\frac{4}{5}2^{2\gamma_{1}M_{t}-2\gamma_{2}M_{t}}(s-t_{1})-\frac{11}{10}|\not{X}(t_{1})|2^{\gamma_{1}M_{t}-\gamma_{2}M_{t}},$
(3.17)
$\Longrightarrow|\not{X}(s)|^{2}-|\not{X}(t_{1})|^{2}\geq\frac{4}{5}2^{2(\gamma_{1}-\gamma_{2})M_{t}}(s-t_{1})^{2}-\frac{22}{10}|\not{X}(t_{1})|2^{(\gamma_{1}-\gamma_{2})M_{t}}(s-t_{1}),$
$\Longrightarrow|\not{X}(s)|\geq
2^{(\gamma_{1}-\gamma_{2})M_{t}-5}(s-t_{1})-2|\not{X}(t_{1})|,\quad\Longrightarrow|t_{2}-t_{1}|\lesssim
2^{a_{i}-(\gamma_{1}-\gamma_{2})M_{t}}.$ (3.18)

From the above estimate of the time difference, the fact that $t_{1},t_{2}\leq
2^{\epsilon M_{t}/100}$, and the estimate (3.23) in Lemma 3.1, the following
estimate holds for any $s\in[t_{1},t_{2}]$,

$\big{|}|\not{V}(s)|-|\not{V}(t_{1})|\big{|}\lesssim\big{(}\sum_{b\in\\{0,2/3,5/6,1\\}}2^{-ba_{i}}2^{b(\gamma_{1}-\gamma_{2})M_{t}+(\gamma_{1}-2\epsilon)M_{t}}\big{)}(s-t_{1})+2^{(3{\alpha}^{\star}+\epsilon)M_{t}/4}\leq
2^{\gamma_{1}M_{t}-3\epsilon M_{t}/2},$
$\big{|}|V(s)|-|V(t_{1})|\big{|}\lesssim\big{[}\big{(}\sum_{b\in\\{0,2/3,5/6,1\\}}2^{-ba_{i}}2^{b(\gamma_{1}-\gamma_{2})M_{t}+(\gamma_{1}-2\epsilon)M_{t}}\big{)}(s-t_{1})+2^{(3{\alpha}^{\star}+20\epsilon)M_{t}/4}\big{]}\leq
2^{\gamma_{2}M_{t}-3\epsilon M_{t}/2}.$

Moreover, if $\not{X}(t_{1})\cdot\not{V}(t_{1})\geq 0$, i.e.,
$d|X(t)|^{2}/dt\big{|}_{t=t_{1}}\geq 0$, and $|\not{X}(t_{1})|\in
I_{i}/I_{i-1}=(2^{a_{i}+5},2^{a_{i}+10}]$, then we can improve the obtained
estimate (3.17) as follows,

$\textup{If $\not{X}(t_{1})\cdot\not{V}(t_{1})\geq
0$},\quad\frac{\not{X}(s)\cdot\not{V}(s)}{\sqrt{1+|V(s)|^{2}}}\geq\frac{4}{5}2^{2(\gamma_{1}-\gamma_{2})M_{t}}(s-t_{1})-2^{-10+a_{i}}2^{(\gamma_{1}-\gamma_{2})M_{t}},$
$\Longrightarrow\forall s\in[t_{1},t_{2}],\quad|\not{X}(s)|\geq
2^{(\gamma_{1}-\gamma_{2})M_{t}-1}(s-t_{1})+2^{-1}|\not{X}(t_{1})|$ (3.19)

Recall (3.13). From the above estimate, we have
$|\not{X}(s)|\in[2^{a_{k}+3},2^{a_{k}+8}]\subsetneq(2^{a_{k}},2^{a_{k}+10}]$,
i.e., the space characteristics will leave $I_{i}$ and enter $I_{i+1}$ instead
of $I_{i-1}$.

To sum up, our desired Fact (i) and Fact (ii) hold from the above discussion.
Now, we are ready to improve the bootstrap assumption (3.12).

Thanks to Fact (i), we only have to show that the trajectory of
$|\not{X}(s)|,s\in[\tau_{\ast},\tau^{\ast}]$, visit intervals $I_{i}\cup
I_{i+1},i\in\\{0,\cdots,K-1\\}$ for at most $CM_{t}$ number of times, which is
only logarithmic.

Since the $d|\not{X}(s)|/ds$ is bounded, the trajectory of $|\not{X}(s)|$ can
only visit intervals $J_{i}:=I_{i}/I_{i-1},i\in\\{0,\cdots,K\\}$ for finite
times. We order the visited intervals within $[\tau,\tau^{\star}]$ with
respect to time as follows $(J_{i_{1}},J_{i_{2}},\cdots,J_{i_{X}})$, $\forall
p\in\\{1,\cdots,X\\},i_{p}\in\\{0,\cdots,K\\}$. Due to the possible revisit
scenario, e.g., $J_{1}\longrightarrow J_{2}\longrightarrow
J_{1}\longrightarrow\cdots$, the size of finite number $X$ is not clear.

Due to the continuity of characteristics, we have
$i_{p}-i_{p+1}\in\\{1,-1\\}$. Let ${i_{\iota_{0}}}$ be the first local minimum
of the ordered set $(i_{1},i_{2},\cdots,i_{X})$, then $\iota_{0}\leq
K+1,i_{\iota+1}=i_{\iota}+1$. Let $\tau_{\iota_{0}}$ be the time such that the
$|\not{X}(\tau_{\iota_{0}})|$leaves $I_{i_{\iota_{0}}}$ and enters
$I_{i_{\iota_{0}}+1}/I_{\iota_{0}}$. As $|\not{X}(s)|$ increases at time
$\tau_{\iota_{0}}$, we know that
$d|\not{X}(t)|^{2}/dt\big{|}_{t=\tau_{\iota_{0}}}\geq 0$. Therefore, from the
obtained Fact (ii), we know that $i_{\kappa}\geq i_{\iota_{0}}$ for any
$\kappa\geq\iota_{0}$. Let $\iota_{1}:=\inf\\{\kappa:i_{\kappa}\geq
i_{\iota_{0}}+2,\kappa\geq\iota_{0}\\}$. We know that
$J_{i_{\iota_{1}}}=I_{i_{\iota_{0}+2}}/I_{i_{\iota_{0}+1}}$ and
$\cup_{\kappa\in[{\iota_{0}},{\iota_{1}}-1]\cap\mathbb{Z}}J_{i_{\kappa}}\subset
I_{i_{\iota_{0}+1}}\cup I_{\iota_{0}}$.

Similarly, let $\tau_{1}$ be the time such that
$|\not{X}(\tau_{\iota_{1}})|$leaves $I_{i_{\iota_{1}}-1}=I_{i_{\iota_{0}+1}}$
and enters $J_{\iota_{\iota_{1}}}=I_{i_{\iota_{0}}+2}/I_{i_{\iota_{0}}+1}$,
from the obtained Fact (ii), we know that $i_{\kappa}\geq i_{\iota_{0}}+1$ for
any $\kappa\geq\iota_{1}$. Therefore, inductively, we can define a sequence of
$\\{\iota_{k}\\}_{k=0}^{m}$ such that the following property holds,

$\\{\kappa:i_{\kappa}\geq i_{\iota_{m}}+1\\}=\emptyset,\quad\forall
k\in\\{0,\cdots,m-1\\},\quad i_{\iota_{k+1}}\geq i_{\iota_{k}}+1,\quad
X_{\iota_{k}}:=\cup_{\kappa\in[\iota_{k},\iota_{k+1}-1]\cap\mathbb{Z}}J_{i_{\kappa}}\subset
I_{i_{\iota_{k}}+1}\cup I_{i_{\iota_{k}}},\\\
X_{\iota_{m}}:=\cup_{\kappa\in[\iota_{m},K]\cap\mathbb{Z}}J_{i_{\kappa}}\subset
I_{i_{\iota_{m}}+1}\cup I_{i_{\iota_{m}}}.$ (3.20)

If $i_{\iota_{m}}=K$, we use the convention that $I_{K+1}:=\emptyset.$

Since $\\{i_{\iota_{k}}\\}_{k=0}^{m}$ is an increasing sequence with upper
bound $K$, we know that $m\leq K+1$. From the above discussion, we regroup the
ordered set $(J_{i_{1}},J_{i_{2}},\cdots,J_{i_{X}})$, as follows,

$(J_{i_{1}},J_{i_{2}},\cdots,J_{i_{\iota_{0}-1}},X_{\iota_{0}},X_{\iota_{1}},\quad
X_{\iota_{m}}),$ (3.21)

in which the order is still consistent with the time order that the
characteristic $|\not{X}(s)|,s\in[\tau_{\ast},\tau^{\ast}]$ travels.

Since the total number of sets in (3.21) is less than $2(K+1)\leq 100M_{t}$,
see (3.13), which is only logarithmic, from the relation in (3.20) and Fact
(i), our bootstrap assumption is improved. Hence finishing the bootstrap
argument, i.e., $\tau^{\ast}=\tau^{\star}$, which further implies that
$\tau^{\ast}=\tau^{\star}=\tau.$

Step $\mathbf{2}$ Estimate of the full velocity characteristics within the
time interval $[0,\tau]$.

As a result of the Step $\mathbf{1}$ and (2.6), we know that
$\alpha_{\tau}M_{\tau}\leq\alpha_{t}(\tau)M_{t}\leq(\alpha^{\star}-2\epsilon)M_{t}$.
Now, our goal is to show that $\tau=t.$ Recall (3.14). For any $(x,v)\in
R_{t}(0)$, $t_{1},t_{2}\in[t^{\ast},\tau]$, the following estimate holds from
the estimate (3.24) in Lemma 3.2

$\big{|}|V(t_{2})|^{2}-|V(t_{1})|^{2}\big{|}\leq
2\big{|}\int_{t_{1}}^{t_{2}}{V}(s)\cdot E(s,X(s))ds\big{|}\lesssim
2^{(2\gamma-5\epsilon)M_{t}}\leq 2^{2\gamma M_{t}-4.5\epsilon M_{t}}.$

Therefore, our bootstrap assumption in (3.10) is improved. Hence finishing the
bootstrap argument, i.e., $\tau=t.$ Recall the definition of the majority set
in (2.5). Since $\tau=t$, after rerunning the argument in Step $\mathbf{1}$,
we have

$\beta_{t}=\sup_{s\in[0,t]}\beta_{t}(s)\leq
1-2\epsilon,\quad\alpha_{t}=\sup_{s\in[0,t]}\alpha_{t}(s)\leq\alpha^{\star}-\epsilon.$

Hence finishing the proof of our desired estimate (3.8). ∎

The key ingredient of the first induction in (3.1) is the following Lemma, in
which we use the smoothing effect by localizing the frequency on Fourier side
and then doing normal form transformation. It worth to remark that the
following Lemma is also valid for the RVP system (1.4) because we will not use
the extra equation in the sRVM system (1.2) in the proof.

######  Lemma 3.1.

Let $t\in[0,T),a_{p}\in\mathbb{Z},\alpha^{\star}=7/10+\iota,\iota:=10^{-2}$,
$t_{1},t_{2}\in[0,t]$ be fixed, s.t., $M_{t}\gg 1,$ $\forall
s\in[t_{1},t_{2}],|\not{X}(s)|\sim
2^{a_{p}},{\alpha}_{s}M_{s}\leq\alpha^{\star}M_{t},|\not{V}(s)|\sim
2^{\gamma_{1}M_{t}},|V(s)|\sim 2^{\gamma_{2}M_{t}}$, where
$\gamma_{1}\in[\alpha^{\star}-2\epsilon,\alpha^{\star}]$,
$\gamma_{2}\leq(1-3\epsilon)$. Assume that
$C(\cdot,\cdot):\mathbb{R}^{2}\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$
is a smooth function s.t., the following estimate holds for any
$s\in[t_{1},t_{2}]$,

$\big{|}C(\not{x},V(s))\big{|}+|\not{X}(s)|\big{|}\nabla_{\not{x}}C(\not{x},V(s))|_{\not{x}=\not{X}(s)}\big{|}+\sum_{|\alpha|\leq
10}|V(s)|^{|\alpha|}\big{|}\nabla_{v}^{\alpha}C(\not{x},v)|_{v=V(s)}\big{|}\lesssim\mathcal{M}(C).$
(3.22)

Then the following estimate holds,

$\big{|}\int_{t_{1}}^{t_{2}}C(\not{X}(s),V(s))\cdot
E(s,X(s))ds\big{|}\lesssim\mathcal{M}(C)\big{[}\big{(}\sum_{b\in\\{0,2/3,5/6,1\\}}2^{-ba_{p}}2^{b(\gamma_{1}-\gamma_{2})M_{t}+(\gamma_{1}-2\epsilon)M_{t}}\big{)}(t_{2}-t_{1})+2^{(3\alpha^{\star}+\epsilon)M_{t}/4}\big{]}.$
(3.23)

###### Proof.

See section 4. ∎

The key ingredient of the second induction in (3.1) is the following Lemma, in
which we use both the smoothing effect and the hyperbolic natural of the sRVM
system (1.2) by using the extra equation.

######  Lemma 3.2.

Let $t\in[0,T),a_{p}\in\mathbb{Z},\alpha^{\star}=7/10+\iota,\iota:=10^{-2}$,
$\gamma\in[1-3\epsilon,1]$, $t_{1},t_{2}\in[0,t]$ be fixed, s.t., $M_{t}\gg
1,$ $\forall s\in[t_{1},t_{2}],|\not{X}(s)|\sim 2^{a_{p}},|V(s)|\sim 2^{\gamma
M_{t}},\alpha_{s}M_{s}\leq\alpha^{\star}M_{t}$. Then the following estimate
holds,

$\big{|}\int_{t_{1}}^{t_{2}}{V}(s)\cdot E(s,X(s))ds\big{|}\lesssim
2^{(2\gamma-5\epsilon)M_{t}}.$ (3.24)

###### Proof.

See section 5. ∎

##  4\. Proof of Lemma 3.1

We first rule out the case $|\not{X}(s)|$ is very small, If $a_{p}\leq
2(\gamma_{1}-\gamma_{2})M_{t}-20\epsilon M_{t}$, then from the estimate (2.19)
in Lemma 2.4, we have

$\big{|}\int_{t_{1}}^{t_{2}}C(\not{X}(s),V(s))\cdot E(s,X(s))ds\big{|}\lesssim
2^{\alpha^{\star}M_{t}+2\epsilon
M_{t}}2^{-a_{p}/2}\mathcal{M}(C)(t_{2}-t_{1})\lesssim
2^{-a_{p}+(\gamma_{1}-\gamma_{2})M_{t}}2^{(\alpha^{\star}-5\epsilon)M_{t}}\mathcal{M}(C)(t_{2}-t_{1}).$

Now, it would be sufficient to consider the case $a_{p}\geq
2(\gamma_{1}-\gamma_{2})M_{t}-20\epsilon M_{t}$. After doing dyadic
decomposition for the size of frequency, the size of $\not{v}$, and $v$, we
have

$\big{|}\int_{t_{1}}^{t_{2}}C(\not{X}(s),V(s))\cdot
E(s,X(s))ds\big{|}\lesssim\sum_{k,j_{2}\in\mathbb{Z}_{+},j_{1}\in[0,j_{2}+2]\cap\mathbb{Z}}\big{|}H_{k;j_{1},j_{2}}(t_{1},t_{2})\big{|},\\\
H_{k;j_{1},j_{2}}(t_{1},t_{2})=\int_{t_{1}}^{t_{2}}C(\not{X}(s),V(s))\cdot
E_{k;j_{1},j_{2}}(s,X(s))ds,$ (4.1)

where $E_{k;j_{1},j_{2}}(\cdot,\cdot)$ is defined in (2.12). From the obtained
estimates (2.20) and (2.21), we know that it would be sufficient to consider
the case fixed $k,j_{1},j_{2}\in\mathbb{Z}_{+}$, s.t.,
$k\in[0,3M_{t}+5\epsilon M_{t}]\cap\mathbb{Z}$,
$j_{2}\in[0,(1+\epsilon)M_{t}]\cap\mathbb{Z}$ and
$j_{1}\in[0,j_{2}+2]\cap\mathbb{Z}$. Moreover, based on the possible sizes of
$k,j_{1},j_{2}$, we split into three sub-cases as follows.

$\bullet$ If $k\leq
j_{2}+(\gamma_{1}-\gamma_{2}+\alpha^{\star}-5\epsilon)M_{t}$ or $k\geq
2j_{1}+j_{2}-(\alpha^{\star}-5\epsilon)M_{t}.$

From the estimate (2.16) in Lemma 2.3, we have

$\big{|}H_{k;j_{1},j_{2}}(t_{1},t_{2})\big{|}\lesssim\min\\{2^{-k+j_{2}+2j_{1}},2^{k-j_{2}+\epsilon
M_{t}}2^{-a_{p}}\\}\mathcal{M}(C)(t_{2}-t_{1})\lesssim\sum_{b\in\\{0,1\\}}2^{-ba_{p}}2^{b(\gamma_{1}-\gamma_{2})M_{t}+(\alpha^{\star}-5\epsilon)M_{t}}.$
(4.2)

$\bullet$ If
$k\in[j_{2}+(\gamma_{1}-\gamma_{2}+\alpha^{\star}-5\epsilon)M_{t},2j_{1}+j_{2}-(\alpha^{\star}-5\epsilon)M_{t}]$,
$j_{1}\leq(\gamma_{1}-\gamma_{2})M_{t}/2+(\alpha^{\star}-6\epsilon)M_{t}.$

From the estimate (2.17) in Lemma 2.3, we have

$\big{|}H_{k;j_{1},j_{2}}(t_{1},t_{2})\big{|}\lesssim
2^{-a_{p}/2+j_{1}+\epsilon M_{t}}\mathcal{M}(C)(t_{2}-t_{1})\lesssim
2^{-a_{p}/2}2^{(\gamma_{1}-\gamma_{2})M_{t}/2+(\alpha^{\star}-5\epsilon)M_{t}}\mathcal{M}(C)(t_{2}-t_{1}).$
(4.3)

$\bullet$ If
$k\in[j_{2}+(\gamma_{1}-\gamma_{2}+\alpha^{\star}-5\epsilon)M_{t},2j_{1}+j_{2}-(\alpha^{\star}-5\epsilon)M_{t}]$
and $j_{1}\geq(\gamma_{1}-\gamma_{2})M_{t}/2+(\alpha^{\star}-6\epsilon)M_{t}$.

Let $\theta_{V(s)}(v):=(\hat{V}(s)-\hat{v})/|\hat{V}(s)-\hat{v}|$. Based on
the possible size of $\theta_{V(s)}(v)\cdot\xi/|\xi|$, $|\hat{V}(s)-\hat{v}|$,
and the angle between $(-X_{2}(s),X_{1}(s),0)/|\not{X}(s)|$ and
$\theta_{V(s)}(v)$, we decompose $H_{k;j_{1},j_{2}}(t_{1},t_{2})$ into three
parts as follows,

$H_{k;j_{1},j_{2}}(t_{1},t_{2})=\sum_{i=1,2,3}\int_{t_{1}}^{t_{2}}H_{k,j_{1},j_{2}}^{i}(s)ds,$
(4.4)

where

$H_{k,j_{1},j_{2}}^{1}(s)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(s)\cdot\xi}iC(\not{X}(s),V(s))\cdot\xi|\xi|^{-2}\widehat{f}(s,\xi,v)\varphi_{k;j_{1},j_{2}}(v,\xi)\varphi_{l_{1},\alpha}(v,X(s),\tilde{V}(s))dvd\xi,$
(4.5)
$H_{k,j_{1},j_{2}}^{2}(s)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(s)\cdot\xi}iC(\not{X}(s),V(s))\cdot\xi|\xi|^{-2}\widehat{f}(s,\xi,v)\varphi_{k;j_{1},j_{2}}(v,\xi)\psi_{\leq
l_{2}}(\theta_{V(s)}(v)\cdot\tilde{\xi})\big{(}1-\varphi_{l_{1},\alpha}(v,X(t),\tilde{V}(s))\big{)}dvd\xi,$
(4.6)
$H_{k,j_{1},j_{2}}^{3}(s)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(s)\cdot\xi}iC(\not{X}(s),V(s))\cdot\xi|\xi|^{-2}\widehat{f}(s,\xi,v)\varphi_{k;j_{1},j_{2}}(v,\xi)\psi_{>l_{2}}(\theta_{V(s)}(v)\cdot\tilde{\xi})\big{(}1-\varphi_{l_{1},\alpha}(v,X(s),\tilde{V}(s))\big{)}dvd\xi,$
(4.7)

where
$\varphi_{k;j_{1},j_{2}}(v,\xi):=\varphi_{j_{1}}(\not{v})\varphi_{j_{2}}(v)\varphi_{k}(\xi)$
the cutoff function $\varphi_{l_{1},\alpha}(v,\tilde{V}(t))$ is defined as
follows,

$\varphi_{l_{1},\alpha}(v,X(s),\tilde{V}(s)):=\psi_{\leq
l_{1}}(|\tilde{v}-\tilde{V}(s)|)+\psi_{>l_{1}}(|\tilde{v}-\tilde{V}(s)|)\psi_{\leq\alpha}(\theta_{V(s)}(v)\times(-X_{2}(s),X_{1}(s),0)/|\not{X}(s)|),$
(4.8)

and the thresholds of cutoff functions are chosen as follows,

$l_{1}:=(\alpha^{\star}M_{t}+(\gamma_{1}-\gamma_{2})M_{t}-a_{p}+k-3j_{2})/2-5\epsilon
M_{t},\quad\alpha=\frac{2}{3}l_{1},\quad
l_{2}:=(\gamma_{1}-\gamma_{2}+\alpha^{\star}-10\epsilon)M_{t}+j_{2}-k+\alpha.$
(4.9)

$\oplus$ The estimate of $H_{k,j_{1},j_{2}}^{1}(s).$

Recall (4.5). As in [16][Lemma 4.3], for any fixed $a,b\in\mathbb{S}^{2},a\neq
b,$ $0<\epsilon\ll 1$, we have

$S_{a,b}:=\\{c:c\in\mathbb{S}^{2},|(a-c)\times
b|\leq\epsilon\\},\quad|S_{a,b}|\lesssim\epsilon^{3/2},\quad|a-c|\lesssim\epsilon^{1/2}.$
(4.10)

From the above estimate and the volume of support of $v$, we have

$\big{|}\int_{t_{1}}^{t_{2}}H_{k,j_{1},j_{2}}^{1}(s)ds\big{|}\lesssim
2^{\epsilon M_{t}}\min\\{2^{-k+3j_{2}+2l_{1}}+2^{-k+3j_{2}+3\alpha}\\}\lesssim
2^{-a_{p}+(\gamma_{1}-\gamma_{2})M_{t}}2^{(\alpha^{\star}-6\epsilon)M_{t}}.$
(4.11)

$\oplus$ The estimate of $H_{k,j_{1},j_{2}}^{2}(s).$

Note that, in terms of kernel, we have

$H_{k,j_{1},j_{2}}^{2}(s)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}f(s,X(s)-y,v)\tilde{K}_{k,l_{2}}(y,v,V(s))\big{(}1-\varphi_{l_{1},\alpha}(v,X(s),\tilde{V}(s))\big{)}\varphi_{j_{1}}(\not{v})\varphi_{j_{2}}(v)dvdy,$
(4.12)

where

$\tilde{K}_{k,l_{2}}(y,v,V(s))=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}i\xi|\xi|^{-2}\varphi_{k}(\xi)\psi_{\leq
l_{2}}(\theta_{V(s)}(v)\cdot\tilde{\xi})d\xi.$

By doing integration by parts in $\theta_{V(s)}(v)$ direction and directions
perpendicular to $(\theta_{V(s)}(v))^{\bot}$, we have the following estimate
for the kernel $\tilde{K}_{k,l_{2}}(y,v,V(s))$,

$|\tilde{K}_{k,l_{2}}(y,v,V(s))|\lesssim
2^{2k+l_{2}}(1+2^{k+l_{2}}|y\cdot\theta_{V(s)}(v)|)^{-N_{0}^{3}}(1+2^{k}|y\times\theta_{V(s)}(v)|)^{-N_{0}^{3}}.$
(4.13)

From the above estimate, we know that “$y$” is localized inside a cylinder
with base in the plane perpendicular to $\theta_{V(s)}(v)$. Due to the cutoff
function $(1-\varphi_{l_{1},\alpha}(v,X(s),\tilde{V}(s)))$ in (4.12), the
angle between $\theta_{V(s)}(v)$ and $(-X_{2}(s),X_{1}(s),0)/|\not{X}(s)|)$ is
greater than $2^{\alpha}$, which means that the intersection of the cylinder
with any $x_{1}x_{2}$ plane is less than $(2^{-k-\alpha})^{2}.$

Note that $a_{p}\geq-k-l_{2}+10\epsilon M_{t}.$ Hence, from the cylindrical
symmetry of solution and the above estimate, we have

$\int_{t_{1}}^{t_{2}}\big{|}H_{k,j_{1},j_{2}}^{2}(s)\big{|}ds\lesssim
2^{2k+l_{2}}\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}(1+2^{k+l_{2}}|y\cdot\theta_{V(s)}(v)|)^{-N_{0}^{3}}(1+2^{k}|y\times\theta_{V(s)}(v)|)^{-N_{0}^{3}}f(s,X(s)-y,v)\varphi_{j_{1}}(\not{v})$
$\times\varphi_{j_{2}}(v)\big{(}1-\varphi_{l_{1},\alpha}(v,X(s),\tilde{V}(s))\big{)}dvdyds\lesssim
1+\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{2^{2k+l_{2}-k-\alpha+\epsilon
M_{t}/10}}{|\not{X}(s)-\not{y}|}f(s,X(s)-y,v)\varphi_{j_{1}}(\not{v})\varphi_{j_{2}}(v)dvdyds$
$\lesssim 1+2^{2k+l_{2}-k-\alpha+\epsilon
M_{t}/10}2^{-j_{2}}2^{-a_{p}}\lesssim
2^{-a_{p}+(\gamma_{1}-\gamma_{2}+\alpha^{\star}-6\epsilon)M_{t}}.$ (4.14)

$\oplus$ The estimate of $H_{k,j_{1},j_{2}}^{3}(s).$

Recall (4.7). Note that, for any $(v,\xi)\in
supp(\psi_{>l_{2}}(\theta_{V(s)}(v)\cdot\tilde{\xi})(1-\varphi_{l_{1},\alpha}(v,X(s),\tilde{V}(s))))$,
c.f., (4.8), we have

$|\hat{V}(s)-\hat{v}|\gtrsim|\tilde{V}(s)-\tilde{v}|-2^{-2M_{t}+4\epsilon
M_{t}}\gtrsim
2^{l_{1}},\quad\Longrightarrow|\hat{V}(t)\cdot\tilde{\xi}-\hat{v}\cdot\tilde{\xi}|=|\hat{V}(t)-\hat{v}||\theta_{V(t)}(v)\cdot\tilde{\xi}|\gtrsim
2^{l_{1}+l_{2}}.$ (4.15)

Let $g(s,x,v):=f(s,x+s\hat{v},v).$ For $H_{k,j_{1},j_{2}}^{3}(s)$, we do
integration by parts in $s$ once. As a result, we have

$\big{|}\int_{t_{1}}^{t_{2}}H_{k,j_{1},j_{2}}^{3}(s)ds\big{|}\lesssim\big{|}End_{k,j_{1},j_{2}}(t_{1},t_{2})\big{|}+\big{|}\widetilde{H}_{k,j_{1},j_{2}}^{1}(t_{1},t_{2})\big{|}+\big{|}\widetilde{H}_{k,j_{1},j_{2}}^{2}(t_{1},t_{2})\big{|},$
(4.16)

where

$End_{k,j_{1},j_{2}}(t_{1},t_{2}):=\sum_{a=1,2}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(t_{a})\cdot\xi-
it_{a}\hat{v}\cdot\xi}C(\not{X}(t_{a}),V(t_{a}))\cdot\xi|\xi|^{-2}\hat{g}(t_{a},\xi,v)\varphi_{k;j_{1},j_{2}}(v,\xi)(\hat{V}(t_{a})\cdot\xi-\hat{v}\cdot\xi)^{-1}$
$\times\psi_{>l_{2}}(\theta_{V(t_{a})}(v)\cdot\tilde{\xi})\big{(}1-\varphi_{l_{1},\alpha}(v,X(t_{a}),\tilde{V}(t_{a}))\big{)}dvd\xi$
$=\sum_{a=1,2}\sum_{n\in[l_{1},2]\cap\mathbb{Z}}\sum_{l\in[l_{2},2]\cap\mathbb{Z}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}f(t_{a},X(t_{a})-y,v)\widetilde{K}^{0}_{k,n,l}(y,X(t_{a}),V(t_{a}),v)\varphi_{j_{1}}(\not{v})\varphi_{j_{2}}(v)dydv,$
(4.17)
$\widetilde{H}_{k,j_{1},j_{2}}^{1}(t_{1},t_{2}):=\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(s)\cdot\xi-
is\hat{v}\cdot\xi}C(\not{X}(s),V(s))\cdot\xi|\xi|^{-2}\varphi_{k,j_{1},j_{2}}(v,\xi)\partial_{s}\hat{g}(s,\xi,v)\big{(}1-\varphi_{l_{1},\alpha}(v,X(s),\tilde{V}(s))\big{)}$
$\times(\hat{V}(s)\cdot\xi-\hat{v}\cdot\xi)^{-1}\psi_{>l_{2}}(\theta_{V(s)}(v)\cdot\tilde{\xi})dvd\xi
ds,$ (4.18)
$\widetilde{H}_{k,j_{1},j_{2}}^{2}(t_{1},t_{2}):=\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(s)\cdot\xi}\partial_{s}\big{(}C(\not{X}(s),V(s))\cdot\xi|\xi|^{-2}\psi_{>l_{2}}(\theta_{V(s)}(v)\cdot\tilde{\xi})(\hat{V}(s)\cdot\xi-\hat{v}\cdot\xi)^{-1}\big{(}1-\varphi_{l_{1},\alpha}(v,X(s),\tilde{V}(s))\big{)}\big{)}$
$\times\varphi_{k;j_{1},j_{2}}(v,\xi)\varphi_{j_{1}}(\not{v})\varphi_{j_{2}}(v)\hat{f}(s,\xi,v)dvd\xi
ds$
$=\sum_{n\in[l_{1},2]\cap\mathbb{Z}}\sum_{l\in[l_{2},2]\cap\mathbb{Z}}\int_{s_{1}}^{s_{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}f(s,X(s)-y,v)\widetilde{K}^{1}_{k,n,l}(y,X(s),V(s),v)\varphi_{j_{1}}(\not{v})\varphi_{j_{2}}(v)dydvds,$
(4.19)

where the kernels
$\widetilde{K}^{i}_{k,l_{1},l_{2}}(y,\not{X}(s),V(s),v),i\in\\{0,1\\},$ are
defined as follow,

$\widetilde{K}^{0}_{k,n,l}(y,X(s),V(s),v):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\frac{\varphi_{k}(\xi)C(\not{X}(s),V(s))\cdot\xi}{|\xi|^{2}(\hat{V}(s)\cdot\xi-\hat{v}\cdot\xi)}\psi_{l}(\theta_{V(t_{a})}(v)\cdot\tilde{\xi})\psi_{n}(|\tilde{v}-\tilde{V}(t_{a})|)\big{(}1-\varphi_{l_{1},\alpha}(v,X(s),\tilde{V}(s))\big{)}d\xi,$
(4.20)
$\widetilde{K}^{1}_{k,n,l}(y,X(s),V(s),v):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\partial_{s}\big{(}\frac{\varphi_{k}(\xi)C(\not{X}(s),V(s))\cdot\xi}{|\xi|^{2}(\hat{V}(s)\cdot\xi-\hat{v}\cdot\xi)}\psi_{>l_{2}}(\theta_{V(s)}(v)\cdot\tilde{\xi})$
$\times\psi_{l}(\theta_{V(s)}(v)\cdot\tilde{\xi})\psi_{n}(|\tilde{v}-\tilde{V}(s)|)\big{(}1-\varphi_{l_{1},\alpha}(v,X(s),\tilde{V}(s))\big{)}\big{)}d\xi.$
(4.21)

By using the same strategy as in the obtained estimate (4.14), we have

$\big{|}End_{k,j_{1},j_{2}}(t_{1},t_{2})\big{|}\lesssim\sum_{a=1,2}\sum_{n\in[l_{1},2]\cap\mathbb{Z}}\sum_{l\in[l_{2},2]\cap\mathbb{Z}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}f(s_{a},X(t_{a})-y,v)2^{k-n}\varphi_{j_{1}}(\not{v})\varphi_{j_{2}}(v)(1+2^{k+l}|y\cdot\theta_{V(t_{a})}(v)|)^{-N_{0}^{3}}$
$\times(1+2^{k}|y\times\theta_{V(t_{a})}(v)|)^{-N_{0}^{3}}\mathcal{M}(C)dydv\lesssim
2^{5\epsilon M_{t}}2^{-l_{1}-\alpha-j_{2}-a_{p}}\mathcal{M}(C)\lesssim
2^{-a_{p}/6+2j_{2}/3-5(\alpha^{\star}+\gamma_{1}-\gamma_{2})M_{t}/3}\mathcal{M}(C)$
$\lesssim 2^{-a_{p}/6+7M_{t}/3-10\alpha^{\star}M_{t}/3+50\epsilon
M_{t}}\mathcal{M}(C)\lesssim 2^{\alpha^{\star}M_{t}/2}\mathcal{M}(C).$ (4.22)

$\dagger$ The estimate of $\widetilde{H}_{k,j_{1},j_{2}}^{1}(t_{1},t_{2})$.

Recall (4.18) and (1.2). We have

$\partial_{s}g(s,x,v)=\nabla_{x}\phi(s,x+s\hat{v})\cdot\nabla_{v}f(s,x+s\hat{v},v).$

Hence, after doing integration by parts in $v$, we have

$\widetilde{H}_{k,j_{1},j_{2}}^{1}(t_{1},t_{2}):=\sum_{n\in[l_{1},2]\cap\mathbb{Z}}\sum_{l\in[l_{2},2]\cap\mathbb{Z}}\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\widetilde{K}^{j_{1},j_{2}}_{k,n,l}(y,\not{X}(s),V(s),v)\cdot
E(s,X(s)-y)f(s,X(s)-y,v)dydvds,$ (4.23)

where the kernel $\widetilde{K}^{j_{1},j_{2}}_{k,n,l}(y,\not{X}(s),V(s),v)$ is
defined as follows,

$\widetilde{K}^{j_{1},j_{2}}_{k,n,l}(y,\not{X}(s),V(s),v):=\int_{\mathbb{R}^{3}}e^{ix\cdot\xi}\nabla_{v}\big{(}\varphi_{j_{1}}(\not{v})\varphi_{j_{2}}(v)\big{(}1-\varphi_{l_{1},\alpha}(v,X(s),\tilde{V}(s))(\hat{V}(s)\cdot\xi-\hat{v}\cdot\xi)^{-1}\psi_{l}(\theta_{V(s)}(v)\cdot\tilde{\xi})$
$\times\psi_{n}(|\tilde{v}-\tilde{V}(s)|)\big{)}C(\not{X}(s),V(s))\cdot\xi|\xi|^{-2}\varphi_{k}(\xi)d\xi.$
(4.24)

As a result of direct computation, by doing integration by parts in
$\theta_{V(s)}(v)$ direction and directions perpendicular to
$(\theta_{V(s)}(v))^{\bot}$, we have

$|\widetilde{K}^{j_{1},j_{2}}_{k,n,l}(y,\not{X}(s),V(s),v)|\lesssim
2^{k+l}\big{(}2^{-2l-2n-j_{2}}+2^{-l-n-
j_{1}}\big{)}(1+2^{k+l}|y\cdot\theta_{V(s)}(v)|)^{-N_{0}^{3}}(1+2^{k}|y\times\theta_{V(s)}(v)|)^{-N_{0}^{3}}\mathcal{M}(C).$
(4.25)

From the above estimate of kernel in (4.25) and the rough estimate of the
electric field (2.19) in Lemma 2.4, the following estimate holds after using
the same strategy as in the obtained estimate (4.14),

$\big{|}\widetilde{H}_{k,j_{1},j_{2}}^{1}(t_{1},t_{2})\big{|}\lesssim\sum_{n\in[l_{1},2]\cap\mathbb{Z}}\sum_{l\in[l_{2},2]\cap\mathbb{Z}}\int_{t_{1}}^{t_{2}}\mathcal{M}(C)2^{k+l}\big{(}2^{-2l-2n-j_{2}}+2^{-l-n-
j_{1}}\big{)}\|\nabla_{x}\phi(s,\cdot)\|_{L^{\infty}_{x}}$
$\times\varphi_{j_{1}}(\not{v})\varphi_{j_{2}}(v)(1+2^{k+l}|y\cdot\theta_{V(s)}(v)|)^{-N_{0}^{3}}(1+2^{k}|y\times\theta_{V(s)}(v)|)^{-N_{0}^{3}}f(s,X(s)-y,v)dydvds$
$\lesssim(2^{-2l_{1}-l_{2}-j_{2}}+2^{-l_{1}-j_{1}})2^{-\alpha-
j_{2}-a_{p}}2^{\alpha^{\star}M_{t}-a_{p}/2+10\epsilon
M_{t}}\mathcal{M}(C)(t_{2}-t_{1})$ $\lesssim 2^{30\epsilon
M_{t}}\big{(}2^{14M_{t}/3-17\alpha^{\star}M_{t}/3}+2^{-2a_{p}/3+7M_{t}/3-10\alpha^{\star}M_{t}/3-(\gamma_{1}-\gamma_{2})M_{t}/2}\big{)}\mathcal{M}(C)(t_{2}-t_{1}).$
(4.26)

$\dagger$ The estimate of $\widetilde{H}_{k,j_{1},j_{2}}^{2}(s_{1},s_{2})$.

Recall (4.19) and the definition of kernel
$\widetilde{K}^{1}_{k,n,l}(y,X(s),V(s),v)$ in (4.21). As a result of direct
computations, the following estimate holds for the kernel after doing
integration by parts in $\xi$ in $\theta_{V(s)}(v)$ direction and
$(\theta_{V(s)}(v))^{\bot}$ directions,

$\big{|}\widetilde{K}^{1}_{k,n,l}(y,X(s),V(s),v)\big{|}\lesssim
2^{k+l-l-n}\big{(}2^{-l-n-\gamma_{2}M_{t}}|\nabla_{x}\phi(s,X(s))|+2^{-\alpha+(\gamma_{1}-\gamma_{2})M_{t}-a_{p}}\big{)}$
$\times(1+2^{k+l}|y\cdot\theta_{V(s)}(v)|)^{-N_{0}^{3}}(1+2^{k}|y\times\theta_{V(s)}(v)|)^{-N_{0}^{3}}\mathcal{M}(C).$
(4.27)

From the above estimate of kernel and the rough estimate of the electric field
(2.19) in Lemma 2.4, we have

$\big{|}\widetilde{H}_{k,j_{1},j_{2}}^{2}(s_{1},s_{2})\big{|}\lesssim\sum_{n\in[l_{1},2]\cap\mathbb{Z}}\sum_{l\in[l_{2},2]\cap\mathbb{Z}}\int_{t_{1}}^{t_{2}}\mathcal{M}(C)2^{k-n}\big{(}2^{-l-n-\gamma_{2}M_{t}}\|\nabla_{x}\phi(s,\cdot)\|_{L^{\infty}_{x}}+2^{-\alpha+(\gamma_{1}-\gamma_{2})M_{t}-a_{p}}\big{)}2^{-k-\alpha-
j_{2}-a_{p}+10\epsilon M_{t}}ds$
$\lesssim\big{(}2^{-2l_{1}-l_{2}+(\alpha^{\star}-\gamma_{2})M_{t}-a_{p}/2}+2^{-5l_{1}/3+(\gamma_{1}-\gamma_{2})M_{t}-a_{p}}\big{)}2^{-2l_{1}/3-j_{2}-a_{p}+15\epsilon
M_{t}}\mathcal{M}(C)(t_{2}-t_{1})$ $\lesssim 2^{20\epsilon
M_{t}}\big{(}2^{-2k/3+3j_{2}-8\alpha^{\star}M_{t}/3-5(\gamma_{1}-\gamma_{2})M_{t}/3+a_{p}/6}+2^{-7k/6+5j_{2}/2-7\alpha^{\star}M_{t}/6-(\gamma_{1}-\gamma_{2})M_{t}/6-5a_{p}/6}\big{)}\mathcal{M}(C)(t_{2}-t_{1})$
$\lesssim 2^{30\epsilon
M_{t}}\big{(}2^{14M_{t}/3-17\alpha^{\star}M_{t}/3+a_{p}/6}+2^{8M_{t}/3-11\alpha^{\star}M_{t}/3-5a_{p}/6}\big{)}\mathcal{M}(C)(t_{2}-t_{1}).$
(4.28)

Recall (4.16). To sum up, after combining the obtained estimates (4.22),
(4.26), and (4.28), we have

$\big{|}\int_{t_{1}}^{t_{2}}H_{k,j_{1},j_{2}}^{3}(s)ds\big{|}\lesssim\mathcal{M}(C)\big{(}\sum_{b\in\\{0,2/3,5/6\\}}2^{-ba_{p}+b(\gamma_{1}-\gamma_{2})M_{t}+(\alpha^{\star}-10\epsilon)M_{t}}(t_{2}-t_{1})\big{)}+2^{\alpha^{\star}M_{t}/2}\mathcal{M}(C).$

After combining the above estimate, the estimate (4.11), and (4.14), we have

$\big{|}H_{k;j_{1},j_{2}}(t_{1},t_{2})\big{|}\lesssim\mathcal{M}(C)\big{(}\sum_{b\in\\{0,2/3,5/6,1\\}}2^{-ba_{p}+b(\gamma_{1}-\gamma_{2})M_{t}+(\alpha^{\star}-6\epsilon)M_{t}}(t_{2}-t_{1})\big{)}+2^{\alpha^{\star}M_{t}/2}\mathcal{M}(C).$

Hence finishing the proof of our desired estimate (3.23).

##  5\. Proof of Lemma 3.2

From the obtained estimates (2.16), (2.20) and (2.21), we know that it would
be sufficient to consider the case fixed $k\in[4M_{t}/5-20\epsilon
M_{t},3M_{t}+5\epsilon M_{t}]\cap\mathbb{Z}$. Moreover, from the obtained
estimates (2.22) and (2.20), we have

$\int_{t_{1}}^{t_{2}}\big{|}{{\not{V}(s)}}\cdot
P_{k}(\not{E})(s,X(s))\big{|}ds\lesssim
2^{(1+10\epsilon)M_{t}}2^{\alpha^{\star}M_{t}+10\epsilon M_{t}}\lesssim
2^{9M_{t}/5}.$ (5.1)

Hence, to control the LHS of (3.24), it would be sufficient to control the
following term for fixed $k\in[4M_{t}/5-20\epsilon M_{t},3M_{t}+5\epsilon
M_{t}]\cap\mathbb{Z}$,

$\big{|}\int_{t_{1}}^{t_{2}}{V_{3}(s)}\big{(}P_{k}{E}(s,X(s))\big{)}_{3}ds\big{|}.$

Based on the size of $\not{\xi}$, we decompose
$\big{(}{E}_{k;j_{1},j_{2}}(s,X(s))\big{)}_{3}$ further as follows,

$\big{(}P_{k}{E}(s,X(s))\big{)}_{3}=\sum_{\tilde{k}\in[0,k+2]\cap\mathbb{Z}}\tilde{E}_{k,\tilde{k}}(s,x),\quad\tilde{E}_{k,\tilde{k}}(s,x)=\sum_{j_{2}\in[0,(1+\epsilon)M_{t}]\cap\mathbb{Z},j_{1}\in[0,j_{2}+2]\cap\mathbb{Z}}\tilde{E}_{k,\tilde{k}}^{j_{1},j_{2}}(s,x),\\\
\tilde{E}_{k,\tilde{k}}^{j_{1},j_{2}}(s,x)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}{K}_{k,\tilde{k}}(y)f(s,x-y,v)\varphi_{j_{1}}(\not{v})\varphi_{j_{2}}(v)dydv,\quad{K}_{k,\tilde{k}}(y)=\int_{\mathbb{R}^{3}}e^{ix\cdot\xi}\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\frac{i\xi_{3}}{|\xi|^{2}}d\xi.$
(5.2)

By doing integration by parts in $\xi$, we have

$|{K}_{k,\tilde{k}}(y)|\lesssim
2^{2\tilde{k}}(1+2^{\tilde{k}}|\not{y}|)^{-N_{0}^{3}}(1+2^{{k}}|y_{3}|)^{-N_{0}^{3}}.$
(5.3)

Based on the possible size of $\tilde{k}$, we split into two cases as follows.

$\bullet$ If $\tilde{k}\leq k-30\epsilon M_{t}.$

From the estimate of kernel in (5.3), the cylindrical symmetry of solution,
and the estimate (2.14) in Lemma 2.2, we have

$\int_{t_{1}}^{t_{2}}\|\tilde{E}_{k,\tilde{k}}^{j_{1},j_{2}}(s,\cdot)\|_{L^{\infty}_{x}}ds\lesssim
2^{-10M_{t}}+\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}\int_{|y|\leq
2^{-\tilde{k}+\epsilon
M_{t}}}\min\\{2^{2\tilde{k}},\frac{2^{\tilde{k}+\epsilon
M_{t}/2}}{|\not{x}-\not{y}|}\\}f(s,x-y,v)\varphi_{j_{1}}(\not{v})\varphi_{j_{2}}(v)dydvds$
$\lesssim 2^{-10M_{t}}+2^{\tilde{k}-2j_{1}+j_{2}+6\epsilon M_{t}}.$ (5.4)

Alternatively, if we use the volume of support of $v$, the following estimate
holds from the estimate of kernel in (5.3),

$\int_{s_{1}}^{s_{2}}\|\tilde{E}_{k,\tilde{k}}^{j_{1},j_{2}}(s,\cdot)\|_{L^{\infty}_{x}}ds\lesssim
2^{\epsilon
M_{t}}\min\\{2^{-k+2j_{1}+j_{2}},2^{2\tilde{k}-j_{2}},2^{2\tilde{k}-nj_{2}+(n-1)M_{t}}\\}.$
(5.5)

After using the estimates (5.4) and (5.5) for the case
$j_{2}\in[0,(1+\epsilon)M_{t}]\cap\mathbb{Z}$ and using the same argument used
in obtaining the estimate (2.20) for the case
$[1+\epsilon)M_{t},\infty)\cap\mathbb{Z}$, we have

$\sum_{j_{2}\in\mathbb{Z}_{+},j_{1}\in[0,j_{2}+2]\cap\mathbb{Z}}\int_{s_{1}}^{s_{2}}\|\tilde{E}_{k,\tilde{k}}^{j_{1},j_{2}}(s,\cdot)\|_{L^{\infty}_{x}}ds\lesssim\sum_{j_{2}\in[0,(1+\epsilon)M_{t}]\cap\mathbb{Z}_{+},j_{1}\in[0,j_{2}+2]\cap\mathbb{Z}}\min\\{2^{-10M_{t}}+2^{\tilde{k}-2j_{1}+j_{2}+6\epsilon
M_{t}},2^{\epsilon M_{t}}2^{-k+2j_{1}+j_{2}}\\}$ $+1\lesssim 1+2^{3.5\epsilon
M_{t}+(\tilde{k}-k)/2+(1+\epsilon)M_{t}}\lesssim 2^{(1-10\epsilon)M_{t}}.$
(5.6)

Therefore, recall (5.2), from the above estimate, we have

$\int_{t_{1}}^{t_{2}}\big{|}{V_{3}(s)}\tilde{E}_{k,\tilde{k}}(s,X(s))\big{|}ds\lesssim
2^{(2\gamma-5\epsilon)M_{t}}.$ (5.7)

$\bullet$ If $\tilde{k}\in[k-30\epsilon M_{t},k+2]\cap\mathbb{Z}$.

Only for this case, we exploit the hyperbolic nature of the sRVM system (1.2).
Recall (1.2) and (5.2). Note that, the following wave equation holds for
$\tilde{E}_{k,\tilde{k}}(s,x),$

$\partial_{s}^{2}\tilde{E}_{k,\tilde{k}}-\Delta\tilde{E}_{k,\tilde{k}}=-4\pi\displaystyle{\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}{K}_{k,\tilde{k}}(y)\partial_{s}f(s,x-y,v)\hat{v}_{3}dvdy}-4\pi\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}{K}_{k,\tilde{k}}(y)\partial_{x_{3}}f(s,x-y,v)dvdy$
$=\mathcal{N}_{k,\tilde{k}}^{1}(s,x)+\mathcal{N}_{k,\tilde{k}}^{2}(s,x),$
(5.8)

where

$\mathcal{N}_{k,\tilde{k}}^{1}(s,x)=4\pi\big{(}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}{K}_{k,\tilde{k}}(y)\hat{v}_{3}\hat{v}\cdot\nabla_{x}f(s,x-y,v)dvdy-\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}{K}_{k,\tilde{k}}(y)\partial_{x_{3}}f(s,x-y,v)dvdy\big{)},$
(5.9)
$\mathcal{N}_{k,\tilde{k}}^{2}(s,x)=4\pi\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}{K}_{k,\tilde{k}}(y)E(s,x-y)\cdot\nabla_{v}\hat{v}_{3}f(s,x-y,v)dvdy.$
(5.10)

Let

$\mathcal{E}_{k,\tilde{k}}(s):=(\partial_{s}+i|\nabla|)\tilde{E}_{k,\tilde{k}}(s),\quad\tilde{E}_{k,\tilde{k}}(s)=(2i|\nabla|)^{-1}(\mathcal{E}_{k,\tilde{k}}(s)-\overline{\mathcal{E}_{k,\tilde{k}}(s)}).$
(5.11)

From the equality (5.8), we have

$(\partial_{s}-i|\nabla|)\mathcal{E}_{k,\tilde{k}}(t)=\mathcal{N}_{k,\tilde{k}}^{1}(s)+\mathcal{N}_{k,\tilde{k}}^{2}(s),\quad\Longrightarrow\mathcal{E}_{k,\tilde{k}}(s)=e^{is|\nabla|}\mathcal{E}_{k,\tilde{k}}(0)+\int_{0}^{s}e^{i(s-\tau)|\nabla|}\big{(}\mathcal{N}_{k,\tilde{k}}^{1}(\tau)+\mathcal{N}_{k,\tilde{k}}^{2}(\tau)\big{)}d\tau.$
(5.12)

After localizing the size of $v$, we have

$\forall
i\in\\{1,2\\},\quad\int_{0}^{s}e^{i(s-\tau)|\nabla|}|\nabla|^{-1}\mathcal{N}_{k,\tilde{k}}^{i}(\tau)d\tau=\sum_{j\in\mathbb{Z}_{+}}T^{j;i}_{k,\tilde{k}}(s,x),\\\
T^{j;1}_{k,\tilde{k}}(s,x)=\int_{0}^{s}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+i(s-\tau)|\xi|}\frac{\nabla_{v}\hat{v}_{3}}{|\xi|}\cdot\hat{E}(\tau,\eta)\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)\hat{f}(\tau,\xi-\eta,v)d\eta
d\xi dvd\tau,\\\
T^{j;2}_{k,\tilde{k}}(s,x)=\int_{0}^{s}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+i(s-\tau)|\xi|}\frac{i(\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3})}{|\xi|}\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)\hat{f}(\tau,\xi,v)d\xi
dvd\tau.$ (5.13)

From the above equality, (5.11) and the fact that $V_{3}(s)\in\mathbb{R}$, we
have

$\big{|}\int_{t_{1}}^{t_{2}}{V_{3}(s)}\tilde{E}_{k,\tilde{k}}(s,X(s))ds\big{|}\lesssim
1+\sum_{i=1,2}\sum_{j\in\mathbb{Z}}\big{|}\int_{t_{1}}^{t_{2}}{V_{3}(s)}T^{j;i}_{k,\tilde{k}}(s,X(s))ds\big{|}.$
(5.14)

We first rule out the case $j\geq(1+\epsilon)M_{t}$. From the volume of
support of $\xi$ and the rough estimate of the electric field (2.19) in Lemma
2.4, we have

$\sum_{j\geq(1+\epsilon)M_{t}}\|T^{j;1}_{k,\tilde{k}}(s,\cdot)\|_{L^{\infty}_{x}}+\|T^{j;2}_{k,\tilde{k}}(s,\cdot)\|_{L^{\infty}_{x}}\lesssim\sup_{\tau\in[0,t]}\sum_{j\geq(1+\epsilon)M_{t}}\|{f}(\tau,x,v)\varphi_{j}(v)\|_{L^{1}_{x}L^{1}_{v}}\big{(}2^{3k+\epsilon
M_{t}}+2^{2k-j+\epsilon M_{t}}\|E(\tau,\cdot)\|_{L^{\infty}_{x}}\big{)}$
$\lesssim\sum_{j\geq(1+\epsilon)M_{t}}2^{9M_{t}+2\epsilon
M_{t}}2^{-nj+(n-1)M_{t}}\lesssim 1.$ (5.15)

Now, we focus on the case $j\in[0,(1+\epsilon)M_{t}]\cap\mathbb{Z}$. Recall
(5.13). Note that,

$e^{iX(s)\cdot\xi+i(s-\tau)|\xi|}=\frac{1}{i|\xi|+i\hat{V}(s)\cdot\xi}\partial_{s}e^{iX(s)\cdot\xi+i(s-\tau)|\xi|}.$

Hence, after doing integration by parts in $s$, we have

$\int_{t_{1}}^{t_{2}}{V_{3}(s)}T^{j;1}_{k,\tilde{k}}(s,X(s))ds=\mathfrak{H}^{j;1}_{k,\tilde{k}}(t_{1},t_{2})+\mathfrak{H}^{j;2}_{k,\tilde{k}}(t_{1},t_{2})+\mathfrak{High}^{j}_{k,\tilde{k}}(t_{1},t_{2})),$
(5.16)
$\int_{t_{1}}^{t_{2}}{V_{3}(s)}T^{j;2}_{k,\tilde{k}}(s,X(s))ds=H^{j;1}_{k,\tilde{k}}(t_{1},t_{2})+H^{j;2}_{k,\tilde{k}}(t_{1},t_{2})+High^{j}_{k,\tilde{k}}(t_{1},t_{2}),$
(5.17)

where

$\mathfrak{H}^{j;1}_{k,\tilde{k}}(t_{1},t_{2})=\int_{t_{1}}^{t_{2}}\int_{(\mathbb{R}^{3})^{3}}e^{iX(s)\cdot\xi}\frac{i{V_{3}(s)}\nabla_{v}\hat{v}_{3}\cdot\hat{E}(s,\eta)}{|\xi|(|\xi|+\hat{V}(s)\cdot\xi)}\hat{f}(s,\xi-\eta,v)\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)d\eta
d\xi dvds,$ (5.18)
$\mathfrak{H}^{j;2}_{k,\tilde{k}}(t_{1},t_{2})=\sum_{a=1,2}\int_{0}^{t_{a}}\int_{(\mathbb{R}^{3})^{3}}e^{iX(t_{a})\cdot\xi+i(t_{a}-\tau)|\xi|}\frac{i(-1)^{a-1}{V_{3}(t_{a})}\nabla_{v}\hat{v}_{3}}{|\xi|(|\xi|+\hat{V}(t_{a})\cdot\xi)}\cdot\hat{E}(\tau,\eta)\hat{f}(\tau,\xi-\eta,v)\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)d\eta
d\xi dvd\tau,$ (5.19)
$\mathfrak{High}^{j}_{k,\tilde{k}}(t_{1},t_{2})=\int_{t_{1}}^{t_{2}}\int_{0}^{s}\int_{(\mathbb{R}^{3})^{3}}e^{iX(s)\cdot\xi+i(s-\tau)|\xi|}\partial_{s}\big{[}\frac{i{V_{3}(s)}\nabla_{v}\hat{v}_{3}}{|\xi|(|\xi|+\hat{V}(s)\cdot\xi)}\big{]}\cdot\hat{E}(\tau,\eta)\hat{f}(\tau,\xi-\eta,v)\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)d\eta
d\xi dvd\tau ds.$ (5.20)
$H^{j;1}_{k,\tilde{k}}(t_{1},t_{2})=\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(s)\cdot\xi}\frac{V_{3}(s)(\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3})}{|\xi|(|\xi|+\hat{V}(s)\cdot\xi)}\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)\hat{f}(s,\xi,v)d\xi
dvds,$ (5.21)
$H^{j;2}_{k,\tilde{k}}(t_{1},t_{2})=\sum_{a=1,2}\int_{0}^{t_{a}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(t_{a})\cdot\xi+i(t_{a}-\tau)|\xi|}\frac{(\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3})}{|\xi|}\frac{(-1)^{a-1}V_{3}(t_{a})}{|\xi|+\hat{V}(t_{a})\cdot\xi}\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)\hat{f}(\tau,\xi,v)d\xi
dvd\tau,$ (5.22)
$High^{j}_{k,\tilde{k}}(t_{1},t_{2})=\int_{t_{1}}^{t_{2}}\int_{0}^{s}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(s)\cdot\xi+i(s-\tau)|\xi|}\partial_{s}\big{[}\frac{V_{3}(s)(\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3})}{|\xi|(|\xi|+\hat{V}(s)\cdot\xi)}\big{]}\varphi_{k}(\xi)\varphi_{j}(v)\varphi_{\tilde{k}}(\not{\xi})\hat{f}(\tau,\xi,v)d\xi
dvd\tau ds.$ (5.23)

$\oplus$ The estimate of $\mathfrak{H}^{j;1}_{k,\tilde{k}}(s_{1},s_{2})$.

Recall (5.18). In terms of kernel, we have

$\mathfrak{H}^{j;1}_{k,\tilde{k}}(t_{1},t_{2})=\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}{V_{3}(s)}\mathfrak{K}^{j;0}_{k,\tilde{k}}(y,v,V(s))\cdot
E(s,X(s)-y)f(s,X(s)-y,v)dydvds,\\\
\mathfrak{K}^{j;0}_{k,\tilde{k}}(y,v,V(s)):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\frac{i\nabla_{v}\hat{v}_{3}}{|\xi|(|\xi|+\hat{V}(s)\cdot\xi)}\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)d\xi.$
(5.24)

Note that

$\forall v\in
supp(\varphi_{\tilde{j}}(\not{v})\varphi_{j}(v)),\qquad|\nabla_{v}\hat{v}_{3}|=|(\frac{-v_{3}v_{1}}{\langle
v\rangle^{3}},\frac{-v_{3}v_{2}}{\langle
v\rangle^{3}},\frac{v_{1}^{2}+v_{2}^{2}}{\langle v\rangle^{3}})|\lesssim
2^{\tilde{j}-2j}.$ (5.25)

After doing integration by parts in $\xi$ many times, from the above estimate,
the following estimate holds for the kernel
$\mathfrak{K}^{j;0}_{k,\tilde{k}}(y,v,V(s))$,

$|\mathfrak{K}^{j;0}_{k,\tilde{k}}(y,v,V(s))|\varphi_{\tilde{j}}(\not{v})\lesssim
2^{k+\tilde{j}-2j+90\epsilon M_{t}}(1+2^{k-30\epsilon
M_{t}}|y|)^{-N_{0}^{3}}\varphi_{\tilde{j}}(\not{v})\varphi_{{j}}(v).$

From the above estimate of kernel and the estimate (2.18) in Lemma 2.4, we
have

$|\mathfrak{H}^{j;1}_{k,\tilde{k}}(t_{1},t_{2})|\lesssim\sum_{\tilde{j}\in[0,j+2]\cap\mathbb{Z}}\int_{t_{1}}^{t_{2}}2^{k+\tilde{j}-2j+\gamma
M_{t}+100\epsilon
M_{t}}\min\\{2^{-3k+2\tilde{j}+j},2^{-j}\\}\|E(s,\cdot)\|_{L^{\infty}_{x}}ds\lesssim
2^{-k/2+2M_{t}+200\epsilon M_{t}}\lesssim 2^{7M_{t}/4}.$ (5.26)

$\oplus$ The estimate of $\mathfrak{H}^{j;2}_{k,\tilde{k}}(t_{1},t_{2})$.

Recall (5.19). From the Kirchhoff’s formulas in Lemma 2.1, in terms of kernel,
we have

$\mathfrak{H}^{j;2}_{k,\tilde{k}}(t_{1},t_{2})=\sum_{\tilde{j}\in[0,j+2]\cap\mathbb{Z}}\mathfrak{H}^{\tilde{j},j;2}_{k,\tilde{k}}(t_{1},t_{2}),\quad\mathfrak{H}^{\tilde{j},j;2}_{k,\tilde{k}}(t_{1},t_{2}):=\sum_{a=1,2}\int_{0}^{t_{a}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}(-1)^{a-1}V_{3}(t_{a})\big{(}(t_{a}-\tau)\mathfrak{K}^{j;1}_{k,\tilde{k}}(y,v,V(t_{a}),\omega)$
$+\mathfrak{K}^{j;2}_{k,\tilde{k}}(y,v,V(t_{a}),\omega)\big{)}E(\tau,X(t_{a})-y+(t_{a}-\tau)\omega)f(\tau,X(t_{a})-y+(t_{a}-\tau)\omega,v)\varphi_{\tilde{j}}(\not{v})d\omega
dydvd\tau,$ (5.27)

where the kernels
$\mathfrak{K}^{j_{1},j_{2};i}_{k,\tilde{k}}(y,v,V(s)),i\in\\{1,2\\}$, are
defined as follows,

$\mathfrak{K}^{j;1}_{k,\tilde{k}}(y,v,V(s),\omega)=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\frac{(\xi\cdot\omega+|\xi|)\nabla_{v}\hat{v}_{3}}{|\xi|(|\xi|+\hat{V}(s)\cdot\xi)}\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)d\xi,$
(5.28)
$\mathfrak{K}^{j;2}_{k,\tilde{k}}(y,v,V(s),\omega)=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\frac{\nabla_{v}\hat{v}_{3}}{|\xi|(|\xi|+\hat{V}(s)\cdot\xi)}\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)d\xi.$
(5.29)

By doing integration by parts in $\xi$ many times, from the estimate (5.25),
we have

$\big{(}|\mathfrak{K}^{j;1}_{k,\tilde{k}}(y,v,V(s),\omega)|+2^{k}|\mathfrak{K}^{j;2}_{k,\tilde{k}}(y,v,V(s),\omega)|\big{)}\varphi_{\tilde{j}}(\not{v})\varphi_{j}(v)\lesssim
2^{2k+\tilde{j}-2j+90\epsilon M_{t}}(1+2^{k-30\epsilon
M_{t}}|y|)^{-N_{0}^{3}}\varphi_{\tilde{j}}(\not{v})\varphi_{j}(v).$

From the above estimates of kernels, the following estimate holds after using
the volume of support of $v$ and the estimate (2.18) in Lemma 2.4,

$\big{|}\mathfrak{H}^{\tilde{j},j;2}_{k,\tilde{k}}(t_{1},t_{2})\big{|}\lesssim\sum_{a=1,2}\int_{0}^{t_{a}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{0}^{2\pi}\int_{0}^{\pi}2^{k+\tilde{j}-2j+\gamma
M_{t}+90\epsilon M_{t}}((t_{a}-\tau)2^{k}+1)(1+2^{k-30\epsilon
M_{t}}|y|)^{-N_{0}^{3}}\|E(\tau,\cdot)\|_{L^{\infty}_{x}}$ $\times
f(\tau,X(t_{a})-y+(t_{a}-\tau)(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta),v)\varphi_{\tilde{j}}(\not{v})\varphi_{j}(v)\sin\theta
d\theta d\phi dydvd\tau$ (5.30)
$\lesssim\sum_{a=1,2}2^{-k+2\tilde{j}+j}2^{\tilde{j}-2j+\gamma
M_{t}+40\epsilon
M_{t}}\int_{0}^{s_{a}}\|E(\tau,\cdot)\|_{L^{\infty}_{x}}d\tau\lesssim
2^{-k+3\tilde{j}-j+2\gamma M_{t}+100\epsilon M_{t}}.$ (5.31)

Alternatively, if we use the cylindrical symmetry of the distribution
function, the rough estimate (2.19) in Lemma 2.4 for the electric field, and
the estimate (2.14) in Lemma 2.2 and change coordinates $\theta\rightarrow
X_{3}(t_{a})-y_{3}+(t_{a}-\tau)\cos\theta$, the following estimate holds,

$\big{|}\mathfrak{H}^{\tilde{j},j;2}_{k,\tilde{k}}(t_{1},t_{2})\big{|}\lesssim\sum_{a=1,2}\int_{0}^{t_{a}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{0}^{2\pi}\int_{0}^{\pi}2^{\tilde{j}-2j+\gamma
M_{t}+40\epsilon
M_{t}}((t_{a}-\tau)2^{k}+1)\varphi_{\tilde{j}}(\not{v})\varphi_{j}(v)(1+2^{k-20\epsilon
M_{t}}|y_{3}|)^{-N_{0}^{3}}$
$\times\frac{2^{4{\alpha}_{\tau}M_{\tau}/3+M_{t}/3+5\epsilon
M_{t}}}{|(z_{1},z_{2})|^{1-2\epsilon}}f(\tau,z_{1},z_{2},X_{3}(t_{a})-y_{3}+(t_{a}-\tau)\cos\theta)\sin\theta
d\theta d\phi dz_{1}dz_{2}dydvd\tau+2^{-10M_{t}}$ $\lesssim
2^{4\alpha^{\star}M_{t}/3+M_{t}/3+\gamma M_{t}+100\epsilon
M_{t}}2^{\tilde{j}-2j}2^{-2\tilde{j}+j}+2^{-10M_{t}}.$ (5.32)

After combining the estimates (5.31) and (5.32), we have

$\big{|}\mathfrak{H}^{j;2}_{k,\tilde{k}}(t_{1},t_{2})\big{|}\lesssim\sum_{\tilde{j}\in[0,j+2]\cap\mathbb{Z}}2^{\gamma
M_{t}+100\epsilon M_{t}}\min\\{2^{-k+2\tilde{j}+\gamma
M_{t}},2^{4\alpha^{\star}M_{t}/3+M_{t}/3}2^{-\tilde{j}-j}+2^{-10M_{t}}\\}$
$\lesssim 2^{-k/2+5\gamma M_{t}/3+2\alpha^{\star}M_{t}/3+200\epsilon
M_{t}}\lesssim 2^{1.9\gamma M_{t}}.$ (5.33)

$\oplus$ The estimate of $\mathfrak{High}^{j}_{k,\tilde{k}}(s_{1},s_{2})$.

Recall (5.20). As a result of direct computations, we have

$\partial_{s}\big{[}\frac{i{V_{3}(s)}}{|\xi|(|\xi|+\hat{V}(s)\cdot\xi)}\big{]}=\langle
V(s)\rangle^{-1}E(s,X(s))\cdot\mathfrak{M}(\xi,v,V(s)),$ (5.34)
$\mathfrak{M}(\xi,v,V(s))=\frac{\mathbf{e}_{3}\langle
V(s)\rangle}{|\xi|(|\xi|+\hat{V}(s)\cdot\xi)}-\frac{V_{3}(s)}{|\xi|(|\xi|+\hat{V}(s)\cdot\xi)^{2}}\big{(}\xi-\hat{V}(s)\hat{V}(s)\cdot\xi\big{)}.$

Hence

$\mathfrak{High}^{j}_{k,\tilde{k}}(t_{1},t_{2})=\int_{t_{1}}^{t_{2}}\langle
V(s)\rangle^{-1}E(s,X(s))\cdot\widetilde{\mathfrak{High}}^{j}_{k,\tilde{k}}(s)ds,$

where

$\widetilde{\mathfrak{High}}^{j}_{k,\tilde{k}}(s):=\int_{0}^{s}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(s)\cdot\xi+i(s-\tau)|\xi|}\nabla_{v}\hat{v}_{3}\mathfrak{M}(\xi,v,V(s))\cdot\hat{E}(\tau,\eta)\hat{f}(\tau,\xi-\eta,v)\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)d\eta
d\xi dvd\tau.$ (5.35)

Note that, recall (5.19), the difference between
$\widetilde{\mathfrak{High}}^{j}_{k,\tilde{k}}(s)$ and
$\mathfrak{H}^{j;2}_{k,\tilde{k}}(t_{1},t_{2})$ lies only in the symbol and
the characteristic time, which don’t play much role in the proof of the
obtained estimate (5.33). With minor modifications, we have

$\forall
s\in[t_{1},t_{2}],\quad|\widetilde{\mathfrak{High}}^{j}_{k,\tilde{k}}(s)|\lesssim
2^{1.9\gamma M_{t}}.$

From the above estimate and the estimate (2.18) in Lemma 2.4, we have

$\big{|}\mathfrak{High}^{j_{1},j_{2}}_{k,\tilde{k}}(t_{1},t_{2})\big{|}\lesssim\int_{t_{1}}^{t_{2}}2^{0.9\gamma
M_{t}+5\epsilon M_{t}}\|E(s,\cdot)\|_{L^{\infty}_{x}}ds\lesssim 2^{1.9\gamma
M_{t}+40\epsilon M_{t}}.$ (5.36)

$\oplus$ The estimate of $H^{j;1}_{k,\tilde{k}}(t_{1},t_{2}).$

Recall (5.21). In terms of kernel, we have

$H^{j;1}_{k,\tilde{k}}(t_{1},t_{2})=\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\mathcal{K}_{\tilde{k},k}(y,v,V(s))f(s,X(s)-y,v)\varphi_{j}(v)dydvds,$

where

$\mathcal{K}_{\tilde{k},k}(y,v,V(s)):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\frac{V_{3}(s)(\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3})}{|\xi|(|\xi|+\hat{V}(s)\cdot\xi)}\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)d\xi.$

Due to the facts that $|\not{V}(s)|\leq 2^{\alpha^{\star}M_{t}},|V(s)|\gtrsim
2^{(1-3\epsilon)M_{t}}$, and $|\not{\xi}|/|\xi|\gtrsim 2^{-30\epsilon M_{t}}$,
for any $(v,\xi)\in
supp(\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{\tilde{j}}(\not{v})\varphi_{j}(v))$,
we have

$|\tilde{\xi}\times\tilde{V}(s)|\gtrsim 2^{-30\epsilon
M_{t}},\quad|\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3}|\leq|\xi_{3}|(1-\hat{v}_{3}^{2})+|\hat{v}_{1}\xi_{1}|+|\hat{v}_{2}\xi_{2}|\lesssim
2^{\tilde{j}-j+k}.$ (5.37)

Therefore, after doing integration by parts in $\xi$ along $V(s)$ and
directions perpendicular to $V(s)$, the following estimate holds for the
kernel,

$\big{|}\mathcal{K}_{\tilde{k},k}(y,v,V(s))\big{|}\varphi_{\tilde{j}}(\not{v})\varphi_{j}(v)\lesssim
2^{2{k}+200\epsilon M_{t}+\tilde{j}-j}(1+2^{{k}-30\epsilon
M_{t}}|y|)^{-N_{0}^{3}}\varphi_{\tilde{j}}(\not{v})\varphi_{j}(v).$ (5.38)

From the above estimate of the kernel, the volume of support of $v$ and the
estimate (2.8) if $|\not{v}|\geq 2^{(\alpha^{\star}+\epsilon)M_{t}}$, after
rerunning the argument used in obtaining the estimate (5.5), we have

$\big{|}H^{j;1}_{k,\tilde{k}}(t_{1},t_{2})\big{|}\lesssim\sum_{\tilde{j}\in[0,j+2]\cap\mathbb{Z}}2^{\gamma
M_{t}+300\epsilon
M_{t}+\tilde{j}-j}\min\\{2^{-\tilde{k}}2^{2(\alpha^{\star}+\epsilon)M_{t}+j},2^{\tilde{k}-2\tilde{j}+j}\\}\lesssim
2^{1.8\gamma M_{t}}.$ (5.39)

For better presentation, we defer the estimate of remaining terms
$H^{j;2}_{k,\tilde{k}}(t_{1},t_{2})$ and $High^{j}_{k,\tilde{k}}(t_{1},t_{2})$
to Lemma 5.1 and Lemma 5.2 respectively.

To sum up, recall (5.14), (5.16) and (5.17), our desired estimate (3.24) holds
from the obtained estimates (5.1), (5.7), (5.14), (5.15), (5.26), (5.33),
(5.59), (5.39), the estimate in Lemma 5.40, and the estimate (5.66) in Lemma
5.2.

∎

Now, we give the deferred estimate of $H^{j;2}_{k,\tilde{k}}(t_{1},t_{2})$ in
the following Lemma.

######  Lemma 5.1.

Let $j\in[0,(1+\epsilon)M_{t}]\cap\mathbb{Z}$, $k\in[4M_{t}/5-20\epsilon
M_{t},3M_{t}+5\epsilon M_{t}]\cap\mathbb{Z}$ and
$\tilde{k}\in[0,k+2]\cap\mathbb{Z}$, s.t., $\tilde{k}\geq k-30\epsilon M_{t}$.
Under the assumption of Lemma 3.2, we have

$\big{|}H^{j;2}_{k,\tilde{k}}(t_{1},t_{2})\big{|}\lesssim 2^{1.9\gamma
M_{t}+10\epsilon M_{t}}.$ (5.40)

###### Proof.

Recall (5.22). Note that $t\leq 2^{\epsilon M_{t}-10}$. We localize further
the size of $t_{a}-\tau$ and the angle between $\xi/|\xi|$ and $v/|v|$ as
follows,

$H^{j;2}_{k,\tilde{k}}(t_{1},t_{2})=\sum_{m\in[-10M_{t},\epsilon
M_{t}]\cap\mathbb{Z}}\sum_{l\in[\bar{l},2]\cap\mathbb{Z}}H^{j;2}_{k,\tilde{k};m,l}(t_{1},t_{2}),$
(5.41)

where $\bar{l}:=-j+5M_{t}/18$ and $H^{j;2}_{k,\tilde{k};l}(s_{1},s_{2})$ is
defined as follows,

$H^{j;2}_{k,\tilde{k};m,l}(t_{1},t_{2})=\sum_{a=1,2}(-1)^{a-1}V_{3}(t_{a})\int_{0}^{t_{a}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(t_{a})\cdot\xi+i(t_{a}-\tau)|\xi|}\frac{(\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3})}{|\xi|(|\xi|+\hat{V}(t_{a})\cdot\xi)}\varphi_{l;\bar{l}}(\frac{\xi\times
v}{|\xi||v|})\varphi_{m;-10M_{t}}(t_{a}-\tau)$
$\times\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)\hat{f}(\tau,\xi,v)d\xi
dvd\tau.$ (5.42)

From the Kirchhoff’s formula in Lemma 2.1, after representing
$H^{j_{1},j_{2};2}_{k,\tilde{k};m,l}(t_{1},t_{2})$ in terms of kernel, the
following estimate holds from the volume of support of $v$ and $\tau$,

$\big{|}H^{j;2}_{k,\tilde{k};m,l}(t_{1},t_{2})\big{|}\lesssim 2^{m+200\epsilon
M_{t}}2^{3j}.$

From the above estimate, we can rule out the case $m=-10M_{t}$ and focus on
the case $m\in(-10M_{t},\epsilon M_{t}]\cap\mathbb{Z}$, in which we have
$t_{a}-\tau\sim 2^{m}.$

Based on the possible size of $l$, we separate into two cases as follows.

$\oplus$ If $l=\bar{l}=-j+5M_{t}/18$.

Recall the equality (2.11) in the proof of Kirchhoff’s formula. From the
stationary point of view, we know that the angle between $\theta$ and $\xi$
are localized around the size of $2^{-(m+k)/2}$. Therefore, after localizing
the angle between $v$ and $\theta$, the following equality holds from the
Kirchhoff’s formulas in Lemma 2.1,

$H^{j;2}_{k,\tilde{k};m,l}(t_{1},t_{2})=\sum_{\ast\in\\{\textup{ess},\textup{err}\\}}\mathcal{H}^{\ast;j;1}_{k,\tilde{k};m,l}(t_{1},t_{2})+\mathcal{H}^{\ast;j;2}_{k,\tilde{k};m,l}(t_{1},t_{2}),\quad\mathcal{H}^{\ast;j;i}_{k,\tilde{k};l}(t_{1},t_{2})=\sum_{a=1,2}\int_{0}^{t_{a}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}(-1)^{a-1}(t_{a}-\tau)\\\
\times
V_{3}(t_{a})\varphi_{m;-10M_{t}}(t_{a}-\tau)\mathcal{K}^{\ast;i}_{k,\tilde{k};l}(y,v,V(t_{a}),\omega)f(\tau,X(t_{a})-y+(t_{a}-\tau)\omega)\varphi_{j}(v)d\omega
dydvd\tau,\quad i\in\\{1,2\\},$ (5.43)

where the kernels
$\mathcal{K}^{i}_{k,\tilde{k}}(y,v,V(t_{a}),\omega),i\in\\{1,2\\},$ are
defined as follows,

$\mathcal{K}^{\ast;1}_{k,\tilde{k};l}(y,v,V(t_{a}),\omega):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\frac{i(\xi\cdot\omega+|\xi|)(\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3})}{4\pi|\xi|(|\xi|+\hat{V}(t_{a})\cdot\xi)}\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{l;\bar{l}}(\frac{\xi\times
v}{|\xi||v|})\phi_{\ast}(\omega,v)d\xi,$
$\mathcal{K}^{\ast;2}_{k,\tilde{k};l}(y,v,V(t_{a}),\omega):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\frac{(\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3})}{4\pi|\xi|(|\xi|+\hat{V}(t_{a})\cdot\xi)}\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{l;\bar{l}}(\frac{\xi\times
v}{|\xi||v|})\phi_{\ast}(\omega,v)d\xi,$

and the cutoff functions
$\phi_{\ast}(\omega,v),\ast\in\\{\textup{ess},\textup{err}\\}$, are defined as
follows,

$\phi_{\textup{ess}}(\omega,v):=\psi_{\leq\max\\{l,-(m+k)_{+}/2\\}+\epsilon
M_{t}}(\omega\times\tilde{v}),\quad\phi_{\textup{err}}(\omega,v)=1-\phi_{\textup{ess}}(\omega,v).$
(5.44)

We first rule out the error type terms. Due to the fact that
$\angle(\xi,\pm\omega)\geq 2^{-(k+m)/2+\epsilon M_{t}/2}$, we know that we
gain at least $2^{-\epsilon M_{t}/2}$ each time we do integration by parts in
$\omega$ once. Hence, after doing integration by parts in $\omega$ many times
and then do integration by parts in $\xi$, the following estimate holds for
the error type kernels
$\mathcal{K}^{err;i}_{k,\tilde{k};l}(\cdot,\cdot,\cdot),i\in\\{1,2\\}$,

$|\mathcal{K}^{err;1}_{k,\tilde{k};l}(y,v,V(t_{a}),\omega)|+2^{k}|\mathcal{K}^{err;2}_{k,\tilde{k};l}(y,v,V(t_{a}),\omega)|\lesssim
2^{3k+2l-100M_{t}}(1+2^{k+l-30\epsilon M_{t}}|y|)^{-N_{0}^{3}}.$ (5.45)

From the above estimate of kernels, the following estimate holds from the
volume of support of $v$,

$\big{|}H^{err;j;1}_{k,\tilde{k};m,l}(t_{1},t_{2})\big{|}+\big{|}H^{err;j;2}_{k,\tilde{k};m,l}(t_{1},t_{2})\big{|}\lesssim
2^{-50M_{t}+2j_{1}+j_{2}}\lesssim 1.$ (5.46)

Now, we focus on the essential type terms. Note that, for any $(\xi,v)\in
supp(\varphi_{k}(\xi)\varphi_{l;\bar{l}}(\tilde{v}\times\tilde{\xi})$, we have

$\big{|}\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3}\big{|}\lesssim\min_{\mu\in\\{+,-\\}}|\xi|\big{(}|\hat{v}_{3}-\mu\xi_{3}/|\xi||+|\hat{v}\cdot\xi/|\xi|-\mu|\big{)}\lesssim
2^{k+l}.$

After doing integration by parts in $\xi$ in $v$ direction and directions
perpendicular to $v$, from the above estimate and the estimate (5.37), we have

$\big{(}|\mathcal{K}^{ess;1}_{k,\tilde{k}}(y,v,V(t_{a}),\omega)|+2^{k}|\mathcal{K}^{ess;2}_{k,\tilde{k}}(y,v,V(t_{a}),\omega)|\big{)}\varphi_{\tilde{j}}(\not{v})\varphi_{j}(v)\lesssim
2^{3k+2l+120\epsilon
M_{t}}\min\\{2^{\tilde{j}-j},2^{l}\\}\phi_{\textup{ess}}(\omega,v)$
$\times(1+2^{k-30\epsilon M_{t}}|y\cdot v|)^{-N_{0}^{3}}(1+2^{k+l-30\epsilon
M_{t}}|y\times v|)^{-N_{0}^{3}}\varphi_{\tilde{j}}(\not{v})\varphi_{j}(v).$
(5.47)

From the above estimate of kernels and the volume of support of $\omega$ and
$v$, we have

$\big{|}H^{ess;j;1}_{k,\tilde{k};l}(t_{1},t_{2})\big{|}+\big{|}H^{ess;j;2}_{k,\tilde{k};l}(t_{1},t_{2})\big{|}\lesssim
2^{\gamma M_{t}+l+150\epsilon
M_{t}}2^{m}2^{2\max\\{l,-(m+k)_{+}/2\\}}\min\\{2^{3j},2^{3k+2l-j}\\}\lesssim
2^{\gamma M_{t}+160\epsilon M_{t}}\big{(}2^{3l}+2^{-k+l}\big{)}$
$\times\min\\{2^{3j},2^{3k+2l-j}\\}\lesssim 2^{\gamma M_{t}+160\epsilon
M_{t}}\big{(}2^{3l+3j}+2^{l+2j}2^{(2l-j)/3}\big{)}\lesssim
2^{M_{t}+160\epsilon M_{t}}(2^{3l+3j}+2^{5l/3+5j/3})\lesssim 2^{1.9\gamma
M_{t}}.$ (5.48)

Recall the decomposition (5.43). After combining the above estimate with the
estimate (5.46), we have

$\big{|}H^{j;2}_{k,\tilde{k};m,l}(t_{1},t_{2})\big{|}\lesssim 2^{1.9\gamma
M_{t}}.$ (5.49)

$\oplus$ If $l>\bar{l}=-j+5M_{t}/18$.

Recall (5.42). For this case, to exploit the oscillation in $\tau$, we do
integration by parts in $\tau$ once. As a result, we have

$\big{|}H^{j;2}_{k,\tilde{k};m,l}(t_{1},t_{2})\big{|}\leq\sum_{a=1,2}\big{|}\widetilde{End}^{j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})\big{|}+\big{|}\tilde{\mathcal{H}}^{j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})\big{|},$
(5.50)

where

$\widetilde{End}^{j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})=\big{|}\int_{0}^{t_{a}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(t_{a})\cdot\xi+i(t_{a}-\tau)|\xi|}\frac{V_{3}(t_{a})(\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3})\varphi_{l;\bar{l}}(\tilde{v}\times\tilde{\xi})\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)}{|\xi|(|\xi|+\hat{V}(t_{a})\cdot\xi)(|\xi|+\hat{v}\cdot\xi)}$
$\times\partial_{\tau}\varphi_{m;-10M_{t}}(t_{a}-\tau)\hat{f}(\tau,\xi,v)d\xi
dvd\tau\big{|}+\big{|}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(t_{a})\cdot\xi+i(t_{a}-\tau)|\xi|}\frac{V_{3}(t_{a})(\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3})\varphi_{l;\bar{l}}(\tilde{v}\times\tilde{\xi})\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)}{|\xi|(|\xi|+\hat{V}(t_{a})\cdot\xi)(|\xi|+\hat{v}\cdot\xi)}$
$\times\varphi_{m;-10M_{t}}(t_{a}-\tau)\hat{f}(\tau,\xi,v)d\xi
dv\big{|}_{\tau=0}^{t_{a}}\big{|},$ (5.51)
$\tilde{\mathcal{H}}^{j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})=\int_{0}^{t_{a}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(t_{a})\cdot\xi+i(t_{a}-\tau)|\xi|-i\tau\hat{v}\cdot\xi}\frac{V_{3}(t_{a})(\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3})\varphi_{l;\bar{l}}(\tilde{v}\times\tilde{\xi})\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)}{|\xi|(|\xi|+\hat{V}(t_{a})\cdot\xi)(|\xi|+\hat{v}\cdot\xi)}$
$\times\varphi_{m;-10M_{t}}(t_{a}-\tau)\partial_{\tau}\hat{g}(\tau,\xi,v)d\xi
dvd\tau.$ (5.52)

From the Kirchhoff’s formula in Lemma 2.1, we can represent
$\widetilde{End}^{j_{1},j_{2};a}_{k,\tilde{k};m,l}(t_{1},t_{2})$ in terms of
kernel as follows,

$\big{|}\widetilde{End}^{j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})\big{|}\lesssim
1+\big{|}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\tilde{\mathcal{K}}^{0}_{k,\tilde{k};l}(y,v,V(t_{a}))f(t_{a},X(t_{a})-y,v)\varphi_{j}(v)dydv\big{|}+\big{|}\int_{0}^{t_{a}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\partial_{\tau}\varphi_{m;-10M_{t}}(t_{a}-\tau)$
$\times\big{(}\tilde{\mathcal{K}}^{0}_{k,\tilde{k};l}(y,v,V(t_{a}))+(t_{a}-\tau)\tilde{\mathcal{K}}^{1}_{k,\tilde{k};l}(y,v,V(s_{a}),\omega)\big{)}\varphi_{j}(v){f}(\tau,X(t_{a})-y+(t_{a}-\tau)\omega,v)d\omega
dydvd\tau\big{|},$ (5.53)

where the kernels $\tilde{\mathcal{K}}^{0}_{k,\tilde{k};l}(y,v,V(t_{a}))$ and
$\tilde{\mathcal{K}}^{1}_{k,\tilde{k};l}(y,v,V(t_{a}),\omega)$ are defined as
follows,

$\tilde{\mathcal{K}}^{0}_{k,\tilde{k};l}(y,v,V(t_{a})):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\frac{V_{3}(t_{a})(\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3})}{|\xi|(|\xi|+\hat{V}(t_{a})\cdot\xi)(|\xi|+\hat{v}\cdot\xi)}\varphi_{l;\bar{l}}(\tilde{v}\times\tilde{\xi})\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)d\xi,\\\
\tilde{\mathcal{K}}^{1}_{k,\tilde{k};l}(y,v,V(t_{a}),\omega):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\frac{V_{3}(t_{a})(|\xi|+\xi\cdot\omega)(\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3})}{|\xi|(|\xi|+\hat{V}(s_{a})\cdot\xi)(|\xi|+\hat{v}\cdot\xi)}\varphi_{l;\bar{l}}(\tilde{v}\times\tilde{\xi})\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)d\xi.$
(5.54)

Recall (1.2). We have

$\partial_{t}g(t,x-t\hat{v},v)=(\partial_{t}+\hat{v}\cdot\nabla_{x})f(t,x,v)=-\nabla_{v}\cdot\big{(}E(t,x)f(t,x,v)\big{)}.$

Hence, from the above equality, the Kirchhoff’s formula in Lemma 2.1, after
doing integration by parts in $v$, we can represent
$\tilde{\mathcal{H}}^{j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})$ in terms of kernel
as follows,

$\tilde{\mathcal{H}}^{j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})=\int_{0}^{t_{a}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}E(\tau,X(t_{a})-y+(t_{a}-\tau)\omega)\cdot\nabla_{v}\big{[}\big{(}\tilde{\mathcal{K}}^{0}_{k,\tilde{k};l}(y,v,V(t_{a}))+(t_{a}-\tau)\tilde{\mathcal{K}}^{1}_{k,\tilde{k};l}(y,v,V(t_{a}))\big{)}\varphi_{j}(v)\big{]}$
$\times\varphi_{m;-10M_{t}}(t_{a}-\tau)f(\tau,X(t_{a})-y+(t_{a}-\tau)\omega,v)d\omega
dydvd\tau.$ (5.55)

Recall (5.54). After doing integration by parts in $\xi$ in $v$ direction and
directions perpendicular to $v$, we have

$|\tilde{\mathcal{K}}^{1}_{k,\tilde{k};l}(y,v,V(t_{a}),\omega)|+2^{k}|\tilde{\mathcal{K}}^{0}_{k,\tilde{k};l}(y,v,V(t_{a}))|$
$\lesssim 2^{2k+l+\gamma M_{t}+150\epsilon M_{t}}(1+2^{k-30\epsilon
M_{t}}|y\cdot\tilde{v}|)^{-N_{0}^{3}}(1+2^{k+l-30\epsilon
M_{t}}|y\times\tilde{v}|)^{-N_{0}^{3}},$ (5.56)
$\big{(}|\nabla_{v}\tilde{\mathcal{K}}^{1}_{k,\tilde{k};l}(y,v,V(t_{a}),\omega)|+2^{k}|\nabla_{v}\tilde{\mathcal{K}}^{0}_{k,\tilde{k};l}(y,v,V(t_{a}))|\big{)}\varphi_{j}(v)$
$\lesssim 2^{2k+l+\gamma M_{t}+150\epsilon M_{t}}2^{-j-l}(1+2^{k-30\epsilon
M_{t}}|y\cdot\tilde{v}|)^{-N_{0}^{3}}(1+2^{k+l-30\epsilon
M_{t}}|y\times\tilde{v}|)^{-N_{0}^{3}}\varphi_{j}(v).$ (5.57)

Recall (5.53). Similar to the decomposition we did in (5.43) and the obtained
estimate (5.46), we can also rule out the case when the angle between $\omega$
and $\pm v$ is relatively big, i.e., the error type. As a result, from the
estimate of kernels in (5.56) and the estimate (2.8) if $|\not{v}|\geq
2^{(\alpha^{\star}+\epsilon)M_{t}}$, we have

$\big{|}\widetilde{End}^{j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})\big{|}\lesssim
1+2^{\gamma M_{t}+400\epsilon
M_{t}}\min\\{2^{k+l-j},2^{-2k-l+\alpha^{\star}M_{t}+2j}\\}+\big{|}\int_{0}^{t_{a}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\varphi_{j}(v)\partial_{\tau}\varphi_{m;-10M_{t}}(t_{a}-\tau)$
$\times\big{(}\tilde{\mathcal{K}}^{0}_{k,\tilde{k};l}(y,v,V(t_{a}))+(t_{a}-\tau)\tilde{\mathcal{K}}^{1}_{k,\tilde{k};l}(y,v,V(s_{a}),\omega)\big{)}{f}(\tau,X(t_{a})-y+(t_{a}-\tau)\omega,v)\phi_{\textup{ess}}(\omega,v)d\omega
dydvd\tau\big{|}.$

From the above estimate and the Jacobian of changing coordinates
$(y_{1},y_{2},\theta)\longrightarrow X(t_{a})-y+(t_{a}-\tau)\omega$ for the
above integral with the kernel
$\tilde{\mathcal{K}}^{1}_{k,\tilde{k};l}(y,v,V(s_{a}),\omega)$, where
$\omega=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$, we have

$|\widetilde{End}^{j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})|\lesssim 1+2^{\gamma
M_{t}+400\epsilon M_{t}}2^{\alpha^{\star}M_{t}/3}+2^{\gamma M_{t}+400\epsilon
M_{t}}\min\\{2^{k-j},2^{-k-l+2\alpha^{\star}M_{t}+j}(2^{m+2l}+2^{-k})\big{)}\\}$
$\lesssim 1+2^{\gamma M_{t}+\alpha^{\star}M_{t}+400\epsilon M_{t}}\lesssim
2^{1.8\gamma M_{t}}.$ (5.58)

Now, we focus on the estimate of
$\tilde{\mathcal{H}}^{j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})$. Recall (5.55).
After doing dyadic decomposition for the size of $\not{v}$, from the estimate
of kernels in (5.57), we have

$|\tilde{\mathcal{H}}^{j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})|\lesssim\sum_{\tilde{j}\in[0,j+2]\cap\mathbb{Z}}|\tilde{\mathcal{H}}^{\tilde{j},j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})|,$
(5.59)
$\tilde{\mathcal{H}}^{\tilde{j},j;a}_{k,\tilde{k};m,l}(t_{1},t_{2}):=\int_{0}^{t_{a}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{0}^{2\pi}\int_{0}^{\pi}2^{k-j+\gamma
M_{t}+200\epsilon
M_{t}}\big{(}2^{m+k}+1\big{)}\varphi_{m;-10M_{t}}(t_{a}-\tau)\|E(\tau,\cdot)\|_{L^{\infty}_{x}}(1+2^{k-30\epsilon
M_{t}}|y\cdot\tilde{v}|)^{-N_{0}^{3}}$ $\times(1+2^{k+l-40\epsilon
M_{t}}|y\times\tilde{v}|)^{-N_{0}}f(\tau,X(t_{a})-y+(t_{a}-\tau)(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta),v)\varphi_{\tilde{j}}(\not{v})\varphi_{j}(v)\sin\theta
d\theta d\phi dydvd\tau.$ (5.60)

Similar to the obtained estimate (5.32), by using the cylindrical symmetry of
solution, the following estimate holds from changing coordinate
$\theta\rightarrow X_{3}(t_{a})-y_{3}+(t_{a}-\tau)\cos\theta$, the space-time
estimate (2.14) in Lemma 2.2 and the rough estimate of the electric field
(2.19) in Lemma 2.4,

$\big{|}\tilde{\mathcal{H}}^{\tilde{j},j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})\big{|}\lesssim\int_{0}^{t_{a}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{0}^{2\pi}\int_{0}^{\pi}2^{\gamma
M_{t}+210\epsilon
M_{t}}2^{-j-l}\big{(}2^{k}(t_{a}-\tau)+1\big{)}\|E(\tau,\cdot)\|_{L^{\infty}_{x}}\varphi_{\tilde{j}}(\not{v})\varphi_{j}(v)\varphi_{m;-10M_{t}}(t_{a}-\tau)$
$\times\frac{(1+2^{k+l-40\epsilon
M_{t}}|y_{3}|)^{-N_{0}^{3}}}{|(z_{1},z_{2})|^{1-2\epsilon}}f(\tau,z_{1},z_{2},X_{3}(t_{a})-y_{3}+(t_{a}-\tau)\cos\theta)\sin\theta
d\theta d\phi dz_{1}dz_{2}dy_{3}dvd\tau+2^{-10M_{t}}$ $\lesssim
2^{4{\alpha}^{\star}M_{t}/3+M_{t}/3+220\epsilon
M_{t}}2^{-j-2l}2^{-2\tilde{j}+j+\gamma M_{t}}+2^{-10M_{t}}.$ (5.61)

Alternatively, from the volume of support of “$v$”, the estimate (2.8) if
$|\not{v}|\geq 2^{(\alpha^{\star}+\epsilon)M_{t}}$, and the estimate (2.18) in
Lemma 2.4, we have

$\textup{(\ref{june13eqn51})}\lesssim 2^{-k-l+\gamma M_{t}+210\epsilon
M_{t}}2^{-j-l}\min\\{2^{2\tilde{j}+j},2^{2\alpha^{\star}M_{t}+j}\\}\int_{0}^{t_{a}}\|E(\tau,\cdot)\|_{L^{\infty}_{x}}d\tau\lesssim
2^{-k-2l+(2\gamma+220\epsilon)M_{t}}\min\\{2^{2\tilde{j}},2^{2\alpha^{\star}M_{t}}\\}.$
(5.62)

Based on the size of $l$, we split further into two sub-cases as follows.

$\dagger$ If $l\in[-j+5M_{t}/18,-40\epsilon M_{t}]\cap\mathbb{Z}$.

Note that, for any $(v,\xi)\in
supp(\varphi_{\tilde{j}}(\not{v})\varphi_{j}(v)\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi))$,
we have

$\big{|}\frac{\not{\xi}}{|\xi|}\big{|}\gtrsim 2^{-30\epsilon
M_{t}},\quad|\tilde{v}\times\tilde{\xi}|\sim 2^{l},\Longrightarrow
2^{\tilde{j}-j}\sim\big{|}\frac{\not{\xi}}{|\xi|}\big{|}\gtrsim 2^{-30\epsilon
M_{t}}-2^{l}\gtrsim 2^{-30\epsilon M_{t}}.$

From the above estimate and the obtained estimate (5.61), we have

$\big{|}\tilde{\mathcal{H}}^{\tilde{j},j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})\big{|}\lesssim
2^{4{\alpha}^{\star}M_{t}/3+4M_{t}/3+300\epsilon M_{t}-2l-2\tilde{j}}\lesssim
2^{2j-2\tilde{j}+4{\alpha}^{\star}M_{t}/3+4M_{t}/3-5M_{t}/9+400\epsilon
M_{t}}\lesssim 2^{1.8\gamma M_{t}}.$ (5.63)

$\dagger$ If $l\in[-40\epsilon M_{t},2]\cap\mathbb{Z}$.

After combining the obtained estimates (5.61) and (5.62), we have

$\big{|}\tilde{\mathcal{H}}^{\tilde{j},j;a}_{k,\tilde{k};m,l}(t_{1},t_{2})\big{|}\lesssim\min\\{2^{-k-2l+(2\gamma+220\epsilon)M_{t}}\min\\{2^{2\tilde{j}},2^{2\alpha^{\star}M_{t}}\\},2^{4{\alpha}^{\star}M_{t}/3+4M_{t}/3+200\epsilon
M_{t}-2l-2\tilde{j}}\\}$ $\lesssim
2^{-2l-k/2+2{\alpha}^{\star}M_{t}/3+2M_{t}/3+(\gamma+220\epsilon)M_{t}}\lesssim
2^{1.9\gamma M_{t}}.$ (5.64)

Recall (5.50) and (5.59). After combining the estimates (5.58), (5.63), and
(5.64), we have

$\big{|}H^{j;2}_{k,\tilde{k};m,l}(t_{1},t_{2})\big{|}\lesssim 2^{1.9\gamma
M_{t}+5\epsilon M_{t}}.$ (5.65)

Recall the decomposition (5.41). Our desired estimate (5.40) holds from the
above estimate and the estimate (5.49). ∎

Lastly, we give the deferred estimate of $High^{j}_{k,\tilde{k}}(t_{1},t_{2})$
in the following Lemma.

######  Lemma 5.2.

Let $j\in[0,(1+\epsilon M_{t})]\cap\mathbb{Z}$, $k\in[4M_{t}/5-20\epsilon
M_{t},3M_{t}+5\epsilon M_{t}]\cap\mathbb{Z}$ and
$\tilde{k}\in[0,k+2]\cap\mathbb{Z}$, s.t., $\tilde{k}\geq k-30\epsilon M_{t}$.
Under the assumption of Lemma 3.2, we have we have

$\big{|}High^{j}_{k,\tilde{k}}(t_{1},t_{2})\big{|}\lesssim
2^{(1.9\gamma+20\epsilon)M_{t}}.$ (5.66)

###### Proof.

Recall (5.23) and (5.34). We have

$High^{j}_{k,\tilde{k}}(t_{1},t_{2})=\int_{t_{1}}^{t_{2}}\langle
V(s)\rangle^{-1}E(s,X(s))\cdot\widetilde{High}^{j}_{k,\tilde{k}}(s)ds,\\\
\widetilde{High}^{j}_{k,\tilde{k}}(s):=\int_{0}^{s}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(s)\cdot\xi+i(s-\tau)|\xi|}\mathfrak{M}(\xi,v,V(s))(\hat{v}_{3}\hat{v}\cdot\xi-\xi_{3})\varphi_{\tilde{k}}(\not{\xi})\varphi_{k}(\xi)\varphi_{j}(v)\hat{f}(\tau,\xi,v)d\xi
dvd\tau.$ (5.67)

Recall (5.22). Note that, the difference between
$\widetilde{High}^{j}_{k,\tilde{k}}(s)$ and
$H^{j;2}_{k,\tilde{k}}(t_{1},t_{2})$ lies only in the symbols and the
characteristic time evaluated, which don’t play much role in the proof of the
estimate (5.40). With minor modifications, we have

$\forall
s\in[s_{1},s_{2}],\quad|\widetilde{High}^{j}_{k,\tilde{k}}(s)|\lesssim
2^{1.9\gamma M_{t}+10\epsilon M_{t}}.$

Recall (5.67). From the above estimate and the estimate (2.18) in Lemma 2.4,
we have

$\big{|}High^{j}_{k,\tilde{k}}(s_{1},s_{2})\big{|}\lesssim\int_{s_{1}}^{s_{2}}2^{-\gamma
M_{t}}\|E(s,\cdot)\|_{L^{\infty}}2^{1.9\gamma M_{t}+10\epsilon
M_{t}}ds\lesssim 2^{(1.9\gamma+20\epsilon)M_{t}}.$ (5.68)

Hence finishing the proof of our desired estimate (5.66).

∎

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