*BoltzmannThe Boltzmann Theorem
*PovznerLemmaPovzner Lemma
problem2Loss to Gain Moment Rate  Property

'f 'f
'φ 'φ

mthm[thm]Main Theorem

^d ×^d

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\begin{document}
\title[Boltzman equation, coerciveness, exponential tails ans Lebesgue integrability ]
%Kinetic models
%for statistical mechanics\\
%\ \\
%An Introduction to the Boltzmann equation for Diluted Gases\\
%\ \\
%{Revisiting the Cauchy problem for the Boltzmann equation for hard potentials with integrable cross section: from generation of moments to propagation of $L^p$-bounds}
{The Boltzmann equation for hard potentials with integrable angular transition: Coerciveness, exponential tails rates, and Lebesgue integrability}

\author{Ricardo J. Alonso}
\curraddr{{\bf Ricardo J. Alonso}; Texas A\&M, Science Department, Education City, Doha, Qatar;   and Departamento de Matemática, PUC-Rio,
Rio de Janeiro, Brasil.}
<EMAIL_ADDRESS>
\author{Irene M. Gamba}
\curraddr{{\bf Irene. M. Gamba}; Department of Mathematics and Oden Institute for Computational Science and Engineering, The University of Texas at Austin.}
<EMAIL_ADDRESS>\thanks{}

%    \subjclass is required.

\subjclass[2020] {Boltzmann equations, Interacting particle systems and Kinetic theory of gases in time-dependent statistical mechanics, Singular nonlinear integral equations, Functional analytic methods in summability,  polynomial and exponential moments and $L^p$ theory methods,  	abstract ODE theory in Banach spaces. \\ {\bf MSC:} 35Q20,  82C22, 82C40,45G05, 40H05.}

%%    Abstract is required.

%    Abstract is required.
\begin{abstract}
This manuscript  focus on an extensive survey with new techniques on the problem of  solving the Boltzmann flow by bringing a unified approach to the Cauchy problem to homogeneous kinetic equations with Boltzmann-like collision operators under integrability assumption of the scattering profile in the particle-particle interaction mechanism.  The work focuses on the relevant hard potential case where the solution properties are studied with a modern take.  While many of the discussed results can be found the literature spread over several papers along the years, we bring a complete program that includes a new approach to the existence and uniqueness theorem securing the well-posedness theory,  to moments estimates, and integrability propagation for the homogeneous Boltzmann flow.   In particular,  a detailed calculation of classical polynomial  moments upper bounds as function of the coerciveness  is described, which characterized the rate of exponential moments obtained by summability of the polynomial ones. In addition, a  proof of  uniform propagation of $L^\infty$ regularity under general integrable scattering kernels is performed.  Along the way, constants  appearing in estimates are carefully calculated,  improving most of  previous exiting results in the literature.
For the non expert reader we also include a general discussion of the basic elements of the Boltzmann model and important key results for the understanding of the mathematical discussion of the equation and include an extensive  set of references that enrich  and motivate  further discussions on Boltzmann flows for broader gas modeling configuration such as  gas mixtures systems, polyatomic gases, multilinear collisional forms such as, ternary or quartic, to  those derived from symmetry braking  quantum mean field theories or weak turbulence models from  spectral energy  waves in classical fluid.  \end{abstract}

\maketitle

\tableofcontents

\vfill\newpage
\section{Fundamentals of binary elastic collisional flows}

\subsection{Introduction}

The notion of statistical mechanics and physics   models  were independently introduced by Ludwig Boltzmann and Charles Maxwell during last quarter of the nineteen century motivated  by the search of finding rigorous questions and answers  to the   theory of thermodynamics of a gas.
The resulting statistical viewpoint essentially   expresses a gas evolution  model  as the chaotic motion of huge numbers of molecules rushing here and there at large speeds interacting, or colliding, and rebounding
according to  a given law, being of elementary mechanics nature or other attributes from probabilistic considerations.
These models nowadays have applications beyond statistical mechanics and physics. Then, without loss of generalization, the $N$ velocity states may also be interpreted as the space of attributes associated to the model on which a probability density evolves in time as the attribute change values. That is the case in recent applications that range from opinion dynamics, wealth distribution as much as multi agent dynamics operating with given interactions laws.
However,  we will focus in this manuscript mostly in the original in the classical framework for particle physics, revisiting a framework developed in the last forty years and introducing new techniques and improved results that unified the  the theory of the Boltzmann equation in the space homogeneous setting that will enable   clear improvements in the space inhomogeneous case under specific boundary conditions in bounded domains or at infinity.

The space homogeneous Boltzmann equation is model of a statistical flow. It  describes the evolution of a probability density  according to a law given by   a non-local multi linear operator acting on the probability density, the so called collision operator. Such operator  determines the evolution of the   probability density evaluated at different states of velocities, or attributes,  interacting  according to the basic laws of thermodynamics in physical sciences.
In classical thermodynamics and mechanics pressure is identify as the mechanical effect of the impact of the moving
molecules as they strike against a solid wall  or a region of different molecular properties.  For the starting of the derivation of collisional theory by the classical Boltzmann flow,  molecules are assumed to be
modeled  by spheres that move according to the laws of classical elastic or inelastic mechanics.
If no external forces (such as gravity) are assumed to act on the molecules, each of them will move along a
straight line unless they strike another sphere or the wall. In the classical mechanics interaction, these are called  billiard models.

The {\em Boltzmann Transport Equation} (BTE) models an attempt to describe the an evolution process from a
statistical point of view: {\em ``it determines the state of the gas using some statistical mechanics laws from knowing the statistical mechanical initial state"}.  In this sense, the  Boltzmann Transport Equation is an {statistical  flow  model  whose solution describe the evolution of a probability density of finding a particle at a given position, with a given velocity, at a given time}.

\subsection{The Boltzmann Transport Equation}
The model for a rarefied mono atomic gases under binary elastic interactions, originally introduced  by L. Boltzmann in 1872 [38], and analytically studied by many authors including Carleman, Wild, Chapman, Cowling, Cercignani, Illner, Pulvirenti  [44, 110, 46, 47, 48] to name a few,  is written as follows
\begin{equation}\label{BTE}
{\frac{\partial f}{\partial t} + v\cdot \nabla_x f
= a^2 N \int_{\real^3\times \mathbb{S}^{2}_{+}}(\fprime \fprime_* - ff_*) B(|u|, \hat u\cdot\eta ) \,dv_*d\eta}
\end{equation}
\begin{equation}\label{Elastic Interaction}
\vprime=v - (u\cdot\eta)\, \eta ,\ \ \vprime_*=v_* + (u\cdot\eta)\, \eta ,\ \ \quad u=v-v_*
\end{equation}

\nn for $\eta\in\mathbb{S}^2$ a vector in the direction of $v-\vprime=\vprime_*-v_*$;
where $f(x,v,t)$ describes the evolution of the probability distribution
function (pdf) of finding a particle at position $x$, velocity $v$ and time $t$,
for a billiard model of hard spheres and prescribed initial probability
$f(x,v,0) = f_0 (x_0,v_0)$. In addition, the notation for the pdf on pre collisional velocities is $\fprime:=f( x,\vprime,t)$ and $\fprime_*:=f( x,\vprime_*,t)$. The corresponding to the ones on post-collisional velocities are simply $f:=f( x,v,t)$ and $f_*:=f( x,v_*,t)$.  \\

Rigorous mathematical derivations of the Boltzmann homogeneous  flow was first developed by Sznitman [97] by probabilistic methods.
In  the space inhomogeneous setting, the  derivations of the Boltzmann flow model under consideration that have been discussed in  several manuscripts  the past twenty years, including the work of  Pulvirenti, Saffirio,  Simonella [95],  Gallagher, Saint Raymond and Texier \cite {Gal-SRay-Texier} among many other groups, are  based in the formal derivation by Landford.   All these results, following strict mathematical rigor, involve  arguments  short time $t$ with short time validity up to  the average of the first collision time.  However, the classical space inhomogeneous Boltzmann initial valued problem, has beed solved  global in time, first for short range potentials  with initial data near vacuum following the pioneer work  by Kaniel and Shimbrot, and Illner and Shimbrot [82, 80],  and later by for long range potentials by  Ukai and Asanov, Bellomo, Toscani and Palczewski,  \cite{ukai-asano-82, bellomo-toscani, toscani-ARMA-86}. Toscani  \cite{toscani-ARMA-88, pal-tosc-JMP89} address the first existence for large data, near a fixed Maxwellian state. Hamdache, in [76, 77], solved the near vacuum problem  by fixed point theorem's arguments; and Goudon [74] extended the Hamdache techniches for long range potentials, to data near a fixed Maxwellian state. A couple of decades later,   Glassey [72] found global solutions for the relativistic Boltzmann flow near vacuum; followed by the work of Alonso and Gamba [10, 11] motivated by the work $W^{m,p}_k(\R^d)$ from Alonso and Carneiro and Alonso Carneiro and Gamba [6, 7],  revisited the Kaniel-Shimbrot and Illner-Shimbrot [82, 80] iteration by a modified approach and improved global global in time estimates not valid for near vacuum  data for short rage potentials, but enlarged the initial data set subject to different Maxwellian distribution with closeness in $L^\infty$-exponentially weighted, and including the $L^p$-differentiability.   Most recently, the problem of longtime behavior of the solution near vacuum was address for long-range potentials by Bardos, Gamba,  Golse and Levermore [26] showing  that, in the vicinity of global Maxwellians with finite mass, the dispersion due to the advection operator quenches the dissipative effect of the Boltzmann collision integral,   resulting in a typical scattering mechanisms where the  large time limit of solutions of the Boltzmann flow  is given by noninteracting, freely transported states.

The initial value problem associated to the space homogeneous equation \eqref{BTE} in the space homogeneous setting has been proven to be globally well-posed under general physically conditions and rather broad initial data in size.  Furthermore, the model enjoys properties such as propagation and generation of polynomial and exponential statistical moments or observables, as well as propagation of Sobolev norms of any order.  These particular properties of higher Lebesgue integrability and Sobolev regularity propagation has been the attention of several studies, perhaps starting with Carleman [45] and continued with Arkeryd [24, 25], Gustaffson [75], Lions [84], and Toscani and Villani [104].

In the last few years, the advance of these theories have enable new techniques involving  the notion of angular averaging lemmas that permitted further studies of lower estimates in the search of coerciveness, summability of  moments to obtained the propagation and generation of exponential moments, as well as exponential wighted $L^p_\ell(\R^d)$ estimates for $p\in[1,\infty]$ and $\ell>2.$

In general,   moment estimates  is a fundamental tool that enables the existence and uniqueness theory  associated to the Cauchy problem for the the spatially homogeneous Boltzmann equation. One can roughly separate the problem characterizing the transition probability  measure rates (or collision kernels) on whether  depending of the following characteristics: they may depend on potential rates associated intramolecular potential  modeled by functions of the relative  speed associated  binary interaction, and on the angular transition measure that quantifies the transition rate with respect to the deflection angle before or after the interactions, as envisioned by H. Grad [73].
That is, these transition rates may or may not depend on the relative speed of the local colliding velocities as much as in the integrability condition of the angular part of such interactions.

Historically, these problems started to be addressed, starting  before the mid twenty's century, starting by the work of Carleman, Wild  and later  Arkeryd  and Elmroth [44, 45, 24, 25, 59, 94],  for  transition rates with  bounded  intramolecular rates and bounded angular transition functions.   Di Blassio [54]contributed by finding the proof of uniqueness to the Boltzmann flow for hard potentials, but just having a few moments bounded. The estimates needed to obtain these results were under the assumption that the angular transition function is bounded.

Around the same time,  the  pioneering work of Bobylev [30, 29] fully solved the problem for transition probability rates independent of the intramolecular potentials commonly referred as to {\em Maxwell Molecule}  or  {\em Maxwell type models}, where the Boltzmann flow can be fully solved in Fourier space with the $C^\infty(\R^d)$ topology associated to generating master equations to the evolutions  probabilistic or  stochastic flows, with profound discussions on convergence rates to the Boltzmann-Maxwell statistical equilibrium state, as shown in the work of  Toscani and Villani, and Carlen, Carvalho and Lu and  [104, 43] and references therein.

The problem of hard or soft intramolecular potential interactions, that is the higher the relative speed  the more or less collision rates, respectively, started to be addressed on whether   the integrability of the angular transition function was bounded or not.

By the last decade of the twentieth century the Boltzmann flow model enters in maturity, starting by  the work of Desvilletes [51]  who proved interesting point that  it can propagate any moments, and shortly after,  Wennberg [109] showed that moments can be generated if the initially data has bounded entropy in addition to bounded  mass and energy.  In the same year of Wennberg's work, a groundbreaking result of Bobylev [31] showed that the generated moments are summable paving the way to the understanding of propagation and generation of $L^1(\R^d)$ exponentially weighted estimates in several frameworks such us inelastic theories for hard spheres by
Gamba Panferov and Bobylev, Alonso and Lods for hard spheres in [37, 5, 15] and later with Panferov, Gamba and Villani  [64] showed the extension of this estimates to hard potentials and show the technique extends to angular transition functions in $L^1(\S^d)$ as well establishing that $L^\infty(\R^d)$-Maxwellian weighted estimates for  solutions as constructed by Arkeryd [25], later  extended by Bobylev and Gamba [36] to the Maxwell type of interactions.
These results inspired the work of Mouhot [89] on generation of  $L^1(\R^d)$ exponentially weighted estimates,  Alonso and Gamba [9] solutions derivatives,  and  Canizo and Mouhot, in  collaboration with the authors, [8], to study their summability by showing the convergence of moments's partial sums.

Further studies of these results were extended to hard potentials with the classical Grad angular non-cut-off condition  [85, 101, 91]  valid for type of solutions constructed by Morimoto, Wang and Yang [88]. In addition  Gamba, Pavlovic, and Taskovic [68] showed pointwise exponential estimates for conditional solutions and  recent results of Fournier [60] further clarifies and improves $L^1(\R^d)$ exponentially weighted estimates, both paper under  angular non-cut-off conditions.

These manuscripts also expands and improves the existing results of properties such as the propagation of higher Lebesgue integrability,  initiated by  the work Gustaffson,  and later by Wennberg  in [75, 107], obtaining estimates for  the collision operator in  Banach spaces   $L^r(\R^d)$  for $1 \leq r < \infty$. Lions [84]  introduced the breakthrough concept of {\em gain of integrabilty}, by meaning that it is possible to control   the gain operator  estimates for hard potentials in  $L^r(\R^d)$ weighted spaces having lower growth when compared  to the corresponding one  obtained by  the loss operator. This property, which still requires a bounded angular transition rate functions,  may be viewed as a smoothing effect obtained  under  special conditions of the potential and angular transition rates or collision kernels.  Later by Toscani,  Villani and Mouhot [104, 90] used these Lions' results to obtain not only the propagation and Sobolev regularity in the classical sense for the space homogenous Boltzmann flows.  In the last decade, Carneiro and the authors revisited the work of  Gustaffson [7, 7] that inspired the work of the authors [13] to obtained significant improvements removing conditions from the work of Lions on the transition probability kernels, but also quantified the rate of integrability estimates
by parameters  depending on the Cauchy problem data and   the $L^r(\R^d)$ norm of the initial data.

Thus, the new results presented in the current  manuscript provides a unified set of  fundamental  technical improvements and corrections  related to the collision operator estimation,  removing many of the constrains  angular transition and potential rate growth extended now to  any integrable angular transition function and any hard potential behavior, bounded at zero and with a growth rate corresponding to supper hard spheres.    Such improvements include a new complete proof of existence of unique solutions in $C^1(0,T, L^1_k(\R^d))$ enabling   explicit computations of  coercive constants and their dependence of the model parameters, and obtaining a priori estimates on the collision operator  enabling the implementation  of a proof theorem for Ordinary Differential flows in Banach spaces for  existence and uniqueness of  solutions propagation and generation of  on moments (or expectations) associated to the solution $f(v,t)$ of the Boltzmann flow, globally in time. In addition we developed bound, explicitly as function of the coercive constants. In addition a new proof  for the summability of moments globally in time   is presented,  securing the the propagation and generation of  exponential tails whose rate and exponent order are explicit and written in terms of the problem data. In particular one can easily characterize   the exponential rates by the Cauchy Problem  data.  The existence and uniqueness theorem follows the lines of the proposed approach by Bressan [39], in an unpublished note, after being completed by a complete proof  for the collisional integral satisfying the  sub tangent condition property  in Lemma~\ref{prop_subtan}.

The second part of the manuscript focus on the applications of tools developed for the $L^p_\ell(\R^d)$  inspired by the estimates from  the authors developed in [7, 7] revisit  Young's inequalities and Carleman integral representation in obtaining explicit tares of gain of integrability estimates.  New results, such us the lower bound  Proposition \ref{lem-cancel}, depending inly on the Banach space norm and the choice of potential rates in the transition probability kernel,  enables  $L^p_\ell(\R^d)$-norm propagation  for general transition probabilities   rates for arbitrary hard potentials and just angular intregability in the scattering direction with an explicit coercive rate.  These results yield,  most significantly, not only the generalization of Arkeryd's pioneering work on the propagation of $L^{\infty}$-integrability property for very constrained conditions of the potential and angular scattering to  such rather general scattering form  achieved in a very elementary manner to any  $L^{\infty}_\ell$, but also     exhibits how the coercive constant,  calculated in the first part giving global in time control to propagation and generation of  $L^1_{2\ell}(\R^d)$,  regulates the rate of exponential weights for any  $L^p_\ell(\R^d)$. As a consequence, the  control  of spectral gap estimates can be explicitly characterized as a function of the coerciveness, as it  will be shown in a future work by the authors.

A remarkable point of this manuscript is that entropy is not needed for any of these results, since the proof of  lower bounds only require statistical moment considerations just from the Banach space without invoking an a priori estimate to a solution of the Boltzmann flow, as seen in Lemma \ref{lblemma}.

%Additional comments will be made in sections 6, 7, and 8.

This manuscript is lay down as follows.
Section~\ref{elementary-properties} focus on Elementary properties of the Boltzmann Equation,   Section~\ref{hardpotsection} states the Cauchy problem for Maxwell and hard potentials  in Theorem~\ref{CauchyProblem}, and  develops lower bounds in  Theorems~\ref{lblemma},  a priori estimates  on moments of the collisional integral   Theorem~\ref{mom-coll-op},   and Lebesgue polynomial  moments  propagation  and generation  of global in time bounds fully characterized by the coerciveness of the problem data Theorem~\ref{propagation-generation} that enable    the proof of  existence and uniqueness Theorem~\ref{CauchyProblem} in Section~\ref{existence-uniqueness}.
Section~\ref{exponential-tails} considerably    improves the Propagation and generation of exponential moments with a detailed characterization of the exponential rates.
The following two Sections~\ref{convo-ine+gain int} and  Section~\ref {Lr-propagation} focus on
novel  proofs of  convolution inequalities and gain of integrability properties for the collision operator, and develop en new energy method to show  $L^r_\ell$  propagation theory for any  $r\in[1,\infty]$, with $\ell>2^+.$

The manuscript culminates with  a short Section~\ref{fine properties} addressing a revision on fine properties of the collision operator, followed by
an Appendix ~\ref{appendix}  included  in order
to reproduced the proof of
Theorem~\ref{Theorem_ODE} following the same lines for  solving  ODEs in Banach spaces as  proposed by A. Bressan in  an unpublished manuscript posted in [39].

In the last few years,  these techniques and there results have been recently extended to the case of vector value solutions to gas mixture system for  particle with disparate masses, and for the scalar model,  polyatomic gases, in the works of Gamba, Pavic-Colic and De la Canal [66, 67, 41]  ,  by Strain and Yu for relativistic particles, and later Strain and  Taskovic
[100, 99] for the the relativistic Fokker Plank  Landau flow.
The $L^p_\ell(\R^d)$ vector valued solutions for to gas mixture system is being addressed in [42] and for an arbitrary system   for of monoatomic and polyatomic gases is being addressed  in  [12].

In addition several kinetic collisional frameworks have implemented the summability of moments estimates to calculate  $L^1(\R^d)$ polynomial and exponentially weighted estimates, like in the work  of
Alonso, Bagland and Lods [4] implemented these techniques to construct solutions to one dimensional Boltzmann flow modeling polymers;   Alonso, Gamba and Tran [16] in the solution of system of the Quantum Boltzmann gas coupled to the Bose Einstein condensed  system at cold temperatures, as well as Gamba, Smith and Tran [69] applied to the estimates for the kinetic wave equation derived by Luov and Zacharov for water waves applied to stratified flows in deep oceans.
Most recently,  the Illner, Kaniel and  Shimbrot  iteration schemes have been extended   to global in time solutions to  binary-ternary Boltzmann flows  with initial data near vacuum,  while the global in time solutions for its space homogeneous problems for large data with  propagating and generating  moments estimates  resulted in lovable problems whose solutions also exhibit global in time $L^1(\R^d)$-polynomial and exponential moments, as shown by Ampatzoglou, Gamba, Pavlovic and Taskovi\v{c} [19, 20].

These type of propagation of  moments estimates were also developed for the space homogeneous Fokker Plank Landau equation for hard potentials by Desvillettes and Villani [52, 53] in their studies of $L^1_\ell(\R^d)$-norm propagation properties as much as higher $H^\alpha(\R^d)$ polynomial and exponential  weighted regularity enabling the spectral analysis of the Landau flow for hard potentials.

Finally, in a unique application,  $L^1(\R^d)$ and  $H^\alpha(\R^d)$  polynomial and exponential weighted propagation estimates have been very instrumental to developed error estimates for conservative spectral schemes for both the numerical error, consistency and stability to numerical solutions of the homogeneous Boltzmann flow by and Tharkabhushaman and authors  [17], and also by  Pennie and one of the authors  for the homogeneous Landau flows [92].

\section{Elementary Properties of the Boltzmann Equation}\label{elementary-properties}

\subsection{Elastic binary (particle-particle) interaction:}\label{items elastic}
The standard glossary associated to the collision of
two particles located at positions in space centered  at points $x$ and $x_*$ in $\R^d$ with velocities  $v$ and $v_*$ encountering a collisional interaction and leaving with post collisional velocities $v'$ and $v'_*$, consists on

%\begin{itemize}%\label{items elastic}
${\bf 1.}$  Their relative velocities are
\begin{equation*}%\label{elasticcollision.eps}
u:= v - v_* \  \ \text{  \it before, \ \ \ and }\
u':= v' - v'_*  \ \  \text{\it  after the binary interaction.}
\end{equation*}

${\bf 2.}$\  The unitary vector $\eta:=\frac{x-x*}{|x-x_* |}\in\mathbb{S}^2$ is the {\bf impact direction}, and $\frac{u\cdot \eta}{|u|} = \cos(\pi-\phi)=-\cos\phi$, with {\it impact angle} $\phi$.
\smallskip%

${\bf 3.}$ \ The unitary vector  $\sigma:= \frac{u'}{|u|}\in\mathbb{S}^2$ is the {\bf scattering direction} given by
the {\em post-collisional relative velocity} $u'$ with respect to an elastic collision, from which $|u|=|u'|$ naturally follows.
\smallskip

${\bf 4.}$  \ Consequently, the  identity $\cos \theta= \frac{u\cdot \sigma}{|u|} = \frac{u\cdot u'}{|u|^2}$ defines the {\it scattering angle} $\theta$.   The relationship  between impact and scattering angles is $2\phi = \pi - \theta.$
\smallskip

In addition,  when  the angles $\phi=0$ or $\theta=\pi$,  the interaction is called a {\it head-on} or {\it knock on} collision relevant for hard to infinite (short range) intramolecular  potentials. When $\phi=\pi/2$ or $\theta=0$, it is
called a {\it grazing} or {\em glancing} collision. The later one is relevant for soft (long range) interactions such as in the modeling of Coulomb interactions.

\subsection{Post-collisional velocities using the impact direction}
From items 1 and 2 from above, the elastic binary interaction laws are fully determined by local conservation of momentum and kinetic energy following from collisional specular reflection laws with respect to the impact direction $\eta$, that is
\begin{equation}\label{eq:6.1}		%%\tag{6.1}
{u\cdot \tau = u' \cdot \tau}\ \text{ and }\
{u'\cdot\eta = -u\cdot \eta} \quad \text{with}\ \ \tau\cdot\eta=0\,, \qquad \text{implying}\quad u' = u - 2( u\cdot\eta) \eta\,,
\end{equation}
and the equivalent relations for the pairs $\{v', v'_*\}$ determined by  by their pre interactions ones $\{v,v_*\}$ by
\begin{equation}\label{eq:6.3}		%% \tag{6.3}
\begin{array}{l}
v' = v - \left( (v - v_*) \cdot\eta\right) \eta  \ =\  v - (u\cdot \eta)  \eta\\
\noalign{\vskip6pt}
v'_* = v_* + \left( (v - v_*) \cdot\eta\right) \eta  \ =\  v_* +(u\cdot \eta)  \eta \,.
\end{array}
\end{equation}

% \centerline{\epsfysize=2.75truein\epsfbox{elastic-collision2.eps}}}
%  % Requires \usepackage{graphicx}
%  \caption{Elastic binary interactions in relative velocity $u$ - impact direction $\eta$ frame}\label{fig-1}

\nn Elastic binary interactions mean that relations \eqref{eq:6.1} and \eqref{eq:6.3} are
{\bf reversible}

\subsection{Post-collisional velocities using the scattering direction and the notation $u^{\pm}$.}  Equivalently, the relation between pre to post collisional velocities can also be expressed in the center of mass $V=\frac{v+v_*}2$ and relative velocity $u=v-v_*$ framework through the scattering direction $\sigma$, defined in item 3 above which in light of the latter equation in \eqref{eq:6.1} reads
\begin{equation*}
\sigma = \widehat{u} - 2( \widehat{u}\cdot\eta) \eta \in \mathbb{S}^{2}\,,
\end{equation*}
by the interaction law
\begin{equation}\label{eq:9.3-Vu}		%%\tag{7.3}
\begin{array}{l}
v' \ =\  V +\frac12 |u|\sigma  \ =\  v + \frac12(-u+|u| \sigma) =: v +u^- = v_* + u^+\, ,\\
\noalign{\vskip6pt}
v'_* \ = \ V - \frac12 |u|\sigma   \ =\  v_* +  \frac12(u-|u| \sigma)  =: v_* - u^- = v- u^+\,  ,
\end{array}
\end{equation}
after introducing the notation   $u^{\pm} = \frac{ |u|\sigma \pm u}{2}$. Therefore, setting the vectors $u^-=v'-v$, while $u^+=v- v'_*$, clearly  $u^{-}+u^{+}=u'=|u|\sigma$. It follows that  $|u|=|u'|$ and  clearly  $u^{-}\cdot u^{+}=0$  by direct computations.\\

\nn We left to the readers to check that  these  elastic binary constitutive relations  are  equivalent to the following {\em local conservation} identities
\begin{align}\label{eq:8}
v + v_* &=  v' + v'_* \ ,  \qquad \qquad \text{ center of mass  or  local momentum}  \\
|v|^2 + |v_*|^2 &= |v'|^2 + |v'_*|^2\ , \qquad \text{ local energy.}  \nonumber
\end{align}
making indistinguishable the `pre' from the `post' interaction at the particle velocity level.  \\

\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=2.5]
\node (v) at (1.93,0.52) [preaction={shade, ball color=violet},
%pattern=vertical lines, pattern color=green!30!black
,circle,scale=1.5] {};
\node (vs) at (-1.93,-0.52) [preaction={shade, ball color=white!70!gray},circle,scale=1.5, ,
%pattern=crosshatch dots, pattern color=black
] {};
%\node (V) at (1.158,0.312)  [circle,scale=1.7, shade,ball color=blue] {};
\node (vp) at (1.414,-1.414)%(1.72,-0.25)
[preaction={shade,ball color=violet},circle,scale=1.5, ,pattern=crosshatch dots, pattern color=black] {};
\node (vps) at (-1.414,1.414) %(-1.1,2.57)
[preaction={shade,ball color=white!70!gray}, circle,scale=1.5,pattern=crosshatch dots, pattern color=black ] {};
\node (V) at (0,0)	 [circle,scale=1, shade,ball color=green] {};
\node (start) at (0,-3)  [circle,scale=0.1, gray] {};
%\node (vp old) at (1.414,-1.414)   [circle,scale=1, shade,ball color=green!30!black!70!white] {};
%\node (vps old) at (-1.414,1.414)   [circle,scale=1, shade,ball color=green!30!black!70!white] {};

\begin{scope}[>=latex]
\draw[->,line width=0.4ex,gray!80!black] (V) -- (0.6071,-0.6071) node[near end, above,black] {$\sigma$};
\draw[->,line width=0.2ex] (start) --  (vs) node[left, pos=1.07] {$v_*$};%[left,pos=0.98]
\end{scope}
%\draw (0,0) node{$\bullet$} node[left]{$O$};
%\draw (0,0) circle(0.6cm);
%\draw[->] (0,0)--(-0.707,0.707 ) coordinate (X) node[right]{$x$};
%\draw[->] (0,0)--(0,2.5) coordinate (Y) node[right]{$y$};
\begin{scope}[>=latex]
\draw[->,line width=0.4ex,gray!80!black] ( V) -- ([turn]345:0.8cm) node[midway, above=-0.7cm, black] {$\eta$};
%\draw[->,blue!50!black] (0,0)--($(0,0)!0.6cm!(0,-1)$) node[midway,above left, black] {$r$};
%\draw ($(0,0)!2cm!(4,3)$) coordinate (P) node{$\bullet$} node[above right](P){$P$};
%\draw[->,thick, Orchid!50!black] ($(0,0)!2cm!(4,3)$) -- ([turn]90:1cm) node[above, Orchid!50!black]{$							\overrightarrow{v_e}$};
\draw    pic["$\theta$", draw=red, ->,  angle eccentricity=1.2,  angle radius=1cm]{angle=vp--V--v};
\end{scope}
%		%
\begin{scope}[>=latex, on background layer]
\draw[densely dotted,color=green!50!black!50!white,line width=0.2ex] (0,0) circle (2cm);
\draw[densely dotted,->,color=green!30!black!70!white,line width=0.2ex] (vps) -- (vp) node[densely dotted, color=green!30!				black,pos=0.25,above right=-0.2cm,line width=0.2ex] {};
\draw[->,color=violet!80!white, line width=0.2ex] (start) -- (v)  node[pos=1.0,right=0.4cm] {\color{violet}$v$};
\draw[->,line width=0.2ex] (vs) -- (v) node[near end,sloped, black,above left] {$u$};
\draw[densely dashed,->,color=green!80!black,line width=0.2ex] (start) -- (V) node[color=green!50!black, above=0.03cm] {$V\!=\!			\frac{v\!+\!v_*}{2}$};
\draw[->,color=blue!60!white!40!,line width=0.2ex]  (vps)--(vp) node[color=green!30!black, right=-1.2cm, near start, sloped, above] 			{\color{blue}$\!\!\! \! \ \ \ \  u'\!=\!|u|\sigma$};
\draw[densely dotted,->,color=violet!50!black!50!white,line width=0.2ex] (start) -- (vp) node[color=violet!50!black,below right,line 		width=0.2ex] {$ v'\!=\!V + \frac{1}{2} \left|u\right| \sigma$};
\draw[densely dotted,->,color=black!45!white,line width=0.2ex] (start) -- (vps) node[color=black!90!white, above=0.1cm, left=0.32cm]
{$ V \!-\! \frac{1}{2} \left|u \right| \sigma\!=\!v'_*$};
%\draw[->,densely dashed,blue,line width=0.6ex] (V old) --  (V) node[midway, sloped,above, blue] {$\!\frac12({r_{ij}\!-\!r_{ji})}u_{ij}$};
%\draw[dashdotted,->,color=blue,line width=0.2ex] (start) -- (V) node[pos=0.98,below right=-0.05cm];{\color{blue}$V$;
%\draw[dashdotted,->,color=blue,line width=0.2ex] (start) -- (vps) node[pos=1.05,left] {\color{black}$v'_j=v'_*$};
%\draw[dashdotted,->,color=blue,line width=0.2ex] (start) -- (vp) node[pos=1.1,below right=0.12cm] {\color{red}$v'$};
\draw[->,color=red,line width=0.2ex] (vp) -- (v) node[
%pos=0.15, right=0.07cm, ]
midway, sloped, below]{\color{red} {\scriptsize\color{red}$ v\!-\!v'\!=\!-u^-\!:=\! -\frac12 (-u\!+\!\left|u\right| \sigma  \!) $}};
\draw[->,color=red,line width=0.2ex] (vps) -- (vs) node[
%pos=0.2, right=-3.5cm,
midway, sloped, above] {\scriptsize\color{red} {\color{red}$\tiny\ v_*\!-\!v'_*\!=\! u^-\!:=\!\frac12 (-u\!+\!\left|u\right| \sigma  \!) $}};
\draw[->,color=red,line width=0.2ex] (v) -- (vps) node[midway, sloped, above]
%node[pos=0.5, right=-0.4cm]
{\color{red} {\scriptsize \color{red}$\qquad v_*\!-\!v\! =-u^+\!:= \! -\frac12(u \!+\!|u|\sigma)$ }};
\draw[->,color=red,line width=0.2ex] (vs) -- (vp) node[pos=0.01cm, below right=0.08cm,
sloped, above] {\scriptsize\color{red} {\color{red}$\qquad \ \ u^+\!:= \!\frac12 (u \!+\! |u|\sigma)$}};
\end{scope}
\end{tikzpicture}
\end{center}
\caption{elastic interaction diagram and coordinates	}
%	 Collision Diagram in  center of mass and relative velocity coordinates for binary interactions of systems with disparate masses:  Illustration of the collision transformation, with notation $V_{ij}\!:=\!\frac{m_i v_i \! +\! m_j v_j}{m_i \!+\! m_j}\!=\! r_{ij} v_i\!+\! r_{ji} v_j$, $u\!:=\!u_{ij}\!:=\!v_i-v_j$; \,   $u'_{ij}\!:=\!v'_i \!-\!v'_j\!=\!|u_{ij}|\sigma$. The displacement of the center of mass with respect to a single component elastic binary interaction is given by $(r_{ij}\!-\!\frac 12)u_{ij}\!=\! \frac12 (r_{ij}\!-\!r_{ji}) u_{ij}$, if $m_i\!>\!m_j$.   The reference reversed relative velocity  is denoted by $u_{ji}\!:=\!-u\!=:\!\tilde u$
%	Solid lines denote vectors after a collision. Dash-dotted vectors  represent primed (pre-collisional) quantities that can be calculated from the given data and compared to the case $m_i=m_j$, represented by dotted vectors. Dashed vector direction is the displacement along the direction of the relative velocity $u\!=\!u_{ij}$ proportional to the half difference of relative masses.  Note that the scattering direction $\sigma$  is preserved as the pre-collisional relative velocity $u'_{ij}$ keeps the same magnitude as the post-collisional $u_{ij}$,   $u'_{ij}$ is parallel to the reference elastic pre-collisional relative velocity $|u|\sigma$.
\label{coll-diagram}
\end{figure}
%\caption{elastic interaction diagram and coordinates}
This wording induces the notion of time associated to collisional events, in terms of what is the conformation of  velocity states pairs,   before or after the interaction.
In particular,  while the notation exchange is admissible for binary interacting  pairs
$\{v', v'_*\}$ by $\{\vprime,\vprime_*\}$ for as long the collision is {reversible} or elastic, it will  soon become transparent that  this time  reversibility will be lost  at the level of  Boltzmann equation due  to the Molecular Chaos Assumption which distinguishes time.
Because of properties to be studied soon, this previous comment prompts the immediate quite useful representation given in Section~\ref{pre-notation} on the distinction in the notation of {\it pre} and {\it post} interacting velocities though out this manuscript.\\

\subsection{Time reference notation}\label{pre-notation}
Relations \eqref{eq:6.3} express the post collisional velocities in terms of pre collisional velocities.  It is possible to reverse such presentation to express the pre collisional velocities $\{\vprime,\vprime_*\}$ and its relative velocity $\uprime$ in terms of the post collisional velocities $\{v,v_*\}$ and its relative velocity $u$.  In the impact direction $\eta$ framework the expression for the relative velocity is given, in light of \eqref{eq:6.1}, by the relations
\begin{equation}\label{eq:9.1}%% \tag{7.1}
{u\cdot\tau = \uprime\cdot\tau}\,,\qquad
{\uprime\cdot\eta = -u\cdot\eta}\,, \quad \text{ and }\quad \uprime = u - 2(u\cdot \eta)  \eta\,.
\end{equation}
Now, using equations \eqref{eq:6.3} and \eqref{eq:9.1}, the pair $\{\vprime,\vprime_*\}$ can be written as
\begin{equation}\label{eq:9.3}		%%\tag{7.3}
\begin{array}{l}
\vprime \ =\  v +  ( \uprime \cdot \eta )\eta   \ =\  v - ( u \cdot \eta)  \eta\\
\noalign{\vskip6pt}
\vprime_*\  = \  v_* + ( \uprime \cdot \eta )\eta   \ =\  v_* - (u\cdot \eta)  \eta .
\end{array}
\end{equation}
Comparing expressions \eqref{eq:6.3} and \eqref{eq:9.3} one observes the reversibility of the collisional laws, being the only difference the conceptual interpretation of the pair $\{v,v_*\}$; for the former represents pre collisional velocities and for the later the pos collisional velocities.\\

Consequently, the pre collisional velocities representation in terms of the post collisional velocities in the scattering direction $\sigma$ framework is identical to \eqref{eq:9.3-Vu}, namely for $u^{\pm} = \frac{u \pm |u|\sigma }{2}$
\begin{equation}\label{eq:9.3-Vu-post}		%%\tag{7.3}
\vprime \ = \ v \,+ u^-  ,\qquad \vprime_* \ =\  \ v \, - u^+ \,,\qquad \text{so}  \ \ \uprime :=\vprime-\vprime_*= -|u|\sigma
\end{equation}
The difference lies in the interpretation of the $\sigma$ direction is reversed with respect of the direction of  $\uprime$,  or simply $'\!\sigma=-\widehat{\uprime}$ in light of the last equation in \eqref{eq:9.1}.

\medskip

While this observation may be irrelevant in the definition of local  elastic interactions, {\em it actually is very significant for the derivation of the Boltzmann equation  as it sets the stability properties for the Boltzmann flow}.  In addition, this notation is naturally relevant for the study of the inelastic kinetic theory, where   {\em local inelastic interaction are not reversible,} as shown in [65]

%v + v_* = v' + v'_* \\
%|v|^2 + |v_*|^2 = |v'|^2 + |v'_*|^2
%\end{array}  \qquad
%\text{conservation of}\\ \text{center of mass} \\ \noalign{\vskip12pt}
%\text{conservation of}\\ \text{kinetic energy}
% \includegraphics[angle=0,width=0.8\textwidth,height=0.60\textwidth]{figures/Ex1/Ex1_Density_Damping.png}
%\label{fig:Adv Maxwellian}
% $$\epsfysize=3.0truein \epsfbox{elastic-collision_400x400_p1.eps}$$
%Figure 2 shows the elastic interaction, where, instead of denoting the incoming velocities with $v, v_*$,
%we now use standard notation $v, v_*$ with the corresponding post collisional velocities  $v, v'_*$ univocally determined by the elastic relations describe above.

%  % Requires \usepackage{graphicx}
%  \includegraphics[width=150mm]{Elastic-particle-interaction.pdf}\\
%  \caption{Elastic binary collisions between two particles with incoming velocities $v$ and $v_*$}\label{fig-2}

\smallskip

\vspace{8pt}
\noindent The following lemma rigorously calculates the Jacobian transformation for  elastic interactions  case, and its determinant value, in the impact direction $\eta$ framework, which is
a crucial point in the derivation of the Boltzmann equation.
\bigskip

\begin{lem}\label{lemma1}
\noindent  After an elastic interaction, the determinants of the Jacobian of the variables transformation for the elastic interactions for a pair $\{v,v_*\}$ viewed as either before or after the interaction, are both the same for fixed $\eta\in\mathbb{S}^{2}$.  That is $ \det{J_{(v',v'_*)/(v,v_*)}}\ =\ -1 \ =\
\det{J_{(\vprime,\vprime_*)/(v,v_*)}}$.
\end{lem}
\begin{proof}
In order to show this property,   we use the exchange of coordinates defined in
(\ref{eq:6.1} - \ref{eq:6.3}) for  $(v_,v_*)$ before the interaction, or (\ref{eq:9.1} - \ref{eq:9.3})  after the interaction.  Indeed starting from  \eqref{eq:6.3} for a fixed unitary vector $\eta$, we can subtract the equations to obtain for $u=v - v_*$ and $v_*$, that
\begin{equation}\label{eq:10.1.1}		%% \tag{6.3}
u' = u- 2\left( u \cdot\eta\right) \eta\,,\hspace{1cm} v'_* = v_*  +(u\cdot \eta)  \eta\,.
\end{equation}

\medskip

\nn Since  $J_{(u',v'_{*})/(u,v_{*})} = J_{(v',v'_{*})/(v,v_{*})}$, it suffices to find the Jacobian for the linear transformation \eqref{eq:10.1.1}.  This transformation can be simply written as

{\large \begin{equation}\label{eq:10.1.2}
\begin{bmatrix}
u' \\
\end{bmatrix} \ = \
\begin{bmatrix}
\mathcal{R}_{\eta} &0 \\
\mathcal{P}_{\eta}  & \text{1}_{3\times 3}  \\
\end{bmatrix} \, .
\begin{bmatrix}
\end{bmatrix} \, ,
\end{equation}}
where $\mathcal{R}_{\eta}$ is the simple reflection along $\eta$ and $\mathcal{P}_{\eta}$ the simple projection along $\eta$.  In particular,
\begin{equation}\label{eq:10.1.2}
\det J_{_{{ (u',v'_*)}/{(u,v_*)} } }\ =  \ \det \mathcal{R}_{\eta}  \ = \  -1 \, .
\end{equation}  }

\nn  Due to the nature of elastic interaction laws, a similar proof (left for the reader) shows
{\large \begin{equation}\label{eq:10.1.2}
\det J_{_{{(\vprime,\vprime_*)}/{(v,v_*)} } }\ =      J_{_{ {(\uprime,\vprime_*)}/{(u,v_*)} } }   \ =\  -1 \, .
\end{equation}
\end{proof}
%\section{The Boltzmann argument in modern perspective}
%\noindent The transport Liouville equation is impractical for near equilibrium phenomena ($6 N = 10^{20}$ per cubic cm).
%Instead, heuristically, the Boltzmann equation gives a `scalar" equation for a \textbf{one particle
%probability density function} (pdf) in 7 dimensions, say $f^{(1)}(x,v,t)$. Justification is by means
%of \textbf{statistical independence}.
%\vspace{20pt} \noindent \textbf{Heriarchy reduction.}\\
%\noindent {\bf Step 1:}  Label with $x_1$ and $v$ the `first' particle moving with velocity $v$, and let $f^{(1)}$ be the probability
%of finding this first particle in a certain state no matter what the states of the particles $2, \dots, N$ are.  These statistical hierarchical reduction is done by taking statistical
%{\bf marginals} of the $3N$-point probability density function with respect to velocities $v=v_i,  i=2, \dots, N.$
%Then, the {\bf one-point} probability density function is defined by
% f^{(1)}(x_1,v,t) = \int_{(\mathcal{D}\times \mathbb{R})^{3N-3}}
%P(x_1,v,\dots,x_N,v_N,t)\, dx_2 \, dv_* \,\dots\, dx_N \, dv_N.
%\vspace{6pt} \noindent {\bf Step 2:} \
% Write an evolution equation for $f^{(1)}$ such that it satisfies
% \item [{\bf i} - ] In the absence of collisions (i.e. $N=1$), the one-point density function $f^{(1)}=f$ satisfies the transport Liouville equation.
% \item [{\bf ii} - ] The probability of occurrence of a collision is related to finding a second particle $(x_2,v_*)$
%with a center $x_2$ exactly one diameter from the center $x_1$ of the particle whose probability density  is $f^{(1)}$.

%\vspace{20pt} \noindent Hence, in order  to model the probability of a particle centered at $x_2$, interacting with the one
%centered at $x_1$,  requires  to introduce the {\bf two-point} probability density function:
% f^{(2)}(x_1,v,x_2,v_*,t) =
%\int_{(\mathcal{D}\times \mathbb{R})^{3N-6}} P(x,v,t) \, dx_3 \, dv_3 \,\dots \, dx_N \, dv_N.
%\vspace{8pt} \noindent Therefore, following the free flow dynamics \eqref{freeflow} we may write the balance, infinitesimally in time,
% f^{(1)}\left( x_1 + v \delta t, \, v, \, t+\delta t \right) =
% f^{(1)}(x_1,v,t) + \left( \frac{\partial f^{(1)}}{\partial t} \right)_{Coll} \delta t
%that, as $\delta t \to 0$, implies
%\begin{equation*}  \left( \frac{\partial f^{(1)}}{\partial t} \right)_{Coll} =
% \frac{\partial f^{(1)}}{\partial t} + v \nabla_{x_1} f^{(1)},
% \end{equation*}
%where the collective interactions $\left( \frac{\partial f^{(1)}}{\partial t} \right)_{Coll} $
% must satisfy \textbf{i} and \textbf{ii}  conditions from above.
%\vspace{20pt} \noindent Clearly for $N=1$, the interval of time needed to encounter a collision
%is $\infty$ and so $\left( \frac{\partial f^{(1)}}{\partial t} \right)_{Coll} \equiv 0$. Thus,
%$f^{(1)}$ satisfies the transport Liouville equation and \textbf{i} above is satisfied. In order to
%satisfy both, \textbf{i} and {\bf ii}, it is enough to set
%$\left( \frac{\partial f^{(1)}}{\partial t} \right)_{Coll} $ to be
%the following where $G$ is \textbf{total gain} and $L$ is \textbf{total loss}, yielding:
% \left( \frac{\partial f^{(1)}}{\partial t} \right)_{Coll} =
% (G - L) (f^{(1)}),
%\noindent  Infinitesimally, these two terms $G$ and $L$, respectively,  mean\\
%    \item[Rate of {\bf Loss}] \    $ L\, dx_1 \, dv \, dt $
% is the      expected $\#$ of particles with (position, velocity)$=(x_2,v_*) \in (x_1, x_1 + dx_1) \times (v, v + dv)$
%     that \textbf{disappear (or are knocked out)} from this range of values because of a collision at times in the interval
%     $ (t, t+dt)$   in the  $ v$  direction, that is   {\bf $u \cdot \eta <0$}, or equivalently,  $(v-v_*) \cdot (x_1-x_2) = \frac{d}{dt}(x_1-x_2)^2<0.$
% \item[Rate of {\bf Gain}] \ $G\, dx_1 \, dv \, dt $
% is the    expected $\#$ of particles with (position, velocity)$=(x_2,v_*) \in  (x_1, x_1 + dx_1) \times (v, v + dv)$
%   that \textbf{enter} in this range of values because of a collision at times in the interval
%    $(t, t+dt)$ { in the } $v-$direction, that is  $u \cdot \eta > 0$ , or equivalently,  $(v-v_*) \cdot (x_1-x_2) = \frac{d}{dt}(x_1-x_2)^2>0.$

%  % Requires \usepackage{graphicx}
%  \includegraphics[width=100mm]{elastic-collision1.jpg}\\
%  \caption{Momentum transfer at a binary collision, with $\theta $ is the scattering angle, $b$ is the  impact parameter
%$\gamma= (\pi - \theta)/2${ impact angle} and  $a$ is the diameter of the \em{independent, identically distributed (iid)} spheres}\label{fig-4}
%Since there are $N-1$, disregarding multiple
%collisions and identical point masses, then
%$$G = (N-1) g\qquad\text{ and }\qquad L = (N-1) \kmax$$
% where $g$ and $\kmax$ are (now) the contribution to the collision
%from a {\em fixed} particle, say $x_2$.\\
%Thus,  the momentum transfer as the volume element from
%{\em scattering} in the collision process is  calculated from the  total volume element differential   for a scattering event of a {\bf hard sphere binary} collision  follows from  {\em differential of volume of a cylinder of base $dS = a^2 d\eta$ with
%height $\approx |u_2\cdot\eta|\,dt$}.   Hence   $dx_2 dt \approx a^2 |u_2\cdot\eta|\, d\eta\,dt  $. \\
% \centerline{\epsfxsize=6truein\epsfbox{elastic-collision1.eps}}
%  % Requires \usepackage{graphicx}
%  \includegraphics[width=100mm]{elastic-collision1.jpg}\\
%  \caption{Momentum transfer at a binary collision}\label{fig-4}
%  % Requires \usepackage{graphicx}
%  \includegraphics[width=70mm]{infinite-cylinder.jpg}\\
%  \caption{Differential volume element at a hard sphere scattering of binary collisions }\label{fig-5}
% $$\lower.45truein\hbox{\epsfxsize=2.0truein\epsfbox{infinite-cylinder.eps}}

%Next, define  $f^{(2)} (x_1,v,x_2,v_*,t)\,dx_1\,dv\,dv_*\, a^2 |u_2\cdot \eta|\,d\eta\,
%dt \equiv $ probability of a collision of particle $x_2$ with particle $x_1$
%in the ranges $(x_1,x_1+dx_1)(v,v+dv)$, $(x_2,x_2+dx_2)(v_*,v_*+dv_*)$
%and $(t,d+dt)$ occurring a points $dS$.\\
%Then, fixing the particle with center at $x_1$, and moving the one at $x_2$,
%or equivalently the direction $\eta = \frac{x_2-x_1}{|x_1-x_2|}$,
%along the sphere (of radius $a$) with all possible velocities, the infinitesimal contribution (from the material derivative
%$\partial t + v\nabla_{x_1}$)
%loss term is modeled by\\
%$$\kmax \, dx_1\,dv\,dt =
%\left[ \int_{\real^3} \int_{S^-} f^{(2)} (x_1,v,x_1 + a\eta,v_*,t)
%|u_2\cdot  \eta| a^2 \,d\eta dv_* \right]
%\, dx_1\,dv\,dt$$
%with $S^- := \{u_2\cdot\eta <0\}$, so that,
%taking into account all possible $N-1$ particles colliding with $x_1$
%(i.e. globally in $x_1, v$ and $t$),   one obtains   the gain collisional part is \\
%$$G =
%\left\{ (N-1) a^2 \int_{\real^3\times S^+} \mkern-36mu
%f^{(2)}  (x_1,v,x_1 + a\eta,v_* ,t)
%|(v_*-v)\cdot \eta|\, d\eta dv_* \right\} \,$$\\
%%Gain := integration of particles that {\em moves away} each other before the
%\nn Similarly, the loss collisional part is\\
%$$L  =
%\left[(N-1) a^2 \int_{\real^3\times S^-} \mkern-36mu
%f^{(2)} (x_1,v,x_1 +a\eta,v_*,t) |(v_*-v)\cdot\eta|  \,d\eta dv_* \right]
%\, .$$
%%Loss := integration of particles that {\em moves towards} from each other after
%%the collision.
%\nn Hence
%$$\frac{\partial}{\partial t} f^{(1)} + v \nabla_{x_1} f^{(1)} =
%(N-1) a^2 \mathop{\int\!\!\!\!\int}\limits_{\real^3\times S}
%f^{(2)} (x_1,v,x_1 +a\eta,v_*,t) \left((v_*-v) \cdot\eta\right)
% \,d\eta dv_* .$$
% However this reduction is not very useful.  But, if we use continuity of $P(\bx,\bv,t)$ at the collisions event, in agreement with the collisional law in (1.9)
%v' = v + \eta (\eta\cdot u_2)\ ,\quad
%v'_* = v_* -\eta (\eta\cdot u_2)\ ,\quad
%u_2= v_ 2- v \ , \quad   a\eta=x_2-x_1 \, ;
%then writing the two-point correlation function
%$$f^{(2)}(x_1,v,x_2,v_*,t) =
%f^{(2)}  (x_1,v+\eta(\eta\cdot u_2),x_2,v_* - \eta (\eta\cdot u_2) ,t), $$
%and replacing  by this identity {\bf only in the  gain term}
%G   =
%\bigg( (N-1) a^2\int_{\real^3\times S^+} \mkern-6mu
%f^{(2)} (x_1,\vprime,x_1 + a\eta, \vprime_2,t)
%|(\vprime_2 - \vprime)\cdot \eta | \,d\eta d\vprime_2\bigg)\, .
%Therefore, exchanging  the angular  coordinate $\eta$ by $-\eta$, so that the angular impact direction is  now $\eta=(x_1-x_2)/|x_1-x_2 |$,  reverses the angular integration from post to pre changing the integration domain from $S^+$ to $S^-$, with a $(-1)$ factor from the angular Jacobian. Then,  the gain term $G$ in \eqref{gain_jaco1} becomes

% G &=
% (N-1) a^2\int_{_{\real^3\times S^-}} \mkern-6mu
%\   \frac{-1}{J_{{(\vprime_2,\vprime)}/{(v_*, v)}} } \, f^{(2)} (x_1,\vprime,x_1 - a\eta, \vprime_2,t)
%|(v_* - v)\cdot \eta |  \, d\eta dv_*  \nonumber\\[6pt]
%&\ \\[6pt]
%  &=  (N-1) a^2\int_{_{\real^3\times S^-}} \mkern-6mu
%  \,f^{(2)} (x_1,v - \eta (\eta\cdot u),
%x_1 - a\eta, v_* +\eta (\eta\cdot u), t)
%|(v_* - v)\cdot \eta | \,\eta dv_* dv,   \nonumber
%\nn since  by Lemma~\ref{lemma1}   the exchange for reversible elastic interactions has a Jacobian  $J_{(\vprime,\vprime_2)/(v,v_*)}=-1$,  with now  the relative velocity becomes  $u=v -v_*$.
%\begin{remark}  \ \ The exchange of coordinates at an elastic collision, in addition to the properties discussed earlier, also  satisfy the following additional ones:
% \begin{itemize}
%\item Note that we have used   the Jacobian matrix  of the transformation satisfies
%\break $\det J_{(\vprime,\vprime_*)/(v,v_*)}=-1$ , as shown in Lemma,    which combined
%with the change of $\eta$ to $-\eta$ and the symmetry
%$|(\vprime_2 - \vprime)\cdot \eta | =|(v_* - v)\cdot \eta |$ with respect to this change in impact direction.
%\item  The exchanged of coordinates $\eqref{elas-collis}$ maps $S^+=\{ u\cdot\eta>0\}$ into
%$S^-=\{ u\cdot\eta<0\}.$
%\item One may choose to denote $v-\eta(u\cdot\eta)= v'$ or $\vprime$ as they are
%indistinguishable for a reversible interaction. Similarly $v_*-\eta(u\cdot\eta)= v_*'$ or
% \nn {Now we can understand the Boltzmann argument from a scaling perspective:}
%Take a box  $\Omega$ containing a rarefied gas at room temperature consisting of $N$ identical spherical particles with radius $a$. Let with $N$  be sufficiently large and  $a$  small, such that the difference between a center at $x_1$ and $x_1+a\eta$ is of $\O(a)$ negligible. Then for
%\begin{picture}(300,90)		%% straight cube
%\put(0,10){\line(0,3){25}}	%% left side
%\put(0,10){\line(3,0){25}}	%% bottom side
%\put(0,35){\line(3,0){25}}	%% top side
%\put(25,10){\line(0,3){25}}	%% right side
%\put(14,19){\line(0,3){25}}	%% left side
%\put(14,19){\line(3,0){25}}	%% bottom side
%\put(14,44){\line(3,0){25}}	%% top side
%\put(39,19){\line(0,3){25}}	%% right side
%\put(60,40){$N\approx 10^{20}$}
%\put(60,25){$a = 10^{-8}$ cm}
%\put(60,10){$(N-1) a^2 \cong Na^2 = 10^4 \text{ cm}^2 = 1\text{m}^2$}
% $N\approx 10^{20}$, $a = 10^{-8}$ cm, and  $(N-1) a^2 \cong Na^2 = 10^4 \text{ cm}^2 = 1\text{m}^2$,
%since the {\em volume} occupied by the particles is $N \, a^3 \approx
%10^{-4} \text{cm}^3$, that means collisions between two {\em pre-selected} particles
%are rare. This implies that, in such {\em rarefied gas state}, two spheres that collide are actually two {\em randomly
%chosen} particles (or equivalently  two {\em uncorrelated}  positions and velocities {\em
% before the interaction}).

%\subsection{Molecular chaos hypothesis and the Boltzmann equation}
%A very fundamental assumption  of propagation of decorrelation before the interactions is invoked:
% f^{(2)} (x_1,v,x_2,v_*,t)
%= f^{(1)} (x_1,v,t) f^{(1)} (x_2,v_*,t) \qquad \text{ for two particles that are {\em about to collide}}.
%\noindent {\bf It is important to stress that
%this assumption means that is  applied  only to the loss term, } and therefore to the gain term
%when written in the {\em pre-collisional} velocities $\vprime$ and $\vprime_2$.
%Thus, as $a\to 0$, $N\to\infty$, $x_1+a\eta \to x_1$,   the transport  of probability densities is reduced to a scalar non-local (integral)  non-linear  equation for the {\bf single-point probability density function (or pdf)}  $f^{(1)}(x,v,t)$,  given by
%$$\left( \frac{\partial f^{(1)}}{\partial t} + \frac{\vec v}{m}
%\nabla_{x_1} f^{(1)}\right) (x_1,v_*,t)
%\ = \  Na^2 Q (f^{(1)}) (x_1,v,t)$$
%\nn where the  collision operator   becomes
%Q(f^{(1)}) (x_1,v,t) &=
%\iint_{\real^3 \times S^+} \mkern-6mu
%f^{(1)} (x_1,\vprime,t) f^{(1)} (x_1,\vprime_2,t)
%|(\vprime_2 - \vprime)\cdot \eta| \, d\eta d\vprime_2  \nonumber \\
%&\qquad\qquad - \int_{\real^3\times S^-} \mkern-6mu
%f^{(1)} (x_1,v,t) f^{(1)} (x_1,v_*,t) | (v_*-v)\cdot\eta|  \,d\eta dv_*  \label{mol-chaos}
%&=\int_{\real^3\times S^-}\mkern-6mu
%\left[ f^{(1)} (x_1,\vprime,t) f^{(1)} (x_1,\vprime_2,t)
%-    f^{(1)} (x_1,v,t) f^{(1)} (x_1,v_*,t)\right] |(v_*-v)\cdot \eta|
% \, d\eta dv_*
%with $u=v -v_*$ and $\eta$  which is now an {\em unitary
%vector} in the direction of either $v- \vprime$, or $\vprime_2- v_*$ or   $u'-u$, as can be easily see  in the collision diagram from  figure~\ref{fig-2}.  \\
%\nn Denoting, from now on,
%$v$ by $v$, $v_*$ by $v_*$ and
%$f^{(1)}(x,v,t) =f$, $f^{(1)}(x,v_*,t) = f_*$ and the corresponding
%$f^{(1)}(x,v'_*,t) = f'$ and $f^{(1)}(x,v'_*,t) = f'_*$ as well as the corresponding
%precollisional notations, and  assuming a   symmetry of the {\em angular}
%contribution of the collision kernel  $|u\cdot \eta|$ as $\eta \in S^+$ or $\eta \in S^-$,
%the choice of the angular integrating domain between either $S^+$ or $S^-$ is irrelevant.
%The traditional convention choses the former one. \\

\subsection{The collision kernels}
Transition probability rates quantifying the scattering rates of  collisional transfer between before and after the interactions as  a function of the relative speeds and   impact or scattering directions,  are often referred  collision kernels taking the form
\begin{equation}\label{coll-ker}                             %{coll-ker}
B\left(|u|, \frac{|u\cdot\eta|}{|u|}\right)
\end{equation}
where $|u|$ the relative speed determined by the intramolecular potentials,  and the angular transition function dependence on $\hat u\cdot\eta$ the impact (or  $\hat u\cdot\sigma$ scattering) angle determined by the strength of the collision with respect to the angular dependence of such interaction, with $\hat u$ denoting the renormalized unit vector in the direction of $u$

\medskip

\nn It is common  to assumed that, for instance, $ B(|u|, \hat u\cdot\eta)$ takes the form
\begin{equation}\label{coll-ker-2}                             %{coll-ker}
B(|u|, \hat u\cdot\eta)= \Phi_\gamma(|u|) \, b(\hat u\cdot\eta)\, ,
\end{equation}
where $\Phi(|u|)$ is called the  potential calculated from intramolecular potential laws for binary interactions,  and  $b(\hat u\cdot\eta)$ the angular transition probability or  differential cross section.
They are usually assumed to be power laws but this is not necessary. The growth  of $\Phi(|u|)$ and  integrability  of $b$ conditions are the most important properties to consider since they sensibly affect the fundamental characteristics of solutions to the equation.  Typical collision kernels and designations found in the literature are, for example,  given by a potential transition rate function $\Phi_\gamma(|u|)$  to be non-negative and to satisfy the following  $\gamma$-order homogeneity condition
$c_{\Phi}  |u|^\gamma \le  \Phi_\gamma (|u|)  \le C_{\Phi} |u|^\gamma,$, with $c_{\Phi}$ and $C_{\Phi}$  positive constants, independent of the relative velocity  $u$. The potential rate $\gamma$ is a crucial characteristic of, not only the physics modeling of the interaction, but also the profound impact in the analytical properties of the Boltzmann flow.

These rates are commonly classify by referring to these potential rates as to  {\it hard sphere model} if $ \gamma=1$;   as {\it hard potentials model}  if $0< \gamma<1$; as {\it  Maxwell type interactions model}  if $ \gamma=0$ and $ c_{\Phi}= C_{\Phi}$ constant;
as  {\it soft potentials model}  if $-d< \gamma<0$;  and  as { \it Coulomb potential model}  if  $\gamma^{-d}$, in which case the Boltzmann model fails to be well posed problem if  the angular transition $b(\hat u\cdot\eta)\in L^1(\mathbb {S}^{d-1}).$\\

\subsection{Scattering direction representation for the collision transition probability or cross section}
%\textcolor{red}{Should we give the discussion of section 1.3 and on in $d$ dimension given this section?}
The angular coordinates exchange from the impact direction $\eta$ to the scattering direction $\sigma$ is carefully described as follows.  Let $u= v-v_*$ the relative velocity associated to an elastic  interaction and let $P$ be the orthogonal plane to $\widehat{u}$.  Taking spherical coordinates to represent the coordinate system $\{\widehat{u},P\}$ generating $\mathbb{R}^{d}$,  set $\{r , \phi,\ep_1,\ldots,\ep_{d-2}\}$, where $r>0$ is the radial coordinate, $\phi \in [0,\pi)$ the polar angle (being the zenith defined by $\widehat{u}$), and $\{\ep_1,\ldots,\ep_{u-2}\}$ the $d-2$ azimuthal angular variables associated to $P$, then,
% {\parbox{2.0truein}{\epsfxsize=55truemm\epsfbox{BTE.eps}}
%  % Requires \usepackage{graphicx}
%  \includegraphics[width=100mm]{BTE.jpg}\\
%  \caption{}\label{fig-11}
since the scattering direction  $\sigma$ is unit vector in the direction deflecting from specular reflection of $u$ in the plane orthogonal to the $\eta$ direction,  it is clear that $|u| \sigma = u - 2(u\cdot \eta)\eta = u'_{elastic}$ is the specular reflection relation,  or, equivalently,
$\sigma = \hat{u} - 2(\hat{u}\cdot\eta)\eta\,.$

Gathering all the angular information, the Boltzmann collisional integral is modeled
in terms of  the scattering direction  $\sigma$ instead of  the impact direction $\eta$, as a consequence it is needed to  perform the exchange of coordinates $(\eta, v_*) \to (\sigma,v_*)$ for  both $\eta$ and $ \sigma$ in $\mathbb{S}^{d-1}$,  proceeding as in the following steps.

First, starting form  item 2 in subsection~\ref{items elastic}, and using that $
\hat u\cdot \widehat P=0$,  the impact $\phi$ and scattering $\theta$ angles are related through the identities
\begin{equation}\label{cos theta sigma}
\cos\phi = -\hat{u}\cdot\eta\quad\text{and }\quad \cos\theta = \hat{u}\cdot\sigma = 1-2|\hat{u}\cdot\eta|^2= 1- 2\cos^2\phi\,.
\end{equation}

\medskip

%Starting  from item 2 in subsection~\ref{items elastic},  the relation between the impact angle of $\phi$ and direction $\eta$ are determined by  $\frac{u\cdot \eta}{|u|} = \cos(\pi-\phi)=-\cos\phi$.
%In addition, the impact  and  scattering angles  satisfy the relation  $2\phi = \pi - \theta\,. $
From the second identity in \eqref{cos theta sigma}, clearly the cosine of the impact angle is
\begin{equation}\label{scat-direc.2}
\cos\phi = \big|\hat u\cdot \eta\big| \ = \ \sqrt{ \frac{1 - {\hat u\cdot\sigma} }{2} }  \, ,
\end{equation}
and the unitary impact $\eta$  is represented  by the decomposition in the impact spherical  framework
$\eta = -\cos(\phi)\widehat{u} + \sin(\phi)\widehat{P}$.
%and so  the representation of the scattering direction $\sigma$ takes the form
%\sigma =  \hat{u} - 2(\hat{u}\cdot\eta)\eta=\widehat{u} + 2\cos(\phi)\big(-\cos(\phi)\widehat{u} + \sin(\phi)\widehat{P}\big) = (1-\cos(2\phi))\hat{u} + \sin(2\phi)\widehat{P}\,.
%Next, taking the dot product of the scattering and $\hat u$, using \eqref{cos theta sigma}, yields
% \begin{align*}
%\cos\theta =\sigma\cdot \eta &=  \hat{u}\cdot \eta  - 2(\hat{u}\cdot\eta) = -\hat{u}\cdot \eta  =
% (1-\cos(2\phi))\hat{u}\cdot\eta. =  (1-\cos(2\phi)) \cos\phi +  \sin(2\phi)\sin\theta.

%Next, taking the dot product of the scattering and impact direction, using \eqref{cos theta sigma}, yields
% \begin{align*}
%\cos\theta =\sigma\cdot \eta &=  1-\cos(2\phi)  +  \sin(2\phi)\widehat P \cdot \hat u = 1-\cos(2\phi)
%In other words, the transformation simply doubles the impact angle.

Next,  Under this framework,
the differential of area element of  impact direction $\eta\in \mathbb{S}^{d-1}$
as a function of the reference  coordinates $(\phi, \ep)$ is given by
\begin{equation}\label{angle.2}
{\rm d}\eta =  \sin^{d-2} (\phi)\, {\rm d}\phi\,{\rm d}\ep\,,
\end{equation}

Correspondingly, the differential of area element of  scattering direction $\sigma$
as a function of the impact direction $\eta$, in the reference  coordinates $(\phi, \ep)$, is given by
\begin{equation}\label{angle.3}
{\rm d}\sigma = 2 \sin^{d-2} (2\phi)\, {\rm d}\phi\,{\rm d}\ep\,.
\end{equation}

As a consequence of the relations \eqref{angle.2} and \eqref{angle.3},
\begin{equation}\label{angle.4}
{\rm d}\sigma = \frac{2 \sin^{d-2}(2\phi)}{\sin^{d-2}(\phi) } \, {\rm d}\eta = 2^{d-1} \cos^{d-2}(\phi) \, {\rm d}\eta \, .
\end{equation}
%\item[{4.}] Now we can ``dot $\widehat{u}\,$" the expression \eqref{scat-direc} to obtain
%\cos(\phi) = \big|\hat u\cdot \eta\big| \ = \ \sqrt{ \frac{1 - {\hat u\cdot\sigma} }{2} }  \, .

\medskip

\nn In particular, in the case   of hard spheres collisional symmetric kernel in three dimensions $B(|u|, {\hat u\cdot\eta} ) = (u\cdot\eta)_+$, relation \eqref{angle.4} yields the identity $(u\cdot\eta)_+  {\rm d}\eta\, =  |u| \cos\phi\,  {\rm d}\eta\, =  \frac{|u|}4\, {\rm d}\sigma$.  That is,
for any integrable function $g$ on the sphere $\mathbb{S}^{2}$
\begin{equation}\label{angular.1}
\int_{\mathbb{S}^{2}} \ (u\cdot\eta)_+\, g\big((u\cdot\eta)\eta\big) {\rm d}\eta =
\frac{ |u|}4 \int_{\mathbb{S}^{2}_{+}} g\Big( \frac{u-|u|\sigma}{2} \Big) {\rm d}\sigma  \, .
\end{equation}
For more general kernels in $d$-dimension, the exchange of angular coordinates becomes
\begin{equation}\label{angular.2}\begin{split}
\int_{\mathbb{S}^{d-1}}   g\big((u\cdot\eta)\eta\big)\, B_{\eta} (|u|, (\widehat u\cdot\eta)_{+} ) {\rm d}\eta = \int_{\mathbb{S}^{d-1}} g\Big( \frac{u-|u|\sigma}{2} \Big) \, B_{\sigma}(|u|,\widehat{u}\cdot \sigma )\, {\rm d}\sigma \,,
\end{split}\end{equation}
where, from relation \eqref{angle.4}  and \eqref{scat-direc.2} in $d$-dimensions
\begin{equation}\label{angular.3}
B_\sigma (|u|,{\widehat u\cdot \sigma} ) \ =\ \frac{1}{2^{d/2}} \big( 1 - \widehat{u}\cdot\sigma \big)^{-\frac{d-2}{2}}\,B_\eta \Big( |u|, \sqrt{\tfrac{1-\widehat{u}\cdot\sigma}{2}}\Big)\,.
\end{equation}

\bigskip

\subsection{Integrability of the angular transition}
The Grad's cut-off assumption [73] is simply the condition that the transition probability  kernel $B(|u|,\widehat{u}\cdot\sigma)\in L^\infty(\mathbb{S}^{d-1})$, that is  the angular transition density being bounded in $\widehat{u}\cdot\sigma$ with $\sigma\in \mathbb{S}^{d-1}$. While some literature has referred as to Grad's cut-off assumption having an  integrable the angular part, this manuscript will not use the {\it cut-off} wording.
More precisely, this   work extends all known results for the existence, uniqueness and long time stability  to the weaker condition of assuming  that the angular transition density  is integrable, exactly meaning $B(|u|,\widehat{u}\cdot\sigma)\in L^1(\mathbb{S}^{d-1}, d\sigma), $ without any constraints on the pointwise behavior of  $B(\cdot,y)$ with respect to $y$.  That is,
\begin{align}\label{grad-cut-off.1}
&\int_{\mathbb{S}^{d-1}} B\big(|u|,{\hat u\cdot \sigma}\big) \,{\rm d}\sigma = \omega_{d-2}   \int_{-\pi}^\pi B (|u|,\cos(\theta))\sin\theta^{{d-2}} \,{\rm d}\theta  \ =\ \omega_{n-2}   \int_{-1}^1 B (|u|,z)({1-z^2})^{\frac{d-3}2} \,{\rm d}z
\end{align}
is an a.e. finite function of $|u|$, with $\omega_{d-2}$ the surface measure of the $d-2$ dimensional sphere.  In general, the integrand on \eqref{grad-cut-off.1} is not bounded.  In the cases when $B(|u|,\cos(\theta)) \sim |u|^\gamma \,  b(\cos(\theta))$
%\begin{align*}B(|u|,\cos(\theta)) \sim |u|^\gamma \,  b(\cos(\theta))
the integrability assumption simply becomes
$\int_0^\pi b(\cos(\theta)) \sin^{d-2} \theta\, {\rm d}\theta\  < \infty.$

The techniques proposed in this manuscript to overcome the  lack of boundedness of  $b(\hat u\cdot \sigma)$ at  pointwise behavior in
$\hat u\cdot \sigma$, is overcome by decompositions qualifying the amount of  mass of the function $b(\hat u\cdot \sigma)$ at points of singularities.

%\subsection{Fundamental properties of the collisional integral}
This section is discuss under the assumption  that there, is at least, a positive solution $f$ of the elastic Boltzmann equation that has the regularity and decay at infinity properties that are needed to perform all the operations that follow.
There are several fundamental  properties of the {\em elastic collision integral operator} $Q(f,f)((x,v,t)$ when focusing on it as a function of  velocity space $v$.  A more general form of the Boltzmann equation is given by, for $d\ge 2$
\begin{equation}\label{eq:elementary2}
{Q(f,f)(v) = \int_{\real^d} \int_{\mathbb{S}^{d-1}} \Big( \frac1{{J}}\fprime\fprime_* - ff_* \Big)  \,B(|u|, \hat u\cdot\eta)
d\eta dv_*} \, .
\end{equation}
where  $u=v-v_*$ is the relative molecular velocity for an interacting pair  $\vprime = v- \eta (\eta\cdot u)$, $\vprime_* = v_* + \eta(\eta\cdot u)$, respectively, whose exchange of molecular  velocity  coordinates transformation from pre to post binary interaction is given by the Jacobian  ${{J}}= | \det J_{(v,v_*)/(\vprime,\vprime_*)} |$.   In addition,  if $B(|u|, \hat u\cdot\eta)\in L^1(\mathcal S^{d-1}, d \sigma)$ then
\begin{equation}\label{eq:elementary2}
Q(f,f)(v) = \int_{\real^d\times \mathbb{S}^{d-1}} \frac1{J}f(\vprime)f(\vprime_*) \,B(|u|, \hat u\cdot\eta)
d\eta dv_* \ -\ f(v)  \nu(v),
\end{equation}
where $\nu(v) =   \int_{\real^d\times \mathbb{S}^{d-1}}f(v_*)  \,B(|u|, \hat u\cdot\eta)
d\eta dv_* $ is the collision frequency.  Of course, in the case of elastic interactions Lemma \ref{lemma1} shows that $J=1$.

\subsection{Weak or Maxwell formulation}\label{weak-maxell-form}
The weak or Maxwell formulation of the Boltzmann collision operator is a fundamental tool in the basic  analysis of the dynamics of the Boltzmann flow.  In fact moment estimates are obtained by averaging properties of the collisional integral when multiplied and integrated by a test function in $v$-space.

\smallskip
\noindent
We start introducing the symmetrized form of the collisional integral \eqref{BTE} is defined as
\begin{equation*}\label{eq:sym-4.1}
Q_s (f,g) (v):=\frac12 \iiint_{\real^d\times\times \mathbb{S}^{d-1}}
\left[ \frac1J\left( \fprime\gprime_* + \gprime\fprime_* \right) - fg_* - f_*g\right] \,   B(|u|, \hat u\cdot\eta )\,{\rm d}\eta {\rm d}v_* \,,
\end{equation*}
with, clearly,  $Q_s(f,f)=Q(f,f)$.  The objective is to use the interchange of velocity and angular coordinates to yield an integral formulation where the exchange of molecular velocity laws only appears in the test function.  Thus, let $\varphi (v)$ be a scalar value function $\varphi(v) :\real^d \to\real$, it holds that
\begin{equation}\label{eq:4.1}  	%% \tag{4.1}
\begin{split}
\int_{\real^d} & Q_s (f,g) (v) \varphi (v)\, {\rm d}v\\
& = \frac12 \iiint_{\real^d\times\real^d \times \mathbb{S}^{d-1}}
\left[ \frac1J\left( \fprime\gprime_* + \gprime\fprime_* \right) - fg_* - f_*g\right]  \varphi (v)\,   B(|u|, \hat u\cdot\eta )\,{\rm d}\eta {\rm d}v_* {\rm d}v\,.
\end{split}
\end{equation}
\begin{thm}\label{Maxwellwf} The weak or Maxwell formulation of the collisional integral, written in symmetric form is given by
\begin{equation}\label{maxwell-weak}  	%% \tag{4.1.}
\begin{split}
\int_{\real^d}  & Q_s (f,g) (v) \varphi (v)\, {\rm d}v\\
& = \frac14 \iiint_{\real^d\times\real^d \times \mathbb{S}^{d-1}}
\left( fg_* + f_*g\right)  \big( \varphi (v')+\varphi (v'_*)- \varphi (v)-\varphi (v_*) \big) B(|u|, \hat u\cdot\eta ) \,{\rm d}\eta {\rm d}v_* {\rm d}v\,,
\end{split}
\end{equation}
where  now $v'$ and  $v'_*$ are the {\bf post-collisional velocities} to $v$ and $v_*$, respectively.
\end{thm}
\begin{proof} Note that $J\, {\rm d}\vprime_* {\rm d}\vprime = {\rm d}v_* {\rm d}v$.  Therefore, for the terms related to the gain collisional part it holds that
\begin{equation}
\begin{split}\label{eq:4.2}  	%% \tag{4.2}
A_1 :&= \frac12  \iiint_{\real^d\times\real^d \times \mathbb{S}^{d-1}}\frac1J \big(  \fprime_*\,\gprime + \gprime_*\,\fprime\big)\varphi (v) B(|u|, \hat u\cdot\eta )\,{\rm d}\eta {\rm d}v_* {\rm d}v\\
&=\frac12  \iiint_{\real^d\times\real^d \times \mathbb{S}^{d-1}} \big(  \fprime_*\,\gprime + \gprime_*\,\fprime\big)\varphi (\vprime -  ( \uprime \cdot \eta )\eta) B(|'\!u|, -\widehat {'\!u}\cdot\eta ){\rm d}\eta {\rm d}\vprime_* {\rm d}\vprime\,.
\end{split}
\end{equation}
We used, for the second equality, \eqref{eq:9.3} and \eqref{eq:9.1} to express $v$ and $u\cdot\eta$ in terms of the pre collisional velocities and the fact that $|u|=|'\!u|$ due to the elastic nature of the interaction.  Consequently, performing the change $\eta\rightarrow-\eta$ it follows that
\begin{equation}
\begin{split}\label{eq:4.21}  	%% \tag{4.2}
A_1 &=\frac12  \iiint_{\real^d\times\real^d \times \mathbb{S}^{d-1}} \big(  \fprime_*\,\gprime + \gprime_*\,\fprime\big)\varphi (\vprime -  ( \uprime \cdot \eta )\eta) B(|'\!u|, \widehat {'\!u}\cdot\eta ){\rm d}\eta {\rm d}\vprime_* {\rm d}\vprime\\
&=\frac12  \iiint_{\real^d\times\real^d \times \mathbb{S}^{d-1}} \big(  f_*\,g + g_*\,f\big)\varphi (v') B(|u|, \widehat {u}\cdot\eta ){\rm d}\eta {\rm d}v_* {\rm d}v\,,
\end{split}
\end{equation}
where in the latter we simply renamed the pre collisional variables as $(v,v_*)$.  Of course, $v'$ is the post collisional velocity defined in \eqref{eq:6.3}.  Additionally, performing the interchange $v\leftrightarrow v_* $ which produces the interchange $\vprime\leftrightarrow \vprime_* $ and, then, the change $\eta\rightarrow-\eta$ it follows that
\begin{equation}
\begin{split}\label{eq:4.22}  	%% \tag{4.2}
A_1 &=\frac12  \iiint_{\real^d\times\real^d \times \mathbb{S}^{d-1}} \big(  f_*\,g + g_*\,f\big)\varphi (v'_*) B(|u|, \widehat {u}\cdot\eta ){\rm d}\eta {\rm d}v_* {\rm d}v\,.
\end{split}
\end{equation}
Then, adding \eqref{eq:4.21} and \eqref{eq:4.22} it holds that
\begin{equation}\label{eq:4.23}
A_1 =  \frac14\iiint_{\real^d\times\real^d \times \mathbb{S}^{d-1}} \big(  f_*\,g + g_*\,f\big)\big(\varphi(v') + \varphi (v'_*)\big) B(|u|, \widehat {u}\cdot\eta ){\rm d}\eta {\rm d}v_* {\rm d}v\,.
\end{equation}
For the term related to the loss part one simply uses the interchange $v\leftrightarrow v_*$ and, then, the change $\eta\rightarrow-\eta$ to obtain that
\begin{equation}
\begin{split}\label{eq:4.24}  	%% \tag{4.2}
A_2 :&=\frac12  \iiint_{\real^d\times\real^d \times \mathbb{S}^{d-1}} \big(  f_*\,g + g_*\,f\big)\varphi (v) B(|u|, \widehat {u}\cdot\eta ){\rm d}\eta {\rm d}v_* {\rm d}v\\
&=\frac12  \iiint_{\real^d\times\real^d \times \mathbb{S}^{d-1}} \big(  f_*\,g + g_*\,f\big)\varphi (v_*) B(|u|, \widehat {u}\cdot\eta ){\rm d}\eta {\rm d}v_* {\rm d}v\,.
\end{split}
\end{equation}
Then, adding the two lines in \eqref{eq:4.24} one has that
\begin{equation}\label{eq:4.25}  	%% \tag{4.2}
A_2 =\frac14  \iiint_{\real^d\times\real^d \times \mathbb{S}^{d-1}} \big(  f_*\,g + g_*\,f\big)\big(\varphi (v) + \varphi(v_*)\big) B(|u|, \widehat {u}\cdot\eta ){\rm d}\eta {\rm d}v_* {\rm d}v\,.
\end{equation}
The result follows after subtracting \eqref{eq:4.25} from \eqref{eq:4.23}.
\end{proof}
The following corollary is a direct consequence for the quadratic case $g=f$.
\begin{cor}\label{coro4}   {\bf  Weak form or Maxwell form  of the collisional integral}
\begin{align}
\begin{split}\label{eq:maxwell}
\int_{\real^3} Q(f,f) \varphi (v) \,{\rm d}v
&= \frac12\iint_{\real^d\times\real^d }
ff_*  \left(\int_{\mathbb{S}^{d-1}} \big(\varphi (v') + \varphi (v'_*) - \varphi (v) - \varphi (v_*)\big)
B(|u|, \hat u\cdot\eta ) {\rm d}\eta \right) {\rm d}v_* {\rm d}v\\
&= \frac12\iint_{\real^d\times\real^d }f(v)f(v-u) G_{i}(v,u) {\rm d}u {\rm d}v\, ,  \quad i=1,2, \quad \text{with} \\
G_{1}(v,u) :&=  \int_{\mathbb{S}^{d-1}} \big( \varphi (v') + \varphi (v' - u') - \varphi (v) - \varphi (v-u) \big)
B(|u|, \hat u\cdot\eta ) {\rm d}\eta\,,\\
G_{2}(v,u):&= 2\int_{\mathbb{S}^{d-1}} \big( \varphi (v') - \varphi (v) \big)
B(|u|, \hat u\cdot\eta ) {\rm d}\eta\,.
\end{split}
\end{align}
\end{cor}
\begin{remark} Corollay \ref{coro4} exhibit a {\sl  weighted  double convolution form} for the weak form of the collisional integral.
This structure is due to the bilinear (or multilinear) mixing associated to these operators.
Such structures are   actually  very convenient forms for analytical and computational purposes related to the Boltzmann equation.
\end{remark}
Something very unique associated to elastic interactions is the micro irreversibility of collisions, that is, $v' = \vprime$ and $v'_* = \vprime_*$.  A consequence of such irreversibility and Theorem \ref{Maxwellwf} is the following symmetric weak form which consequently produce the dissipation of entropy for the Boltzmann model.
\begin{cor}\label{coro5}{\bf  Symmetric weak form or Maxwell form of the collisional integral}
\begin{align*}
\int_{\real^d} Q(f,f) \varphi(v){\rm d}v
& = \frac14 \int_{\real^d\times\real^d\times \mathbb{S}^{d-1}}
(f'f'_* \!-\! ff_*) \big(\varphi(v)\!+\varphi(v_*) -  \varphi(v'_*) - \varphi(v'_*) \big)B(|u|, \hat u\cdot\eta ){\rm d}v_*{\rm d}v{\rm d}\eta\,.
\end{align*}
\end{cor}
\begin{proof}
Due to irreversibility $\varphi (v') + \varphi (v'_*) = \varphi (\vprime) + \varphi (\vprime_*)$.  Recall also that $J=1$ due to elasticity, consequently one has that
\begin{align*}
\int_{\real^d}& Q(f,f) \varphi(v){\rm d}v\\
&=\frac12\iint_{\real^d\times\real^d\times\mathbb{S}^{d-1} }
ff_* \big(\varphi (v') + \varphi (v'_*) - \varphi (v) - \varphi (v_*)\big)
B(|u|, \hat u\cdot\eta ) {\rm d}\eta {\rm d}v_* {\rm d}v\\
&=\frac12\iint_{\real^d\times\real^d\times\mathbb{S}^{d-1} }
ff_* \big( \varphi (\vprime) + \varphi (\vprime_*) - \varphi (v) - \varphi (v_*)\big)
B(|u|, \hat u\cdot\eta ) {\rm d}\eta {\rm d}v_* {\rm d}v\\
&= \frac12\iint_{\real^d\times\real^d\times\mathbb{S}^{d-1} }
f'f'_* \big( \varphi (v) + \varphi (v_*) - \varphi (v') - \varphi (v'_*)\big)
B(|u|, \hat u\cdot\eta ) {\rm d}\eta {\rm d}v_* {\rm d}v\,.
\end{align*}
The result follows adding the first and last equalities.
\end{proof}

\subsection{Collision invariants  of $Q(f,f)$ }

It is very easy to see  from \eqref{eq:maxwell}  that
testing the elastic collisional integral with any linear combination of test function
$\varphi(v)\in \text{span}\{ 1, v, |v|^2 \}$,  with  arbitrary coefficients $A,B$ and $C$, the elastic interacting pairs $(v,v_{*})$ and $(v',v'_{*})$ satisfying
\begin{equation}\label{elastic-col}
\begin{aligned}
v+v_* \  & = \ v'+v'_*   \qquad \ \ \   \text{local momentum conservation} \\
|v|^2 +|v_*|^2 \ & = \  |v'|^2+|v'_*|^2     \qquad \text{local energy conservation,}
\end{aligned}
\end{equation}
cleary, by Corollary \ref{coro4}, the following equivalence hold
\begin{equation}\label{eq:collision2}		%% \tag{7.1}
\int_{\real^d} Q(f,f) \varphi (v) \,{\rm d}v = 0\quad\text{ if and only if }
\quad {\varphi (v'_*) + \varphi (v') = \varphi (v) + \varphi (v_*)},
\quad \text{a.e. in} (v,v_*).
\end{equation}
In particular, any $\varphi (v)$ satisfying \eqref{eq:collision2} for any vectors $(v,v_{*})$ and $(v',v'_{*})$ satisfying \eqref{elastic-col},  is referred to as a {\em collision invariant}.
The three invariants of the collisional integral are referred as conservation of mass, momentum and kinetic energy conditions, which  follows from the transport equation when integrating in velocity space and in physical space under boundary conditions without external sources of energy. \\

Next, the {\it the Boltzmann Theorem} resolves the question on whether it is possible to show that  elastic collisional law is enough to show that, if $\int_{\real^d} Q(f,f) \varphi (v) \,{\rm d}v=0$  for any multiplier  $\varphi(v)$,  then  $\varphi(v) = A +B\cdot v + C|v|^2$ for $v\in\real^3$, under the assumptions that  $f$ is assumed non-negative, so $ff_* > 0$ in $\real^{2d} \times \mathbb{S}^{d-1}$,  and the transition probability or collision cross section $ B(|u|, u \cdot \eta)=0$ has positive  measure zero in $\real^d \times \mathbb{S}^{d-1}.$  \\

\begin{thm}[The Boltzmann Theorem]\label{BoltzTheo}
The function  $\varphi(v)$ is a collision invariant if and only if
\begin{equation}\varphi (v) = \varphi (v) = A+\vec B\cdot v +  C|v|^2,\qquad \text{or all}\ v\in\real^3\,.
\end{equation}
\end{thm}

This statement was first proven by   Boltzmann in 1890 for  $\varphi$ twice differentiable, Cercignani and Arkeryd 1970 for
$\varphi$ in $L^1_{\loc}$, and  Wennberg in 1992 for $\varphi$ just a distribution on $v$. We show here a different proof.

\begin{proof}
Let any distribution $\varphi \in \mathcal{D}^1 (\real^d)$
be a collision invariant satisfying the relation  \eqref{eq:collision2}
%\varphi (v') + \varphi(v'_*) = \varphi (v) + \varphi(v_*)
for all pairs $(v,v_*)$ and $(v' ,v'_*)$ generating an elastic interaction
\begin{equation}\label{B2}
v+v_* = v' + v'_* \qquad\text{ and }\qquad
|v|^2 + |v_*|^2 = |v'|^2 + |v'_*|^2,
\end{equation}
%it is  equivalent to  having a quadratic polynomial of the form
%\begin{center}$\varphi (v) = A+\vec B\cdot v +  C|v|^2$ \ \ \ for all $v \in \real^d$.
%That means $\varphi (v)$ is a ``collision invariant''  of the elastic Boltzmann collisional operator if it is generated by a $N+2$ dimensional polynomial space give by the $\Span \{1,\vec v,|v|^2\}$.

Take any arbitrary $V\in \real^d$, and consider any two arbitrary vectors ${\bf v_1}$ and ${\bf v_2}$ with {\em the same magnitude}.
These vectors can be thought to be taking an arbitrary center of mass  $V$, and two arbitrary relative velocities ${\bf v_1}:=u$ and ${\bf v_2}:=|u|\sigma$, respectively, as it will be shown next.

Set the interacting pairs     $(v,v_*)$ and $(v' ,v'_*)$ defined  the arbitrary vectors, $V$, ${\bf v_1}$ and ${\bf v_2}$ as defined by the choice   the center of mass and relative velocity coordinates \eqref{eq:9.3-Vu},  that is
\begin{align}\label{four-point}
(v, v_*)&= (V+{\frac{\bf v_1}2} ,V-{\frac{\bf v_1}2}) = (V+{\frac{u}2} ,V-{\frac{u}2})  \qquad \qquad\ \mathrm{ and} \nonumber \\
(v',v'_*)&= (V+\frac{\bf v_2}2, V-{\frac{\bf v_2}2}) = (V+{\frac{|u|\sigma}2} ,V-{\frac{|u|\sigma}2}),
\end{align}
which clearly satisfy the local conservation properties relations in \eqref{B2}, since, using  $|{\bf v_1}|=|{\bf v_2}|$,
\begin{align*}\label{ce-rv-coord}
\frac{v+v_*}2 &= \frac{v' + v'_* }2 =V,    \ \  \text{conservation of center of mass} \\
|v|^2 + |v_*|^2 &= \ 2|V|^{2} + \frac{|{\bf v_1}|^{2}}{2} =|v'|^2 + |v'_*|^2,
\ \ \text{energy identity},
\end{align*}
Moreover,   $v'-v=\frac12 ({\bf v_2}-{\bf v_1})= \frac12(|u|\sigma -u) = -u^- $ and $v'_*-v_*=-\frac12 ({\bf v_2}-{\bf v_1})= -\frac12(|u|\sigma -u)=u^-.$\\

Next, let any function $\varphi (v)$  in $C^2 (\real^d)$ to
satisfy the relation \eqref{eq:collision2} on the pairs $(v, v_*)$ and $(v', v'_*)$  given by \eqref{four-point}, respectively. Then, subtract
$2\varphi (w)$ on both sides of identity \eqref{eq:collision2} and divide by $|v|^2$  the left hand side, and  by $|v_*|^2$ the right hand side, respectively (as  both arbitrary vectors $v$ and $v_*$ have the same magnitude).  The resulting identity approximates  the linear wave equation by taking second variations by finite differences
\begin{equation}\label{fpapprox}
\frac1{|{\bf v_1}|^2} \left\{ \varphi \Big( V+\frac{\bf v_1}2 \Big)
+ \varphi \Big( V-\frac{\bf v_1}2\Big) - 2\varphi (V)\right\}
= \frac1{|{\bf v_2}|^2}\left\{ \varphi \Big( V+\frac{\bf v_2}2 \Big)
+ \varphi \Big( V-\frac{\bf v_2}2\Big) -2\varphi (V)\right\},
\end{equation}
defined for any arbitrary vectors $v$ and $v_*$ with the same magnitude $|\bf v_1|= |\bf v_2|$.  Therefore, invoking the $v_i$-directional Taylor expansions, for $i=1,2$,  and taking the limit as $|\bf v_1| = |\bf v_2| \to 0^+$,  yields the identity
\begin{equation*}
\langle \mathcal{H}(\varphi)(V)\widehat{\bf v_1},\widehat{\bf v_1}\rangle
= \langle \mathcal{H}(\varphi)(V)\widehat{\bf v_2},\widehat{\bf v_2}\rangle,
\end{equation*}
where  $\mathcal{H}(\varphi)$ is the Hessian matrix of $\varphi$ and $\widehat{y}$ is the  unit vectors in the direction of $y$.  This property suffices to show that the second derivatives of $\varphi$, in any direction, must be constant.  Indeed, choosing
\begin{equation*}
\widehat{\bf v_1} = \frac{e_{i}+e_{j}}{\sqrt{2}}\,, \qquad  \widehat{\bf v_2} = \frac{e_{i} - e_{j}}{\sqrt{2}},\quad  \text{implying}\   \ \widehat{\bf v_1}\cdot \widehat{\bf v_2} =0, \  \ \text{for any}\ \ j\neq i,  \ 1\le i,j\le d.
\end{equation*}
It follows
\begin{equation*}
\big(\partial^{2}_{ij}\varphi\big)(V) = \langle \mathcal{H}(\varphi)(V)e_{i},e_{j}\rangle = 0\, \quad\text{and}\quad
\big(\partial^{2}_{ii}\varphi\big)(V) = \langle \mathcal{H}(\varphi)(V)e_{I},e_{I}\rangle=\langle \mathcal{H}(\varphi)(V)e_{j},e_{j}\rangle = \big(\partial^{2}_{jj}\varphi\big)(V)\,,
\end{equation*}
and since  the vector $w\in\mathbb{R}^{d}$ is arbitrary this leads to $\partial^{2}_{ij}\varphi = 0$ and $\partial^{2}_{ii}\varphi = \partial^{2}_{jj}\varphi$ for any $i\neq j$, means that mixed derivatives vanish, and   imply $\varphi(v)$ separates as
\varphi(v) = \sum^{d}_{i=1} f_{i}(v^{i})\,,\qquad \text{on any arbitrary}\ \  v=(v^{1},v^{2},\cdots,v^{d})\,,
for some scalar functions $f_{i}:\mathbb{R}\rightarrow\mathbb{R}$.

Furthermore, since second pure derivatives are equal, yields
$f''_{1}(v^{1}) = f''_{2}(v^{2}) = \cdots = f''_{d}(v^{d}) = \text{constant}$.  That is, $f_{i}(v^{i}) = a^{i} + b^{i}v^{i} + c^{i} (v^{i})^{2}$, for some real numbers $a^{i},\,b^{i},\,c^{i}$.  Or equivalently,
$$\varphi(v) = \frac{C|v|^2}2 + \vec B \cdot v + A\  \text{ for any }\ v\in\real^d\ .$$
This proves the statement of the theorem for any $\varphi \in C^2 (\real^d)$.

\medskip

To complete the proof take $\varphi \in \mathcal{D}^1 (\real^d)$ a collision invariant and use the mollification $\varphi *\rho_\ep$, where $\rho_\ep \in C^\infty_o$-mollifier.  Then, since the mass and energy conserving pre and post collisional pairs $(v, v_*)$ and $(v', v'_*)$, defined in \eqref{four-point}, remain  the mass and energy conserving under any translation $\tau_z \varphi(v)=\varphi(v-z)$, then the  chosen collision invariant $\varphi(v)\in \mathcal{D}^1 (\real^d)$ satisfies
\begin{align*}
(\varphi * \rho_\ep) (v) +  (\varphi * \rho_\ep)(v_*) &= \int_{\mathbb{R}^{d}}\left(\tau_z \varphi(v) + \tau_z \varphi(v_*)\right) \rho_{\ep}(z) {\rm d}z \\
&= \int_{\mathbb{R}^{d}} \left(\tau_z \varphi(v') + \tau_z \varphi(v'_*)\right)  \rho_{\ep}(z) {\rm d}z = (\varphi * \rho_\ep) (v') +  (\varphi * \rho_\ep)(v'_*)\,.
\end{align*}

Consequently, $(\varphi *\rho_\ep)$ satisfies the five-point approximating rule \eqref{fpapprox} for the couple of pairs defined in \eqref{four-point},  implying
$$(\varphi *\rho_\ep) (v) = \frac{C^{\ep}}{2} |v|^2 +\vec B^{\ep}\cdot v +A^{\ep}\,, $$
Hence, $\lim_{\ep\downarrow 0} (\varphi *\rho_\ep) (v) =\varphi$ induces the limit s$A^{\ep},\,\vec B^{\ep},\, C^{\ep}$ to some $A,\,\vec B,\,C$, and so
$\varphi (v) = \frac{C}2 |v|^2 +\vec B\cdot v +A\,.$
This last statement completes the proof.
\end{proof}

\subsection{The Boltzmann inequality and the $\cH$-Theorem}

From the weak-Maxwell formulation we can  obtain the following fundamental results for the conservative (elastic) Boltzmann collisional form
\begin{thm}[The Boltzmann inequality and the $\cH$-Theorem]
If $f$ is non-negative and $Q(f,f)\,\log f$ is integrable and
$\log f = \varphi (v)$, then
\begin{equation}\label{eq:boltzmann}
\begin{aligned}
&{\rm i)} \quad\text{Boltzmann Inequality: }\quad
\int_{\real^d} Q(f,f)\log f\,{\rm d}v \le 0 \, , \quad \text{and}\\
&{\rm ii)}\quad \text{The $\cH$-Theorem:}\quad \ \ \displaystyle \int_{\real^d}Q(f,f)\, \log f \, {\rm d}v  = 0 \quad   \text{ iff } \quad f(v) = \exp (A+B\cdot v - C|v|^2)\,.
\end{aligned}
\end{equation}
\end{thm}

\begin{proof}
For $\varphi (v) = \log f(v)$ as a test function, from the symmetric weak Maxwell
form Corollary~\ref{coro4}
\begin{equation}\label{eq:10.1}		%% \tag{10.1}
\begin{split}
\int_{\real^d} Q(f,f) \log f(v){\rm d}v
& = \frac14 \int_{\real^d\times\real^d\times \mathbb{S}^{d-1}}\mkern-48mu
(f'f'_* \!-\! ff_*) [\log f\!+\!\log f_* \!-\! \log f'_* \!-\!\log f']B(|u|, \hat u\cdot\eta ){\rm d}v_*{\rm d}v{\rm d}\eta\\
\noalign{\vskip6pt}
&= \frac14\int_{\real^d\times\real^d\times \mathbb{S}^{d-1}}\mkern-48mu
(f'f'_* - ff_*) (\log ff_* - \log f'_* f') B(|u|, \hat u\cdot\eta )\,{\rm d}v_*\,{\rm d}v\,{\rm d}\eta\\
\noalign{\vskip6pt}
%&= -\frac14 \int_{\real^d\times\real^d\times \mathbb{S}^{d-1}}\mkern-48mu
%(f'f'_* - ff_*)(\log f'_* f' - \log ff_*) B(|u|, \hat u\cdot\eta )\,{\rm d}v_*\,{\rm d}v\,{\rm d}\eta\\
& =: - \mathcal{D} (f) \le 0\qquad \text{\em Entropy dissipation rate}\,.
\end{split}
\end{equation}
The sign in last inequality holds because the $\log z$ is monotone increasing function, satisfying
$(\log z-\log y) (z-y) > 0\; \text{ for any }\; z \ne y>0\; \text{ and }\;  (\log z-\log y) (z-y) = 0\; \text{ iff }\; z= y\,.$

Therefore, if there is an probability density functions $M(v)$ that nullifies the entropy dissipation rate, then it must satisfy
$\mathcal{D}(M) = 0$ if , and only if  $M'M'_* = MM_*$
and hence,  the Boltzmann Theorem~\ref{BoltzTheo} yields
\begin{equation*}
{\log M(v) = A+B\cdot v +C |v|^2}     \text{ or, equivalently }\     {M(v) = \exp (A+B\cdot v +  C|v|^2)}\,.
\end{equation*}

Finally, since any solution of the Boltzmann equation is a probability density in all  $v \in \R^d$, then its integral must be finite,
and hence the constant $C<0$.  That means  equilibrium solution  $M(v):=f_{eq}(v)$  must be a Gaussian distribution in $v$-space.
\end{proof}
In particular, setting $\vec B= {\bf v}, \  C = -\frac{m}{2k_BT},\ \exp (A) = \left(\frac{m}{2\pi k_BT} \right)^{3/2},$
the equilibrium state $M(v)$ becomes
\begin{equation}\label{eq:boltzmann-maxwell}
M(v)=M_{m,{\bf v}, T}(v)= \left( \frac{m}{2\pi k_BT}\right)^{3/2}
e^{-\frac{m|v -{\bf v}|^2}{2k_BT}}  \qquad
\text{Boltzmann-Maxwell molecular distribution}\,,
\end{equation}
that is also  expected  to be a stable stationary state for the single particle probability density function that solves the Boltzmann equation.   This state  $M_{m(x,t),{\bf v}(x,t), T(x,t)}(v)$ is often called a  {\em Local Maxwellians} or {\em  Local Equilibrium Statistical States (L-ESS)} for  the case of the spatially inhomogeneous problem.

Additionally, when these  states,  solve both the transport and the collision part, that is
\begin{equation}
\partial_t {M} + v\cdot \nabla_x {M} = 0 \ \ \ \text{and}  \ \ \ Q({M},{M})=0
\end{equation}
they are called  {\em Global Maxwellians} or  {\em  Global Equilibrium Statistical States (G-ESS)}.  If the parameters  $\bf u$ and $T$ are time independent  (may still depend on $x$-space) the  state $f_{eq}$  is referred as {\em Stationary Statistical Equilibrium  States (SESS)}.  {\em SESS} are stable and the H-Theorem is crucial for proving such stability as much as the functional structure of the solution space and the prescribed spatial boundary conditions.   In addition the {\em Global Maxwellians} are natural  {\em similarity} solutions   and can be proved to be attractors of the flow, even when they still are time dependent.  In such case one can referred to them as {\em stable Global Equilibrium Statistical States}.

\subsubsection{Entropy functionals and thermalization} Define the Boltzmann entropy functional as
\begin{equation}\label{eq:Htheorem.5.1}
\cH (f) = \int_{\real^d} f\, \log f\,{ d}v\,,
\end{equation}
and the relative Boltzmann entropy functional, defines for an equilibrium Maxwellian $M(v)$, meaning the for any $f(v)\in L^1_2(\mathbb{R}^d)\cap L\log L)(\mathbb{R}^d)$, such that $\int_{\mathbb{R}^d} f(v,t) \log M(v) dv= \int_{\mathbb{R}^d} f_0(v) \log M(v) dv$
\begin{equation}\label{eq:Htheorem.5.2}
\cH (f\, | \,M)  :=    \int_{\real^d} \tfrac{f}{M(v)} \log\Big(\tfrac{f}{M}\Big) M(v) dv =\int_{\real^d} f\, \log\Big(\tfrac{f(v,t)}{M(v,t)}\Big)\,{\rm d}v =  \int_{\real^d}f \log f dv -\int_{\mathbb{R}^d} f_0(v) \log M(v) dv.
\end{equation}
Then, from  calculation \eqref{eq:10.1} and the $\mathcal{H}$-theorem, it follows
\begin{equation}\label{eq:Htheorem.6}
\frac{{\rm d}}{{\rm d}t}\cH (f\,|\,M)  = - \mathcal{D} (f) \le 0\,,
\end{equation}
where the entropy dissipation functional was given in the calculation \eqref{eq:10.1}.
%\mathcal{D} (f) (t) &= -  \int_{\real^d} Q (f,f) \log f\,{\rm d}v \\
%& = - \frac12 \int_{\real^d\times\real^d\times \mathbb{S}^{d-1}_+} (f'f'_* - ff_*) (\log (f'f'_*) -
%\log (ff_*)) | u\cdot\eta |\,{\rm d}\eta\, {\rm d}v_*\, {\rm d}v\, .

\begin{remark}
This last theorem stresses the {\em irreversible in time} nature of the
Boltzmann equation, even though we started with an $N$-particle collisional
system with {\em time reversible elastic collisions}.
\end{remark}

The reason for this ``loss of reversibility'' lies on the molecular chaos
assumption: ``velocities of the particles {\em about to collide} are
combined that we model the gain operator as acting on the pre-collisional
Looking back at the Boltzmann inequality \eqref{eq:10.1}
that presets the sign of $(\log z-\log y) (z-y)$ expression that defines

{% \tcr
This preliminary section concludes with the anticipation of new results sprouting from the work in this manuscript.
More precisely, the addressing of the question of   whether is
possible to use \eqref{eq:Htheorem.6} to conclude, at least for solutions $f(v,t)$ of the homogeneous Boltzmann problem, that the long time thermalization limit
$\lim_{t\to\infty^+ }f(v,t) = M_{\rho, \rho u,\rho e} (v)$ occurs
occurs in some functional sense, and there this is a Banach space that not only the limit exits but also it is possible to establish the decay rate to zero of the $f(v,t) - M_{\rho, \rho u,\rho e} (v)$ in a related functional sense.
While  question has been addressed en the last decades, the following statements offer an anticipation of how to improve existing results.

\smallskip

%  % Requires \usepackage{graphicx}
%  \includegraphics[width=60mm]{Htheorem.jpg}
%  \caption{ }\label{fig-6}
%\begin{proof}[Proof of {\em Proposition~\ref{lem-CK2}}]
%Since $\log x$ is convex down, then the segment joining $(1,0)$ and
%$(2,\log 2)$ is under the graph of $\log x$.
%  % Requires \usepackage{graphicx}
% \begin{figure}[h]
%  \includegraphics[width=60mm]{ck-fig3.jpg}\\
%  \caption{ }\label{fig-9}

%  % Requires \usepackage{graphicx}
%  \includegraphics[width=60mm]{ck-fig1.jpg}\\
%  \caption{The function $g(\tau)|\tau|:= \g(\tau)|\tau|$, defined in Proposition~\ref{lem-CK2}  provides a pointwise bound from below to  $\psi(\tau)$ from \eqref{CK1}. }\label{fig-7}
%% $$\epsfbox{ck-fig1.eps}$$

Starting from the key step in solving this problem, discussed and proven in [104, 105], is the inequality
\begin{equation}\label{entropy-dissipation-control}
c_f\,\mathcal{H}(f|M)^{\alpha}\leq \mathcal{D}(f)
\end{equation}
with $M$ the thermal equilibrium associated to $f$ and for some explicit constants $c_f>0$ and $\alpha>1$ depending on properties of the solution $f$ itself.  Of particular importance is the appearance of higher moments, discussed in a later section, since the tail of a solution plays a central role in the equilibrium convergence rate which is related to $\alpha$.  Interestingly the aforementioned inequality is valid for soft and hard potentials under similar regularity properties of $f$, but in practice soft potential models have much ticker tails which can produce very slow thermalisation rates [43].  We refer to [105] for a extensive discussion of this inequality including an illuminating short draft of the proof.\\

Continuing with the discussion, from \eqref{eq:Htheorem.6} and \eqref{entropy-dissipation-control} we are led to the inequality
\begin{equation*}
\frac{{\rm d}}{{\rm d}t}\cH (f|M)   + c_{f}\,\cH (f|M)^{\alpha} \le 0\,,
\end{equation*}
which in turn leads to
\begin{equation}\label{eq:Htheorem.16}
\cH (f(t)|M) \leq \frac{\cH (f_0|M)}{\big(1+(\alpha-1)\cH (f_0|M)^{\alpha-1}\,c_f\,t\big)^{\frac{1}{\alpha-1}}}\,.
\end{equation}
A elegant application of the {\em Cszisar-Kullback inequality}, developed for information theory, shows that
\begin{equation}\label{eq:Htheorem.17}
\| f - M\|_{L^{1}(\real^d)} \leq \int_{\real^d} |f-M|\,dv \le \sqrt{2\,\cH (f \mid M)}\,.
\end{equation}
Therefore, from \eqref{eq:Htheorem.16} it follow that
\begin{equation*}
\| f(t) - M\|^{2}_{L^{1}(\real^d)}  \ \leq \frac{2\,\cH (f_0|M)}{\big(1+(\alpha-1)\cH (f_0|M)^{\alpha-1}\,c_f\,t\big)^{\frac{1}{\alpha-1}}}\,.
\end{equation*}
So $f(v,t) \xrightarrow[t\to\infty]{} M(v)$ strongly in $L^1(\real^d)$ at algebraic rate $\frac{1/2}{\alpha-1}$.

%\section{Inelastic Collisions - invariants for hard sphere kernels}
%Here we consider the Boltzmann equation as derived in Section 1 and 2, but now the interaction
%corresponds to inelastic collisions.  Thus we write
%\frac{\partial}{\partial t} f + v\cdot\nabla_x f = Q(f,f)\\
%Q (f,f) = \int_{\real^3} \int_{S^{2}}
%\left\{ \frac1{e|J_e|} \fprime \fprime_* - ff_*\right\} |u\cdot \eta|\,d\eta\,dv_*  \\
%\hskip.5in u = v   -v _*\quad \text{\em relative velocity};
% v' = v   - \frac{(1+e)}2 (u\cdot \eta)\eta  \qquad \text{and}\quad
%\qquad v' = v   + \frac{(1+e)}2 (u\cdot \eta)\eta \\
%\text{or\ equivalently }\\
% u' \cdot \eta = -e \ u\cdot\eta,\ (\text{or}\  u\cdot\eta = - e \, \uprime\cdot\eta ), \ \text{and}\   u' \cdot \eta^{\perp} =  \ u\cdot  \eta^{\perp},
%$0< e\le 1$ {\em restitution coefficient} is the measure of inelasticity. It may depend on
%a parametrization of $|u\cdot\eta|$, but for the time being we assume it constant.
%%$$\epsfysize=3.0truein \epsfbox{inelastic-collision_400x400_p1.eps}$$
%  % Requires \usepackage{graphicx}
%  \includegraphics[width=150mm]{inelastic-collisions.jpg}\\
%  \caption{}\label{fig-10}
%The corresponding determinant of the Jacobian of the variable transformation, as presented in the case of elastic hard sphere
%interactions as well, is given by
%J_e = \left| \frac{\partial (v',v'_*)}{\partial (v ,v_*)  }\right| \, ,
%and it is computed as follows.
%In order to compute this determinant Jacobian,
%define first the matrix associated to the tensor product of the impact direction
%$$N = \left[ \begin{matrix}
%\eta_1^2&\eta_1\eta_2&\eta_1 \eta_3\\
%\eta_3\eta_1&\eta_3\eta_2&\eta_3^2\end{matrix} \right] = \eta\otimes \eta\, .$$
%Therefore the following calculation follows
%$\det J_e \ =\ \left| \frac{\partial (v',v'_*)}{\partial (v ,v_*)  }\right|  \ =\  -\, e$.
%Noting that $\eta_1^2 +\eta_2^2 + \eta_3^2 =1$,
%the determinant of Jacobian transformation is given by the block matrix
%J_e = \left| \frac{\partial (v',v'_*)}{\partial (v ,v_*)  }\right|
%& = \left|
%\begin{matrix} 1 - \frac{1+e}2 N & \frac{1+e}2 N\\
%\frac{1+e}2 N & 1- \frac{1+e}2N \end{matrix}
% \right|    \\
%&\text{ adding the two columns and replacing in the first one}\\
%& \approx \left|  \begin{matrix} 1&\frac{1+e}2N\\
%1&1 -\frac{1+e}2 N\end{matrix}  \right| \\
%&\text{ subtracting  the first row from the second one and replacing in the second row}\\
%&\approx \left|   \begin{matrix} 1&\frac{1+e}2 N\\
%0&1- \frac{(1+e)}2N\end{matrix} \right|
%= \det (1-(1+e)N)  = -\, e
%%= e(z) + z e_z(z)  = \theta_z(z)
%%with $\theta(z) = z\, e(z)$.
% \iiint f'f'_*  |u\cdot \eta| \,d\eta\, dv_* \, dv &\ =\
%   \iiint f'f'_*   \frac1e |u' \cdot \eta | \,  \left|\frac{\partial (v, v_*)}{\partial (v', v'_*)}\right|
%dv' \,dv'_*  \\
%&\  =\    \iiint  \frac 1{e\,J_e} \, f'f'_*   |u' \cdot \eta |  \, dv' \,dv'_* \ =\   \iiint \frac 1{e\,J_e} ff_*    |u \cdot \eta |  \, dv\,dv_*
% \end{split}\end{equation}
% \\
%{$\ds \iiint f'f'_*  \mkern-24mu
%\mathop{|n\cdot u|}\limits_{\substack{\Bigg\downarrow\\
%\ds |n\cdot u| = \frac 1e |u'\cdot u|}}
%\mkern-16mu  \mathop{d\eta\,dv_* \,dv}\limits_{\substack{\big\downarrow\\ }}
%\put(166,6){$\ds dv_* \,dv
%= \left|\frac{\partial (v, v_*)}{\partial (v', v'_*)}\right|
%dv' \,dv'_* = \frac1{J_{e}} dv' \,dv'_*$}

\section{The Cauchy problem for Maxwell and hard potentials}\label{hardpotsection}

A natural space to solve the homogeneous Boltzmann equation is the space of probability measures.  In particular, we develop the theory in the space of integrable functions, whose observables, or moments, may be bounded.
from the functional viewpoint, we recall  classical weighted Banach space $L^1_\ell(\real^d)$ associated to statistical processes describing the evolution weighted integrable functions,  defined by
\begin{equation}\label{k-norm}
L^{1}_{\ell}(\real^d)\  = \  \Big\{ \ g \; \text{measurable} \ : \  \|g\|_{L^1_\ell} \  = \ \int_{\real^d} |g(v)| \langle v \rangle^{\ell}\,dv \ < \ \infty\Big\}\,,
\end{equation}
any $\ell\geq0$.

As usual, we referred as the Lesbegue weight $\langle v \rangle = \sqrt{1+|v|^{2}}$.  Note that $L^1_{\ell_2} \hookrightarrow L^1_{\ell_1}$, since  $\|f\|_{L^1_{\ell_1}} \le \|f\|_{L^1_{\ell_{2}}} $, whenever  $\ell_2 \ge \ell_1$.

A closely related concept is the one of $k^{th}$-Lebesgue moment or observable of $g(v,t)$
\begin{equation}\label{k-moment}
m_{k}[g](t):=\int_{\real^d}\ g(v,t)\,\langle v \rangle^{2k}\ dv\,, \qquad   k \in  \overline{\mathbb{R}^+}  \,.
\end{equation}
Note that $\|g\|_{L^1_{2k}}\equiv m_{k}[g]$ whenever the scalar valued function takes only positive values.

We present a revision of an approach first introduced in the unpublished notes  by A. Bressan  [39] in the context of classical Boltzmann  equations for elastic binary collisions for hard spheres in three  dimensions with angular cross section.  The approach is based on abstract ODE theory and it is used here to solve the Cauchy problem for the  Boltzmann equation for Maxwell type of interactions as well as for hard potentials for initial data with finite mass and energy, that is,
\begin{equation}\label{energy}
0\leq f_0\,,\qquad m_{1}(f_0)<\infty\,.
\end{equation}
%The problem for Maxwell type of interactions  is easy to show that moments will satisfy a linear Ordinary Differential Inequality (ODI),  where it can be shown it is possible to generate a Picard Iteration for contractive Lipschitz maps in the Banach space $L^{1}_{k}(\real^d)$ that solve the Cauchy problem globally.
%However for the hard potential interactions case,   one need to solve the evolution of an

Thus, the associated classical space homogeneous Boltzmann collisional model for elastic binary interactions induces an evolution flow in the space $L^{1}_{\ell}(\real^d)$ by setting the identity
\begin{equation}\label{k-moment-identity}
\frac{d}{dt}m_{k}[f]:=\int_{\real^d}\ Q(f,f)(v,t)\,\langle v \rangle^{2k}\ dv\,,\qquad k\geq0\,,
\end{equation}
where the right had side may or may not  be positive.

Therefore, we focus first in an existence theory for the Cauchy problem in a general framework that allows for solution to the Boltzmann flow by solving an Ordinary Differential Inequality in the Banach space$L^{1}_{\ell}(\real^d)$, namely the following theorem will be rigorously  addressed
after recalling the weak formulation of the Boltzmann collision operator
\begin{equation}\label{bina-weak2}
\begin{split}
\frac{d}{dt} \int_{\R^N}  f(v,t) \varphi (v)\, dv\,
=  \int_{\R^d \times \R^d}  f(v,t)\, f(v_*,t)
G_{\varphi}(v_*,v)  dv_*\, dv\, ,
\end{split}
\end{equation}
for any test function $\varphi(v,t)$, where  the binary weights function $G_{\varphi}(v_*,v) $ is determined by the $\sigma$-average of the $\mathbb{S}^{d-1}$ sphere, decomposed in a possitive and negativeparts.
\begin{equation}\label{weightuv2}
\begin{split}
&G_{\varphi}(v_*,v)  := \left(G_{\varphi}^+ - G_{\varphi}^-\right)(v_*,v):=\int_{\mathbb{S}^{d-1}} \big(\varphi (v') + \varphi (v'_*) - \left(\varphi (v) + \varphi (v_*)\right) \big) B(|v-v_*|, \sigma) d\sigma\\
\noalign{\vskip8pt}
&v'=v - \tfrac12(|u|\sigma-u)\, , \qquad v'_*= v_* +\tfrac12(|u|\sigma-u)\, ,  \qquad u=v-v_*\, ,
\end{split}
\end{equation}
where the transition probability  is assumed to take the  form $B(|v-v_*|, \sigma) = \Phi(u)\,b(\hat{u}\cdot\sigma).$  The angular part of the scattering angular kernel $b:=b(\hat{u}\cdot\sigma)$ is  assumed to be integrable with respect to the measure $d\sigma$. Conditions on the potential transition rate function $\Phi_\gamma(|u|)$ are assumed to be non-negative and to satisfy the following  $\gamma$-order homogeneity condition
\begin{align}\label{pot-Phi_1}
c_{\Phi}  |u|^\gamma \le  \Phi_\gamma (|u|)  \le C_{\Phi} |u|^\gamma\, ,
\end{align}

\begin{thm}[\bf{Cauchy problem for the homogeneous binary Boltzmann equation}] \label{CauchyProblem}
Let $\gamma\in [0,2]$, $b(\hat{u}\cdot\sigma, d\sigma)$ integrable, and initial data $f_0$ with finite mass and energy $\|f_0\|_{L^1_2}<\infty.$

If $\gamma\in(0,2]$ and the initial data  $\|f_0\|_{L^1_{2+\eps}}<\infty$ for any arbitrary $\epsilon>0$, then there exists a unique nonnegative function
\begin{equation*}
f(v,t) \in \mathcal{C} \big( [0,\infty); L^{1}_{2}( \mathbb{R}^d ) \big) \cap  \mathcal{C}^1\big( (0,\infty);L^{1}_{\ell}(\mathbb{R}^{d})\big)\,,\quad \forall\
\ell\ge 2,
\end{equation*}
\begin{equation}\label{collision3}
\left\{\begin{array}{l}
f_t = Q(f,f) =  \int_{\R^d \times S^{d-1}} \big( \fprime\, \fprime_*  - f\, f_*)\, \Phi(|v-v_*|)\, b( \hat u\cdot \sigma) \, d\sigma dv_* \,, \\ \noalign{\vskip6pt}
f(v,0) = f_0 (v)\,.
\end{array}\right.
\end{equation}
Furthermore, the conservation laws hold
\begin{equation}\label{collision3.2}
\int f(v,t)\text{d}v = \int f_0(v)\text{d}v,\quad \int f(v,t)v\text{d}v = \int f_0(v)v\text{d}v,\quad \int f(v,t)|v|^{2}\text{d}v =\int f_0(v)|v|^{2}\text{d}v\,.
\end{equation}
In the case $\gamma\in[0,2]$,   $\|f_0\|_{L^1_{\ell}}<\infty$, then    $f(v,t) \in \mathcal{C} \big( [0,\infty); L^{1}_{\ell}( \mathbb{R}^d ) \big) \cap  \mathcal{C}^1\big( (0,\infty);L^{1}_{\ell}(\mathbb{R}^{d})\big)$ for all
$\ell\ge 2.$
\end{thm}

The success of such global existence solely relies in two fundamental sufficient conditions. One is the upper control of the positive part associated to the collision operator $k^{th}$-moment in the right hand side of identity \eqref{k-moment-identity} by a linear or constant form in the solution $k^{th}$-moment, whose constant parameters may only depend of the conserved constant quantities  $m_0[f]$ and $m_1[f_0]$, mass and Lebesgue energy respectively.  This upper control is obtained firstly by an energy identity that splits the local energy conservation in a convex combination of the potential energy and the center of mass; and secondly by calculating the $\sigma$-averages of the positive part of $G^+_{\langle v\rangle^\ell}= \int_{\mathbb{S}^{d-1}} (\langle v\rangle^\ell+\langle v_*\rangle^\ell)b( \hat u\cdot \sigma) d\sigma$, which    decays in $\ell$, proportionally to  $\ell^{th}$-Lebesgue energy. These estimates, inspired in the work of Bobylev [31],  extend the classical Povzner estimates to a large larger class of angular transition functions $b(\hat{u}\cdot\sigma)$.  Such upper estimates are now referred  by Angular Averaging Lemma for the control of the positive contributions to the weight  $G_{\langle v\rangle^k}$.

The second  fundamental sufficient condition is a lower bound.  We look into two possible venues. Under the posed assumption on the angular averaging of $b(\hat{u}\cdot\sigma)$ being finite, the first one simple  looks into a  lower pointwise estimate of the negative contribution $G^-_{\langle v\rangle^\ell}= \|b\|_{L^1(\mathbb{S}^{d-1})}\, (\langle v\rangle^\ell+\langle v_*\rangle^\ell)$.
This estimate is sufficient to prove Theorem~\ref{CauchyProblem}.
However, to work out the propagation of the $L^p(\real^d)$ theory $1\le p\le\infty$ we need a stronger version of the the lower bound, which can be obtain by a functional inequality for elements in $L^1(\real^d)$ that, under suitable conditions  controls the convolution of such an element by $\langle v\rangle^\gamma$, by a finite constant proportional to  $\langle v\rangle^\gamma$, for $0<\gamma\leq 2$. This is all worked out in  the Lower Bound Lemma~\ref{lblemma}.

These estimates also enable the control of propagation and generation of exponential tails.

Further, we will show that, if  the initial data satisfies   $ \| f_0\|^1_{\ell} \cap   \| f_0\|^p_{\ell}<\infty$, with $1<p\leq\infty$, then this property propagates in time.
This result requires an extra moment requirement $m_{1+\epsilon}(f_0)<\infty$, for some $\epsilon>0$ to secure a lower bound that warrants   a coerciveness property  needed to show the $  \| f\|^p_{\ell}(t)$ boundedness for all time (i.e. propagation of initial `regularity'.)  This is the  crucial Lemma~\ref{lblemma}  shown in  in subsection~\ref{lower-bounds}.

\smallskip

While the wellposedness theory of the Boltzmann equation has been addressed by several authors [45, 24], with the most general result is given in [87], where the initial data is assumed only with finite mass and energy,  their arguments are based on the moment propagation and generation theory [59, 51, 109, 31]   that is based on an estimate of integration on each $|v'|^\ell$ and
$|v'_*|^\ell$ weights  evaluated in the postcollisional molecular velocities proportional to the angular transition  element $b( \hat u\cdot \sigma) d\sigma$.
All these results assumed that not only the function $b( \hat u\cdot \sigma)$ was bounded but also needed to introduced  a cut-off near $\hat u\cdot \sigma=0$.    Such estimates are referred as Povzner lemma and was originally introduced by Povzner [94] in 1960s and the show that such angular integration is proportional to $(|v| + |v_*|)^\ell$-power.
That estimate yields the propagation and generation of moments for which an iteration map, constructed by Arkeryd in [24] that yields compactness sufficiently enough to construct a unique solution with $2^+$-moments.

In 1997 a groundbreaking technique was introduced by Bobylev   [31], applied to the hard sphere model in three  dimensions with a constant angular function $b( \hat u\cdot \sigma)$. This new approach proposes that uses the  local energy conservation  to write the integration of postcollisional molecular velocities in the center of mass-relative velocity framework.  That way the integration  readily proves a control of the decay rate of  such angular averaging for the case of hard spheres in three dimensions with a constant angular cross section.
An important step to obtain evolution estimates for moments is a fundamental lemma that controls the function $G_{\varphi}(v_*,v)$  Later, it was extended in [37] for inelastic collisions.  More extension related to the generality of $b$ where made later on in [64, 5].  Finally, in [85, 101] the lemma was extended to non cutoff scattering.\\

\subsection{Lower bound estimates}\label{lower-bounds}
A functional estimate for lower bound of convolution function with concave power law functions can be obtain  by the following lower bound estimate, that will become essential later on the calculation of propagation of  $L^p$-Banach norms for any $1<p\leq \infty$.  This is a fundamental result related to a lower control for the convolution operator associated to the collision frequency corresponding to the negative contributions of the moments associated to the collision operator $Q(f,f)(v)$.  This a new result, developed for probability densities and independent of being solution of the Boltzmann equation. A first notion of this lower bound was introduced by the authors for the analysis of stability and error estimates for numerical approximations to the Boltzmann equation by conservative spectral schemes as shown in [17], and it is reproduced here in a more analytical framework. Modification of this results were employed in to obtain  existence results and qualitative results  for the one dimensional dissipative Boltzmann type equation for polymers [4], and more  recently used in the well-posedness of of solution of a gas mixture modeled by a system of Boltzmann equation with disparate masses [66]  and for polyatomic gas taking into account the exchange of internal energies [67] \\

\subsubsection{Pointwise estimates}
Conditions on the potential transition rate function $\Phi_\gamma(|u|)$ are assumed to be non-negative and to satisfy the following  $\gamma$-order homogeneity condition
\begin{align}\label{pot-Phi_1}
c_{\Phi}  |u|^\gamma \le  \Phi_\gamma (|u|)  \le C_{\Phi} |u|^\gamma\, ,
\end{align}
for which,   invoking convex and concave inequalities for non negative scalar quantities the term  $|u|^\gamma $ can be  estimated by the Lebesgue weights $ \langle v\rangle^{\gamma}:= (1+|v|^2)^{\gamma/2}$ from above and below, respectively,   as follows
\begin{equation}\label{pot-Phi_2.0}
C_{\gamma} \langle v\rangle^{\gamma} - \langle v_{*}\rangle^{\gamma} \leq  \Phi_\gamma(|u|)  \leq \tilde C_{\gamma} \left( \langle v\rangle^{\gamma} + \langle v_{*}\rangle^{\gamma}\right)
\end{equation}
with the lower and upper bound constants $C_{\gamma} =c_{\Phi} \min\{ 2^{-\gamma/2}, 2^{(2-\gamma)/2}\}$ and
$\tilde C_{\gamma} =
C_{\Phi} \max\{ 2^{\gamma/2}, 2^{\gamma/2-1}\}$, respectively,  with $c_{\Phi}\leq C_{\Phi}. $

Since in this work we focus on values of $\gamma \in(0,2]$, we only consider the potential part of the transition probability  for   $C_{\gamma} =c_{\Phi}  2^{-\gamma/2}$ and $\tilde C_{\gamma}= C_{\Phi} 2^{\gamma/2}$  to be
\begin{align}\label{pot-Phi_2}
c_{\Phi}  (2^{-\gamma/2}\, \langle v\rangle^{\gamma} - \langle v_{*}\rangle^{\gamma}) \leq  \Phi_\gamma(|u|)
\leq  C_{\Phi} 2^{\gamma/2}\left( \langle v\rangle^{\gamma} + \langle v_{*}\rangle^{\gamma}\right)\, .
\end{align}

In particular, the  lower constant bound $c_{\Phi}$   is part of the evaluation of the coercivity property associated to the proof of existence of global in time Cauchy problem solution of the classical elastic Boltzmann equation for binary elastic interactions in the Banach space $C(0,\infty, L^1_k(\mathbb{R}^d) )$, $k>2$, as much as the characterization of propagation and generation of exponential high energy tails   associated to  solutions in $ L^1(\mathbb{R}^d)$.

\subsubsection{A functional  lower bound estimate for convolution of functions in $L^1_k(\mathbb{R}^d)$ spaces with concave power law functions}\label{coercive-2}
This is a functional  estimate in $L^1_\ell(\mathbb{R}^d)$ Banach spaces that provides a fundamental  necessary control   to obtain  a coercive estimate that enables  global in time estimates  to the Boltzmann flow in  $L^p_\ell(\mathbb{R}^d)$, for $1< p \le \infty$.
It consists in finding a lower bound inequality  for
any probability density function   $f(x,\cdot,t) \in L^1_\ell(\mathbb{R}^d)$,  in  $v\in \mathbb{R}^d$ and $\ell>2^+$,  convolved with a concave potential   function $\Phi(v)$ satisfying a condition like  \eqref{pot-Phi_1} by   the $\gamma$-Lebesgue polynomial weight proportional to  a  constant that depends on the semi-norms  $\dot{L}_\ell^1(\mathbb{R}^d)$, for $\ell=0,1,2$ and $2+\beta$, for any $\beta>0$. The proof, as it will be shown next, is independent of the the function being a solution of the Boltzmann flow,  controls from below the collision frequency term $\nu(f)(v,t)$  associated to a potential rate   $\Phi(|u|)$, i.e. the convolution of any function $f(v,t)$.

%{\tt Move this paragraph to the introduction}
\medskip
\begin{remark}
This result actually convolves the transition probability rate, or collision kernel, with a probability density satisfying the condition of the Lower Bound Lemma~\ref{lblemma}.
It must be noted that this  Lemma was not used for the existence theory. It is not needed since the transition probability can be  estimated pointwise  using \eqref{pot-Phi_1} and \eqref{pot-Phi_2}.  However the reader may have noticed that the existence of solutions  relied of the moment estimates of the collisional integral shown in Theorem\ref{mom-coll-op},  proven  in this Section~\ref{hardpotsection}, and the application an existence Theorem~\ref{Theorem_ODE}  and Lemma~\ref{Lemma_ODE-extension2} making use of the   upper solutions of the moments ordinary inequalities  as global in times bounds found in   Theorem~\ref{propagation-generation} to obtain solutions in $ \mathcal{C}\big([0,\infty); L^{1}_{2}(\mathbb{R}^{d}) \big)\ \cap\ \mathcal{C}^{1}\big((0,\infty); L^{1}_{\ell}(\mathbb{R}^{d}) \big),$ with $2\le \ell<2(1+\gamma), $ solving  the Cauchy problem with initial data $f_0\in L^{1}_{2+\epsilon}(\mathbb{R}^{d}).$
In particular the strategy developed in this manuscript, produces a solution with
lower upper global bounds for large times than  the one obtained from the classical upper global bound estimates obtained by Bernoulli's equations, as written in [51] and [109].    Thus,  while this lower bound is not strictly necessary for the argument given in the proof of Theorem~\ref{thm:exp-moment-gen},  it considerably simplifies its proof without sparing the explicit characterization of the exponential rates, which is one of the goals of this review.

Nevertheless, we found the Lower Bound Lemma necessary for study of  $L^{2}$-integrability propagation, that   allows for  the considerably improvement and completeness  properties of  $L^p_\ell(\real^d)$ propagation norms  proof,  for $p\in (0,\infty]$,  from the original argunent developed in [90].
\end{remark}
\begin{lem}[\bf{Lower Bound Lemma}]\label{lblemma}
Fix $\gamma\in(0,2]$, and assume $0\leq  f:=f(\cdot, v)$ satisfies
%C\geq \int_{\mathbb{R}^{d}}f(v,t)dv \geq c\,,\quad  C\geq\int_{\mathbb{R}^{d}}f(v,t)|v|^{2}dv \geq c\,,\quad \int_{\mathbb{R}^{d}}f(v,t)\,v\,dv=0\,,
\begin{equation*}
C\ge \max\left\{ \int_{\mathbb{R}^{d}}f\, dv, \, \int_{\mathbb{R}^{d}}f \,|v|^{2}dv  \right\}  \geq
\min\left\{ \int_{\mathbb{R}^{d}}f\, dv, \, \int_{\mathbb{R}^{d}}f\,|v|^{2}dv  \right\}   \ge  c >0\,,\quad \int_{\mathbb{R}^{d}}f\,v\,dv=0\,,
\end{equation*}
for some positive constants $C$ and $c$.  Assume also that for some $\beta>0$
\begin{equation*}
0<\int_{\mathbb{R}^{d}}f\, |v|^{2+\beta}dv\leq B_{\beta}.
\end{equation*}
\bigskip
\nn Then, there exists $c_{lb}:=c_{lb}(B,C,c, \beta,\gamma)>0$ such that
\begin{equation}\label{lower-bound}
\big(f \ast_v|\cdot|^{\gamma}\big)(v)\geq c_{lb}\langle v \rangle^{\gamma}\,.
\end{equation}
with $0<c_{lb}$ explicitly defined by
\begin{equation}\label{c-lb}
c_{lb}    := \frac {c}{4}\,
\left(2\left(\frac{2^{2/\beta}}c \max\{C,B_{\beta}\}\langle \left(4C/c\right)^{\frac{1}{\gamma}} \rangle^{2+\beta} \right)^{\frac1\beta} \right)^{-\frac{1}{2-\gamma}  } \left( 1+\big(4C/c\big)^{\frac{2}{\gamma}}\right)^{-\frac{\gamma}2}
\end{equation}
\end{lem}

\begin{proof}
The case $\gamma=0$ is trivial, thus, assume $\gamma\in(0,2]$.  Take $r>0$, $v\in B(0,r)\subset{\mathbb{R}^{d}}$, the open ball centered at the origin and radius $r$.
Note that for any $R>0$,
\begin{align}\label{lbe1}
\begin{split}
\int_{\{|v-w|\leq R\}}f(\cdot,w)|v-w|^2&\text{d}w=\int_{\mathbb{R}^d}f(\cdot,w)|v-w|^2\text{d}w-\int_{\{|v-w|\geq R\}}f(\cdot,w)|v-w|^2\text{d}w\\
&\geq c\,\langle v \rangle^{2} - \int_{\{|v-w|\geq R\}}f(\cdot,w)|v-w|^2\text{d}w\\
&\geq c\, \langle v \rangle^{2} - \frac{1}{R^{\beta}}\int_{\{|v-w|\geq R\}}f(\cdot,w)|v-w|^{2+\beta}\text{d}w\,.
\end{split}
\end{align}

Since $|v| < r$, then
\begin{equation*}
\int_{\{|v-w|\geq R\}}f(\cdot,w)|v-w|^{2+\beta}\text{d}w\leq 2^{1+\beta}\max\{C,B_{\beta}\}\langle v \rangle^{2+\beta}\leq 2^{1+\beta}\max\{C,B_{\beta}\}\langle r \rangle^{2+\beta}.
\end{equation*}

Next, taking    $R=R(r,c, C ,\beta)$ sufficiently large to satisfy \\
\begin{equation}\label{R-clb}
\frac{2^{1+\beta}}{R^{\beta}}\max\{C,B_{\beta}\}\langle r \rangle^{2+\beta}\leq \frac{c}{2}\,,
\end{equation}\\
\nn or, equivalently,\\
\begin{equation}\label{largeR}
2\left(\frac{2^{2/\beta}}c \max\{C,B_{\beta}\}\langle r \rangle^{2+\beta} \right)^{\frac1\beta} \leq {R}\,,
\end{equation}\\
\nn it follows from estimate \eqref{lbe1} that\\
\begin{equation}\label{lbe2}
\int_{\{|v-w|\leq R\}}f(\cdot,w)|v-w|^2\text{d}w\geq c\, \langle v \rangle^2 - \frac{2^{1+\beta}}{R^{\beta}}\max\{C,B_{\beta}\}\langle r \rangle^{2+\beta}\geq \frac{c}{2}\,, \quad \forall\; v \in B(0,r)\,.
\end{equation}

Hence,  one can also infer from \eqref{lbe2} that, for any $\gamma\in(0,2]$ and $R$ satisfying \eqref{largeR}, it follows\\
\begin{align}\label{lbe3}
\begin{split}
\int_{\mathbb{R}^d}f(\cdot,w)|v-&w|^{\gamma}\text{d}w  \geq \int_{\{|v-w|\leq R\}}f(\cdot,w)|v-w|^{\gamma}\text{d}w\\
&\geq\frac{1}{R^{2-\gamma}}\int_{\{|v-w|\leq R\}}f(\cdot,w)|v-w|^{2}\text{d}w\geq\frac{c}{2R^{2-\gamma}}\,, \quad \forall\; v \in B(0,r)\,.
\end{split}
\end{align}

Additionally, for any $v\in\mathbb{R}^{d}$ and $\gamma\in(0,2]$,
\begin{align}\label{lbe4}
\int_{\mathbb{R}^d}f(\cdot,w)|v-w|^{\gamma}\text{d}w \geq \int_{\mathbb{R}^d}f(\cdot,w)\big(\tfrac{1}{2}|v|^{\gamma}-|w|^{\gamma}\big)\text{d}w\geq \big(\tfrac{c}{2}|v|^\gamma - C\big),\qquad \,.
\end{align}
As a consequence of \eqref{lbe3}  and \eqref{lbe4},
\begin{equation*}
\int_{\mathbb{R}^d}f(\cdot,w)|v-w|^{\gamma}\text{d}w\geq\Big(\frac{c}{2R(r)^{2-\gamma}}\,\textbf{1}_{B(0,r)}(v)+\big(\tfrac{c}{2}|v|^\gamma - C\big)\,\textbf{1}_{B(0,r)^c}(v) \Big).
\end{equation*}

Choosing $r:=r_{*}=\big(4C/c)^{\frac{1}{\gamma}}\geq1$ one ensures that $\frac{c}{2}|v|^\gamma - C\geq \frac{c}{4}|v|^{\gamma}$ for any $|v|\geq r$.  Then, combining with the election of $R=R(r_{*})\geq1$ from \eqref{largeR},
\begin{align*}
\int_{\mathbb{R}^d}f(\cdot,w)|v-w|^{\gamma}\text{d}w\geq\Big(\frac{c}{2R(r_{*})^{2-\gamma}}\,\textbf{1}_{B(0,r_{*})}(v)+\frac{c}{4}\,|v|^\gamma\,\textbf{1}_{B(0,r_{*})^c}(v) \Big)\\
\geq \frac {c}{4\,R^{2-\gamma}(r_*)}\Big( \textbf{1}_{B(0,r_{*})}(v) + |v|^\gamma\,\textbf{1}_{B(0,r_{*})^c}(v) \Big)\geq \frac {c\,\langle v \rangle^{\gamma}}{4\,R^{2-\gamma}(r_*)\langle r_{*}\rangle^{\gamma}}\,.
\end{align*}

Thus, the lower bound \eqref{lower-bound} holds with $c_{lb} $ as
\begin{equation*}
c_{lb}  = \frac {c}{4\,R^{2-\gamma}(r_*)\langle r_{*}\rangle^{\gamma}}\,
\end{equation*}

Therefore, using that $r_{*}=\big(4C/c)^{\frac{1}{\gamma}}$ and  invoking \eqref{R-clb} to calculate $R(r_*)$, the exact value of the lower bound constant $c_{lb}$ representation \eqref{c-lb}  is obtained.
\end{proof}

\smallskip

\begin{remark}\label{poincare}
This lower estimate is a functional inequality that control a convolution with a probability density  from below.
In this context, the constant $c_{lb}$  can be viewed as the analog to the Poincare constant fundamental to obtain  the coerciveness property needed to develop an  existence and uniqueness property in a suitable classical Sobolev  space.
\end{remark}

\smallskip

\subsection{moment-estimates for binary interactions}
We start by proving the following  identity, that can be found in [31],  giving a relation between local energies written  in the scattering angle framework  for center of mass and relative velocity coordinates from \eqref{weightuv2}.

\noindent
The following elementary estimate naturally follows  from the binomial identity

\begin{lem}\label{lem:binom}
The following estimate holds
\begin{equation}\label{povz18}
(z+y)^{q}-z^{q}-y^{q}  \le q2^{q-3}\big(z^{q-1}y + y^{q-1}z\big)\,,\qquad\text{for all} \quad  2 \le q\in\mathbb{R}.
\end{equation}
\end{lem}

\begin{proof}
Note the identity
\begin{equation*}
(z+y)^{q}-z^{q}-y^{q} = \int_0^z \int_0^y q(q-1) (t+\tau)^{q - 2} \,d\tau\,dt\,.
\end{equation*}
For any $q\geq2$, the function $|\cdot|^{q-2}$ is convex, therefore,
\begin{equation*}
(t+\tau)^{ q - 2} \leq 2^{q-3}\big( t^{q-2} + \tau^{q-2} \big)\,.
\end{equation*}
Thus, we obtain
\begin{equation*}
(z+y)^{q}-z^{q}-y^{q} \le
2^{q-3}\int_0^z \int_0^y q(q - 1) \big( t^{ q - 2 } + \tau^{ q - 2 } \big) \,d\tau\,dt =
q2^{(q-3)}\big(zy^{ q - 1 } + z^{ q - 1 }y \big)\,.
\end{equation*}
\end{proof}

In addition, in order to   study of  $sk$-moments  of the solution to the Boltzmann equation with $0<s\le1$, it is also useful to invoke the following Lemma whose proof can be found in  reference [37], Lemma~2. The statement is transcribe here, adjusted to the current notation  for the ease of the reader.

\begin{lem}\label{BGP04-bino-sum} Let the {\em floor function} associated to a  real number, be denoted by $
\lfloor q \rfloor := \mbox{integer part of} \; q.$ Then, for any $1<n\in \mathbb{N}$ and $s\in (0,1]$, let  $1<sn\in\mathbb{R}$  and $J_{sn}= \lfloor \frac{sn}2\rfloor$,  the Newton's generalized  binomial expansion upper estimate for binomial forms with real valued exponents, is given by
\begin{equation}\label{binsum}
\sum^{J_{sn}-1}_{j=1} { sn \choose j}\left(x^{sn-j}y^{j} + x^{j} y^{sn-j}\right)
\le (x+y)^{sn} - x^{sn} -y^{sn}\! \le \! \sum^{J_{sn}}_{j=1} { sn \choose j}\left(x^{sn-j}y^{j} + x^{j} y^{sn-j}\right)
\end{equation}
where the binomial coefficients for non-integer $sn$ are
defined as
\begin{equation}\label{binsum-coeff}
{ sn \choose j}:=\frac{sn(sn-1) \dots (sn-j + 1)}{j!}, \ j\ge 1; \quad { sn \choose 0}:=1.
\end{equation}
In addition,  the case when $sn$ is an odd integer the second inequality  in \eqref{binsum}
becomes an equality, which coincides with the binomial expansion of $(x+y)^{sn}.$
\end{lem}

A natural consequence from the two inequalities in \eqref{binsum} are the following two estimates
\begin{align}\label{binsum-2}
&\sum^{J_{sn}}_{j=0} { sn \choose j}\left(x^{sn-j}y^{j} + x^{j} y^{sn-j}\right) \le K (x+y)^{sk},  \qquad \text{for} \ 0<k\le2\nonumber \\
\text{and}\qquad&\ \\