# The relativistic quantum Boltzmann equation near equilibrium

Gi-Chan Bae  Department of mathematics, Sungkyunkwan University, Suwon 16419,
Republic of Korea<EMAIL_ADDRESS>,  Jin Woo Jang  University of Bonn,
Institute for Applied Mathematics, Endenicher Allee 60, 53115 Bonn, Germany
<EMAIL_ADDRESS>and  Seok-Bae Yun  Department of mathematics,
Sungkyunkwan University, Suwon 16419, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

The relativistic quantum Boltzmann equation (or the relativistic Uehling-
Uhlenbeck equation) describes the dynamics of single-species fast-moving
quantum particles. With the recent development of the relativistic quantum
mechanics, the relativistic quantum Boltzmann equation has been widely used in
physics and engineering such as in the quantum collision experiments and the
simulations of electrons in graphene. In spite of such importance, there has
been no mathematical theory on the existence of solutions for the relativistic
quantum Boltzmann equation to the best of authors’ knowledge. In this paper,
we prove the global existence of a unique classical solution to the
relativistic Boltzmann equation for both bosons and fermions when the initial
distribution is nearby a global equilibrium.

###### Key words and phrases:

Relativistic quantum Boltzmann equation, Relativistic Uehling-Uhlenbeck
equation, Bose-Einstein distribution, Fermi-Dirac distribution, Nonlinear
energy methods

##  1\. Introduction

The dynamics of single-species fast-moving quantum particles is described by
the relativistic quantum Boltzmann equation:

(1.1)
$\displaystyle\begin{split}p^{\mu}\partial_{\mu}F=p^{0}\partial_{t}F+p\cdot\nabla_{x}F=C(F,F,F,F),\end{split}$

where $F(x,p,t)$ is a momentum distribution function on the phase point
$(x,p)\in\mathbb{T}^{3}\times\mathbb{R}^{3}$ at time $t\in[0,\infty)$. The
relativistic quantum collision operator is given by

(1.2)
$C(F_{1},F_{2},F_{3},F_{4})=\int_{\mathbb{R}^{3}}\frac{dq}{q^{0}}\int_{\mathbb{R}^{3}}\frac{dp^{\prime}}{p^{\prime
0}}\int_{\mathbb{R}^{3}}\frac{dq^{\prime}}{q^{\prime
0}}W(p,q|p^{\prime},q^{\prime})\cr\times\big{[}F_{1}(p^{\prime})F_{2}(q^{\prime})(1+\tau
F_{3}(p))(1+\tau F_{4}(q))-(1+\tau F_{1}(p^{\prime}))(1+\tau
F_{2}(q^{\prime}))F_{3}(p)F_{4}(q)\big{]},$

where $\tau=+1$ for bosons, and $\tau=-1$ for fermions. When the functions
$F_{1}$, $F_{2}$, $F_{3}$ and $F_{4}$ are identical, we denote $C(F,F,F,F)$ as
$C(F)$. The transition rate $W(p,q|p^{\prime},q^{\prime})$ is defined as

(1.3) $\displaystyle
W(p,q|p^{\prime},q^{\prime})=\frac{c}{2}s\sigma(g,\theta)\delta^{(4)}(p^{\mu}+q^{\mu}-p^{\prime\mu}-q^{\prime\mu}),$

where $\sigma(g,\theta)$ denotes the differential cross-section which
describes the collisions between particles. The $4$-dimensional Dirac-delta
function implies the conservation laws of momentum and energy. The precise
definitions of $s$ and $\sigma(g,\theta)$ are given in Section 1.2.

###  1.1. Motivations and a brief history

By the recent advances in the relativistic quantum theory, there has been
increasing interest in the relativistic quantum Boltzmann equation in various
fields of physics and engineering. Li et al. [53] announced several
experimental and theoretical results about the interaction between heavy-ion
and neutron using the relativistic quantum Boltzmann equation. In [11], Buss
et al. provided several experiments of quantum interaction such as pion-
nucleus interactions, heavy-ion reactions, electron-nucleus collisions and
neutrino-nucleus interactions using an equation called Giessen Boltzmann-
Uehling-Uhlenbeck (GiBUU) transport model. Since the speed of the electron in
graphene is close to that of light, Lapitski developed the lattice Boltzmann
method to simulate the stream of the electrons in graphene in [51]. We also
introduce some other experiments on the relativistic quantum Boltzmann model
[49, 54, 78].

Despite such modern advances in physics and engineering, there have been quite
few mathematical studies on the relativistic quantum Boltzmann equation. Akama
[1] considered several types of relativistic quantum kinetic models such as
the Boltzmann, the Fokker-Planck, the Landau equations, and this was the first
appearance of the relativistic quantum Boltzmann equation. The unique
determination of the relativistic quantum equilibrium satisfying $H$-theorem
and conservation laws has been considered by Escobedo et al. in [26, 25]. To
the best of authors’ knowledge, the mathematical theory on the existence of
solutions to the equation has not been studied yet.

Regarding the Newtonian non-quantum Boltzmann equation, a brief list of the
classic literature includes [2, 4, 13, 14, 17, 19, 29, 39, 38, 37, 81, 20, 9,
8, 12, 28, 34, 33, 41, 43, 50, 69, 70].

The mathematical studies on the relativistic non-quantum Boltzmann equation
are also not rich. In 1940, Lichnerowicz and Marrot then suggested the first
relativistic Boltzmann equation in [56]. In [23], Dudyński and Ekiel-Jeżewska
studied the linearized relativistic Boltzmann equation and obtained the
estimates for the linear term in the hard and soft potential cases. The
authors also constructed the global existence of mild solution in [22].
Glassey and Strauss obtained some estimates on the derivatives of the
collision map between pre-post collisional momenta including the upperbound on
the average of $\partial p^{\prime}/\partial p$ and $\partial
q^{\prime}/\partial p$ in [32]. Very recently, Chapman et al. [16] provided an
analytic and numerical evidence that the lower bound for the Jacobian
determinant for the collision map $p\mapsto p^{\prime}$ or $q^{\prime}$ in the
center-of-momentum frame is zero. In the nearby-equilibrium regime, Glassey
and Strauss [30] established a unique global solution in $\mathbb{T}^{3}$ for
the hard potential case, and it was extended to the whole space case via the
fourteen-moment compensating function in [31]. For the soft potential case,
the global existence and the asymptotic behavior were constructed in torus in
[71] and extended to the whole space in [77]. In [40], Guo and Strain obtained
the stability of the relativistic Vlasov-Maxwell-Boltzmann system via the use
of two different coordinate represenatations for the post-collisional momenta
in order to resolve the issue of the momentum singularity. In [21], Duan and
Yu provided the global existence of solutions in the weighted $L^{\infty}$
framework. Recently, Wang [82] showed the global wellposedness of the
relativistic Boltzmann equation with large amplitude initial data. In the case
of Coulombic interaction, Strain and Guo [75] constructed a unique global-in-
time classical solution for the relativistic Landau-Maxwell system. The
reduction of the relativistic collision operator using the center-of-momentum
frame can be found in [26, 72]. Andréasson et al. [3] proved the finite-time
blowup of the gain term when the initial data starts with the characteristic
ball. The uniform $L^{1}$ stability of mild solution is established in [42]
when the initial data is sufficiently small and it decays exponentially fast.
In [46], Jang and Yun proved the regularizing estimate of the gain term. For
the spatially homogeneous case, Strain and Yun obtained useful inequalities
about the relativistic pre-post collisional velocity and proved the existence
of solution in [76]. The uniform $L^{\infty}$ bounds of the solution are
established in [45] and the $L^{p}$ bounds are established in [47]. For the
Newtonian limit of the relativistic particles, we refer to [73].

The mathematical studies on the non-relativistic quantum Boltzmann equation
are also not rich. We would like to mention that Nordheim in $1928$ and
Uehling - Uhlenbeck in $1933$ discussed the non-relativistic quantum Boltzmann
equation in [48] and [79], respectively. Benedetto et al. [6, 7] showed the
rigorous validity of the quantum Boltzmann equation from the $N$-body
Schrödinger equation in the weak coupling regime. In spatially homogeneous
case for the Fermi-Dirac particles, Lu classified quantum equilibrium in [61].
In [67], Lu and Wennberg established the strong stability in $L^{1}$ and
proved the convergence of the solution to the equilibrium. The weak solution
to the equation is constructed in the soft potential case [62, 63]. For the
spatially homogeneous case, several studies on bosons can be found in [10, 27,
55, 58, 60, 66, 64, 65, 59, 68]. For general mathematical and physical
reviews, we refer to [4, 13, 14, 15, 17, 19, 29, 39, 38, 37, 74, 81].

###  1.2. Notations

Before introducing our main results, we define several notations on the
relativistic quantities. First of all, we remark that we normalize all
physical constants to $1$ throughout the paper including the speed of light
$c$. We denote the energy-momentum $4$-vector as $p^{\mu}$ and usually write
it as $p^{\mu}=(p^{0},p^{1},p^{2},p^{3})$. The energy-momentum $4$-vector with
the lower index is written as a product in the Minkowski metric
$p_{\mu}=\eta_{\mu\nu}p^{\nu}$, where the Minkowski metric is given by
$\eta_{\mu\nu}=diag(-1,1,1,1)$. The inner product of energy-momentum
$4$-vectors $p^{\mu}$ and $q_{\mu}$ is defined via the Minkowski metric:

$\displaystyle
p^{\mu}q_{\mu}=p^{\mu}\eta_{\mu\mu}q^{\mu}=-p^{0}q^{0}+\sum_{i=1}^{3}p^{i}q^{i}.$

The energy of a relativistic particle is given by $p^{0}=\sqrt{1+|p|^{2}}$.
Then we can see that the inner product of an energy-momentum $4$-vector with
itself is $p^{\mu}p_{\mu}=-1$. We note that the inner product of energy-
momentum $4$-vectors is Lorentz invariant $p^{\mu}q_{\mu}=\Lambda
p^{\mu}\Lambda q_{\mu}$, where $\Lambda$ is a Lorentz transform which will be
defined below.

We define the relative energy $s$ between two energy-momentum 4-vectors
$p^{\mu}$ and $q^{\mu}$ as

(1.4) $\displaystyle
s(p^{\mu},q^{\mu})=-(p^{\mu}+q^{\mu})(p_{\mu}+q_{\mu})=-2p^{\mu}q_{\mu}+2,$

and the relative momentum $g$ between them as

(1.5) $\displaystyle
g(p^{\mu},q^{\mu})=\sqrt{(p^{\mu}-q^{\mu})(p_{\mu}-q_{\mu})}=\sqrt{2(-p^{\mu}q_{\mu}-1)}.$

Note that $s=g^{2}+4$. In the same sense, we define the relative momentum
between $p^{\mu}$ and another momentum $p^{\prime\mu}$ and $q^{\prime\mu}$ as
$\bar{g}=g(p^{\mu},p^{\prime\mu})$ and $\tilde{g}=g(p^{\mu},q^{\prime\mu})$,
respectively. Also, we define $\bar{s}=s(p^{\mu},p^{\prime\mu})$ and
$\tilde{s}=s(p^{\mu},q^{\prime\mu})$, so that $\bar{s}=\bar{g}^{2}+4$ and
$\tilde{s}=\tilde{g}^{2}+4$.

###  1.3. Conservation laws and the Boltzmann $H$-theorem

The conservation law of the pre-post collisional energy-momentum $4$-vectors
is given by

(1.6) $\displaystyle p^{\mu}+q^{\mu}=p^{\prime\mu}+q^{\prime\mu}.$

The inner product with itself gives

$\displaystyle(p^{\mu}+q^{\mu})(p_{\mu}+q_{\mu})=(p^{\prime\mu}+q^{\prime\mu})(p^{\prime}_{\mu}+q^{\prime}_{\mu}).$

Using
$p^{\mu}p_{\mu}=p^{\prime\mu}p^{\prime}_{\mu}=q^{\mu}q_{\mu}=q^{\prime\mu}q^{\prime}_{\mu}=-1$,
we have $p^{\mu}q_{\mu}=p^{\prime\mu}q^{\prime}_{\mu}$. Similarly, we can have
$p^{\mu}p^{\prime}_{\mu}=q^{\mu}q^{\prime}_{\mu}$ and
$p^{\mu}q^{\prime}_{\mu}=p^{\prime\mu}q_{\mu}$, which implies

(1.7)
$\displaystyle\begin{split}g&=g(p^{\mu},q^{\mu})=g(p^{\prime\mu},q^{\prime\mu}),\cr\bar{g}&=g(p^{\mu},p^{\prime\mu})=g(q^{\mu},q^{\prime\mu}),\cr\tilde{g}&=g(p^{\mu},q^{\prime\mu})=g(p^{\prime\mu},q^{\mu}).\end{split}$

Then it turns out [47, Proposition 2.7] that (1.6) can further imply the
Pythagorean theorem as

(1.8) $\displaystyle g^{2}=\bar{g}^{2}+\tilde{g}^{2}.$

The angle of the scattering kernel $\sigma(g,\theta)$ in (1.3) is given by

(1.9)
$\displaystyle\cos\theta=\frac{(p^{\mu}-q^{\mu})(p^{\prime}_{\mu}-q^{\prime}_{\mu})}{g^{2}}.$

By (1.7) and (1.8), $\cos\theta$ can also be written in the following form
[18, page 277, (A.11)]:

(1.10) $\displaystyle\cos\theta=1-2\frac{\bar{g}^{2}}{g^{2}}.$

Now we introduce the center-of-momentum framework. The relativistic pre-post
collisional momenta $p$, $q$ and $p^{\prime}$, $q^{\prime}$ satisfying the
conservation law (1.6) can be expressed as

(1.11)
$\begin{split}p^{\prime}&=\frac{p+q}{2}+\frac{g}{2}\left(w-(\gamma-1)(p+q)\frac{(p+q)\cdot
w}{|p+q|^{2}}\right),\cr
q^{\prime}&=\frac{p+q}{2}-\frac{g}{2}\left(w-(\gamma-1)(p+q)\frac{(p+q)\cdot
w}{|p+q|^{2}}\right),\end{split}$

where $\gamma=(p^{0}+q^{0})/\sqrt{s}$, and $w$ denotes $\mathbb{S}^{2}$
component of the unit sphere:

(1.12) $\displaystyle w=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta),$

on $\theta\in[0,\pi]$ and $\phi\in[0,2\pi]$. The pre-post collisional energy
$p^{\prime 0}$ and $q^{\prime 0}$ is written by

$\displaystyle p^{\prime 0}$
$\displaystyle=\frac{p^{0}+q^{0}}{2}+\frac{g}{2\sqrt{s}}(p+q)\cdot w,$
$\displaystyle q^{\prime 0}$
$\displaystyle=\frac{p^{0}+q^{0}}{2}-\frac{g}{2\sqrt{s}}(p+q)\cdot w.$

Once we divide each side of (1.1) by $p^{0}$, then we can rewrite the
relativistic quantum Boltzmann equation as follows:

(1.13)
$\displaystyle\begin{split}\partial_{t}F+\hat{p}\cdot\nabla_{x}F&=Q(F,F,F,F),\cr
F(x,p,0)&=F_{0}(x,p).\end{split}$

Note that $Q(F,F,F,F)=\frac{1}{p^{0}}C(F)$. We generally denote the normalized
momentum as $\hat{p}$ where

$\displaystyle\hat{p}=\frac{p}{p^{0}}=\frac{p}{\sqrt{1+|p|^{2}}}.$

In this framework, the relativistic quantum collision operator is reduced to
the following form [72]:

(1.14)
$Q(F_{1},F_{2},F_{3},F_{4})=\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}(p^{\mu},q^{\mu})~{}\sigma(g,\theta)\\\
\times\big{[}F_{1}(p^{\prime})F_{2}(q^{\prime})(1+\tau F_{3}(p))(1+\tau
F_{4}(q))-(1+\tau F_{1}(p^{\prime}))(1+\tau
F_{2}(q^{\prime}))F_{3}(p)F_{4}(q)\big{]},$

where the M$\o$ller velocity $v_{\o}$ is given by

$\displaystyle
v_{\o}(p^{\mu},q^{\mu})=\sqrt{\bigg{|}\frac{p}{p^{0}}-\frac{q}{q^{0}}\bigg{|}^{2}-\bigg{|}\frac{p}{p^{0}}\times\frac{q}{q^{0}}\bigg{|}^{2}}=\frac{g\sqrt{s}}{p^{0}q^{0}}.$

One of the properties that the relativistic quantum collision operator
satisfies is the following identity

$\displaystyle\int_{\mathbb{R}^{3}}dp~{}Q(F,F,F,F)\left(\begin{array}[]{c}1\cr
p^{\mu}\end{array}\right)=0,$

which implies the conservation laws of the total mass, momentum and energy:

(1.17)
$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{3}}dx\int_{\mathbb{R}^{3}}dp~{}F\left(\begin{array}[]{c}1\cr
p^{\mu}\end{array}\right)=0.$

The $H$-theorem for the relativistic quantum Boltzmann equation is established
in [26] as

$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{3}}dx\int_{\mathbb{R}^{3}}dp~{}F\ln
F-\tau^{-1}(1+\tau F)\ln(1+\tau F)\leq 0.$

###  1.4. Global equilibria

It has been shown in [25] that the global equilibria for the relativistic
quantum Boltzmann equation (1.1) has the form of

$\mathcal{F}(p)=\frac{1}{e^{\nu(p)}-\tau},\text{ for }\nu(p)=ap^{0}+b\cdot
p+c,$

for some constant $a$, $b,$ and $c.$ In this paper, we rescale the problem and
consider the situation that the macroscopic mean velocity $b=0$ is zero. Then
we can define the global equilibrium $m(p)$ in the form of

(1.18) $\displaystyle m(p)=\frac{1}{e^{ap^{0}+c}-\tau},$

where $a>0$ and $c\geq-a$ in the case of bosons and $a>0$ and $c\in\mathbb{R}$
in the case of fermions. In this paper, we only consider $a>0$ and $c>-a$ in
the case of bosons in order to exclude a possible blow-up of the equilibrium.
We choose and fix the constant $a$ and $c$ and consider the initial
distribution $F_{0}$ nearby the equilibrium $m(p)$. Then we will prove that
the particle distribution $F$ whose initial distribution $F_{0}$ is
sufficiently close to the relativistic quantum global equilibrium $m(p)$ will
converge to $m(p)$ in $H^{N}_{x}L^{2}_{v}$ sense for $N\geq 3$.

We also denote the non-quantum relativistic equilibrium $J(p)$ as

(1.19) $\displaystyle J(p^{0})=e^{-ap^{0}}.$

###  1.5. Spaces

Now we define some notations on norms and inner products which are frequently
used throughout this paper. The constant $C$ is generically used, whose value
can be changed from line to line. Especially, when we want to indicate the
dependency of $a$, we indicate it as $C_{a}$. We define the standard $L^{2}$
norm as

$\displaystyle\|f\|_{L^{2}_{p}}=\left(\int_{\mathbb{R}^{3}}dp~{}|f(p)|^{2}\right)^{\frac{1}{2}},\quad\|f\|_{L^{2}_{x,p}}=\left(\int_{\mathbb{T}^{3}}dx\int_{\mathbb{R}^{3}}dp~{}|f(x,p)|^{2}\right)^{\frac{1}{2}},$

and we define the weighted $L^{2}$ norm as

$\displaystyle\|f\|_{\nu}=\left(\int_{\mathbb{R}^{3}}dp~{}\nu(p)|f(p)|^{2}\right)^{\frac{1}{2}},\quad\|f\|_{x,\nu}=\left(\int_{\mathbb{T}^{3}}dx\int_{\mathbb{R}^{3}}dp~{}\nu(p)|f(x,p)|^{2}\right)^{\frac{1}{2}}.$

The standard $L^{2}$ inner product is given by

$\displaystyle\langle f,g\rangle_{L^{2}_{p}}$
$\displaystyle=\int_{\mathbb{R}^{3}}dp~{}f(p)g(p),\quad\langle
f,g\rangle_{L^{2}_{x,p}}=\int_{\mathbb{T}^{3}}dx\int_{\mathbb{R}^{3}}dp~{}f(x,p)g(x,p).$

We use the multi-index notation

$\displaystyle\alpha=(\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3}),$

to simplify the differential operator:

$\displaystyle\partial^{\alpha}=\partial_{t}^{\alpha_{0}}\partial_{x_{1}}^{\alpha_{1}}\partial_{x_{2}}^{\alpha_{2}}\partial_{x_{3}}^{\alpha_{3}}.$

For a brevity of discussion on the frequently used function $m+\tau m^{2}$, we
often write the variable only at the end of the function as

$\displaystyle(m+\tau m^{2})(p)=m(p)+\tau(m(p))^{2}.$

We define the higher-order energy norm as follows:

$\displaystyle\mathcal{E}(f(t))=\frac{1}{2}\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f(t)\|^{2}_{L^{2}_{x,p}}+\int_{0}^{t}\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f(s)\|^{2}_{x,\nu}ds.$

###  1.6. Hypothesis on the collision kernel

We assume that the differential cross section $\sigma(g,\theta)$ satisfies the
hard potential assumption with an angular cut-off as in [23, 24]:

(1.20) $\displaystyle\sigma(g,\theta)=g\sin\theta.$

This assumption is analogous to the standard hard-sphere assumption for the
Newtonian Boltzmann equation; if $|p+q|$ or $|p-q|$ is sufficiently small or
if $|p-q|$ is much larger than $|p^{0}-q^{0}|$, then it behaves as the
Newtonian hard sphere kernel.

###  1.7. Main results

In this paper, we prove the existence of a unique global-in-time classical
solution of the relativistic quantum Boltzmann equation nearby the global
equilibrium. Before we state our main theorem, we would like to introduce the
reformulation of the relativistic quantum Boltzmann equation via a special
linearization that is relevant to the relativistic and quantum case.

In the non-quantum case, the standard decomposition of the perturbed solution
$f$ near the global equilibrium $\mu$ is $F=\mu+\sqrt{\mu}f$ for
$\mu=e^{-|p|^{2}/2}$ or $\mu=e^{-p^{0}}$ in the Newtonian and the relativistic
cases, respectively. But in the quantum case, the previous decomposition
$F=\mu+\sqrt{\mu}f$ does not guarantee the non-negativity of $\langle
Lf,f\rangle_{L^{2}_{v}}$ as in Lemma 2.2. Thus inspired by the previous work
related to the quantum kinetic models in [5, 80, 52, 57], we choose the
following decomposition of $F$:

$F(x,p,t)=m(p)+\sqrt{m(p)+\tau m^{2}(p)}f(x,p,t),$

where the global equilibrium $m(p)$ is defined as in (1.18):

$\displaystyle m(p)=\frac{1}{e^{ap^{0}+c}-\tau}.$

We plug the decomposition into (1.13) and divide each side of the equation by
$\sqrt{m+\tau m^{2}}$ to have

(1.21)
$\displaystyle\begin{split}\partial_{t}f+\hat{p}\cdot\nabla_{x}f+Lf&=\Gamma(f)+T(f),\cr
f(x,p,0)&=f_{0}(x,p).\end{split}$

The linear term $Lf$ is given by

$Lf=\nu(p)f+K_{1}f-K_{2}f,$

where $\nu(p)$ is the collision frequency of a relativistic quantum particle,
and $K_{1}$ and $K_{2}$ are compact operators. The right-hand side of (1.21)
consists of nonlinear terms $\Gamma(f)$ and $T(f)$ where $\Gamma(f)$ consists
of all the second-order nonlinear terms and $T(f)$ consists of all the third-
order nonlinear terms. We can easily check that the fourth-order nonlinear
terms disappear by cancellation. In the linearization process, we observe that
the collision operator does not satisfy the quad-linearity
$Q(k+h,f,f,f)=Q(k,f,f,f)+Q(h,f,f,f)$. For more of the detailed linearization
process, see Section 2. Now we are ready to state our main theorem.

######  Theorem 1.1.

Let $N\geq 3$. Suppose that the initial data $F_{0}$ satisfies

$\displaystyle\left\\{\begin{array}[]{ll}0\leq F_{0}(x,p)\leq 1\quad\mbox{for
fermions,}\\\ 0\leq F_{0}(x,p)\qquad\hskip 8.53581pt\mbox{for
bosons,}\end{array}\right.$

and the global equilibrium $m(p)$ shares the same total mass, momentum and
energy with the initial data:

(1.26)
$\displaystyle\int_{\mathbb{T}^{3}\times\mathbb{R}^{3}}dxdp~{}F_{0}(x,p)\left(\begin{array}[]{c}1\cr
p^{\mu}\end{array}\right)=\int_{\mathbb{T}^{3}\times\mathbb{R}^{3}}dxdp~{}m(p)\left(\begin{array}[]{c}1\cr
p^{\mu}\end{array}\right).$

Then there exist $\delta>0$ and $C>0$ such that if
$\mathcal{E}(f_{0})\leq\delta$ then there exists a unique global-in-time
solution of (1.21) such that

  1. (1)

The distribution function $F(x,p,t)$ has the following bounds:

$\displaystyle\left\\{\begin{array}[]{ll}0\leq F(x,p,t)\leq 1\quad\mbox{for
fermions,}\\\ 0\leq F(x,p,t)\qquad\hskip 8.53581pt\mbox{for
bosons.}\end{array}\right.$

  2. (2)

The energy norm is bounded globally in time:

$\sup_{t\in\mathbb{R}^{+}}\mathcal{E}(f(t))\leq C\mathcal{E}(f_{0}).$

  3. (3)

There exists a uniform constant $\epsilon>0$ such that the perturbation decays
exponentially:

$\sum_{|\alpha|\leq N}\|\partial^{\alpha}f(t)\|^{2}_{L^{2}_{x,p}}\leq
Ce^{-\epsilon t}.$

  4. (4)

Let $f$ and $\bar{f}$ be the solutions with the initial data $f_{0}$ and
$\bar{f}_{0}$, respectively. Then there exists a positive constant $\delta>0$
such that

$\|f-\bar{f}\|_{L^{2}_{x,p}}\leq e^{-\delta
t}\|f_{0}-\bar{f}_{0}\|_{L^{2}_{x,p}}.$

To prove the main theorem, we need a coercivity estimate of the linear
operator $L$. The linear operator $L$ satisfies the following dissipation
property:

$\displaystyle\langle Lf,f\rangle_{L^{2}_{v}}$
$\displaystyle\geq\delta\|(I-P)f\|_{\nu}^{2},$

for some positive $\delta>0$. The macroscopic projection $Pf$ denotes the
orthonormal projection onto $L^{2}_{p}$ space with respect to the following
$5$-basis:

(1.27) $\displaystyle\left\\{\sqrt{m+\tau m^{2}},p_{1}\sqrt{m+\tau
m^{2}},p_{2}\sqrt{m+\tau m^{2}},p_{3}\sqrt{m+\tau m^{2}},p^{0}\sqrt{m+\tau
m^{2}}\right\\}.$

The $5$-dimensional basis above constitutes the kernel of $L$. Then it is
crucial to obtain some upper-bound estimates of the nonlinear terms $\Gamma$
and $T$ in order to construct the solution. Some parts of the nonlinear terms
can be estimated by a simple change of variables and the Hölder inequality.
However, regarding some second-order nonlinear terms whose integrands consist
of the product of $f(p^{\prime})$ (or $f(q^{\prime})$) and $f(p)$, we need to
take a change of variables of either $p\mapsto p^{\prime}$ (or $p\mapsto
q^{\prime}$) and there occur some non-trivial difficulties. Different from the
non-relativistic case, a uniform positive lower bound for the Jacobian for the
change of variables $|\partial p^{\prime}/\partial p|$ (or $|\partial
q^{\prime}/\partial p|$) does not exist as shown in [16], and hence we need
another way to deal with this difficulty. One way is to proceed the estimates
of the second-order nonlinear terms by lifting the $dq$ integral to energy-
momentum four vector integral $dq^{\mu}$ imposing additional Dirac-delta
function. This technique has been introduced in [18, 44, 45]. Then we reduce
the integral by computing the Dirac-delta function. In this process, there
appears a singularity of $1/\bar{g}$ and the exponential growth with respect
to the $p$ variable from $\exp(p^{0}-p^{\prime 0})$. We will explain this more
in detail in Section 1.8.2.

Once we obtain the estimates of the linear terms and the nonlinear terms, we
define an iteration scheme to construct the local-in-time solution as

(1.28)
$\displaystyle(\partial_{t}+\hat{p}\cdot\nabla_{x})F^{n+1}=Q(F^{n},F^{n},F^{n+1},F^{n}).$

Regarding the case of fermions, the solution $F$ has to be bounded by $1$ from
above. Therefore, we also need to prove that the function $F^{n+1}$ is bounded
in the closed interval $[0,1]$ in each iteration scheme based on the induction
hypothesis that $F^{n}$ is in $[0,1]$. Then we obtain that the collision
operator $Q(F^{n},F^{n},F^{n+1},F^{n})$ is well-defined. On the right-hand
side of (1.28), note that we place $F^{n+1}$ in the $p$ variable position of
the collision operator $Q$ (i.e., at the third input of
$Q(\cdot,\cdot,\cdot,\cdot)$). This is by the specific structure of the
nonlinear operator $Q$ from (1.14). Once we place $F^{n+1}$ as a third input
of $Q$ then we can rewrite (1.28) in a more clear structure

(1.29) $\displaystyle\\{\partial_{t}+\hat{p}\cdot\nabla_{x}-\tau
G(F^{n})+R(F^{n})\\}F^{n+1}=G(F^{n}),$

where $G$ and $R$ is now defined as

$\displaystyle G(F_{1},F_{2},F_{4})$
$\displaystyle=\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)F_{1}(p^{\prime})F_{2}(q^{\prime})(1+\tau
F_{4}(q)),$ $\displaystyle R(F_{1},F_{2},F_{4})$
$\displaystyle=\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)(1+\tau
F_{1}(p^{\prime}))(1+\tau F_{2}(q^{\prime}))F_{4}(q).$

Also, we can show that the boundedness $0\leq F^{n}\leq 1$ implies that $0\leq
G(F^{n})$ and $0\leq R(F^{n})$, which guarantees the boundedness of $F^{n+1}$.
Then the additional linearization of $F^{n+1}=m+\sqrt{m+\tau m^{2}}f^{n+1}$
results in creating the dissipation term $\nu(p)f^{n+1}$ in the left-hand side
of (1.29), which would then result in the exponential decay of the
perturbation $f^{n+1}$ in the $L^{2}$ sense. By the induction argument, we
obtain the uniform boundedness of the energy locally in time.

Then the standard way of extending the local existence to the global one is to
eliminate the dissipation of the linear part $L$. Similarly to the Newtonian
case, we substitute $f=(I-P)f+Pf$ on each side of (1.21) where $Pf$ is defined
by the orthonormal projection of (1.27). Then the expansion of the linear term
$(\partial_{t}+\hat{p}\cdot\nabla_{x})Pf$ yields a linear combination with
respect to the $14$-basis. Thus we can achieve the following coercivity
estimate:

$\displaystyle\sum_{|\alpha|\leq N}\langle
L\partial^{\alpha}f,\partial^{\alpha}f\rangle_{L^{2}_{x,p}}$
$\displaystyle\geq\delta\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f\|_{x,\nu}^{2},$

for some positive constant $\delta>0$. This full coercivity estimate enables
the extension of the local-in-time solution to a global-in-time solution.

###  1.8. Main difficulties and our strategy

In this subsection, we present the difficulties that arise from the estimates
of the nonlinear terms. In the relativistic quantum case, there appear new
types of nonlinear terms involving both pre- and post-collisional momenta at
the same time as $f(p)f(p^{\prime})$ and $f(p)f(q^{\prime})$. In general, this
kind of terms has been expected to appear only in the non-cutoff Boltzmann
theory in the non-quantum case. In the non-cutoff Boltzmann theory, it has
been considered very crucial to understand the change of variables $q\mapsto
p^{\prime}$ or $q^{\prime}$ and the cancellation lemma to understand the
fractional diffusive behavior [36, 44]. In the non-relativistic case, these
new nonlinear terms can be handled because $|\partial p^{\prime}/\partial p|$
and $|\partial q^{\prime}/\partial q|$ are bounded from below, as in [2, 19].
However, in the relativistic case under the center-of-momentum frame, such a
positive uniform lower-bound of the Jacobian of the collision map does not
exist [16].

The remedy of the issue has been introduced in [18, 31, 71] in the center-of-
momentum frame (1.11) where the linear term $K_{2}$ of the difficulty has been
calculated. Namely, the authors lift the $dq$ integral to the energy-momentum
four vector integral $dq^{\mu}$ by imposing an additional Dirac-delta
function, and they take a change of variables via a suitable Lorentz
transform. In this paper, we also follow a similar technique to deal with the
nonlinearity that occurs in dealing with the terms $\Gamma(f)$ and $T(f)$. In
this direction, we however still encounter several other additional
difficulties on the nonlinear term involving $f(p)f(p^{\prime})$ as below. Let
us first denote the nonlinear term $\Gamma_{2,1}$ involving
$f(p)f(p^{\prime})$ as

(1.30)
$\big{|}\langle\Gamma_{2,1}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}\leq\int_{\mathbb{R}^{3}}\frac{dp}{p^{0}}\int_{\mathbb{R}^{3}}\frac{dq}{q^{0}}\int_{\mathbb{R}^{3}}\frac{dp^{\prime}}{p^{\prime
0}}\int_{\mathbb{R}^{3}}\frac{dq^{\prime}}{q^{\prime
0}}s\sigma(g,\theta)\cr\times\delta^{(4)}(p^{\mu}+q^{\mu}-p^{\prime\mu}-q^{\prime\mu})J(q^{0})J(p^{\prime
0}/2)|f(p)||h(p^{\prime})||\eta(p)|,$

and the integral part with respect to the measure $dqdq^{\prime}$ as

(1.31) $\displaystyle
B=\int_{\mathbb{R}^{3}}\frac{dq}{q^{0}}\int_{\mathbb{R}^{3}}\frac{dq^{\prime}}{q^{\prime
0}}s\sigma(g,\theta)\delta^{(4)}(p^{\mu}+q^{\mu}-p^{\prime\mu}-q^{\prime\mu})J(q^{0}).$

Then we present the three main difficulties that arise regarding the nonlinear
terms.

####  1.8.1. The Lorentz-invariant measure $\frac{dp}{p^{0}}$ and the
nonlinear structure of $s\sigma(g,\theta)$

The first difficulty arises from the Lorentz-invaraint measure
$\frac{dp}{p^{0}}$ including the fraction of the energy and the nonlinear
structure of the differential cross section $s\sigma(g,\theta)$ with respect
to the collision variables $p$ and $q$. A good way to remove the fraction of
the energy is to lift $dq$ and $dq^{\prime}$ integrals to the energy momentum
$4$-vector $dq^{\mu}$ and $dq^{\prime\mu}$ integrals by considering extra
Dirac-delta and unit step functions as

$B=\int_{\mathbb{R}^{4}}dq^{\mu}\int_{\mathbb{R}^{4}}dq^{\prime\mu}s\sigma(g,\theta)\delta^{(4)}(p^{\mu}+q^{\mu}-p^{\prime\mu}-q^{\prime\mu})J(q^{0})u(q^{0})u(q^{\prime
0})\delta(q^{\mu}q_{\mu}+1)\delta(q^{\prime\mu}q^{\prime}_{\mu}+1).$

If we simply eliminate the $dq^{\prime\mu}$ integral by computing the Dirac-
delta function of $\delta^{(4)}(p^{\mu}+q^{\mu}-p^{\prime\mu}-q^{\prime\mu})$,
the rest of the integral becomes highly complicated to deal with. Thus,
motivated by de Groot et al. in [18], we instead apply a symmetric change of
variables $\bar{q}^{\mu}=q^{\mu}+q^{\prime\mu}$ and
$\bar{q}^{\prime\mu}=q^{\mu}-q^{\prime\mu}$. Despite the nonlinear structure
of the $s$, $g$ and $\cos\theta$ in (1.4), (1.5) and (1.9), respectively, we
can also represent them as the terms that depend only on the variables
$p^{\mu}$, $p^{\prime\mu}$, $\bar{q}^{\mu}$ and $\bar{q}^{\prime\mu}$ (See
Lemma 3.5) as follows:

$\displaystyle
g_{c}^{2}=\bar{g}^{2}-\frac{1}{2}(p^{\mu}+p^{\prime\mu})(\bar{q}_{\mu}-p_{\mu}-p^{\prime}_{\mu}),\quad
s_{c}=g_{c}^{2}+4,\quad\cos\theta_{c}=1-2\frac{\bar{g}^{2}}{g_{c}^{2}}.$

Then we can have $B$ in the following form

$\displaystyle B$
$\displaystyle=\frac{1}{4}\int_{\mathbb{R}^{4}\times\mathbb{R}^{4}}d\Theta(\bar{q}^{\mu},\bar{q}^{\prime\mu})s_{c}\sigma(g_{c},\theta_{c})\delta^{(4)}(p^{\mu}-p^{\prime\mu}+\bar{q}^{\prime\mu})J\left(\frac{\bar{q}^{0}+\bar{q}^{\prime
0}}{2}\right),$

where

$\displaystyle d\Theta(\bar{q}^{\mu},\bar{q}^{\prime\mu})$
$\displaystyle=d\bar{q}^{\mu}d\bar{q}^{\prime\mu}u(\bar{q}^{0})u(\bar{s}-4)\delta((\bar{q}^{\mu}\bar{q}_{\mu}+\bar{q}^{\prime\mu}\bar{q}^{\prime}_{\mu})+4)\delta(\bar{q}^{\mu}\bar{q}^{\prime}_{\mu}).$

By substituting $\bar{q}^{\prime\mu}=p^{\prime\mu}-p^{\mu}$, we now reduce the
$4$-dimensional Dirac-delta function with the energy-momentum $4$-vector
$\delta^{(4)}(p^{\mu}+q^{\mu}-p^{\prime\mu}-q^{\prime\mu})$ and obtain a more
intuitive form of $B$ as

(1.32)
$B=\frac{1}{4}\int_{\mathbb{R}^{4}}d\bar{q}^{\mu}u(\bar{q}^{0})u(\bar{s}-4)\delta(\bar{q}^{\mu}\bar{q}_{\mu}+(p^{\prime\mu}-p^{\mu})(p^{\prime}_{\mu}-p_{\mu})+4)\cr\times\delta(\bar{q}^{\mu}(p^{\prime}_{\mu}-p_{\mu}))s_{c}\sigma(g_{c},\theta_{c})J\left(\frac{\bar{q}^{0}+p^{\prime
0}-p^{0}}{2}\right).$

####  1.8.2. Singularity with respect to $\bar{g}$

Then another difficulty arises from the singularity in the relative momentum
$\bar{g}$, which occurs from the reduction of the second Dirac-delta function.
We first remark that the second Dirac-delta function of (1.32) consists of an
inner product between energy momentum $4$-vectors. Motivated by the explicit
form of the Lorentz transform by Strain [71], we apply the Lorentz trasnform
which converts $p^{\prime}_{\mu}-p_{\mu}$ to $(0,0,0,\bar{g})$ (i.e.
$\Lambda(p^{\prime}_{\mu}-p_{\mu})=(0,0,0,\bar{g})$). Then we obtain

$\displaystyle\delta(\bar{q}^{\mu}(p^{\prime}_{\mu}-p_{\mu}))=\delta(\Lambda\bar{q}^{\mu}\Lambda(p^{\prime}_{\mu}-p_{\mu}))=\delta(\bar{q}^{3}\bar{g})=\frac{1}{\bar{g}}\delta(\bar{q}^{3}),$

where we used that the Lorentz transform is invariant under the inner-product
and that $\delta(ax)=\frac{1}{a}\delta(x)$. We can now see that one $\bar{g}$
in the Dirac-delta function comes out as $1/\bar{g}$ and creates an additional
singularity. However, motivated by (1.10), we use the half angle formula [18,
page 277, (A.11)] of the $\cos\theta$ and observe that

$\sin^{2}(\theta/2)=\frac{1-\cos\theta}{2}=\frac{\bar{g}^{2}}{g^{2}}.$

Combining with the assumption of the differential cross section, we obtain

$\displaystyle\sigma(g,\theta)=g\sin\theta=2g\sin\frac{\theta}{2}\cos\frac{\theta}{2}=2g\sqrt{\frac{1-\cos\theta}{2}}\cos\frac{\theta}{2}=2\bar{g}\cos\frac{\theta}{2}.$

This allows us to eliminate the singularity of $\bar{g}$.

####  1.8.3. Exponential growth

The last difficulty is regarding the last multiplier on the right-hand side of
the equation (1.32). Since the function $J(\bar{q}^{0}/2)$ is contained in
$d\bar{q}^{0}$ integral, we have an exponential decay in $p^{\prime 0}$ and an
exponential growth in $p^{0}$ from $J\left((p^{\prime 0}-p^{0})/2\right)$ at
the same time. The decay for $p^{\prime 0}$ is beneficial for the upper-bound
estimate, but there is a problematic term of the exponential growth of $p^{0}$
even in (1.30). However, we prove that the remaining part of the right-hand
side of (1.32) includes the following exponential decaying factor

$\displaystyle\exp\left(-\sqrt{\frac{(p^{0}+p^{\prime
0})^{2}}{4}-\frac{|p\times p^{\prime}|^{2}}{\bar{g}^{2}}}\right).$

Then, by the estimates in [30, Lemma 3.1, (iii) and (iv)] of

$\displaystyle\frac{(p^{0}+p^{\prime 0})^{2}}{4}-\frac{|p\times
p^{\prime}|^{2}}{\bar{g}^{2}}=|p-p^{\prime}|^{2}\frac{\bar{g}^{2}+4}{4\bar{g}^{2}}\geq\max\left\\{\frac{\bar{g}^{2}}{4}+1,\frac{1}{4}|p-p^{\prime}|^{2}\right\\},$

we can have an exponential decay of $\exp\left(-|p-p^{\prime}|/2\right)$.
Since the difference of the energy $|p^{\prime 0}-p^{0}|$ can further be
bounded by the difference of the momentum $|p^{\prime}-p|$, the exponential
growth can be absorbed by the exponential decay as

$\displaystyle J\left(\frac{p^{\prime
0}-p^{0}}{2}\right)\exp\left(-\sqrt{\frac{(p^{0}+p^{\prime
0})^{2}}{4}-\frac{|p\times p^{\prime}|^{2}}{\bar{g}^{2}}}\right)\leq
e^{-\frac{1}{2}(p^{\prime 0}-p^{0})}e^{-\frac{1}{2}|p-p^{\prime}|}\leq 1.$

Then we can have properly weighted $L^{2}$ bounds for the nonlinear terms.

###  1.9. Outline of the paper

This paper is organized as follows. In Section 2, we linearize the collision
operator of the relativistic quantum Boltzmann equation nearby a global
equilibrium. In Section 3, we establish several estimates on the linear and
the nonlinear terms. Section 4 is devoted to constructing the unique local-in-
time classical solution. In the last section, we prove the coercivity estimate
and establish the global-in-time classical solution.

##  2\. Linearization of the relativistic quantum Boltzmann equation

In this section, we introduce the reformulation of the equation (1.1) via the
linearization of the relativistic quantum Boltzmann equation nearby the global
equilibrium:

$\displaystyle m(p)=\frac{1}{e^{ap^{0}+c}-\tau}.$

######  Proposition 2.1.

If we substitute $F=m+\sqrt{m+\tau m^{2}}f$ in (1.13), then we have

$\displaystyle\partial_{t}f+\hat{p}\cdot\nabla_{x}f+Lf$
$\displaystyle=\Gamma(f)+T(f)$

where the linear term $Lf$ is decomposed as following form:

$\displaystyle Lf$ $\displaystyle=\nu(p)f+K_{1}f-K_{2}f,$

where the collision frequency $\nu(p)$ is given by

(2.1) $\displaystyle\nu(p)$ $\displaystyle=\frac{1}{1+\tau
m(p)}\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(q)(1+\tau
m(p^{\prime}))(1+\tau m(q^{\prime})),$

and the compact operator $K_{1}$ and $K_{2}$ are defined by

(2.2)
$\displaystyle\begin{split}K_{1}f(p)&=\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)\sqrt{m+\tau
m^{2}(p^{\prime})}\sqrt{m+\tau m^{2}(q^{\prime})}f(q),\cr
K_{2}f(p)&=2\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)\sqrt{m+\tau
m^{2}(q)}\sqrt{m+\tau m^{2}(q^{\prime})}f(p^{\prime}).\end{split}$

The nonlinear term $\Gamma(f)$ is represented as follows:

$\displaystyle\Gamma(f)$ $\displaystyle=\sum_{i=1}^{6}\Gamma_{i}(f,f),\quad
T(f)=\sum_{i=1}^{4}T_{i}(f,f,f).$

We denote the precise definition of nonlinear terms at the end of this proof.

###### Proof.

We substitute $F=m+\sqrt{m+\tau m^{2}}f$ in (1.13) to have

$\displaystyle\sqrt{m+\tau m^{2}}\partial_{t}f+\sqrt{m+\tau
m^{2}}\hat{p}\cdot\nabla_{x}f=Q(m+\sqrt{m+\tau m^{2}}f).$

Dividing $\sqrt{m+\tau m^{2}}$ on each side gives an equation for the
perturbation $f$:

(2.3) $\displaystyle\partial_{t}f+\hat{p}\cdot\nabla_{x}f$
$\displaystyle=\frac{1}{\sqrt{m+\tau m^{2}}}Q(m+\sqrt{m+\tau m^{2}}f).$

To decompose the right-hand side into linear and nonlinear terms, we first
define the zeroth-order-in-$f$ term $Q_{0}$:

$\displaystyle Q_{0}=Q(m,m,m,m).$

As we can see in (1.14), the quantum collision operator includes $(1+\tau F)$
terms. Because of these terms, we cannot have the quad-linear property of $Q$.
In other words, we have $Q(k+h,f,f,f)\neq Q(k,f,f,f)+Q(h,f,f,f)$. So we define
the following four first-order-in-$f$ terms $Q_{1}$, $Q_{2}$, $Q_{3}$ and
$Q_{4}$:

$\displaystyle Q_{1}=Q(m+\sqrt{m+\tau m^{2}}f,m,m,m)-Q_{0},$ $\displaystyle
Q_{2}=Q(m,m+\sqrt{m+\tau m^{2}}f,m,m)-Q_{0},$ $\displaystyle
Q_{3}=Q(m,m,m+\sqrt{m+\tau m^{2}}f,m)-Q_{0},$ $\displaystyle
Q_{4}=Q(m,m,m,m+\sqrt{m+\tau m^{2}}f)-Q_{0}.$

In view of this notation, the collection of the zeroth and the first-order
terms in $f$ can be written as

$\displaystyle\frac{1}{\sqrt{m+\tau
m^{2}}}\left(\sum_{i=1}^{4}Q_{i}+Q_{0}\right).$

Thus we divide the right-hand side of (2.3) into the collection of zeroth,
first-order terms and other terms as

$\displaystyle\frac{1}{\sqrt{m+\tau m^{2}}}Q(m+\sqrt{m+\tau
m^{2}}f)=-Lf+\Gamma(f),$

where

$\displaystyle Lf=-\frac{1}{\sqrt{m+\tau
m^{2}}}\sum_{i=1}^{4}Q_{i}-\frac{1}{\sqrt{m+\tau m^{2}}}Q_{0},$

and

$\displaystyle\Gamma(f)=\frac{1}{\sqrt{m+\tau m^{2}}}Q(m+\sqrt{m+\tau
m^{2}}f)+Lf.$

We first calculate the linear part $Lf$. By definition of $Q_{0}$, we have

$Q_{0}=\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)\bigg{[}m(p^{\prime})m(q^{\prime})(1+\tau
m(p))(1+\tau m(q))\cr-(1+\tau m(p^{\prime}))(1+\tau
m(q^{\prime}))m(p)m(q)\bigg{]}.$

We observe from the conservation of energy

$p^{0}+q^{0}=p^{\prime 0}+q^{\prime 0},$

that

(2.4) $\displaystyle m(p^{\prime})m(q^{\prime})(1+\tau m(p))(1+\tau
m(q))=(1+\tau m(p^{\prime}))(1+\tau m(q^{\prime}))m(p)m(q),$

which gives

$Q_{0}=0.$

For the first-order linear term, an explicit computation gives

$Q_{1}=\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)\bigg{[}\sqrt{m+\tau
m^{2}(p^{\prime})}m(q^{\prime})(1+\tau m(p))(1+\tau m(q))\cr-\tau\sqrt{m+\tau
m^{2}(p^{\prime})}(1+\tau m(q^{\prime}))m(p)m(q)\bigg{]}f(p^{\prime}),$

By (2.4) on the first-order term $Q_{1}$, we have

$Q_{1}=\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(p)m(q)(1+\tau
m(p^{\prime}))(1+\tau m(q^{\prime}))\cr\quad\times\left(\frac{\sqrt{m+\tau
m^{2}(p^{\prime})}}{m(p^{\prime})}-\tau\frac{\sqrt{m+\tau
m^{2}(p^{\prime})}}{1+\tau m(p^{\prime})}\right)f(p^{\prime}),$

which is equal to

$\displaystyle Q_{1}$
$\displaystyle=\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(p)m(q)(1+\tau
m(p^{\prime}))(1+\tau m(q^{\prime}))\frac{f(p^{\prime})}{\sqrt{m+\tau
m^{2}(p^{\prime})}}.$

With similar computations, combining with $Q_{2}$,$Q_{3}$ and $Q_{4}$ yields

(2.5) $\begin{split}Lf=\frac{1}{\sqrt{m+\tau
m^{2}(p)}}\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(p)m(q)(1+\tau
m(p^{\prime}))(1+\tau m(q^{\prime}))\cr\times\bigg{(}\frac{f(p)}{\sqrt{m+\tau
m^{2}(p)}}+\frac{f(q)}{\sqrt{m+\tau
m^{2}(q)}}-\frac{f(p^{\prime})}{\sqrt{m+\tau
m^{2}(p^{\prime})}}-\frac{f(q^{\prime})}{\sqrt{m+\tau
m^{2}(q^{\prime})}}\bigg{)}.\end{split}$

We can easily see that the first term of $Lf$ is equal to $\nu f$. We define
the second term of $Lf$ as $K_{1}f$:

(2.6) $K_{1}f=\frac{m(p)}{\sqrt{m+\tau
m^{2}(p)}}\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)\frac{m(q)}{\sqrt{m+\tau
m^{2}(q)}}\\\ \times(1+\tau m(p^{\prime}))(1+\tau m(q^{\prime}))f(q).$

We observe

(2.7) $\displaystyle\frac{m(p)}{\sqrt{m+\tau
m^{2}(p)}}=e^{-\frac{1}{2}(ap^{0}+c)}.$

Combining with the energy conservation law gives

(2.8) $\displaystyle\frac{m(p)}{\sqrt{m+\tau
m^{2}(p)}}\frac{m(q)}{\sqrt{m+\tau
m^{2}(q)}}=\frac{m(p^{\prime})}{\sqrt{m+\tau
m^{2}(p^{\prime})}}\frac{m(q^{\prime})}{\sqrt{m+\tau m^{2}(q^{\prime})}}.$

Applying it to (2.6), we have

$\displaystyle K_{1}f$
$\displaystyle=\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)\sqrt{m+\tau
m^{2}(p^{\prime})}\sqrt{m+\tau m^{2}(q^{\prime})}f(q).$

We define the collection of the third and the fourth term of $Lf$ in (2.5) as
$-K_{2}f$:

(2.9) $K_{2}f=\frac{m(p)}{\sqrt{m+\tau
m^{2}(p)}}\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(q)\frac{1+\tau
m(p^{\prime})}{\sqrt{m+\tau m^{2}(p^{\prime})}}(1+\tau
m(q^{\prime}))f(p^{\prime})\cr\quad+\frac{m(p)}{\sqrt{m+\tau
m^{2}(p)}}\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(q)(1+\tau
m(p^{\prime}))\frac{1+\tau m(q^{\prime})}{\sqrt{m+\tau
m^{2}(q^{\prime})}}f(q^{\prime}).$

We would call the first line of the $K_{2}f$ in (2.9) as $K_{2,1}f$ and the
second line of (2.9) as $K_{2,2}f$. We then write the $dw$ integral of
$K_{2,2}$ in the spherical coordinate $w\mapsto(\phi,\theta)$ as in (1.12) as
follows:

$K_{2,2}f=\frac{m(p)}{\sqrt{m+\tau
m^{2}(p)}}\int_{\mathbb{R}^{3}}dq\int_{0}^{2\pi}d\phi\int_{0}^{\pi}\sin\theta~{}d\theta~{}v_{\o}\sigma(g,\theta)\\\
\times m(q)(1+\tau m(p^{\prime}))\frac{1+\tau m(q^{\prime})}{\sqrt{m+\tau
m^{2}(q^{\prime})}}f(q^{\prime}).$

Then we apply the change of variables $\theta\rightarrow\pi-\theta$ and
$\phi\rightarrow\pi+\phi$. The change of variables would result in the
exchanged roles of $p^{\prime}$ and $q^{\prime}$, since $w$ in (1.12) changes
into $-w$. Thus we have

$K_{2,2}f=\frac{m(p)}{\sqrt{m+\tau
m^{2}(p)}}\int_{\mathbb{R}^{3}}dq\int_{\pi}^{3\pi}d\phi\int_{\pi}^{0}\sin\theta~{}(-d\theta)\cr\times
v_{\o}\sigma(g,\pi-\theta)m(q)(1+\tau m(q^{\prime}))\frac{1+\tau
m(p^{\prime})}{\sqrt{m+\tau m^{2}(p^{\prime})}}f(p^{\prime}).$

By the assumption of the differential cross section $\sigma(g,\theta)$ in
(1.20), we have $\sigma(g,\theta)=g\sin\theta=\sigma(g,\pi-\theta)$. This
further gives

$\displaystyle K_{2,2}f$ $\displaystyle=\frac{m(p)}{\sqrt{m+\tau
m^{2}(p)}}\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(q)(1+\tau
m(q^{\prime}))\frac{1+\tau m(p^{\prime})}{\sqrt{m+\tau
m^{2}(p^{\prime})}}f(p^{\prime}).$

This shows that $K_{2,2}=K_{2,1}$. Thus we have

(2.10) $\displaystyle\begin{split}K_{2}f&=2K_{2,1}f\\\
&=2\frac{m(p)}{\sqrt{m+\tau
m^{2}(p)}}\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(q)\frac{1+\tau
m(p^{\prime})}{\sqrt{m+\tau m^{2}(p^{\prime})}}(1+\tau
m(q^{\prime}))f(p^{\prime}).\end{split}$

Similarly, we observe that

(2.11) $\displaystyle\frac{1+\tau m(p)}{\sqrt{m+\tau
m^{2}(p)}}=e^{\frac{1}{2}(ap^{0}+c)}.$

Combining with (2.7), we have

(2.12) $\displaystyle\frac{m(p)}{\sqrt{m+\tau m^{2}(p)}}\frac{1+\tau
m(p^{\prime})}{\sqrt{m+\tau m^{2}(p^{\prime})}}=\frac{1+\tau
m(q)}{\sqrt{m+\tau m^{2}(q)}}\frac{m(q^{\prime})}{\sqrt{m+\tau
m^{2}(q^{\prime})}}.$

Substituting it in (2.10), we have

$\displaystyle K_{2}f$
$\displaystyle=2\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)\sqrt{m+\tau
m^{2}(q)}\sqrt{m+\tau m^{2}(q^{\prime})}f(p^{\prime}).$

This completes the derivation of the linear term $\nu f$, $K_{1}f$ and
$K_{2}f$.

Now we consider the nonlinear terms. Since the quantum collision operator is
not quad-linear (Recall that $Q(k+h,f,f,f)\neq Q(k,f,f,f)+Q(h,f,f,f)$), the
second-order nonlinear term is highly complicated to be represented. Thus, we
first observe one of the second-order nonlinear terms involving
$f(p^{\prime})$ and $f(q^{\prime})$:

$\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}\frac{v_{\o}\sigma(g,\theta)}{\sqrt{m+\tau
m^{2}(p)}}\\\ \times\big{[}\sqrt{m+\tau
m^{2}(p^{\prime})}f(p^{\prime})\sqrt{m+\tau
m^{2}(q^{\prime})}f(q^{\prime})(1+\tau m(p))(1+\tau m(q))\cr-\sqrt{m+\tau
m^{2}(p^{\prime})}f(p^{\prime})\sqrt{m+\tau
m^{2}(q^{\prime})}f(q^{\prime})m(p)m(q)\big{]}.$

There are 6 second-order nonlinear terms of the same kind similar to the term
above where the number 6 is coming from the number of choices for choosing 2
variables among the four variables $p$, $q$, $p^{\prime}$, and $q^{\prime}$.
We represent all of the second-order nonlinear terms ($6$ nonlinear terms) as
follows:

(2.13)
$\displaystyle\begin{split}\Gamma(f,h)&=\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw\frac{v_{\o}\sigma(g,\theta)}{\sqrt{m+\tau
m^{2}(p)}}\cr&\times\bigg{[}(m(p^{\prime})m(q^{\prime})-(1+\tau
m(p^{\prime}))(1+\tau m(q^{\prime})))\sqrt{m+\tau m^{2}(p)}f(p)\sqrt{m+\tau
m^{2}(q)}h(q)\cr&\quad+\tau(m(q^{\prime})(1+\tau m(q))-m(q)(1+\tau
m(q^{\prime})))\sqrt{m+\tau m^{2}(p)}f(p)\sqrt{m+\tau
m^{2}(p^{\prime})}h(p^{\prime})\cr&\quad+\tau(m(q^{\prime})(1+\tau
m(p))-m(p)(1+\tau m(q^{\prime})))\sqrt{m+\tau m^{2}(q)}f(q)\sqrt{m+\tau
m^{2}(p^{\prime})}h(p^{\prime})\cr&\quad+\tau(m(p^{\prime})(1+\tau
m(q))-m(q)(1+\tau m(p^{\prime})))\sqrt{m+\tau m^{2}(p)}f(p)\sqrt{m+\tau
m^{2}(q^{\prime})}h(q^{\prime})\cr&\quad+\tau(m(p^{\prime})(1+\tau
m(p))-m(p)(1+\tau m(p^{\prime})))\sqrt{m+\tau m^{2}(q)}f(q)\sqrt{m+\tau
m^{2}(q^{\prime})}h(q^{\prime})\cr&\quad+(1+\tau m(p))((1+\tau
m(q))-m(p)m(q))\sqrt{m+\tau m^{2}(p^{\prime})}f(p^{\prime})\sqrt{m+\tau
m^{2}(q^{\prime})}h(q^{\prime})\bigg{]}\cr&=\Gamma_{1}(f,h)+\Gamma_{2}(f,h)+\Gamma_{3}(f,h)+\Gamma_{4}(f,h)+\Gamma_{5}(f,h)+\Gamma_{6}(f,h).\end{split}$

We also denote as

$\displaystyle\Gamma(f)$ $\displaystyle=\sum_{1\leq i\leq 6}\Gamma_{i}(f,f),$

when $f=h$. Lastly, we consider the following third-order nonlinear terms
involving $f(p^{\prime})$, $f(q^{\prime})$ and $f(p)$:

$\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)\big{[}\tau\sqrt{m+\tau
m^{2}(p^{\prime})}f(p^{\prime})\sqrt{m+\tau
m^{2}(q^{\prime})}f(q^{\prime})f(p)(1+\tau m(q))\cr-\sqrt{m+\tau
m^{2}(p^{\prime})}f(p^{\prime})\sqrt{m+\tau
m^{2}(q^{\prime})}f(q^{\prime})f(p)m(q)\big{]}.$

Similarly, we represent all of the third-order nonlinear terms ($4$ nonlinear
terms) as

(2.14)
$\displaystyle\begin{split}T_{1}(f,h,\eta)&=-\tau\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)f(p)\sqrt{m+\tau
m^{2}(q)}h(q)\sqrt{m+\tau m^{2}(p^{\prime})}\eta(p^{\prime}),\cr
T_{2}(f,h,\eta)&=-\tau\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)f(p)\sqrt{m+\tau
m^{2}(q)}h(q)\sqrt{m+\tau m^{2}(q^{\prime})}\eta(q^{\prime}),\cr
T_{3}(f,h,\eta)&=\tau\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)f(p)\sqrt{m+\tau
m^{2}(p^{\prime})}h(p^{\prime})\sqrt{m+\tau
m^{2}(q^{\prime})}\eta(q^{\prime}),\cr
T_{4}(f,h,\eta)&=\tau\frac{1}{\sqrt{m+\tau
m^{2}(p)}}\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)\sqrt{m+\tau
m^{2}(q)}f(q)\cr&\quad\times\sqrt{m+\tau
m^{2}(p^{\prime})}h(p^{\prime})\sqrt{m+\tau
m^{2}(q^{\prime})}\eta(q^{\prime}).\end{split}$

Similarly, we define

$\displaystyle T(f)$ $\displaystyle=\sum_{1\leq i\leq 4}T_{i}(f,f,f).$

We can easily check that the fourth-order nonlinear term is cancelled. ∎

By the linearization proposition above and substituting $F=m+\sqrt{m+\tau
m^{2}}f$ in (1.13), we obtain the linearized equation for the relativistic
quantum Boltzmann model (1.13) as follows:

(2.15)
$\displaystyle\begin{split}\partial_{t}f+\hat{p}\cdot\nabla_{x}f+Lf&=\Gamma(f)+T(f),\cr
f(x,p,0)&=f_{0}(x,p).\end{split}$

where $f_{0}(x,p)=(F_{0}(x,p)-m)/\sqrt{m+\tau m^{2}}$. Then the conservation
laws (1.17) can be written as following form:

(2.16)
$\displaystyle\begin{split}\int_{\mathbb{T}^{3}}dx\int_{\mathbb{R}^{3}}dp~{}f(x,p,t)\sqrt{m+\tau
m^{2}}&=\int_{\mathbb{T}^{3}}dx\int_{\mathbb{R}^{3}}dp~{}f_{0}(x,p)\sqrt{m+\tau
m^{2}},\cr\int_{\mathbb{T}^{3}}dx\int_{\mathbb{R}^{3}}dp~{}f(x,p,t)p\sqrt{m+\tau
m^{2}}&=\int_{\mathbb{T}^{3}}dx\int_{\mathbb{R}^{3}}dp~{}f_{0}(x,p)p\sqrt{m+\tau
m^{2}},\cr\int_{\mathbb{T}^{3}}dx\int_{\mathbb{R}^{3}}dp~{}f(x,p,t)p^{0}\sqrt{m+\tau
m^{2}}&=\int_{\mathbb{T}^{3}}dx\int_{\mathbb{R}^{3}}dp~{}f_{0}(x,p)p^{0}\sqrt{m+\tau
m^{2}}.\end{split}$

Now we state several useful properties for the linear term $Lf$.

######  Lemma 2.1.

For any smooth function $\phi$, we have

$\displaystyle\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(p)m(q)(1+\tau
m(p^{\prime}))(1+\tau m(q^{\prime}))$
$\displaystyle\times\bigg{(}\frac{f(p)}{\sqrt{m+\tau
m^{2}(p)}}+\frac{f(q)}{\sqrt{m+\tau
m^{2}(q)}}-\frac{f(p^{\prime})}{\sqrt{m+\tau
m^{2}(p^{\prime})}}-\frac{f(q^{\prime})}{\sqrt{m+\tau
m^{2}(q^{\prime})}}\bigg{)}\phi(p)$
$\displaystyle=\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(p)m(q)(1+\tau
m(p^{\prime}))(1+\tau m(q^{\prime}))$
$\displaystyle\times\bigg{(}\frac{f(p)}{\sqrt{m+\tau
m^{2}(p)}}+\frac{f(q)}{\sqrt{m+\tau
m^{2}(q)}}-\frac{f(p^{\prime})}{\sqrt{m+\tau
m^{2}(p^{\prime})}}-\frac{f(q^{\prime})}{\sqrt{m+\tau
m^{2}(q^{\prime})}}\bigg{)}\phi(q)$
$\displaystyle=\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(p)m(q)(1+\tau
m(p^{\prime}))(1+\tau m(q^{\prime}))$
$\displaystyle\times\bigg{(}\frac{f(p)}{\sqrt{m+\tau
m^{2}(p)}}+\frac{f(q)}{\sqrt{m+\tau
m^{2}(q)}}-\frac{f(p^{\prime})}{\sqrt{m+\tau
m^{2}(p^{\prime})}}-\frac{f(q^{\prime})}{\sqrt{m+\tau
m^{2}(q^{\prime})}}\bigg{)}(-\phi(p^{\prime}))$
$\displaystyle=\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(p)m(q)(1+\tau
m(p^{\prime}))(1+\tau m(q^{\prime}))$
$\displaystyle\times\bigg{(}\frac{f(p)}{\sqrt{m+\tau
m^{2}(p)}}+\frac{f(q)}{\sqrt{m+\tau
m^{2}(q)}}-\frac{f(p^{\prime})}{\sqrt{m+\tau
m^{2}(p^{\prime})}}-\frac{f(q^{\prime})}{\sqrt{m+\tau
m^{2}(q^{\prime})}}\bigg{)}(-\phi(q^{\prime})).$

###### Proof.

Note that $m(p)m(q)(1+\tau m(p^{\prime}))(1+\tau m(q^{\prime}))$ is invariant
under the change of variables $(p,q)\leftrightarrow(p^{\prime},q^{\prime})$
from (2.4). Thus, by the change of variables $p\leftrightarrow q$ and
$(p,q)\leftrightarrow(p^{\prime},q^{\prime})$ introduced in [29], we have the
desired results. ∎

######  Lemma 2.2.

We have the following properties for the linear operator $L$:

  1. (1)

$L$ is a symmetric operator: $\langle Lf,g\rangle_{L^{2}_{p}}=\langle
f,Lg\rangle_{L^{2}_{p}}$.

  2. (2)

$Lf=0$ if and only if $f=Pf$.

where $Pf$ is defined as the orthonormal projection to the $L^{2}_{p}$ space
which is spanned by following $5$-dimensional basis:

$\displaystyle\left\\{\sqrt{m+\tau m^{2}},p_{1}\sqrt{m+\tau
m^{2}},p_{2}\sqrt{m+\tau m^{2}},p_{3}\sqrt{m+\tau m^{2}},p^{0}\sqrt{m+\tau
m^{2}}\right\\}.$

###### Proof.

(1) We take an inner product of $g$ with $Lf$ of (2.5) to have

$\displaystyle\int_{\mathbb{R}^{3}}dp~{}gLf$
$\displaystyle=\int_{\mathbb{R}^{3}}dp\frac{g(p)}{\sqrt{m+\tau
m^{2}(p)}}\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(p)m(q)(1+\tau
m(p^{\prime}))(1+\tau m(q^{\prime}))$
$\displaystyle\times\bigg{(}\frac{f(p)}{\sqrt{m+\tau
m^{2}(p)}}+\frac{f(q)}{\sqrt{m+\tau
m^{2}(q)}}-\frac{f(p^{\prime})}{\sqrt{m+\tau
m^{2}(p^{\prime})}}-\frac{f(q^{\prime})}{\sqrt{m+\tau
m^{2}(q^{\prime})}}\bigg{)}.$

By Lemma 2.1, we substitute

$\phi(p)=\frac{1}{\sqrt{m+\tau m^{2}(p)}}g(p)$

and obtain

$\displaystyle\int_{\mathbb{R}^{3}}dp~{}gLf$
$\displaystyle=\frac{1}{4}\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(p)m(q)(1+\tau
m(p^{\prime}))(1+\tau m(q^{\prime}))$
$\displaystyle\times\left(\frac{f(p)}{\sqrt{m+\tau
m^{2}(p)}}+\frac{f(q)}{\sqrt{m+\tau
m^{2}(q)}}-\frac{f(p^{\prime})}{\sqrt{m+\tau
m^{2}(p^{\prime})}}-\frac{f(q^{\prime})}{\sqrt{m+\tau
m^{2}(q^{\prime})}}\right)$ $\displaystyle\times\left(\frac{g(p)}{\sqrt{m+\tau
m^{2}(p)}}+\frac{g(q)}{\sqrt{m+\tau
m^{2}(q)}}-\frac{g(p^{\prime})}{\sqrt{m+\tau
m^{2}(p^{\prime})}}-\frac{g(q^{\prime})}{\sqrt{m+\tau
m^{2}(q^{\prime})}}\right).$

This proves the symmetricity of $L$.
(2) Once we substitute $g=f$ in the equation above, then we have

$\displaystyle\int_{\mathbb{R}^{3}}dp~{}fLf$
$\displaystyle=\frac{1}{4}\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(p)m(q)(1+\tau
m(p^{\prime}))(1+\tau m(q^{\prime}))$
$\displaystyle\times\bigg{(}\frac{f(p)}{\sqrt{m+\tau
m^{2}(p)}}+\frac{f(q)}{\sqrt{m+\tau
m^{2}(q)}}-\frac{f(p^{\prime})}{\sqrt{m+\tau
m^{2}(p^{\prime})}}-\frac{f(q^{\prime})}{\sqrt{m+\tau
m^{2}(q^{\prime})}}\bigg{)}^{2}$ $\displaystyle\geq 0.$

Therefore $Lf=0$ implies

$\displaystyle\frac{f(p)}{\sqrt{m+\tau m^{2}(p)}}+\frac{f(q)}{\sqrt{m+\tau
m^{2}(q)}}=\frac{f(p^{\prime})}{\sqrt{m+\tau
m^{2}(p^{\prime})}}+\frac{f(q^{\prime})}{\sqrt{m+\tau m^{2}(q^{\prime})}},$

which is satisfied if and only if $f=Pf$. Moreover, $L(Pf)=0$ gives the
desired results. ∎

##  3\. Estimates of the linear and the nonlinear terms

This section is devoted to proving several estimates of the linear and the
nonlinear terms. Especially, we emphasize that the estimates of the nonlinear
term $\Gamma_{2}$ involve the main difficulties as we mentioned in Section
1.8. Before we move onto it, we present some useful properties that are
commonly used throughout this paper.

######  Lemma 3.1.

There exist positive constants $C_{1}>0$ and $C_{2}>0$ such that

$\displaystyle C_{1}J(p^{0})\leq m(p)\leq C_{2}J(p^{0}).$

###### Proof.

In the fermionic case, $C_{1}=1/(1+e^{c})$ and $C_{2}=e^{-c}$ give the desired
estimates:

$\displaystyle\frac{1}{e^{c}+1}\frac{1}{e^{ap^{0}}}\leq\frac{1}{e^{ap^{0}+c}+1}\leq\frac{1}{e^{ap^{0}+c}}.$

In the case of bosons, we choose $C_{1}=e^{-c}$ and $C_{2}=1/(e^{c}-e^{-a})$.
Since $c>-a$, we have $C_{2}<\infty$, and we get

$\frac{1}{e^{ap^{0}+c}}\leq\frac{1}{e^{ap^{0}+c}-1}\leq\frac{1}{e^{c}-e^{-a}}\frac{1}{e^{ap^{0}}}.$

∎

By this Lemma, the relativistic quantum equilibrium $m(p)$ can be treated as
the non-quantum relativistic equilibrium $J(p^{0})$. Now we present the
estimates for collision frequency $\nu(p)$ and operator $K_{1}$ and $K_{2}$.

######  Lemma 3.2.

There exists a positive constant $C>0$ such that

$\displaystyle\frac{1}{C}(p^{0})^{\frac{1}{2}}\leq\nu(p)\leq
C(p^{0})^{\frac{1}{2}}.$

The integral operator $K_{i}(f)$ is a compact operator in $L^{2}_{p}$ for
$i=1,2$.

###### Proof.

By Lemma 3.1,

(3.1) $\displaystyle 1+\tau
m(p)=\frac{e^{ap^{0}+c}}{e^{ap^{0}+c}-\tau}=\frac{e^{c}m(p)}{J(p^{0})}\leq C.$

Then by (3.1), we have the upper bound of $\nu$ as

$\displaystyle\nu(p)$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)m(q).$

Similarly we have

(3.2) $\displaystyle\sqrt{m+\tau
m^{2}(p)}=\frac{e^{\frac{1}{2}(ap^{0}+c)}}{e^{ap^{0}+c}-\tau}\leq
Ce^{\frac{1}{2}ap^{0}}=CJ(p^{0}/2),$

which yields

$\displaystyle K_{1}f(p)$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(p^{\prime
0}/2)J(q^{\prime 0}/2)f(q),\text{and}$ $\displaystyle K_{2}f(p)$
$\displaystyle\leq
C\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{0}/2)J(q^{\prime
0}/2)f(p^{\prime}).$

We note that the upper bounds of $\nu$ and $K_{i}$ are identical to those of
non-quantum relativistic linear terms in [23]. Therefore we obtain the desired
results. ∎

######  Lemma 3.3.

The linear operator $L$ satisfies following dissipation property for some
positive constant $\delta$:

$\langle Lf,f\rangle_{L^{2}_{v}}\geq\delta\|(I-P)f\|_{\nu}^{2}.$

###### Proof.

Since $K_{1}$ and $K_{2}$ are compact and $\nu$ satisfies Lemma 3.1, we have
the desired results by following the same proof of Lemma 3.4 of [71]. ∎

###  3.1. Estimates of the second-order nonlinear terms

In this subsection, we establish the estimates on the second-order nonlinear
terms.

######  Lemma 3.4.

We have

$\displaystyle\big{|}\langle\Gamma(f,h),\eta\rangle_{L^{2}_{p}}\big{|}\leq
C\left(\|f\|_{L^{2}_{p}}\|h\|_{\nu}+\|f\|_{\nu}\|h\|_{L^{2}_{p}}\right)\|\eta\|_{\nu}.$

###### Proof.

As we can see in (2.13), there are six second-order nonlinear terms. By the
change of variables $p^{\prime}\leftrightarrow q^{\prime}$, we can see that
$\Gamma_{2}(f,h)=\Gamma_{4}(f,h)$ and $\Gamma_{3}(f,h)=\Gamma_{5}(f,h)$. Thus
we write the proof in three parts. Firstly, we prove the estimates of
$\Gamma_{1}(f,h)$ and $\Gamma_{6}(f,h)$, since they are similar to the non-
quantum relativistic case. Secondly, we present the estimate of
$\Gamma_{3}(f,h)$. Lastly, we prove the most difficult part $\Gamma_{2}(f,f)$.
For this we need several techniques to deal with the difficulties that arise
from the relativistic integrals.
(1) Estimates of $\Gamma_{1}$ and $\Gamma_{6}$: We first consider
$\Gamma_{1}(f,h)$ term. By (3.1), (3.2), and Lemma 3.1, we have

(3.3)
$\displaystyle\big{|}\langle\Gamma_{1}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{0}/2)|f(p)||h(q)||\eta(p)|.$

By the Hölder inequality, we have

$\displaystyle\big{|}\langle\Gamma_{1}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{S}^{2}}dw\left(\int_{\mathbb{R}^{3}}dq~{}v_{\o}^{2}\sigma^{2}(g,\theta)J(q^{0})\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{3}}dq|h(q)|^{2}\right)^{\frac{1}{2}}|f(p)||\eta(p)|.$

For the first $dq$ integral, we recall the definition of $v_{\o}$ and
$\sigma(g,\theta)$:

$\displaystyle
v_{\o}=\frac{g\sqrt{s}}{p^{0}q^{0}},\qquad\sigma(g,\theta)=g\sin\theta.$

By an explicit computation, we have

$\displaystyle s$ $\displaystyle=2+2p^{0}q^{0}-2p\cdot
q~{}\leq~{}2p^{0}q^{0}+2(1-\cos\theta)|p||q|~{}\leq~{}4p^{0}q^{0},$

and

$\displaystyle g=\sqrt{s-4}\leq\sqrt{s}\leq 2\sqrt{p^{0}q^{0}}.$

Applying above inequalities, we have

$\displaystyle\int_{\mathbb{R}^{3}}dq~{}v_{\o}^{2}\sigma^{2}(g,\theta)J(q^{0})\leq
Cp^{0}.$

We apply the Hölder inequality again to obtain the following estimate:

$\displaystyle\big{|}\langle\Gamma_{1}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\|h\|_{L^{2}_{p}}\int_{\mathbb{R}^{3}}dp\int_{\mathbb{S}^{2}}dw~{}(p^{0})^{\frac{1}{2}}|f(p)||\eta(p)|\leq
C\|f\|_{\nu}\|h\|_{L^{2}_{p}}\|\eta\|_{\nu}.$

Now we consider $\Gamma_{6}(f,h)$ term. We first use the energy conservation
law $p^{0}+q^{0}=p^{\prime 0}+q^{\prime 0}$ to have

$\displaystyle\sqrt{m+\tau m^{2}(p^{\prime})}\sqrt{m+\tau
m^{2}(q^{\prime})}\leq CJ(p^{\prime 0}/2)J(q^{\prime
0}/2)=CJ(p^{0}/2)J(q^{0}/2),$

which gives

(3.4)
$\displaystyle\big{|}\langle\Gamma_{6}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{0}/2)|f(p^{\prime})||h(q^{\prime})||\eta(p)|.$

Applying the Hölder inequality yields

$\displaystyle\big{|}\langle\Gamma_{6}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{S}^{2}}dw\left(\int_{\mathbb{R}^{3}}dq~{}v_{\o}\sigma^{2}(g,\theta)J(q^{0})\right)^{\frac{1}{2}}$
$\displaystyle\quad\times\left(\int_{\mathbb{R}^{3}}dq~{}v_{\o}|f(p^{\prime})|^{2}|h(q^{\prime})|^{2}\right)^{\frac{1}{2}}|\eta(p)|.$

Similarly we use $s\leq 4p^{0}q^{0}$ and $g\leq
2(p^{0})^{\frac{1}{2}}(q^{0})^{\frac{1}{2}}$ to have

$\displaystyle\big{|}\langle\Gamma_{6}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{S}^{2}}dw~{}(p^{0})^{\frac{1}{2}}\left(\int_{\mathbb{R}^{3}}dq~{}v_{\o}|f(p^{\prime})|^{2}|h(q^{\prime})|^{2}\right)^{\frac{1}{2}}|\eta(p)|.$

By the Hölder inequality for the $dpdw$ integral, we have

$\displaystyle\big{|}\langle\Gamma_{6}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\|\eta\|_{\nu}\left(\int_{\mathbb{R}^{3}}dp\int_{\mathbb{S}^{2}}dw\
(p^{0})^{\frac{1}{2}}\int_{\mathbb{R}^{3}}dqv_{\o}|f(p^{\prime})|^{2}|h(q^{\prime})|^{2}\right)^{\frac{1}{2}}.$

The energy conservation law implies

$\displaystyle(p^{0})^{1/2}\leq(p^{\prime 0}+q^{\prime 0})^{1/2}\leq(p^{\prime
0})^{1/2}+(q^{\prime 0})^{1/2},$

which yields

$\displaystyle\big{|}\langle\Gamma_{6}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq\|\eta\|_{\nu}\int_{\mathbb{S}^{2}}dw\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq~{}v_{\o}\left((p^{\prime
0})^{\frac{1}{2}}+(q^{\prime
0})^{\frac{1}{2}}\right)|f(p^{\prime})|^{2}|h(q^{\prime})|^{2}.$

Then we consider the pre-post change of variables
$(p,q)\leftrightarrow(p^{\prime},q^{\prime})$ with the Jacobian in [32]:

(3.5)
$\displaystyle\frac{dpdq}{p^{0}q^{0}}=\frac{dp^{\prime}dq^{\prime}}{p^{\prime
0}q^{\prime 0}},$

and recall from (1.7) that we already have

$\displaystyle s(p^{\mu},q^{\mu})=s(p^{\prime\mu},q^{\prime\mu}),\qquad
g(p^{\mu},q^{\mu})=g(p^{\prime\mu},q^{\prime\mu}).$

Thus we have

$\int_{\mathbb{S}^{2}}dw\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq~{}v_{\o}\left((p^{\prime
0})^{\frac{1}{2}}+(q^{\prime
0})^{\frac{1}{2}}\right)|f(p^{\prime})|^{2}|h(q^{\prime})|^{2}.\cr=\int_{\mathbb{S}^{2}}dw\int_{\mathbb{R}^{3}}dp^{\prime}\int_{\mathbb{R}^{3}}dq^{\prime}~{}\frac{g(p^{\prime\mu},q^{\prime\mu})\sqrt{s(p^{\prime\mu},q^{\prime\mu})}}{p^{\prime
0}q^{\prime 0}}\left((p^{\prime 0})^{\frac{1}{2}}+(q^{\prime
0})^{\frac{1}{2}}\right)|f(p^{\prime})|^{2}|h(q^{\prime})|^{2}.$

Finally by $v_{\o}(p^{\prime\mu},q^{\prime\mu})\leq C$ and the Hölder
inequality, we obtain the desired results:

$\displaystyle\big{|}\langle\Gamma_{6}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\left(\|f\|_{L^{2}_{p}}\|h\|_{\nu}+\|f\|_{\nu}\|h\|_{L^{2}_{p}}\right)\|\eta\|_{\nu}.$

(2) Estimates of $\Gamma_{3}$: We split the $\Gamma_{3}$ term into two parts:

$\displaystyle\Gamma_{3}(f,h)$ $\displaystyle=\frac{\tau}{\sqrt{m+\tau
m^{2}(p)}}\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)\\{m(q^{\prime})(1+\tau
m(p))-m(p)(1+\tau m(q^{\prime}))\\}$ $\displaystyle\quad\times\sqrt{m+\tau
m^{2}(q)}f(q)\sqrt{m+\tau m^{2}(p^{\prime})}h(p^{\prime})$
$\displaystyle=\Gamma_{3,1}(f,h)+\Gamma_{3,2}(f,h).$

By $1+\tau m(p)\leq C$ and $1+\tau m(q^{\prime})\leq C$, we obtain

$\displaystyle|\Gamma_{3,1}(f,h)|$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)\frac{J(q^{\prime
0})J(q^{0}/2)J(p^{\prime 0}/2)}{J(p^{0}/2)}|f(q)||h(p^{\prime})|$
$\displaystyle\leq
C\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{0})J(q^{\prime
0}/2)|f(q)||h(p^{\prime})|,$

and

$\displaystyle|\Gamma_{3,2}(f,h)|$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(p^{0}/2)J(q^{0}/2)J(p^{\prime
0}/2)|f(q)||h(p^{\prime})|,$

where we used $J(p^{\prime 0}/2)J(q^{\prime 0}/2)=J(p^{0}/2)J(q^{0}/2)$ for
the exponential term. In the case of $\Gamma_{3,2}$, there are decays of two
precollisional momentum variables $J(p^{0}/2)J(q^{0}/2)$. Thus we can have the
exponential decay with respect to all the pre-post momentum variables by the
energy conservation:

$\displaystyle J(p^{0}/2)J(q^{0}/2)=J(p^{0}/4)J(q^{0}/4)J(p^{\prime
0}/4)J(q^{\prime 0}/4).$

Thus, the estimate of $\Gamma_{3,2}$ can be absorbed in that of
$\Gamma_{3,1}$. Thus, without loss of generality, we only consider the
estimates of $\Gamma_{3,1}$:

$\displaystyle\big{|}\langle\Gamma_{3,1}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{0})J(q^{\prime
0}/2)|f(q)||h(p^{\prime})||\eta(p)|.$

To separate pre-collisional variable and post-collisional variable, we apply
the Hölder inequality as follows:

$\displaystyle\big{|}\langle\Gamma_{3,1}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\left(\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{0})J(q^{\prime
0}/2)|f(q)|^{2}|\eta(p)|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\times\left(\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{0})J(q^{\prime
0}/2)|h(p^{\prime})|^{2}\right)^{\frac{1}{2}}$ $\displaystyle=II_{1}\times
II_{2}.$

Using $s\leq 4p^{0}q^{0}$ and $g\leq
2(p^{0})^{\frac{1}{2}}(q^{0})^{\frac{1}{2}}$, we have

$\displaystyle(II_{1})^{2}$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}(p^{0})^{\frac{1}{2}}(q^{0})^{\frac{1}{2}}J(q^{0})J(q^{\prime
0}/2)|f(q)|^{2}|\eta(p)|^{2}.$

Since $(q^{0})^{\frac{1}{2}}J(q^{0}/2)$ and $J(q^{\prime 0}/2)$ are bounded by
constants, we obtain

$\displaystyle(II_{1})^{2}$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dq~{}|f(q)|^{2}\int_{\mathbb{R}^{3}}dp~{}(p^{0})^{\frac{1}{2}}|\eta(p)|^{2}\leq
C\|f\|_{L^{2}_{p}}^{2}\|\eta\|_{\nu}^{2}.$

For $II_{2}$, we apply the pre-post momentum change of variables
$(p,q)\leftrightarrow(p^{\prime},q^{\prime})$ similarly to the estimate of
$\Gamma_{6}(f,h)$. Note that $v_{\o}=\frac{g\sqrt{s}}{p^{0}q^{0}}$ is
invariant under the change of variables, and we obtain

$\displaystyle(II_{2})^{2}$
$\displaystyle=\int_{\mathbb{R}^{3}}dp^{\prime}\int_{\mathbb{R}^{3}}dq^{\prime}\int_{\mathbb{S}^{2}}dw~{}\frac{g(p^{\prime\mu},q^{\prime\mu})\sqrt{s(p^{\prime\mu},q^{\prime\mu})}}{p^{\prime
0}q^{\prime
0}}\sigma(g(p^{\prime\mu},q^{\prime\mu}),\theta)J(q^{0})J(q^{\prime
0}/2)|h(p^{\prime})|^{2}.$

Similarly, we use $g(p^{\prime\mu},q^{\prime\mu})\leq 2(p^{\prime
0})^{1/2}(q^{\prime 0})^{1/2}$ and $s(p^{\prime\mu},q^{\prime\mu})\leq
4p^{\prime 0}q^{\prime 0}$ to have

$\displaystyle(II_{2})^{2}$
$\displaystyle\leq\int_{\mathbb{R}^{3}}dp^{\prime}\int_{\mathbb{R}^{3}}dq^{\prime}\int_{\mathbb{S}^{2}}dw~{}(p^{\prime
0})^{\frac{1}{2}}(q^{\prime 0})^{\frac{1}{2}}J(q^{0})J(q^{\prime
0}/2)|h(p^{\prime})|^{2}$
$\displaystyle\leq\int_{\mathbb{S}^{2}}dw\int_{\mathbb{R}^{3}}dq^{\prime}~{}(q^{\prime
0})^{\frac{1}{2}}J(q^{\prime
0}/2)\int_{\mathbb{R}^{3}}dp^{\prime}~{}(p^{\prime
0})^{\frac{1}{2}}|h(p^{\prime})|^{2}$ $\displaystyle\leq C\|h\|_{\nu}^{2}.$

Combining the estimates of $II_{1}$ and $II_{2}$ yields

$\displaystyle\big{|}\langle\Gamma_{3,1}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq C\|f\|_{L^{2}_{p}}\|h\|_{\nu}\|\eta\|_{\nu}.$

(3) Estimates of $\Gamma_{2}$: We first separate $\Gamma_{2}$ by two terms:

$\displaystyle\Gamma_{2}(f,h)$
$\displaystyle=-\tau\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)\big{[}m(q)(1+\tau
m(q^{\prime}))-m(q^{\prime})(1+\tau m(q))\big{]}$
$\displaystyle\quad\times\sqrt{m+\tau m^{2}(p^{\prime})}f(p)h(p^{\prime})$
$\displaystyle=\Gamma_{2,1}(f,h)-\Gamma_{2,2}(f,h).$

Using $1+\tau m(p)\leq C$, we estimate each term as follows:

$\displaystyle\big{|}\Gamma_{2,1}(f,h)\big{|}$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{0})J(p^{\prime
0}/2)|f(p)||h(p^{\prime})|,$

and

$\displaystyle\big{|}\Gamma_{2,2}(f,h)\big{|}$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{\prime
0})J(p^{\prime 0}/2)|f(p)||h(p^{\prime})|.$

Since $\Gamma_{2,2}$ has the exponential decays with respect to two post-
collisional variables $p^{\prime}$ and $q^{\prime}$, we can have the decays
for all pre- and post-collisional momentum variables by using the energy
conservation:

$\displaystyle J(p^{0}/2)J(q^{0}/2)=J(p^{0}/4)J(q^{0}/4)J(p^{\prime
0}/4)J(q^{\prime 0}/4).$

Thus the estimate of $\Gamma_{2,2}$ can be absorbed into that of
$\Gamma_{2,1}$. Thus, we only consider $\Gamma_{2,1}$ term. However, different
from the previous cases, it is difficult to separate $f,h$ and $\eta$ using
the Hölder inequality. Thus, we proceed the estimate via lifting the $dw$
integral to $dp^{\prime}dq^{\prime}$ integral imposing a $4$-dimensional
Dirac-delta function:

(3.6) $\big{|}\langle\Gamma_{2,1}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}\leq
C\int_{\mathbb{R}^{3}}\frac{dp}{p^{0}}\int_{\mathbb{R}^{3}}\frac{dq}{q^{0}}\int_{\mathbb{R}^{3}}\frac{dp^{\prime}}{p^{\prime
0}}\int_{\mathbb{R}^{3}}\frac{dq^{\prime}}{q^{\prime
0}}s\sigma(g,\theta)\cr\times\delta^{(4)}(p^{\mu}+q^{\mu}-p^{\prime\mu}-q^{\prime\mu})J(q^{0})J(p^{\prime
0}/2)|f(p)||h(p^{\prime})||\eta(p)|.$

Then we focus on the following estimate:

(3.7) $\displaystyle B$
$\displaystyle=\int_{\mathbb{R}^{3}}\frac{dq}{q^{0}}\int_{\mathbb{R}^{3}}\frac{dq^{\prime}}{q^{\prime
0}}s\sigma(g,\theta)\delta^{(4)}(p^{\mu}+q^{\mu}-p^{\prime\mu}-q^{\prime\mu})J(q^{0}).$

We also lift the $dq$ and $dq^{\prime}$ integrals to the relativistic energy-
momentum $4$-vector integral $dq^{\mu}$ and $dq^{\prime\mu}$ imposing Dirac-
delta function and unit step function:

(3.8)
$B=\int_{\mathbb{R}^{4}}dq^{\mu}\int_{\mathbb{R}^{4}}dq^{\prime\mu}s\sigma(g,\theta)\delta^{(4)}(p^{\mu}+q^{\mu}-p^{\prime\mu}-q^{\prime\mu})J(q^{0})u(q^{0})u(q^{\prime
0})\delta(q^{\mu}q_{\mu}+1)\delta(q^{\prime\mu}q^{\prime}_{\mu}+1),$

where the unit step function $u(x)$ is defined by $1$ when $x\geq 0$ and $0$
when $x<0$. Note that $B$ has the integration over $dq^{\mu}$ and
$dq^{\prime\mu}$, but $s$ and $g$ are functions depending on $p^{\mu}$ and
$q^{\mu}$. Thus it is convenient to split the $q^{\mu}$ and the
$q^{\prime\mu}$ parts from $g$ as in the lemma below.

Before we proceed furtherly, we first provide a preliminary lemma on the
properties of $g,$ $\bar{g}$ and $\tilde{g}$.

######  Lemma 3.5 ([47, 44, 71]).

Define $g,$ $\bar{g}$ and $\tilde{g}$ as (1.7). Then we have

$\displaystyle(1)~{}g^{2}=\bar{g}^{2}+\tilde{g}^{2},$
$\displaystyle(2)~{}\bar{g}^{2}=-\frac{1}{2}(p^{\mu}+q^{\prime\mu})(q_{\mu}+p^{\prime}_{\mu}-p_{\mu}-q^{\prime}_{\mu}),$
$\displaystyle(3)~{}\tilde{g}^{2}=-\frac{1}{2}(p^{\mu}+p^{\prime\mu})(q_{\mu}+q^{\prime}_{\mu}-p_{\mu}-p^{\prime}_{\mu}).$

###### Proof.

The proofs can be found in [47, 71], but for the readers’ convenience, we
present the proof.
(1) By definition of $g$ in (1.7), we have

$\displaystyle-g^{2}+\bar{g}^{2}+\tilde{g}^{2}$
$\displaystyle=-2+2p^{\mu}q_{\mu}-2p^{\mu}p^{\prime}_{\mu}-2p^{\mu}q^{\prime}_{\mu}.$

Since $p^{\mu}$ is energy momentum $4$-vector, we have $p^{\mu}p_{\mu}=-1$,
which gives

$\displaystyle-g^{2}+\bar{g}^{2}+\tilde{g}^{2}$
$\displaystyle=2p^{\mu}p_{\mu}+2p^{\mu}q_{\mu}-2p^{\mu}p^{\prime}_{\mu}-2p^{\mu}q^{\prime}_{\mu}$
$\displaystyle=2p^{\mu}(p_{\mu}+q_{\mu}-p^{\prime}_{\mu}-q^{\prime}_{\mu}).$

Then the conservation law of energy momentum $4$-vector lead to the desired
results.
(2) We denote the right-hand side as $R$ and expand as follows:

$\displaystyle
R=-\frac{1}{2}(p^{\mu}q_{\mu}+p^{\mu}p^{\prime}_{\mu}+q^{\prime\mu}q_{\mu}+q^{\prime\mu}p^{\prime}_{\mu})+\frac{1}{2}(p^{\mu}p_{\mu}+2p^{\mu}q^{\prime}_{\mu}+q^{\prime\mu}q^{\prime}_{\mu}).$

We recall from (1.7) that $p^{\mu}q_{\mu}=p^{\prime\mu}q^{\prime}_{\mu}$ and
$p^{\mu}p^{\prime}_{\mu}=q^{\mu}q^{\prime}_{\mu}$ to have

$\displaystyle
R=-\frac{1}{2}(2p^{\mu}q_{\mu}+2p^{\mu}p^{\prime}_{\mu})+\frac{1}{2}(-2+2p^{\mu}q^{\prime}_{\mu})=-p^{\mu}q_{\mu}+p^{\mu}q^{\prime}_{\mu}-p^{\mu}p^{\prime}_{\mu}-1.$

Using $p^{\mu}p_{\mu}=-1$, we have

$\displaystyle R=p^{\mu}(-q_{\mu}+q^{\prime}_{\mu}-p^{\prime}_{\mu}+p_{\mu}).$

Then the conservation laws of energy momentum $4$-vector yields

$\displaystyle
R=2p^{\mu}(-p^{\prime}_{\mu}+p_{\mu})=-2p^{\mu}p^{\prime}_{\mu}-2.$

By definition of $\bar{g}$, we derived desired results.
(3) The proof can be obtained by the same way as that of (2). So we omit it. ∎

By Lemma 3.5 (1) and (3) above, we use the following representation of $g$:

$\displaystyle
g^{2}=\bar{g}^{2}-\frac{1}{2}(p^{\mu}+p^{\prime\mu})(q_{\mu}+q^{\prime}_{\mu}-p_{\mu}-p^{\prime}_{\mu}).$

Now we go back to the estimates of $B$ and apply the following change of
variables

(3.9)
$\displaystyle\bar{q}^{\mu}=q^{\mu}+q^{\prime\mu},\qquad\bar{q}^{\prime\mu}=q^{\mu}-q^{\prime\mu}.$

Then the reverse relation can be written by

$\displaystyle q^{\mu}=\frac{1}{2}(\bar{q}^{\mu}+\bar{q}^{\prime\mu}),\qquad
q^{\prime\mu}=\frac{1}{2}(\bar{q}^{\mu}-\bar{q}^{\prime\mu}),$

and the Jacobian is given by

$\displaystyle\frac{\partial(q^{\mu},q^{\prime\mu})}{\partial(\bar{q}^{\mu},\bar{q}^{\prime\mu})}=\frac{1}{16}.$

Then the $g_{c}$, $s_{c}$ and $\theta_{c}$ are now expressed as

(3.10) $\displaystyle
g_{c}^{2}=\bar{g}^{2}-\frac{1}{2}(p^{\mu}+p^{\prime\mu})(\bar{q}_{\mu}-p_{\mu}-p^{\prime}_{\mu}),\quad
s_{c}=g_{c}^{2}+4,\quad\cos\theta_{c}=1-2\frac{\bar{g}^{2}}{g_{c}^{2}}.$

To calculate the Dirac-delta function and the unit step function in (3.8), we
use followings. First we use $\delta(x)\delta(y)=2\delta(x+y)\delta(x-y)$ to
have

$\displaystyle\delta(q^{\mu}q_{\mu}+1)\delta(q^{\prime\mu}q^{\prime}_{\mu}+1)$
$\displaystyle=2\delta(q^{\mu}q_{\mu}+q^{\prime\mu}q^{\prime}_{\mu}+2)\delta(q^{\mu}q_{\mu}-q^{\prime\mu}q^{\prime}_{\mu})$
$\displaystyle=4\delta((\bar{q}^{\mu}\bar{q}_{\mu}+\bar{q}^{\prime\mu}\bar{q}^{\prime}_{\mu})+4)\delta(\bar{q}^{\mu}\bar{q}^{\prime}_{\mu}).$

Note that $q^{0}\geq 0$ and $q^{\prime 0}\geq 0$ are equivalent to
$q^{0}+q^{\prime 0}\geq 0$ and $q^{0}q^{\prime 0}\geq 0$. Also,
$q^{0}q^{\prime 0}\geq 0$ is equivalent to $\bar{g}^{2}=2q^{0}q^{\prime
0}-2q\cdot q^{\prime}-2\geq 0$ under the assumption that $q^{\mu}q_{\mu}+1=0$
and $q^{\prime\mu}q^{\prime}_{\mu}+1=0$. Thus we have

$\displaystyle u(q^{0})u(q^{\prime
0})\delta(q^{\mu}q_{\mu}+1)\delta(q^{\prime\mu}q^{\prime}_{\mu}+1)$
$\displaystyle=u(\bar{q}^{0})u(\bar{s}-4)4\delta((\bar{q}^{\mu}\bar{q}_{\mu}+\bar{q}^{\prime\mu}\bar{q}^{\prime}_{\mu})+4)\delta(\bar{q}^{\mu}\bar{q}^{\prime}_{\mu}).$

Thus we have

$\displaystyle B$
$\displaystyle=\frac{1}{4}\int_{\mathbb{R}^{4}\times\mathbb{R}^{4}}d\Theta(\bar{q}^{\mu},\bar{q}^{\prime\mu})s_{c}\sigma(g_{c},\theta_{c})\delta^{(4)}(p^{\mu}-p^{\prime\mu}+\bar{q}^{\prime\mu})J\left(\frac{\bar{q}^{0}+\bar{q}^{\prime
0}}{2}\right),$

where

$\displaystyle d\Theta(\bar{q}^{\mu},\bar{q}^{\prime\mu})$
$\displaystyle=d\bar{q}^{\mu}d\bar{q}^{\prime\mu}u(\bar{q}^{0})u(\bar{s}-4)\delta((\bar{q}^{\mu}\bar{q}_{\mu}+\bar{q}^{\prime\mu}\bar{q}^{\prime}_{\mu})+4)\delta(\bar{q}^{\mu}\bar{q}^{\prime}_{\mu}).$

Now we substitute $\bar{q}^{\prime\mu}=p^{\prime\mu}-p^{\mu}$ to reduce the
$4$-dimensional integration $d\bar{q}^{\prime\mu}$ by reducing the
$4$-dimensional Dirac-delta function as follows:

$\displaystyle B$
$\displaystyle=\frac{1}{4}\int_{\mathbb{R}^{4}}d\Theta(\bar{q}^{\mu})s_{c}\sigma(g_{c},\theta_{c})J\left(\frac{\bar{q}^{0}+p^{\prime
0}-p^{0}}{2}\right),$

where

$\displaystyle d\Theta(\bar{q}^{\mu})$
$\displaystyle=d\bar{q}^{\mu}u(\bar{q}^{0})u(\bar{s}-4)\delta(\bar{q}^{\mu}\bar{q}_{\mu}+(p^{\prime\mu}-p^{\mu})(p^{\prime}_{\mu}-p_{\mu})+4)\delta(\bar{q}^{\mu}(p^{\prime}_{\mu}-p_{\mu})).$

To remove one more Dirac-delta function, we follow that

$\displaystyle
u(\bar{q}^{0})\delta(\bar{q}^{\mu}\bar{q}_{\mu}+(p^{\prime\mu}-p^{\mu})(p^{\prime}_{\mu}-p_{\mu})+4)$
$\displaystyle=u(\bar{q}^{0})\delta(\bar{q}^{\mu}\bar{q}_{\mu}-2p^{\mu}p^{\prime}_{\mu}+2)$
$\displaystyle=u(\bar{q}^{0})\delta(-(\bar{q}^{0})^{2}+|\bar{q}|^{2}+\bar{s})$
$\displaystyle=\frac{\delta(\bar{q}^{0}-\sqrt{|\bar{q}|^{2}+\bar{s}})}{2\sqrt{|\bar{q}|^{2}+\bar{s}}}.$

Then the $d\bar{q}^{0}$ integral with the delta function above is reduced as
follows:

$\displaystyle B$ $\displaystyle=\frac{1}{4}J\left(\frac{p^{\prime
0}-p^{0}}{2}\right)\int_{\mathbb{R}^{3}}\frac{d\bar{q}}{\bar{q}^{0}}s_{c}\sigma(g_{c},\theta_{c})J\left(\frac{\bar{q}^{0}}{2}\right)u(\bar{s}-4)\delta(\bar{q}^{\mu}(p^{\prime}_{\mu}-p_{\mu})),$

where $\bar{q}^{0}$ is defined by $\sqrt{|\bar{q}|^{2}+\bar{s}}$. Since
$\bar{s}-4=\bar{g}^{2}\geq 0$, the last unit step function is always equal to
$1$. In order to make the final Dirac-delta function even simpler, we consider
the Lorentz transform satisfying

$\displaystyle\Lambda(p_{\mu}+p^{\prime}_{\mu})$
$\displaystyle=(\sqrt{\bar{s}},0,0,0),\qquad\Lambda(p_{\mu}-p^{\prime}_{\mu})=(0,0,0,-\bar{g}).$

We can see that in [71] that the Lorentz transform satisfying the relation
above is uniquely determined by following form:

(3.15) $\displaystyle\Lambda$
$\displaystyle=\left[{\begin{array}[]{cccc}\frac{p^{0}+p^{\prime
0}}{\sqrt{\bar{s}}}&-\frac{p_{1}+p^{\prime}_{1}}{\sqrt{\bar{s}}}&-\frac{p_{2}+p^{\prime}_{2}}{\sqrt{\bar{s}}}&-\frac{p_{3}+p^{\prime}_{3}}{\sqrt{\bar{s}}}\\\
\Lambda^{01}&\Lambda^{11}&\Lambda^{21}&\Lambda^{31}\\\ 0&\frac{(p\times
p^{\prime})_{1}}{|p\times p^{\prime}|}&\frac{(p\times
p^{\prime})_{2}}{|p\times p^{\prime}|}&\frac{(p\times
p^{\prime})_{3}}{|p\times p^{\prime}|}\\\ \frac{p^{0}-p^{\prime
0}}{\bar{g}}&-\frac{p_{1}-p^{\prime}_{1}}{\bar{g}}&-\frac{p_{2}-p^{\prime}_{2}}{\bar{g}}&-\frac{p_{3}-p^{\prime}_{3}}{\bar{g}}\end{array}}\right],$

where

$\displaystyle\Lambda^{i1}=\frac{2(p_{i}\\{p^{0}+p^{\prime
0}p^{\mu}p^{\prime}_{\mu}\\}+p^{\prime}_{i}\\{p^{\prime
0}+p^{0}p^{\mu}p^{\prime}_{\mu}\\})}{\bar{g}\sqrt{\bar{s}}|p\times
p^{\prime}|}.$

Then the exponential part can also be expressed carrying out the Lorentz
transform as follows:

$\displaystyle
J\left(\bar{q}^{0}/2\right)=\exp\left(-\bar{q}_{0}/2\right)=\exp\left(-\bar{q}^{\mu}U_{\mu}/2\right)=\exp\left(-\Lambda\bar{q}^{\mu}\Lambda
U_{\mu}/2\right),$

where we used following simple four-vectors:

$\displaystyle U^{\mu}=(1,0,0,0),\qquad U_{\mu}=(-1,0,0,0).$

Applying this Lorentz transform yields

$\displaystyle B$ $\displaystyle=\frac{1}{4}J\left(\frac{p^{\prime
0}-p^{0}}{2}\right)\int_{\mathbb{R}^{3}}\frac{d\bar{q}}{\bar{q}^{0}}s_{\Lambda}\sigma(g_{\Lambda},\theta_{\Lambda})\exp\left(-\Lambda\bar{q}^{\mu}\Lambda
U_{\mu}/2\right)\delta(\Lambda\bar{q}^{\mu}\Lambda(p^{\prime}_{\mu}-p_{\mu})),$

where $g_{\Lambda}$ is written via the Lorentz transform:

$\displaystyle g_{\Lambda}^{2}$
$\displaystyle=\bar{g}^{2}-\frac{1}{2}\Lambda(p^{\mu}+p^{\prime\mu})\Lambda(\bar{q}_{\mu}-p_{\mu}-p^{\prime}_{\mu})$
$\displaystyle=\bar{g}^{2}+\frac{1}{2}(\sqrt{\bar{s}},0,0,0)(\Lambda\bar{q}_{\mu}-(\sqrt{\bar{s}},0,0,0)).$

We can also represent $s_{\Lambda}$ and $\cos\theta_{\Lambda}$ in terms of
$\bar{g}$ and $g_{\Lambda}$ as

$\displaystyle
s_{\Lambda}=g_{\Lambda}^{2}+4\quad\text{and}\quad\cos\theta_{\Lambda}=1-2\frac{\bar{g}^{2}}{g_{\Lambda}^{2}}.$

We now apply the change of variables $\Lambda\bar{q}^{\mu}=\bar{q}^{\mu}$.
Since $d\bar{q}/\bar{q}^{0}$ is Lorentz invariant, we have

$\displaystyle B$ $\displaystyle=\frac{1}{4}J\left(\frac{p^{\prime
0}-p^{0}}{2}\right)\int_{\mathbb{R}^{3}}\frac{d\bar{q}}{\bar{q}^{0}}s_{\lambda}\sigma(g_{\lambda},\theta_{\lambda})\exp\left(-\bar{q}^{\mu}\Lambda
U_{\mu}/2\right)\delta(\bar{q}^{3}\bar{g}),$

where

(3.16) $\displaystyle g_{\lambda}^{2}$
$\displaystyle=\bar{g}^{2}+\frac{1}{2}\sqrt{\bar{s}}(\bar{q}^{0}-\sqrt{\bar{s}}),\quad
s_{\lambda}=g_{\lambda}^{2}+4,\quad\cos\theta_{\lambda}=1-2\frac{\bar{g}^{2}}{g_{\lambda}^{2}}.$

Now we use the spherical coordinates for the variable $\bar{q}$:

$\displaystyle\bar{q}=|\bar{q}|(\sin\psi\cos\phi,\sin\psi\sin\phi,\cos\psi).$

Then $B$ is equal to

$\displaystyle\frac{1}{4}J\left(\frac{p^{\prime
0}-p^{0}}{2}\right)\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\psi\sin\psi\int_{0}^{\infty}\frac{|\bar{q}|^{2}d|\bar{q}|}{\bar{q}^{0}}s_{\lambda}\sigma(g_{\lambda},\theta_{\lambda})\exp\left(-\bar{q}^{\mu}\Lambda
U_{\mu}/2\right)\delta(|\bar{q}|\cos\psi\bar{g}).$

Using $\delta(ax)=(1/a)\delta(x)$ and substituting $\psi=\pi/2$ reduce the
$d\psi$ integral with $\delta(\cos\psi)$ as follows:

(3.17) $\displaystyle B$ $\displaystyle=\frac{1}{4}J\left(\frac{p^{\prime
0}-p^{0}}{2}\right)\int_{0}^{2\pi}d\phi\int_{0}^{\infty}\frac{|\bar{q}|d|\bar{q}|}{\bar{g}\bar{q}^{0}}s_{\lambda}\sigma(g_{\lambda},\theta_{\lambda})\exp\left(-\bar{q}^{\mu}\Lambda
U_{\mu}/2\right)|_{\psi=\pi/2}.$

We first consider the scattering kernel
$s_{\lambda}\sigma(g_{\lambda},\theta_{\lambda})$. The half-angle formula from
(3.16) gives

$\displaystyle\cos\theta_{\lambda}=1-2\sin^{2}\frac{\theta_{\lambda}}{2}=1-2\frac{\bar{g}^{2}}{g_{\lambda}^{2}},$

which implies

(3.18)
$\displaystyle\sin\frac{\theta_{\lambda}}{2}=\frac{\bar{g}}{g_{\lambda}},$

for $0\leq\theta_{\lambda}\leq 2\pi$. Thus we have

$\displaystyle\sigma(g_{\lambda},\theta_{\lambda})=g_{\lambda}\sin\theta_{\lambda}=2g_{\lambda}\sin\frac{\theta_{\lambda}}{2}\cos\frac{\theta_{\lambda}}{2}=2\bar{g}\cos\frac{\theta_{\lambda}}{2}\leq
2\bar{g}.$

From (3.16) and the definition of $\bar{q}^{0}$, we have

$\displaystyle
s_{\lambda}=g_{\lambda}^{2}+4=\bar{g}^{2}+4+\frac{1}{2}\sqrt{\bar{s}}(\bar{q}^{0}-\sqrt{\bar{s}})=\bar{s}+\frac{1}{2}\sqrt{\bar{s}}(\sqrt{|\bar{q}|^{2}+\bar{s}}-\sqrt{\bar{s}}).$

Then we consider the exponential part of (3.17). Using the Lorentz transform
(3.15), we have

$\displaystyle\Lambda U_{\mu}=\left(\frac{p^{0}+p^{\prime
0}}{\sqrt{\bar{s}}},\frac{2|p\times
p^{\prime}|}{\bar{g}\sqrt{\bar{s}}},0,\frac{p^{0}-p^{\prime
0}}{\bar{g}}\right).$

Combining with the spherically expression of $\bar{q}$, we calculate

$\displaystyle\bar{q}^{\mu}\Lambda U_{\mu}|_{\psi=\pi/2}$
$\displaystyle=(\sqrt{|\bar{q}|^{2}+\bar{s}},|\bar{q}|\cos\phi,|\bar{q}|\sin\phi,0)\left(\frac{p^{0}+p^{\prime
0}}{\sqrt{\bar{s}}},\frac{2|p\times
p^{\prime}|}{\bar{g}\sqrt{\bar{s}}},0,\frac{p^{0}-p^{\prime
0}}{\bar{g}}\right)$
$\displaystyle=-\sqrt{|\bar{q}|^{2}+\bar{s}}\frac{p^{0}+p^{\prime
0}}{\sqrt{\bar{s}}}+|\bar{q}|\cos\phi\frac{2|p\times
p^{\prime}|}{\bar{g}\sqrt{\bar{s}}}.$

Thus, (3.17) is bounded by

$\displaystyle B$ $\displaystyle\leq\frac{1}{2}J\left(\frac{p^{\prime
0}-p^{0}}{2}\right)\int_{0}^{2\pi}d\phi\int_{0}^{\infty}\frac{|\bar{q}|d|\bar{q}|}{\sqrt{|\bar{q}|^{2}+\bar{s}}}s_{\lambda}\exp\left(-\sqrt{|\bar{q}|^{2}+\bar{s}}\frac{p^{0}+p^{\prime
0}}{2\sqrt{\bar{s}}}+|\bar{q}|\cos\phi\frac{|p\times
p^{\prime}|}{\bar{g}\sqrt{\bar{s}}}\right).$

We apply the change of variables $|\bar{q}|=\sqrt{\bar{s}}y$ to have

$\displaystyle B$ $\displaystyle\leq\frac{1}{2}J\left(\frac{p^{\prime
0}-p^{0}}{2}\right)\int_{0}^{2\pi}d\phi\int_{0}^{\infty}\frac{\sqrt{\bar{s}}ydy}{\sqrt{y^{2}+1}}s_{\lambda}\exp\left(-\sqrt{y^{2}+1}\frac{p^{0}+p^{\prime
0}}{2}+y\cos\phi\frac{|p\times p^{\prime}|}{\bar{g}}\right),$

where

$\displaystyle
s_{\lambda}=\bar{s}+\frac{1}{2}\sqrt{\bar{s}}(\sqrt{|\bar{q}|^{2}+\bar{s}}-\sqrt{\bar{s}})=\frac{\bar{s}}{2}(1+\sqrt{y^{2}+1}).$

So we have

$B\leq\frac{\bar{s}^{3/2}}{2}J\left(\frac{p^{\prime
0}-p^{0}}{2}\right)\int_{0}^{\infty}\frac{ydy}{\sqrt{y^{2}+1}}\frac{1+\sqrt{y^{2}+1}}{2}\cr\times\exp\left(-\sqrt{y^{2}+1}\frac{p^{0}+p^{\prime
0}}{2}\right)\int_{0}^{2\pi}d\phi\exp\left(y\cos\phi\frac{|p\times
p^{\prime}|}{\bar{g}}\right),$

For the notational simplicity, we denote

(3.19) $\displaystyle R=\frac{p^{0}+p^{\prime 0}}{2},\qquad r=\frac{|p\times
p^{\prime}|}{\bar{g}},$

then we can write

$\displaystyle B\leq\frac{\bar{s}^{3/2}\pi}{2}J\left(\frac{p^{\prime
0}-p^{0}}{2}\right)\int_{0}^{\infty}dy\left(\frac{y}{\sqrt{y^{2}+1}}+y\right)\exp\left(-R\sqrt{y^{2}+1}\right)I_{0}(ry),$

where $I_{0}$ denotes the modified Bessel function of the first kind:

$\displaystyle I_{0}(y)=\frac{1}{\pi}\int_{0}^{\pi}e^{y\cos\phi}d\phi.$

The formula including above Bessel function can be found in [35] and [30] when
$R>r\geq 0$:

(3.20)
$\displaystyle\begin{split}I&=\int_{0}^{\infty}\frac{y}{\sqrt{y^{2}+1}}e^{-R\sqrt{y^{2}+1}}I_{0}\left(ry\right)dy=\frac{e^{-\sqrt{R^{2}-r^{2}}}}{\sqrt{R^{2}-r^{2}}},\cr
II&=\int_{0}^{\infty}ye^{-R\sqrt{y^{2}+1}}I_{0}\left(ry\right)dy=\frac{R}{R^{2}-r^{2}}\left(1+\frac{1}{\sqrt{R^{2}-r^{2}}}\right)e^{-\sqrt{R^{2}-r^{2}}}.\end{split}$

Therefore, we have

(3.21) $\displaystyle B$
$\displaystyle\leq\frac{\bar{s}^{3/2}\pi}{2}J\left(\frac{p^{\prime
0}-p^{0}}{2}\right)\left[\frac{1}{\sqrt{R^{2}-r^{2}}}+\frac{R}{R^{2}-r^{2}}+\frac{R}{(R^{2}-r^{2})^{3/2}}\right]e^{-\sqrt{R^{2}-r^{2}}}.$

We can also find the useful estimates of $R^{2}-r^{2}$ in [30] as

(3.22) $\displaystyle
R^{2}-r^{2}=|p-p^{\prime}|^{2}\frac{\bar{g}^{2}+4}{4\bar{g}^{2}}\geq\max\left\\{\frac{\bar{g}^{2}}{4}+1,\frac{1}{4}|p-p^{\prime}|^{2}\right\\},$

which lead to

$\displaystyle J\left(\frac{p^{\prime
0}-p^{0}}{2}\right)e^{-\sqrt{R^{2}-r^{2}}}\leq e^{-\frac{1}{2}(p^{\prime
0}-p^{0})}e^{-\frac{1}{2}|p-p^{\prime}|}\leq 1.$

In the last inequality, we used

$\displaystyle|p^{\prime
0}-p^{0}|=\big{|}\sqrt{1+|p^{\prime}|^{2}}-\sqrt{1+|p|^{2}}\big{|}=\frac{|(|p^{\prime}|-|p|)(|p^{\prime}|+|p|)|}{\sqrt{1+|p^{\prime}|^{2}}+\sqrt{1+|p|^{2}}}\leq|p^{\prime}-p|.$

The estimates (3.22) also implies that

$\displaystyle\sqrt{R^{2}-r^{2}}$ $\displaystyle\geq
1,\qquad\frac{1}{R^{2}-r^{2}}\leq\frac{4}{\bar{s}}.$

Thus the three terms inside the large bracket of (3.21) are bounded as
follows:

$\displaystyle\frac{1}{\sqrt{R^{2}-r^{2}}}+\frac{R}{(R^{2}-r^{2})^{3/2}}\leq\frac{2R}{R^{2}-r^{2}}\leq
4\frac{p^{0}+p^{\prime 0}}{\bar{s}}.$

Combining the estimates above yields

(3.23) $\displaystyle B$ $\displaystyle\leq C\bar{s}^{1/2}(p^{0}+p^{\prime
0}).$

Now substituting it in (3.6), we turn back to estimate of $\Gamma_{2,1}$ as

$\displaystyle\big{|}\langle\Gamma_{2,1}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\int_{\mathbb{R}^{3}}\frac{dp}{p^{0}}\int_{\mathbb{R}^{3}}\frac{dp^{\prime}}{p^{\prime
0}}\bar{s}^{\frac{1}{2}}(p^{0}+p^{\prime 0})J(p^{\prime
0}/2)|f(p)||h(p^{\prime})||\eta(p)|.$

We use $\bar{s}\leq 4p^{0}p^{\prime 0}$ to have

$\displaystyle\big{|}\langle\Gamma_{2,1}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dp^{\prime}\left(\sqrt{\frac{p^{0}}{p^{\prime
0}}}+\sqrt{\frac{p^{\prime 0}}{p^{0}}}\right)J(p^{\prime
0}/2)|f(p)||h(p^{\prime})||\eta(p)|.$

Using the boundedness $1\leq p^{0}$ and $\sqrt{p^{\prime 0}}J(p^{\prime
0}/4)\leq C$ yields

$\displaystyle\big{|}\langle\Gamma_{2,1}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dp^{\prime}(p^{0})^{\frac{1}{2}}J(p^{\prime
0}/4)|f(p)||h(p^{\prime})||\eta(p)|.$

Finally, we apply the Hölder inequality and obtain

$\displaystyle\big{|}\langle\Gamma_{2,1}(f,h),\eta\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\|h\|_{L^{2}_{p}}\int_{\mathbb{R}^{3}}dp~{}(p^{0})^{\frac{1}{2}}|f(p)||\eta(p)|$
$\displaystyle\leq C\|f\|_{\nu}\|h\|_{L^{2}_{p}}\|\eta\|_{\nu}.$

∎

Regarding the linear terms $K_{1}f$ and $K_{2}f$ in (2.2), we have the
following estimates.

######  Lemma 3.6.

For $i=1,2$, the compact operator $K_{i}$ satisfies the following estimate.

$\displaystyle\langle K_{i}f,h\rangle_{L^{2}_{p}}$ $\displaystyle\leq
C\|f\|_{\nu}\|h\|_{\nu}.$

###### Proof.

Recall the definition of $K_{1}$ in (2.2). Using the energy conservation law,
we can replace post-collisional variables by pre-collisional variables on the
exponential term:

$\displaystyle K_{1}f$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(p^{0}/2)J(q^{0}/2)|f(q)|.$

Then by $v_{\o}\leq 1$ and $g\leq(p^{0})^{1/2}(q^{0})^{1/2}$ and the Hölder
inequality, we have

$\displaystyle\big{|}\langle K_{1}f,h\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq C\|f\|_{L^{2}_{p}}\|h\|_{L^{2}_{p}}.$

For $K_{2}$, by the simple boundedness $\sqrt{m+\tau m^{2}(p)}\leq
CJ(p^{0}/2)$, we have

$\displaystyle K_{2}f$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{0}/2)J(q^{\prime
0}/2)|f(p^{\prime})|.$

Then we take the inner product with $h$ and write it as

$\displaystyle\langle K_{2}f,h\rangle_{L^{2}_{p}}$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{0}/2)J(q^{\prime
0}/2)|f(p^{\prime})||h(p)|.$

If we substitute $h(q^{\prime})=J(q^{\prime 0}/2)$ in $\Gamma_{6}$ in (3.4),
then we get

$\displaystyle\big{|}\langle\Gamma_{6}(f,J),h\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw\
v_{\o}\sigma(g,\theta)J(q^{0}/2)J(q^{\prime 0}/2)|f(p^{\prime})||h(p)|.$

Thus, the estimates of $\Gamma_{6}(f,h)$ gives

$\displaystyle\big{|}\langle K_{2}f,h\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq C\|f\|_{\nu}\|h\|_{\nu}.$

∎

###  3.2. Estimates of the third-order nonlinear terms

In this subsection, we provide the upper-bound estimates for the third-order
nonlinear terms.

######  Lemma 3.7.

We have

$\displaystyle\big{|}\langle T(f,h,\eta),\xi\rangle_{L^{2}_{p}}\big{|}\leq
C\left(\|f\|_{L^{2}_{p}}\|h\|_{L^{2}_{p}}\|\eta\|_{\nu}+\|f\|_{L^{2}_{p}}\|h\|_{\nu}\|\eta\|_{L^{2}_{p}}+\|f\|_{\nu}\|h\|_{L^{2}_{p}}\|\eta\|_{L^{2}_{p}}\right)\|\xi\|_{\nu}.$

###### Proof.

Recall the definition of third-order nonlinear terms in (2.14). Applying the
change of variables $\phi\rightarrow\pi-\phi$ and
$\theta\rightarrow\pi+\theta$, we have $T_{1}(f,h,\eta)=T_{2}(f,h,\eta)$. Thus
we only consider the proof for $T_{1}$,$T_{3}$ and $T_{4}$.
(1) Estimates of $T_{1}$:  Using the boundedness $\sqrt{m+\tau m^{2}(p)}\leq
CJ(p^{0}/2)$, we have

$\displaystyle|T_{1}(f,h,\eta)|$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(p^{\prime
0}/2)J(q^{0}/2)|f(p)||h(q)||\eta(p^{\prime})|.$

We take the inner product with $\xi$ to have

$\big{|}\langle T_{1}(f,h,\eta),\xi\rangle_{L^{2}_{p}}\big{|}\cr\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(p^{\prime
0}/2)J(q^{0}/2)|f(p)||h(q)||\eta(p^{\prime})||\xi(p)|.$

Using the Hölder inequality, we disunite the integrand as follows:

$\displaystyle\big{|}\langle T_{1}(f,h,\eta),\xi\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\left(\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(p^{\prime
0}/2)J(q^{0}/2)|f(p)|^{2}|h(q)|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\times\left(\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(p^{\prime
0}/2)J(q^{0}/2)|\eta(p^{\prime})|^{2}|\xi(p)|^{2}\right)^{\frac{1}{2}}$
$\displaystyle=T_{11}\times T_{12}.$

We first consider $T_{11}$. Using $v_{\o}\leq C$ and $g\leq
2\sqrt{p^{0}q^{0}}$, we have

$\displaystyle(T_{11})^{2}$
$\displaystyle\leq\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}(p^{0})^{\frac{1}{2}}(q^{0})^{\frac{1}{2}}J(p^{\prime
0}/2)J(q^{0}/2)|h(q)|^{2}|f(p)|^{2}$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dq|h(q)|^{2}\int_{\mathbb{R}^{3}}dp(p^{0})^{\frac{1}{2}}|f(p)|^{2}$
$\displaystyle\leq C\|f\|_{\nu}^{2}\|h\|_{L^{2}_{p}}^{2},$

where we used $(q^{0})^{1/2}J(q^{0}/2)\leq C$. To estimate $T_{12}$, we
rewrite $(T_{12})^{2}$ as in the form of (3.6):

$(T_{12})^{2}=\int_{\mathbb{R}^{3}}\frac{dp}{p^{0}}\int_{\mathbb{R}^{3}}\frac{dq}{q^{0}}\int_{\mathbb{R}^{3}}\frac{dp^{\prime}}{p^{\prime
0}}\int_{\mathbb{R}^{3}}\frac{dq^{\prime}}{q^{\prime
0}}s\sigma(g,\theta)J(p^{\prime 0}/2)J(q^{0}/2)\\\
\times|\eta(p^{\prime})|^{2}|\xi(p)|^{2}\delta^{(4)}(p^{\mu}+q^{\mu}-q^{\prime\mu}-q^{\prime\mu}).$

Then we can find that the $dqdq^{\prime}$ integral has a similar form as of
(3.7):

$\displaystyle B^{\prime}$
$\displaystyle=\int_{\mathbb{R}^{3}}\frac{dq}{q^{0}}\int_{\mathbb{R}^{3}}\frac{dq^{\prime}}{q^{\prime
0}}s\sigma(g,\theta)J(q^{0}/2)\delta^{(4)}(p^{\mu}+q^{\mu}-q^{\prime\mu}-q^{\prime\mu}).$

The only difference is that $J(q^{0})$ is modified by $J(q^{0}/2)$. Thus the
exponential growth $J((p^{\prime 0}-p^{0})/2)$ in (3.21) is changed by
$J((p^{\prime 0}-p^{0})/4)$. But since the $R$ and $r$ in (3.19) are also
changed by $R/2$ and $r/2$, we can have

$\displaystyle J\left(\frac{p^{\prime
0}-p^{0}}{4}\right)e^{-\sqrt{R^{2}-r^{2}}/2}\leq e^{-\frac{1}{4}(p^{\prime
0}-p^{0})}e^{-\frac{1}{4}|p-p^{\prime}|}\leq 1.$

Thus we can apply the estimate in (3.23):

$\displaystyle B$ $\displaystyle\leq C\bar{s}^{1/2}(p^{0}+p^{\prime 0}),$

which gives

(3.24) $\displaystyle(T_{12})^{2}\leq
C\int_{\mathbb{R}^{3}}\frac{dp}{p^{0}}\int_{\mathbb{R}^{3}}\frac{dp^{\prime}}{p^{\prime
0}}\bar{s}^{1/2}(p^{0}+p^{\prime 0})J(p^{\prime
0}/2)|\eta(p^{\prime})|^{2}|\xi(p)|^{2}.$

Using $\bar{s}\leq Cp^{0}p^{\prime 0}$ and $(p^{\prime 0})^{1/2}J(p^{\prime
0}/2)\leq C$, we have

$\displaystyle(T_{12})^{2}\leq
C\int_{\mathbb{R}^{3}}dp(p^{0})^{1/2}|\xi(p)|^{2}\int_{\mathbb{R}^{3}}dp^{\prime}|\eta(p^{\prime})|^{2}\leq
C\|\eta\|_{L^{2}_{p}}^{2}\|\xi\|_{\nu}^{2}.$

Combining the estimates of $T_{11}$ and $T_{12}$ yields

$\displaystyle\big{|}\langle T_{1}(f,h,\eta),\xi\rangle_{L^{2}_{p}}\big{|}\leq
C\|f\|_{\nu}\|h\|_{L^{2}_{p}}\|\eta\|_{L^{2}_{p}}\|\xi\|_{\nu}.$

(2) Estimates of $T_{3}$:  Using $\sqrt{m+\tau m^{2}(p)}\leq J(p^{0}/2)$, we
estimate $T_{3}$ as follows:

$\displaystyle|T_{3}(f,h,\eta)|$
$\displaystyle\leq\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(p^{\prime
0}/2)J(q^{\prime 0}/2)|f(p)||h(p^{\prime})||\eta(q^{\prime})|.$

Via the Hölder inequality, we can split the variable as follows:

$\displaystyle\big{|}\langle T_{3}(f,h,\eta),\xi\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\left(\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(p^{\prime
0}/2)J(q^{\prime 0}/2)|f(p)|^{2}|h(p^{\prime})|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\times\left(\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(p^{\prime
0}/2)J(q^{\prime 0}/2)|\xi(p)|^{2}|\eta(q^{\prime})|^{2}\right)^{\frac{1}{2}}$
$\displaystyle=T_{31}\times T_{32}.$

Via the change of variables $p^{\prime}\leftrightarrow q^{\prime}$, we can see
that $T_{31}$ and $T_{32}$ are indeed identical. Thus we only consider
$T_{31}$ part here. Similarly to the estimates of $\Gamma_{2}$, we rewrite
$(T_{31})^{2}$ as follows:

$\int_{\mathbb{R}^{3}}\frac{dp}{p^{0}}\int_{\mathbb{R}^{3}}\frac{dq}{q^{0}}\int_{\mathbb{R}^{3}}\frac{dp^{\prime}}{p^{\prime
0}}\int_{\mathbb{R}^{3}}\frac{dq^{\prime}}{q^{\prime
0}}s\sigma(g,\theta)J(p^{\prime 0}/2)J(q^{\prime
0}/2)|f(p)|^{2}|h(p^{\prime})|^{2}\delta^{(4)}(p^{\mu}+q^{\mu}-q^{\prime\mu}-q^{\prime\mu}).$

Then we note that the energy conservation law and the exponential decays with
respect to the two post-collisional momenta can together imply the decays with
respect to all pre-post collisional variables:

$\displaystyle J(p^{\prime 0}/2)J(q^{\prime
0}/2)=J(p^{0}/4)J(q^{0}/4)J(p^{\prime 0}/4)J(q^{\prime 0}/4).$

Then, we use the boundedness of $B$ in (3.23) as

$\displaystyle B$
$\displaystyle=\int_{\mathbb{R}^{3}}\frac{dq}{q^{0}}\int_{\mathbb{R}^{3}}\frac{dq^{\prime}}{q^{\prime
0}}s\sigma(g,\theta)J(q^{0}/4)\delta^{(4)}(p^{\mu}+q^{\mu}-q^{\prime\mu}-q^{\prime\mu})\leq
C\bar{s}^{1/2}(p^{0}+p^{\prime 0}),$

which yields

$\displaystyle(T_{31})^{2}\leq
C\int_{\mathbb{R}^{3}}\frac{dp}{p^{0}}\int_{\mathbb{R}^{3}}\frac{dp^{\prime}}{p^{\prime
0}}\bar{s}^{1/2}(p^{0}+p^{\prime 0})J(p^{0}/4)J(p^{\prime
0}/4)|f(p)|^{2}|h(p^{\prime})|^{2}.$

Using $\bar{s}\leq Cp^{0}p^{\prime 0}$ and $(p^{0})^{1/2}J(p^{0}/4)\leq C$, we
have

$\displaystyle(T_{31})^{2}\leq
C\int_{\mathbb{R}^{3}}dp|f(p)|^{2}\int_{\mathbb{R}^{3}}dp^{\prime}|h(p^{\prime})|^{2}\leq
C\|f\|_{L^{2}_{p}}^{2}\|h\|_{L^{2}_{p}}^{2}.$

Since $T_{32}$ has the same form with $T_{31}$, we have

$\displaystyle\big{|}\langle T_{3}(f,h,\eta),\xi\rangle_{L^{2}_{p}}\big{|}\leq
C\|f\|_{L^{2}_{p}}\|h\|_{L^{2}_{p}}\|\eta\|_{L^{2}_{p}}\|\xi\|_{L^{2}_{p}}.$

(3) Estimates of $T_{4}$:  Recall the definition of $T_{4}$ in (2.14). Using
Lemma 3.1, we can have $CJ(p^{0}/2)\leq\sqrt{m+\tau m^{2}(p)}\leq
CJ(p^{0}/2)$, which yields

$\displaystyle|T_{4}(f,h,\eta)|$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)\frac{J(q^{0}/2)J(p^{\prime
0}/2)J(q^{\prime 0}/2)}{J(p^{0}/2)}|f(q)||h(p^{\prime})||\eta(q^{\prime})|.$

Using the energy conservation law $J(p^{\prime 0}/2)J(q^{\prime
0}/2)=J(p^{0}/2)J(q^{0}/2)$, we have

$\displaystyle|T_{4}(f,h,\eta)|$
$\displaystyle\leq\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{0})|f(q)||h(p^{\prime})||\eta(q^{\prime})|.$

Then we take the inner product with $\xi$ to have

$\displaystyle\big{|}\langle T_{4}(f,h,\eta),\xi\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{0})|f(q)||h(p^{\prime})||\eta(q^{\prime})||\xi(p)|.$

Via the Hölder inequality, we divide the variable into two pre-collisional
variables and two post-collisional variables as follows:

$\displaystyle\big{|}\langle T_{4}(f,h,\eta),\xi\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\left(\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{0})|f(q)|^{2}|\xi(p)|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\times\left(\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)J(q^{0})|h(p^{\prime})|^{2}|\eta(q^{\prime})|^{2}\right)^{\frac{1}{2}}$
$\displaystyle=T_{41}\times T_{42}.$

For the estimates of $T_{41}$, we use $v_{\o}\sigma(g,\theta)\leq
C(p^{0})^{1/2}(q^{0})^{1/2}$ to have

$\displaystyle(T_{41})^{2}$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}(p^{0})^{\frac{1}{2}}(q^{0})^{\frac{1}{2}}J(q^{0})|f(q)|^{2}|\xi(p)|^{2}\leq
C\|f\|_{L^{2}_{p}}^{2}\|\xi\|_{\nu}^{2},$

where we used $(q^{0})^{\frac{1}{2}}J(q^{0})\leq C$. Similarly, we have

$\displaystyle(T_{42})^{2}$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}(p^{0})^{\frac{1}{2}}|h(p^{\prime})|^{2}|\eta(q^{\prime})|^{2}.$

We use $(p^{0})^{1/2}\leq(p^{\prime 0})^{1/2}+(q^{\prime 0})^{1/2}$ to have

$\displaystyle(T_{42})^{2}$ $\displaystyle\leq
C\int_{\mathbb{R}^{3}}dp\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}\left((p^{\prime
0})^{\frac{1}{2}}+(q^{\prime
0})^{\frac{1}{2}}\right)|h(p^{\prime})|^{2}|\eta(q^{\prime})|^{2}.$

Then applying the pre-post collisional change of variables
$(p,q)\leftrightarrow(p^{\prime},q^{\prime})$ as in (3.5) yields

$\displaystyle T_{42}$ $\displaystyle\leq
C\left(\|h\|_{L^{2}_{p}}\|\eta\|_{\nu}+\|h\|_{\nu}\|\eta\|_{L^{2}_{p}}\right).$

Combining the estimates of $T_{41}$ and $T_{42}$ yields

$\displaystyle\big{|}\langle T_{4}(f,h,\eta),\xi\rangle_{L^{2}_{p}}\big{|}$
$\displaystyle\leq
C\left(\|h\|_{L^{2}_{p}}\|\eta\|_{\nu}+\|h\|_{\nu}\|\eta\|_{L^{2}_{p}}\right)\|f\|_{L^{2}_{p}}\|\xi\|_{\nu}.$

∎

##  4\. Local existence

In this section, we construct the local-in-time classical solution. Here we
briefly introduce a main difference of the quantum Boltzmann theory from the
Newtonian one. From the statistical description of quantum Boltzmann equation
in [17], we denote that the term $(1-F(p))$ in the collision operator for
fermions is the probability that a fermion is being placed in the $p$ momentum
place after a collision. Thus the ratio $(1-F(p))$ has to be non-negative.
Then, in each iteration scheme in the proof of the local existence, the
collision operator depending on $F^{n+1}$ is well-defined only when $F^{n+1}$
is in the interval $[0,1]$ in the case of fermions. Thus we need to prove the
boundedness of the solution $F^{n+1}$ in each iteration step as well.

######  Theorem 4.1.

Let $N\geq 3$. Suppose that the initial data $F_{0}$ satisfies

$\displaystyle\left\\{\begin{array}[]{ll}0\leq
F_{0}(x,p)=m(p)+\sqrt{m(p)-m^{2}(p)}f_{0}(x,p)\leq 1,\quad\mbox{for
fermion}\\\ 0\leq F_{0}(x,p)=m(p)+\sqrt{m(p)+m^{2}(p)}f_{0}(x,p),\quad\hskip
17.07164pt\mbox{for boson},\end{array}\right.$

Then there exist $M_{0}>0$ and $T_{*}>0$ such that if
$\mathcal{E}(f_{0})\leq\frac{M_{0}}{2}$ then there exists a unique local-in-
time solution f(x,p,t) of (2.15) satisfying

  1. (1)

The higher order energy is uniformly bounded:

$\sup_{0\leq t\leq T_{*}}\mathcal{E}(f(t))\leq M_{0}.$

  2. (2)

The distribution function is bounded in $t\in[0,T_{*}]$:

$\displaystyle\left\\{\begin{array}[]{ll}0\leq
F(x,p,t)=m(p)+\sqrt{m(p)-m^{2}(p)}f(x,p,t)\leq 1,\quad\mbox{for fermions}\\\
0\leq F(x,p,t)=m(p)+\sqrt{m(p)+m^{2}(p)}f(x,p,t),\quad\hskip
17.07164pt\mbox{for bosons},\end{array}\right.$

  3. (3)

The higher order energy norm $\mathcal{E}(f(t))$ is continuous in
$t\in[0,T_{*}]$.

###### Proof.

We first take an iteration scheme as follows:

(4.1)
$\displaystyle(\partial_{t}+\hat{p}\cdot\nabla_{x})F^{n+1}=Q(F^{n},F^{n},F^{n+1},F^{n}),$

with $F^{0}(x,p,t)=F_{0}(x,p)$. Note that $F^{n+1}$ in the operator $Q$ is
placed in the $p$ variable location. We proceed the proof by applying the
induction argument. Let us assume that

(4.4) $\displaystyle\left\\{\begin{array}[]{ll}0\leq F^{n}(x,p,t)\leq
1\quad\mbox{for fermions,}\\\ 0\leq F^{n}(x,p,t)\qquad\hskip
8.53581pt\mbox{for bosons,}\end{array}\ \text{and}\ \sup_{0\leq t\leq
T_{*}}\mathcal{E}(f^{n}(t))\leq M_{0}.\right.$

Note that the collision operator $Q(F^{n},F^{n},F^{n+1},F^{n})$ is well-
defined when $F^{n+1}$ is in the interval $[0,1]$. We define

$\displaystyle G(F)$
$\displaystyle=\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)F(p^{\prime})F(q^{\prime})(1+\tau
F(q)),$ $\displaystyle R(F)$
$\displaystyle=\int_{\mathbb{R}^{3}}dq\int_{\mathbb{S}^{2}}dw~{}v_{\o}\sigma(g,\theta)(1+\tau
F(p^{\prime}))(1+\tau F(q^{\prime}))F(q),$

to have

$\displaystyle Q(F^{n},F^{n},F^{n+1},F^{n})=G(F^{n})(1+\tau
F^{n+1}(p))-R(F^{n})F^{n+1}(p).$

Then we rewrite (4.1) as follows:

(4.5) $\displaystyle(\partial_{t}+\hat{p}\cdot\nabla_{x}-\tau
G(F^{n})+R(F^{n}))F^{n+1}=G(F^{n}).$

In the case of fermions, we additionally consider the upper bound of
$F^{n+1}$. We observe that the induction hypothesis $(\ref{assume})_{1}$
implies

$\displaystyle 0\leq G(F^{n}),\quad 0\leq R(F^{n}).$

Since $R(F^{n})$ is positive, we have

$\displaystyle(\partial_{t}+\hat{p}\cdot\nabla_{x}+G(F^{n})+R(F^{n}))F^{n+1}\leq
G(F^{n})+R(F^{n}).$

The associated ODE for the particle characteristic trajectory is given by
$dX(s)/ds=\hat{p}(s)$ where $X(s)=X(s;t,x,p)$. We integrate over the particle
path to obtain

$F^{n+1}(X(t),p,t)\leq
e^{-\int_{0}^{t}(G(F^{n})+R(F^{n}))d\tau}F_{0}(X(0),p)\cr+\int_{0}^{t}e^{-\int_{s}^{t}(G(F^{n})+R(F^{n}))d\tau}(G(F^{n})+R(F^{n}))(X(s),p,s)ds.$

We observe from

$\displaystyle\frac{d}{ds}\left\\{e^{-\int_{s}^{t}(G(F^{n})+R(F^{n}))d\tau}\right\\}=e^{-\int_{s}^{t}(G(F^{n})+R(F^{n}))d\tau}(G(F^{n})+R(F^{n}))(X(s),p,s),$

that

$\displaystyle F^{n+1}(x,p,t)$ $\displaystyle\leq
e^{-\int_{0}^{t}(G(F^{n})+R(F^{n}))d\tau}F_{0}(x-\hat{p}t,p)+1-e^{-\int_{0}^{t}(G(F^{n})+R(F^{n}))d\tau}$
$\displaystyle=e^{-\int_{0}^{t}(G(F^{n})+R(F^{n}))d\tau}(F_{0}(x-\hat{p}t,p)-1)+1$
$\displaystyle\leq 1,$

where we used the boundedness of the initial data $F_{0}\leq 1$.

We now consider the lower bound of $F^{n+1}$. By $G(F^{n})\geq 0$ on (4.5), we
have

$\displaystyle(\partial_{t}+\hat{p}\cdot\nabla_{x}-\tau
G(F^{n})+R(F^{n}))F^{n+1}\geq 0,$

which yields

$\displaystyle F^{n+1}(x,p,t)\geq e^{-\int_{0}^{t}(-\tau
G(F^{n})+R(F^{n}))dt}F_{0}(x-\hat{p}t,p)\geq 0,$

where we used the boundedness of the initial data $F_{0}\geq 0$. Now the
collision operator of (4.1) is well-defined.

Substituting $F^{n+1}=m+\sqrt{m+\tau m^{2}}f^{n+1}$ on (4.1) gives

$\displaystyle(\partial_{t}+\hat{p}\cdot\nabla_{x}+\nu)f^{n+1}=Kf^{n}+\Gamma(f^{n},f^{n+1}),$

where $K=K_{2}-K_{1}$ and

$\displaystyle\Gamma(f^{n},f^{n+1})$
$\displaystyle=\sum_{i=1,2,4}\Gamma_{i}(f^{n+1},f^{n})+\sum_{i=3,5,6}\Gamma_{i}(f^{n},f^{n})$
$\displaystyle\quad+\sum_{i=1,2,3}T_{i}(f^{n+1},f^{n},f^{n})+T_{4}(f^{n},f^{n},f^{n}).$

We take $\partial^{\alpha}$ on each side to have

$\displaystyle\partial_{t}\partial^{\alpha}f^{n+1}+\hat{p}\cdot\nabla_{x}\partial^{\alpha}f^{n+1}+\partial^{\alpha}(\nu
f^{n+1})$
$\displaystyle=\partial^{\alpha}Kf^{n}+\partial^{\alpha}\Gamma(f^{n},f^{n+1}).$

We take the $L^{2}_{x,p}$ inner product with $\partial^{\alpha}f^{n+1}$, then
the nonlinear estimates Lemma 3.4 and Lemma 3.7 with the induction hypothesis
$(\ref{assume})_{2}$ yield

$\displaystyle(1-C\sqrt{M_{0}}-CM_{0}-CT_{*}^{2}M_{0}-CM_{0}^{2}-CM_{0}^{3})\sup_{0\leq
t\leq T_{*}}\mathcal{E}_{n+1}(t)\leq\frac{M_{0}}{2}$

For sufficiently small $T_{*}$ and $M_{0}$, we have

$\displaystyle\sup_{0\leq t\leq T_{*}}\mathcal{E}_{n+1}(t)\leq M_{0}.$

Taking the limit as $n\rightarrow\infty$ gives a local-in-time classical
solution. The remaining proof is standard as in [39, 38] and we omit it. ∎

##  5\. Global existence

In this section, we extend the local solution constructed in Theorem 4.1 to a
global solution. For this we first recover the full coercivity estimates of
the linear operator $L$.

###  5.1. Coercivity estimate

Recall the definition of $Pf$ in Lemma 2.2. Since $Pf$ is the orthonormal
projection to the $L^{2}_{p}$ space with the following basis,

$\displaystyle\left\\{\sqrt{m+\tau m^{2}},p_{1}\sqrt{m+\tau
m^{2}},p_{2}\sqrt{m+\tau m^{2}},p_{3}\sqrt{m+\tau m^{2}},p^{0}\sqrt{m+\tau
m^{2}}\right\\},$

$Pf$ can be written as follows by the Gram-Schmidt process:

$\displaystyle Pf=\mathcal{A}\sqrt{m+\tau m^{2}}+\mathcal{B}\cdot
p\sqrt{m+\tau m^{2}}+\mathcal{C}p^{0}\sqrt{m+\tau m^{2}},$

where $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ are given by

$\displaystyle\mathcal{A}$
$\displaystyle=\frac{1}{\lambda}\int_{\mathbb{R}^{3}}dp~{}f\sqrt{m+\tau
m^{2}}-\frac{\lambda_{0}}{\lambda}\frac{1}{\lambda_{00}-\frac{\lambda_{0}^{2}}{\lambda}}\left(\int_{\mathbb{R}^{3}}dp~{}fp^{0}\sqrt{m+\tau
m^{2}}-\frac{\lambda_{0}}{\lambda}\int_{\mathbb{R}^{3}}dp~{}f\sqrt{m+\tau
m^{2}}\right),$ $\displaystyle\mathcal{B}_{i}$
$\displaystyle=\frac{1}{\lambda_{i}}\int_{\mathbb{R}^{3}}dp~{}fp^{i}\sqrt{m+\tau
m^{2}},$ $\displaystyle\mathcal{C}$
$\displaystyle=\frac{1}{\lambda_{00}-\frac{\lambda_{0}^{2}}{\lambda}}\left(\int_{\mathbb{R}^{3}}dp~{}fp^{0}\sqrt{m+\tau
m^{2}}-\frac{\lambda_{0}}{\lambda}\int_{\mathbb{R}^{3}}dp~{}f\sqrt{m+\tau
m^{2}}\right),$

and

$\displaystyle\lambda$ $\displaystyle=\int_{\mathbb{R}^{3}}dp~{}m(p)+\tau
m^{2}(p),\qquad\hskip
17.07164pt\lambda_{i}=\int_{\mathbb{R}^{3}}dp~{}(p^{i})^{2}(m(p)+\tau
m^{2}(p)),$ $\displaystyle\lambda_{0}$
$\displaystyle=\int_{\mathbb{R}^{3}}dp~{}p^{0}(m(p)+\tau
m^{2}(p)),\qquad\lambda_{00}=\int_{\mathbb{R}^{3}}dp~{}(p^{0})^{2}(m(p)+\tau
m^{2}(p)),$

for $i=1,2,3$. Since $Pf$ has the exponential decay $\sqrt{m+\tau m^{2}}$, we
can have

(5.1) $\displaystyle\sum_{|\alpha|\leq N}\|\partial^{\alpha}Pf\|_{x,\nu}$
$\displaystyle\leq\sum_{|\alpha|\leq
N}\left(\|\partial^{\alpha}\mathcal{A}\|_{L^{2}_{x}}+\|\partial^{\alpha}\mathcal{B}\|_{L^{2}_{x}}+\|\partial^{\alpha}\mathcal{C}\|_{L^{2}_{x}}\right).$

Now we substitute $f=Pf+(I-P)f$ in the perturbation equation (2.15) to have

$\displaystyle(\partial_{t}+\hat{p}\cdot\nabla_{x})(Pf)=-(\partial_{t}+\hat{p}\cdot\nabla_{x})((I-P)f)-L(I-P)f+\Gamma(f)+T(f),$

where we used $L(Pf)=0$ by Lemma 2.2. We expand the left-hand side as follows:

$\displaystyle\left(\partial_{t}\mathcal{A}+\sum_{i=1}^{3}\partial_{x_{i}}\mathcal{A}\frac{p_{i}}{p^{0}}+\sum_{i=1}^{3}(\partial_{t}\mathcal{B}_{i}+\partial_{x_{i}}\mathcal{C})p_{i}+\sum_{1\leq
i,j\leq
3}\partial_{x_{i}}\mathcal{B}_{j}\frac{p_{i}p_{j}}{p^{0}}+\partial_{t}\mathcal{C}p^{0}\right)\sqrt{m+\tau
m^{2}},$

which is a linear combination of following $14$-basis:

(5.2) $\displaystyle\left\\{\sqrt{m+\tau
m^{2}},\quad\frac{p_{i}}{p^{0}}\sqrt{m+\tau m^{2}},\quad p_{i}\sqrt{m+\tau
m^{2}},\quad\frac{p_{i}p_{j}}{p^{0}}\sqrt{m+\tau m^{2}},\quad
p^{0}\sqrt{m+\tau m^{2}}\right\\},$

for $1\leq i,j\leq 3$. To denote the right-hand side, we define

(5.3) $\displaystyle l$
$\displaystyle=-(\partial_{t}+\hat{p}\cdot\nabla_{x}+L)((I-P)f),\qquad
h=\Gamma(f)+T(f).$

By expanding $l$ and $h$ with respect to the $14$-basis elements in (5.2), we
obtain the following macro-micro system:

(5.4)
$\displaystyle\begin{split}\partial_{t}\mathcal{A}&=l_{a}+h_{a},\cr\partial_{x_{i}}\mathcal{A}&=l_{i}+h_{i},\cr\partial_{t}\mathcal{B}_{i}+\partial_{x_{i}}\mathcal{C}&=l_{bci}+h_{bci},\cr\partial_{x_{i}}\mathcal{B}_{j}+\partial_{x_{j}}\mathcal{B}_{i}&=l_{ij}+h_{ij},\cr\partial_{t}\mathcal{C}&=l_{c}+h_{c},\end{split}$

for $i,j=1,\cdots,3$. On the right-hand side,
$l_{a},l_{i},l_{bci},l_{ij},l_{c}$ and $h_{a},h_{i},h_{bci},h_{ij},h_{c}$ are
the coefficients of expansion of $l$ and $h$ with respect to the basis in
(5.2), respectively. We define the summation of the coefficients of $l$ and
$h$ as follows:

$\displaystyle\tilde{l}$ $\displaystyle=l_{a}+l_{c}+\sum_{1\leq i\leq
3}\left(l_{i}+l_{bci}\right)+\sum_{1\leq i,j\leq 3}l_{ij},$
$\displaystyle\tilde{h}$ $\displaystyle=h_{a}+h_{c}+\sum_{1\leq i\leq
3}\left(h_{i}+h_{bci}\right)+\sum_{1\leq i,j\leq 3}h_{ij}.$

Note that the macro-micro system of (5.4) is identical to the macro-micro
system of the relativistic Landau-Maxwell model (98)-(102) in [75] when the
electromagnetic field is zero $B=E=0$. However, we provide the full proof here
for the readers’ convenience. We first observe following property of
$\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$. The conservation laws of $f$
in (2.16) can be written in the following form.

######  Lemma 5.1.

We have

$\displaystyle\int_{\mathbb{T}^{3}}\mathcal{A}(x,t)dx=\int_{\mathbb{T}^{3}}\mathcal{B}(x,t)dx=\int_{\mathbb{T}^{3}}\mathcal{C}(x,t)dx=0.$

###### Proof.

The conservation laws in (1.17) implies

$\displaystyle\int_{\mathbb{T}^{3}\times\mathbb{R}^{3}}dxdp~{}(F-F_{0})\left(\begin{array}[]{c}1\cr
p^{\mu}\end{array}\right)$ $\displaystyle=0.$

Combining with the assumption (1.26), we have

$\displaystyle\int_{\mathbb{T}^{3}\times\mathbb{R}^{3}}dxdp~{}f\sqrt{m+\tau
m^{2}}\left(\begin{array}[]{c}1\cr p^{\mu}\end{array}\right)=0.$

We derived desired results. ∎

We now establish the estimates of $\mathcal{A},\mathcal{B},\mathcal{C}$. The
proof of the following lemma is motivated by the proof of the estimate of
(108) in [75].

######  Lemma 5.2.

Let $N\geq 3$. We have

$\displaystyle\sum_{|\alpha|\leq
N}\left\\{\|\partial^{\alpha}\mathcal{A}\|_{L^{2}_{x}}+\|\partial^{\alpha}\mathcal{B}\|_{L^{2}_{x}}+\|\partial^{\alpha}\mathcal{C}\|_{L^{2}_{x}}\right\\}\leq
C\sum_{|\alpha|\leq
N-1}\left(\|\partial^{\alpha}\tilde{l}\|_{L^{2}_{x}}+\|\partial^{\alpha}\tilde{h}\|_{L^{2}_{x}}\right).$

###### Proof.

We first prove the estimate of $\mathcal{A}$. Taking $\partial^{\alpha}$ on
the first equation of (5.4) gives

$\displaystyle\partial^{\alpha}\partial_{t}\mathcal{A}$
$\displaystyle=\partial^{\alpha}l_{a}+\partial^{\alpha}h_{a}.$

we take the innerproduct with $\partial^{\alpha}\partial_{t}\mathcal{A}$ and
apply the Hölder inequality to have

$\displaystyle\|\partial^{\alpha}\partial_{t}\mathcal{A}\|_{L_{x}^{2}}^{2}\leq
C\left(\|\partial^{\alpha}l_{a}\|_{L^{2}_{x}}\|\partial^{\alpha}\partial_{t}\mathcal{A}\|_{L_{x}^{2}}+\|\partial^{\alpha}h_{a}\|_{L^{2}_{x}}\|\partial^{\alpha}\partial_{t}\mathcal{A}\|_{L_{x}^{2}}\right).$

Dividing each side by
$\|\partial^{\alpha}\partial_{t}\mathcal{A}\|_{L_{x}^{2}}$, we have

$\displaystyle\|\partial^{\alpha}\partial_{t}\mathcal{A}\|_{L_{x}^{2}}\leq
C(\|\partial^{\alpha}l_{a}\|_{L^{2}_{x}}+\|\partial^{\alpha}h_{a}\|_{L^{2}_{x}}).$

Similarly we take $\partial^{\alpha}$ on the second equation of (5.4) to have

$\displaystyle\partial^{\alpha}\partial_{x_{i}}\mathcal{A}$
$\displaystyle=\partial^{\alpha}l_{i}+\partial^{\alpha}h_{i}.$

Multiplying $\partial^{\alpha}\partial_{x_{i}}\mathcal{A}$ on each side and
applying the Hölder inequality, we have

$\displaystyle\|\partial^{\alpha}\nabla_{x}\mathcal{A}\|_{L_{x}^{2}}\leq
C(\|\partial^{\alpha}l_{i}\|_{L^{2}_{x}}+\|\partial^{\alpha}h_{i}\|_{L^{2}_{x}}).$

We apply the Poincaré inequality on $\mathcal{A}$ to have

$\displaystyle\|\mathcal{A}\|_{L_{x}^{2}}-\left\|\frac{1}{|\mathbb{T}^{3}|}\int_{\mathbb{T}^{3}}\mathcal{A}(x)dx\right\|_{L_{x}^{2}}\leq
C\|\nabla_{x}\mathcal{A}\|_{L^{2}_{x}}.$

Then the conservation law for $\mathcal{A}$ in Lemma 5.1 implies

$\displaystyle\|\mathcal{A}\|_{L_{x}^{2}}\leq
C\|\nabla_{x}\mathcal{A}\|_{L^{2}_{x}}.$

This completes the estimate of $\mathcal{A}$. Since $\mathcal{B}$ and
$\mathcal{C}$ are connected by the third equation of (5.4), we consider the
fourth and the fifth equations of (5.4) first. For the notational brevity, we
use $\partial_{j}$ to denote $\partial_{x_{j}}$. Using the fourth equation of
(5.4), we calculate

$\displaystyle\triangle\mathcal{B}_{i}=\sum_{1\leq j\leq
3}\partial_{jj}\mathcal{B}_{i}$ $\displaystyle=\sum_{j\neq
i}\partial_{jj}\mathcal{B}_{i}+\partial_{ii}\mathcal{B}_{i}$
$\displaystyle=\sum_{j\neq
i}\left(\partial_{j}l_{ij}+\partial_{j}h_{ij}-\partial_{ji}\mathcal{B}_{j}\right)+\frac{1}{2}(\partial_{i}l_{ii}+\partial_{i}h_{ii}).$

We substitute the fourth equation for $i=j$ case to have

$\displaystyle\triangle\mathcal{B}_{i}$ $\displaystyle=\sum_{j\neq
i}\left(\partial_{j}l_{ij}+\partial_{j}h_{ij}-\frac{1}{2}(\partial_{i}l_{jj}+\partial_{i}h_{jj})\right)+\frac{1}{2}(\partial_{i}l_{ii}+\partial_{i}h_{ii}).$

Taking $\partial^{\alpha}$ and multiplying $\partial^{\alpha}\mathcal{B}_{i}$
on each side yield

(5.5) $\displaystyle\|\partial^{\alpha}\nabla_{x}\mathcal{B}\|_{L^{2}_{x}}\leq
C\sum_{1\leq i,j\leq
3}\left(\|\partial^{\alpha}l_{ij}\|_{L^{2}_{x}}+\|\partial^{\alpha}h_{ij}\|_{L^{2}_{x}}\right).$

We take $\partial^{\alpha}$ on the fifth equation of (5.4) to have

(5.6)
$\displaystyle\|\partial^{\alpha}\partial_{t}\mathcal{C}\|_{L_{x}^{2}}\leq
C\|\partial^{\alpha}l_{c}\|_{L^{2}_{x}}+\|\partial^{\alpha}h_{c}\|_{L^{2}_{x}}.$

Now we use the third equation of (5.4). We consider the estimate of
$\mathcal{B}$ when there are more than one temporal derivatives. For this, we
take the temporal derivative $\partial_{t}^{n}$ for $1\leq n\leq N-1$ on the
third equation of (5.4) to have

$\displaystyle\partial_{t}^{n}\partial_{t}\mathcal{B}_{i}$
$\displaystyle=\partial_{t}^{n}l_{bci}+\partial_{t}^{n}h_{bci}-\partial_{t}^{n}\partial_{x_{i}}\mathcal{C}.$

Then the estimate (5.6) gives

(5.7) $\displaystyle\|\partial_{t}^{n+1}\mathcal{B}_{i}\|_{L^{2}_{x}}$
$\displaystyle\leq\|\partial_{t}^{n}l_{bci}\|_{L^{2}_{x}}+\|\partial_{t}^{n}h_{bci}\|_{L^{2}_{x}}+C\left(\|\partial_{x_{i}}\partial_{t}^{n-1}l_{c}\|_{L^{2}_{x}}+\|\partial_{x_{i}}\partial_{t}^{n-1}h_{c}\|_{L^{2}_{x}}\right).$

For the estimate of $\mathcal{B}$ when there are less than two temporal
derivatives, we apply the Poincaré inequality on $\mathcal{B}$ and
$\partial_{t}\mathcal{B}$ to have

$\displaystyle\|\partial_{t}^{n}\mathcal{B}_{i}\|_{L^{2}_{x}}-\left\|\frac{1}{|\mathbb{T}^{3}|}\partial_{t}^{n}\int_{\mathbb{T}^{3}}\mathcal{B}_{i}dx\right\|$
$\displaystyle\leq\|\nabla_{x}\partial_{t}^{n}\mathcal{B}_{i}\|_{L^{2}_{x}},$

for $n=0,1$. Then the conservation law in Lemma 5.1 yields

$\displaystyle\|\partial_{t}^{n}\mathcal{B}_{i}\|_{L^{2}_{x}}$
$\displaystyle\leq\|\nabla_{x}\partial_{t}^{n}\mathcal{B}_{i}\|_{L^{2}_{x}}.$

Combining with (5.5) and (5.7), we conclude that

(5.8) $\displaystyle\sum_{|\alpha|\leq
N}\|\partial^{\alpha}\mathcal{B}\|_{L^{2}_{x}}\leq C\sum_{|\alpha|\leq
N-1}\left(\|\partial^{\alpha}\tilde{l}\|_{L^{2}_{x}}+\|\partial^{\alpha}\tilde{h}\|_{L^{2}_{x}}\right).$

For the estimate of $\mathcal{C}$, we take $\partial^{\alpha}$ on third
equation of (5.4) to have

$\displaystyle\partial^{\alpha}\partial_{x_{i}}\mathcal{C}$
$\displaystyle=\partial^{\alpha}l_{bci}+\partial^{\alpha}h_{bci}-\partial^{\alpha}\partial_{t}\mathcal{B}_{i}.$

Applying (5.8) yields

(5.9)
$\displaystyle\|\partial^{\alpha}\partial_{x_{i}}\mathcal{C}\|_{L^{2}_{x}}$
$\displaystyle\leq
C\sum_{|\beta|=|\alpha|}\left(\|\partial^{\beta}\tilde{l}\|_{L^{2}_{x}}+\|\partial^{\beta}\tilde{h}\|_{L^{2}_{x}}\right).$

Then the Poincaré inequality with Lemma 5.1 gives

$\displaystyle\|\mathcal{C}\|_{L_{x}^{2}}\leq
C\|\nabla_{x}\mathcal{C}\|_{L^{2}_{x}}.$

Combining with the estimates (5.6) and (5.9), we derive the desired result. ∎

######  Lemma 5.3.

Let $N\geq 3$. Suppose that

$\displaystyle\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f\|_{L^{2}_{x,p}}^{2}\leq M_{0}.$

Then we have

$\displaystyle(1)\ \sum_{|\alpha|\leq
N-1}\|\partial^{\alpha}\tilde{l}\|_{L^{2}_{x}}\leq C\sum_{|\alpha|\leq
N}\|(I-P)\partial^{\alpha}f\|_{x,\nu},$ $\displaystyle(2)\ \sum_{|\alpha|\leq
N}\|\partial^{\alpha}\tilde{h}\|_{L^{2}_{x}}\leq
C(\sqrt{M_{0}}+M_{0})\sum_{|\alpha|\leq N}\|\partial^{\alpha}f\|_{x,\nu}.$

###### Proof.

For the Newtonian Boltzmann equation and the relativistic Landau-Maxwell
system, analogous proofs can be found in [38] and [75], respectively. The
difference now is that the estimate of the nonlinear term $h$ includes the
third-order nonlinear terms.
(1) We denote the $14$-basis of (5.2) as $\\{e_{i}\\}_{1\leq i\leq 14}$, and
let $\\{e_{i}^{*}\\}_{1\leq i\leq 14}$ be the corresponding orthonormal basis.
Then the orthonormal basis can be written by a linear combination of the
original basis $\\{e_{i}\\}_{1\leq i\leq 14}$ as follows:

$\displaystyle e_{i}^{*}=\sum_{j=1}^{14}C_{ij}e_{j},$

for $j=1,\cdots,14$. We consider the orthonormal expansion of $l$ as follows:

$\displaystyle l=\sum_{i=1}^{14}\langle
l,e_{i}^{*}\rangle_{L^{2}_{p}}e_{i}^{*}=\sum_{i=1}^{14}\left\langle
l,\sum_{j=1}^{14}C_{ij}e_{j}\right\rangle_{L^{2}_{p}}\sum_{k=1}^{14}C_{ik}e_{k}.$

Then the coefficient of $e_{k}$ can be read as follows:

$\displaystyle\sum_{1\leq i,j\leq 14}C_{ij}C_{ik}\langle
l,e_{j}\rangle_{L^{2}_{p}},$

which correspond to $l_{a},l_{i},l_{bci},l_{ij}$, and $l_{c}$. By the
definition of $l$ in (5.3) and the linear operator $L$ in Proposition 2.1, we
can write $l$ as

$\displaystyle l$
$\displaystyle=-(\partial_{t}+\hat{p}\cdot\nabla_{x}+\nu+K_{1}-K_{2})((I-P)f).$

For $|\alpha|=N-1$, we have

$\bigg{\|}\int_{\mathbb{R}^{3}}dp~{}\partial^{\alpha}l\cdot
e_{i}(p)\bigg{\|}_{L^{2}_{x}}\cr\leq\bigg{\|}\int_{\mathbb{R}^{3}}dp~{}\bigg{(}\sum_{|\beta|=N}(1+\hat{p})(I-P)\partial^{\beta}f+(\nu+K_{1}-K_{2})(I-P)\partial^{\alpha}f\bigg{)}\cdot
e_{i}(p)\bigg{\|}_{L^{2}_{x}}.$

We use the Hölder inequality on the $(1+\hat{p})(I-P)\partial^{\beta}f$ term
and the $\nu(I-P)\partial^{\alpha}f$ term. Then we have

$\displaystyle\int_{\mathbb{R}^{3}}dp~{}\bigg{(}\sum_{|\beta|=N}(1+\hat{p})(I-P)\partial^{\beta}f+\nu(I-P)\partial^{\alpha}f\bigg{)}\cdot
e_{i}(p)\leq\sum_{|\alpha|\leq N}\|(I-P)\partial^{\alpha}f\|_{L^{2}_{x,p}},$

where we used the exponential decay $\sqrt{m+\tau m^{2}}$ of the basis
$e_{i}(p)$ in (5.2) that implies

$\displaystyle\int_{\mathbb{R}^{3}}\left((1+\hat{p})^{2}+\nu(p)\right)e_{i}^{2}(p)dp\leq
C.$

From the estimate of the compact operator $K_{j}$ from Lemma 3.6, we also have

$\displaystyle\langle K_{j}f,e_{i}\rangle_{L^{2}_{p}}$ $\displaystyle\leq
C\|f\|_{\nu},$

for $j=1,2$ and $i=1,\cdots,14$. The estimates above together yield

$\displaystyle\bigg{\|}\int_{\mathbb{R}^{3}}\partial^{\alpha}l\cdot
e_{i}(p)dp\bigg{\|}_{L^{2}_{x}}$ $\displaystyle\leq
C\sum_{|\alpha|=N}\|(I-P)\partial^{\alpha}f\|_{L^{2}_{x,p}}+\|(I-P)\partial^{\alpha}f\|_{x,\nu}$
$\displaystyle\leq C\sum_{|\alpha|\leq N}\|(I-P)\partial^{\alpha}f\|_{x,\nu}.$

(2) In the same manner, the coefficient of $e_{k}$ of expansion of $l$ can be
written as

$\displaystyle\sum_{1\leq i,j\leq 14}C_{ij}C_{ik}\langle
h,e_{i}\rangle_{L^{2}_{p}}.$

For the second-order nonlinear terms, by Lemma 3.4, we have

$\displaystyle\big{|}\langle\partial^{\alpha}\Gamma(f,f),e_{i}\rangle_{L^{2}_{x,p}}\big{|}\leq
C\sum_{|\alpha_{1}|+|\alpha_{2}|\leq|\alpha|}\|\partial^{\alpha_{1}}f\|_{L^{2}_{x,p}}\|\partial^{\alpha_{2}}f\|_{x,\nu}\leq
C\sqrt{M_{0}}\sum_{|\alpha_{1}|\leq|\alpha|}\|\partial^{\alpha_{1}}f\|_{x,\nu}.$

For the third-order nonlinear term, Lemma 3.7 yields

$\displaystyle\big{|}\langle\partial^{\alpha}T(f,f,f),e_{i}\rangle\big{|}$
$\displaystyle\leq
C\sum_{|\alpha_{1}|+|\alpha_{2}|+|\alpha_{3}|\leq|\alpha|}\|\partial^{\alpha_{1}}f\|_{L^{2}_{p}}\|\partial^{\alpha_{2}}f\|_{L^{2}_{p}}\|\partial^{\alpha_{3}}f\|_{\nu}.$

Then the Sobolev embedding $H^{2}(\mathbb{T}^{3})\subset\subset
L^{\infty}(\mathbb{T}^{3})$ implies

$\displaystyle\bigg{\|}\int_{\mathbb{R}^{3}}\partial^{\alpha}T(f,f,f)\cdot
e_{i}(p)dp\bigg{\|}_{L^{2}_{x}}$ $\displaystyle\leq
CM_{0}\sum_{|\alpha_{1}|\leq|\alpha|}\|\partial^{\alpha_{1}}f\|_{x,\nu}.$

So we obtain the desired results. ∎

We now have all the estimates to recover the full coercivity. We combine (5.1)
with Lemma 5.2 and Lemma 5.3 to have

$\displaystyle\sum_{|\alpha|\leq N}\|\partial^{\alpha}Pf\|_{x,\nu}$
$\displaystyle\leq\sum_{|\alpha|\leq
N-1}\left(\|\partial^{\alpha}l\|_{L^{2}_{x}}+\|\partial^{\alpha}h\|_{L^{2}_{x}}\right)$
$\displaystyle\leq C\sum_{|\alpha|\leq
N}\left(\|(I-P)\partial^{\alpha}f\|_{x,\nu}+C\sqrt{M_{0}}\|\partial^{\alpha}f\|_{x,\nu}\right).$

Thus, for a sufficiently small $M_{0}$, there exists $\delta>0$ such that

(5.10) $\displaystyle\sum_{|\alpha|\leq N}\langle
L\partial^{\alpha}f,\partial^{\alpha}f\rangle_{L^{2}_{x,p}}$
$\displaystyle\geq\delta\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f\|_{x,\nu}^{2},$

by Lemma 3.3.

###  5.2. Global existence

We now have all the ingredients for the proof of Theorem 1.1. Extending the
local solution constructed in Theorem 4.1 to a global solution is standard as
in [38, 37]. We only sketch the proof here. Recall that substituting
$F=m+\sqrt{m+\tau m^{2}}f$ on (1.13) yields the following linearized equation:

$\displaystyle\partial_{t}f+\hat{p}\cdot\nabla_{x}f+Lf$
$\displaystyle=\Gamma(f)+T(f),$ $\displaystyle f(x,p,0)$
$\displaystyle=f_{0}(x,p).$

We take $\partial^{\alpha}$ on each side to have

$\displaystyle\partial_{t}\partial^{\alpha}f+\hat{p}\cdot\nabla_{x}\partial^{\alpha}f+L\partial^{\alpha}f$
$\displaystyle=\partial^{\alpha}\Gamma(f)+\partial^{\alpha}T(f).$

Taking the inner product with $\partial^{\alpha}f$ yields

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\partial^{\alpha}f\|_{L^{2}_{x,p}}^{2}+\langle
L\partial^{\alpha}f,\partial^{\alpha}f\rangle_{L^{2}_{x,p}}$
$\displaystyle=\langle\partial^{\alpha}f,\partial^{\alpha}\Gamma(f)\rangle_{L^{2}_{x,p}}+\langle\partial^{\alpha}f,\partial^{\alpha}T(f)\rangle_{L^{2}_{x,p}}.$

Applying the full coercivity estimate (5.10), we have

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\partial^{\alpha}f\|_{L^{2}_{x,p}}^{2}+\delta\|\partial^{\alpha}f\|_{x,\nu}^{2}$
$\displaystyle\leq\langle\partial^{\alpha}f,\partial^{\alpha}\Gamma(f)\rangle_{L^{2}_{x,p}}+\langle\partial^{\alpha}f,\partial^{\alpha}T(f)\rangle_{L^{2}_{x,p}}.$

The estimate of the second-order nonlinear terms in Lemma 3.4 gives

$\displaystyle\langle\partial^{\alpha}f,\partial^{\alpha}\Gamma(f)\rangle_{L^{2}_{x,p}}\leq
C\sum_{|\alpha_{1}|+|\alpha_{2}|\leq
N}\int_{\mathbb{T}^{3}}dx~{}\left(\|\partial^{\alpha_{1}}f\|_{L^{2}_{p}}\|\partial^{\alpha_{2}}f\|_{\nu}+\|\partial^{\alpha_{1}}f\|_{\nu}\|\partial^{\alpha_{2}}f\|_{L^{2}_{p}}\right)\|\partial^{\alpha}f\|_{\nu}.$

Without loss of generality, we assume that $\alpha_{1}$ is less than or equal
to $\alpha_{2}$. Since $N\geq 3$, we have $|\alpha_{1}|+2\leq N$. Thus the
Sobolev embedding $H^{2}(\mathbb{T}^{3})\subset\subset
L^{\infty}(\mathbb{T}^{3})$ implies

$\displaystyle\sum_{|\alpha|\leq
N}\langle\partial^{\alpha}f,\partial^{\alpha}\Gamma(f)\rangle_{L^{2}_{x,p}}$
$\displaystyle\leq C\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f\|_{L^{2}_{x,p}}\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f\|_{x,\nu}\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f\|_{x,\nu}$ $\displaystyle\leq
C\sqrt{\mathcal{E}(t)}\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f\|_{x,\nu}^{2}.$

For the estimate of the third-order nonlinear terms, we apply Lemma 3.7 to
have

$\langle\partial^{\alpha}f,\partial^{\alpha}T(f)\rangle_{L^{2}_{x,p}}\leq
C\sum_{|\alpha_{1}|+|\alpha_{2}|+|\alpha_{3}|\leq
N}\int_{\mathbb{T}^{3}}dx~{}\big{(}\|\partial^{\alpha_{1}}f\|_{L^{2}_{p}}\|\partial^{\alpha_{2}}f\|_{L^{2}_{p}}\|\partial^{\alpha_{3}}f\|_{\nu}\cr+\|\partial^{\alpha_{1}}f\|_{L^{2}_{p}}\|\partial^{\alpha_{2}}f\|_{\nu}\|\partial^{\alpha_{3}}f\|_{L^{2}_{p}}+\|\partial^{\alpha_{1}}f\|_{\nu}\|\partial^{\alpha_{2}}f\|_{L^{2}_{p}}\|\partial^{\alpha_{3}}f\|_{L^{2}_{p}}\big{)}\|\partial^{\alpha}f\|_{\nu}.$

Similarly we assume that $\alpha_{1}$ and $\alpha_{2}$ are less than or equal
to $\alpha_{3}$. Since $N\geq 3$, we have $|\alpha_{1}|+2\leq N$ and
$|\alpha_{2}|+2\leq N$. By the Sobolev embedding
$H^{2}(\mathbb{T}^{3})\subset\subset L^{\infty}(\mathbb{T}^{3})$, we have

$\displaystyle\sum_{|\alpha|\leq
N}\langle\partial^{\alpha}f,\partial^{\alpha}T(f)\rangle_{L^{2}_{x,p}}$
$\displaystyle\leq C\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f\|_{L^{2}_{x,p}}\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f\|_{L^{2}_{x,p}}\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f\|_{x,\nu}\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f\|_{x,\nu}$ $\displaystyle\leq
C\mathcal{E}(t)\sum_{|\alpha|\leq N}\|\partial^{\alpha}f\|_{x,\nu}^{2}.$

Combining these estimates, we conclude that

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\partial^{\alpha}f\|_{L^{2}_{x,p}}^{2}+\delta\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f\|_{x,\nu}^{2}$ $\displaystyle\leq
C\left(\sqrt{\mathcal{E}(t)}+\mathcal{E}(t)\right)\sum_{|\alpha|\leq
N}\|\partial^{\alpha}f\|_{x,\nu}^{2}.$

The remaining proof can be established by the standard continuity argument as
in [38, 37]. This completes the proof.

Acknowledgement: J. W. Jang is supported by CRC 1060 The mathematics of
emergent effects at the University of Bonn funded through the German Science
Foundation (DFG).

S.-B. Yun is supported by Samsung Science and Technology Foundation under
Project Number SSTF-BA1801-02.

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