text:
\section{Introduction}
The acoustic system is the linearization about the homogeneous state of
the compressible Euler system. After a suitable choice of units,
in this model the fluid fluctuations $(\rho, u, \theta)$ satisfy
\begin{equation}\label{acoustic-system}
\begin{aligned}
\partial_t \rho+\nabla_{\! x}\!\cdot\! u=0\,,\qquad &\rho(x,0)=\rho^{0}(x)\,,\\
\partial_t u+\nabla_{\! x}(\rho+\theta)=0\,,\qquad & u(x,0)=u^{0}(x)\,,\\
\partial_t \theta+\tfrac{2}{3}\nabla_{\! x}\!\cdot\! u=0\,,\qquad &\theta(x,0)=\theta^{0}(x)\,.
\end{aligned}
\end{equation}
In this paper, we consider the periodic boundary condition,
i.e $x \in \mathbb{T}^3$.

This is one of the simplest system of fluid dynamical equations
imaginable, being essentially the wave equation. It may be derived
directly from the Boltzman equation as the formal limit of moment
equations for an appropriately scaled family of Boltzmann solutions
as the Knudsen number tends to zero.

The program initiated by Bardos, Golse, and Levermore \cite{BGL2}
was to derive the fluid limits which include incompressible Stokes,
Navier-Stokes, Euler equations, and acoustic system from the {\em
DiPerna-Lions renormalized} solutions. This program has been
developed with great success during the last decade, here we only
mention \cite{BGL2, BGL, GL, GS, LM, LM1, LM2} among others. In
particular, Golse and Saint-Raymond  \cite{GS} justified the first
complete incompressible Navier-Stokes limit from the Boltzmann
equation without any compactness assumption.  On the other hand,
higher order approximations with the unified energy method have been
shown by Guo \cite{G} to give rise to a rigorous passage from the
Boltzmann equations to the Navier-Stokes-Fourier systems beyond the
Navier-Stokes approximations in the framework of classical
solutions.

Surprisingly, the status for rigorously deriving the acoustic system
from DiPerna-Lions solutions of Boltzmann equation is still
incomplete. This is mainly because DiPerna-Lions solutions do not
have some properties which are formally satisfied such as local
conservation laws. In \cite{BGL}, the acoustic limit
 was justified for Maxwell
molecular collisions under some assumption on the amplitude of
fluctuations. The result was significantly improved in \cite{GL} to
a large class of hard potentials and the assumption of the amplitude
of fluctuations was relaxed to the order $\varepsilon^m$ with
$m>\frac{1}{2}$. Recently, the borderline case $m=\frac{1}{2}$ was
covered in \cite{JLM} for soft potentials.

In this paper, we take the first step to establish the acoustic
limit from the Boltzmann equation in the framework of classical
solutions. Working with classical solutions has several advantages
than working with the DiPerna-Lions solutions. For example, the
classical solutions automatically satisfy local conservation laws
and have good regularities; the nonlinear interaction can be
controlled by linear dissipation for small solutions.

We employ the nonlinear energy method developed by Guo \cite{G1, G2,
G} in recent years which has been turned out to be applicable to
other problems, for instance see \cite{J}. We justify the limit for
the case that the amplitude of fluctuation is $\varepsilon$, which is not
being optimal. However, our work has advantages in that we can treat
for a large class of collision kernels in a rather uniform way,
including hard potentials, soft potentials and especially Landau
kernels which were not covered in the framework of the renormalized
solutions. Furthermore, different dissipation mechanisms for
macroscopic parts and microscopic parts in the limit process are
clearly presented by the energy dissipation rate. To our best
knowledge, this is the first global-in-time acoustic limit result in the class of
classical solutions.

The paper is organized as follows: the next section contains the
formulation of the Boltzmann equation for different collision
kernels. Some preliminary lemmas regarding the estimates on the
collision operators are listed in Section 3. Then we give a very
brief formal derivation. Section 5 and 6 are devoted to the energy
estimates.

\section{Formulation and Notations}

Consider the following rescaled Boltzmann equation:
\begin{align}\label{R_B}
\partial_t F_\varepsilon  +v\!\cdot\!\nabla\!_x F_\varepsilon
=\frac{1}{\varepsilon} \mathcal{Q}(F_\varepsilon ,F_\varepsilon )
\end{align}
In this paper, as in \cite{G}, we consider two classes of collision
kernels, the first is given by the standard Boltzmann collision
operator $\mathcal{Q}(G_1,G_2)$:
\begin{align}\label{hard}
&&\mathcal{Q}(G_1,G_2)=\int_{{\mathbb{ R}}^3\times S^2}|u-v|^\gamma B(\theta)
|\{G_1(v^{\prime })G_2(u^{\prime })-G_1(v)G_2(u)\}dud\omega,
\end{align}
where $-3<\gamma\leq 1$, $B(\theta)\leq C|\cos\theta|$, $\text{
}v^{\prime }=v-[(v-u)\cdot \omega ]\omega\text{ }$ and $\text{
}u^{\prime }=u+[(v-u)\cdot \omega ]\omega.$ These collision
operators cover hard-sphere interactions and inverse-power law with
an angular cutoff. The hard potential means $0\leq \gamma\leq 1$,
and the soft potential means $-3<\gamma<0$.

The second class is the Landau collision operator
\begin{equation}\label{landau}
\mathcal{Q}(G_1, G_2)=\sum\limits_{1 \leq i,j \leq 3}\partial^i\int\phi_{ij}(v-u)
\{G_1(u)\partial^j G_2(v)-G_2(v)\partial^j G_1(u)\} \mathrm{d} u\,,
\end{equation}
where $\partial^i=\partial_{v_i}$ and
\begin{equation}
\phi_{ij}\equiv \frac{1}{|v|}\left\{\delta_{ij}-\frac{v_i
v_j}{|v|^2} \right\}\,.
\end{equation}

Let
\[
F_\varepsilon =\mu +\varepsilon \sqrt{\mu}f_\varepsilon
\]
be the perturbation around the global Maxwellian
$$\mu=\frac{1}{(2\pi)^{3/2}}e^{-\frac{|v|^2}{2}}.$$ Define
$\mathcal{L}$, the linearized collision operator, as follows
\begin{align}\label{L}
\mathcal{L}g\equiv -\frac{1}{\sqrt{\mu}}\{\mathcal{Q}(\mu,\sqrt{\mu}g)+
\mathcal{Q}(\sqrt{\mu}g,\mu)\},
\end{align}
and the nonlinear collision operator $\Gamma$ as
\begin{align}\label{N}
\Gamma(g,h)=\frac{1}{\sqrt{\mu}} \mathcal{Q}(\sqrt{\mu}g,\sqrt{\mu}h).
\end{align}
The rescaled Boltzmann equation \eqref{R_B} is written in terms of
the perturbation $f_\varepsilon $ as follows:
\begin{align}\label{P_B}
\partial_t f_\varepsilon  +v\cdot\nabla_xf_\varepsilon
+\frac{1}{\varepsilon} \mathcal{L}f_\varepsilon =\Gamma(f_\varepsilon , f_\varepsilon ).
\end{align}
We first recall that the operator $\mathcal{L}\geq0,$ and for
any fixed $(t,x),$ the null space of $\mathcal{L}$ is generated by $[\sqrt{\mu}%
,v\sqrt{\mu},|v|^{2}\sqrt{\mu}]$. For any function $f(t,x,v)$ we
thus can decompose
\[
f=\mathbf{P}f+(\mathbf{I-P})f
\]
where $\mathbf{P}f$ (the hydrodynamic part) is the $L_{v}^{2}$
projection on the null space for $\mathcal{L}$ for given $(t,x).$ We
can further denote
\begin{equation}
\mathbf{P}f=\{\rho_{f}(t,x)+v\cdot u_{f}(t,x)+(\tfrac{|v|^{2}}{2}-\tfrac{3}%
{2})\theta_{f}(t,x)\}\sqrt{\mu}. \label{hfield}%
\end{equation}
Here we define the \textit{hydrodynamic field} of $f$ as
\[
\lbrack\rho_{f}(t,x),u_{f}(t,x),\theta_{f}(t,x)]
\]
which represents the density, velocity and temperature fluctuations
physically.

In order to state  our results precisely, we introduce the following
norms and notations. We use $\langle\cdot\,,\cdot\rangle$ to denote
the standard $L^{2}$ inner product in $\mathbb{R}_{v}^{3},$ while we
use $(\cdot\,,\cdot)$ to denote  the $L^{2}$ inner product either in
$\mathbb{T}^{3}\times\mathbb{R}^{3}$ or in $\mathbb{T}^{3}$ with
corresponding the $L^{2}$ norm $\|\cdot\|.$ We use the standard
notation $H^{s}$ to denote the Sobolev space $W^{s,2}.$ For the
Boltzmann collision operator (\ref{hard}), we define the collision
frequency as
\begin{equation}
\nu (v)\equiv \int_{\mathbb{R}^3} |v-v'|^\gamma\mu(v')dv',
\label{nu}
\end{equation}
which behaves like $|v|^\gamma$ as $|v|\rightarrow\infty.$ It is
natural to define the following weighted $L^{2}$ norm to
characterize the dissipation rate.
\[
|g|_{\nu}^{2} \equiv\int_{\mathbb{R}^{3}}g^{2}(v)\nu(v)dv,\;\;\;
\;\|g\|_{\nu}^{2}   \equiv\int_{\mathbb{T}^{3}\times\mathbb{R}^{3}}%
g^{2}(x,v)\nu(v)\mathrm{d} v\mathrm{d} x.
\]
For the Landau operator \eqref{landau}. let
\begin{equation}
\sigma_{ij}(v)=\int_{\mathbb{R}^3}\frac{1}{|v-u|}\left\{
\delta_{ij}- \frac{(v-u)_i(v-u)_j}{|v-u|^2}\right\} \mu(u)\,\mathrm{d} u\,.
\end{equation}
The natural norms are given by the $\sigma\mbox{-}$norm
\begin{equation*}
\begin{split}
|g|^2_\sigma &\equiv \sum\limits_{1 \leq i,j \leq 3}
\int_\mathbb{R}^3
\{ \sigma_{ij}\partial^i g\partial^j g +\sigma_{ij} v^i v^j g^2\}\,\mathrm{d} v\,,\\
\|g\|^2_\sigma &\equiv \sum\limits_{1 \leq i,j \leq 3}
\int_{\mathbb{R}^3 \times \mathbb{T}^3} \{ \sigma_{ij}\partial^i g\partial^j g
+\sigma_{ij} v^i v^j g^2\}\,\mathrm{d} v\mathrm{d} x\,.
\end{split}
\end{equation*}
We also use a unified notation for the dissipation as $|g|_D$ and
$\|g\|_D$ to denote either $|g|_\nu$ or $|g|_\sigma$, $\|g\|_\nu$ or
$\|g\|_\sigma$ respectively. Let the weight function $w(v)$ be
\[
 w(v)\equiv |(1+|v|^2)^{\frac{1}{2}}.
\]
For both Boltzmann and Landau kernels we have
\begin{equation}
\|w^{-3/2}g\|\leq C \|g\|_D\,.
\end{equation}
See \cite{G} for the details.

In order to be consistent with the hydrodynamic equations, we define
\begin{equation}
\partial_{\alpha}^{\beta}=\partial_{x_{1}}^{\alpha_{1}}\partial_{x_{2}%
}^{\alpha_{2}}\partial_{x_{3}}^{\alpha_{3}}\partial_{v_{1}}^{\beta_{1}%
}\partial_{v_{2}}^{\beta_{2}}\partial_{v_{3}}^{\beta_{3}} \label{derivative}%
\end{equation}
where $\alpha=[\alpha_{1},\alpha_{2},\alpha_{3}]$ is related to the
space derivatives, while $\beta=[\beta_{1},\beta_{2},\beta_{3}]$ is
related to the velocity derivatives.

We now define instant energy functionals and the dissipation
rate.\\

\textbf{Definition 1 (Instant Energy) }\textit{For
}$N\geq8$,\textit{\ for some constant }$C>0,$\textit{\ an instant
energy functional }$\mathcal{E}_{N,l}(f)(t)\equiv
\mathcal{E}_{N,l}(t)$ \textit{satisfies:}\\

 (i) for hard potentials
with $0\leq \gamma\leq 1$ in \eqref{hard}
\begin{equation}\label{E}%
\frac{1}{C}\mathcal{E}_{N,l}(t)\leq\sum_{|\alpha|\leq
N+1}\|\partial_\alpha f\|^2
+\sum_{|\alpha|+|\beta|\leq N}%
\|w^l\partial^\beta_\alpha f\|^{2}\leq C\mathcal{E}_{N,l}(t)\,;
\end{equation}

(ii) for soft potentials with $-3<\gamma< 0$ in \eqref{hard}
\begin{equation}\label{E2}%
\frac{1}{C}\mathcal{E}_{N,l}(t)\leq
\sum_{|\alpha|\leq N+1}\|\partial_\alpha f\|^2+\sum_{|\alpha|+|\beta|\leq N}%
\|w^{\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha f\|^{2}\leq C\mathcal{E}_{N,l}(t)\,;
\end{equation}

(iii) for the Landau kernel \eqref{landau},
\begin{equation}\label{E3}%
\frac{1}{C}\mathcal{E}_{N,l}(t)\leq
\sum_{|\alpha|\leq N+1}\|\partial_\alpha f\|^2+\sum_{|\alpha|+|\beta|\leq N}%
\|w^{l-|\beta|}\partial^\beta_\alpha f\|^{2}\leq C\mathcal{E}_{N,l}(t).\,,
\end{equation}
for all functions $f(t,x,v)$.\\

\textbf{Definition 2 (Dissipation Rate) }\textit{For
}$N\geq8$,\textit{\ the dissipation rate
}$\mathcal{D}_{N}(t)$\textit{\ is defined as}\\

(i) for hard potentials with $0\leq \gamma\leq 1$ in \eqref{hard}
\begin{equation}\label{D}
\begin{split}
\mathcal{D}_{N,l}(t)=\sum_{|\alpha|\leq N+1}\left(\varepsilon\|
\partial_{\alpha}\mathbf{P}f\|^{2}(t)+\frac{1}{\varepsilon}\|\partial_\alpha
(\mathbf{I-P})f\|^2_\nu\right)
\\+\frac{1}{\varepsilon} \sum_{|\alpha|+|\beta|\leq
N}\|w^l\partial^\beta_\alpha(\mathbf{I-P})f \|_{\nu}^{2};
\end{split}
\end{equation}

(ii) for soft potentials with $-3<\gamma< 0$ in \eqref{hard}
\begin{equation}\label{D2}
\begin{split}
 \mathcal{D}_{N,l}(t)=\sum_{|\alpha|\leq N+1}\left(\varepsilon\|
\partial_{\alpha}\mathbf{P}f\|^{2}(t)+\frac{1}{\varepsilon}\|\partial_\alpha
(\mathbf{I-P})f\|^2_\nu\right)
\\+\frac{1}{\varepsilon} \sum_{|\alpha|+|\beta|\leq
N}\|w^{\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha(\mathbf{I-P})f \|_{\nu}^{2}.
\end{split}
\end{equation}

(iii) for the Landau kernel \eqref{landau},
\begin{equation}\label{D3}
\begin{split}
 \mathcal{D}_{N,l}(t)=\sum_{|\alpha|\leq N+1}\left(\varepsilon\|
\partial_{\alpha}\mathbf{P}f\|^{2}(t)+\frac{1}{\varepsilon}\|\partial_\alpha
(\mathbf{I-P})f\|^2_\nu\right)
\\+\frac{1}{\varepsilon} \sum_{|\alpha|+|\beta|\leq
N}\|w^{l-|\beta|}\partial^\beta_\alpha(\mathbf{I-P})f \|_{\nu}^{2}.
\end{split}
\end{equation}\

Both the instant energy and the dissipation rate are carefully
designed to capture the structure of the rescaled Boltzmann equation
\eqref{R_B} in the acoustic regime. For soft potentials,
$\mathcal{E}_{N,l}$ and $\mathcal{D}_{N,l}$ involve a weight
function in $v$ which depends on the number of velocity derivatives
$\partial^\beta$. This is designed to control the velocity
derivatives for the streaming terms $v\cdot\nabla_x$ by a weak
dissipation rate as proposed in \cite{G}. In particular, the
dissipation rates in \eqref{D}, \eqref{D2}, \eqref{D3} in which the
hydrodynamic part has $\varepsilon$ scale
 reflect that we do not observe the dissipation in the limit, which is
exactly the case of the acoustic system.

We state the main result of this article.

\begin{thm}
\label{Acoustic} Let $N\geq 8$. Let $0<\varepsilon\leq \frac{1}{4}$
be
 given.  Suppose $f_\varepsilon  (0,x,v)=f_0^\varepsilon
(x,v)$ satisfies the mass, momentum, and energy conservation laws
\begin{equation}
 (f_0^\varepsilon,[1,v,|v|^2]\sqrt{\mu})=0,
\end{equation}
and $F_\varepsilon  (0,x,v)=\mu+\varepsilon f_0^\varepsilon (x,v)\geq 0$.
If $\;\mathcal{E}_{N,l}(f^{\varepsilon})(0)$ is sufficiently small,
then there exists a unique global-in-time solution $f_\varepsilon (t,x,v)$
to \eqref{P_B}, and moreover there exists an instant energy
functional $\mathcal{E}_{N,l}(f^{\varepsilon})(t)$ such that
\begin{equation}
\frac{d}{dt}\mathcal{E}_{N,l}(f^{\varepsilon})(t)+\mathcal{D}_{N,l}%
(f^{\varepsilon})(t)\leq0. \label{en}%
\end{equation}
In particular, we have the following global energy bound:
\begin{equation}
\sup_{0\leq t\leq\infty}\mathcal{E}_{N,l}(f^{\varepsilon})(t)\leq
\mathcal{E}_{N,l}(f^{\varepsilon})(0). \label{enbound}%
\end{equation}
\end{thm}

\begin{rem}
 The global existence of solutions $f_\varepsilon $ to \eqref{P_B} follows from
the a priori
 global energy bound \eqref{enbound} by rather standard method.
In this article, we focus on proving the uniform bound.
\end{rem}

\begin{rem} Note that due to the weak dissipation \eqref{D}, we cannot deduce
the time decay estimate from the energy inequality \eqref{en} unlike
the incompressible Navier-Stokes-Fourier case in \cite{G, J}.
Indeed, physically, we do not expect any time decay of our instant
energy $\mathcal{E}_{N,l}(f^{\varepsilon})(t)$, since the acoustic
system preserves the initial energy for all time. See Lemma
\ref{regularity}.
\end{rem}

\section{Basic estimates of collision operators}

In this section, we sum up some basic estimates of collision
operators for various kernels considered in this paper. The proofs
can be found in \cite{G}. The following is the coercivity of
$\mathcal{L}$.

\begin{lem} There exists $\delta >0$ such that for any $f\in L^2
(\mathbb{R}_v^3)$
\begin{align}\label{coer}
\langle \mathcal{L}f,f \rangle \geq  \delta |(\mathbf{I-P})f|_\nu^2.
\end{align}
\end{lem}

\begin{lem}\label{Hard-Est}
For hard potential with $\gamma\geq 0$, there exits $C_{|\beta|}$,
$C>0$ such that
\begin{equation}\label{linear}
(w^{2l}\partial^\beta_\alpha\mathcal{L}f\,,\partial^\beta_\alpha f)\geq \frac{1}{2}\|w^l\partial^\beta_\alpha
f\|^2_\nu-C_{|\beta|}\|f\|^2_\nu\,,
\end{equation}
\begin{equation}\label{nonlinear}
(\partial_\alpha^\beta\Gamma(f, g),\partial_\alpha^\beta h)\leq C
\{\|w^l\partial_{\alpha_1}^{\beta_1}
f\|\cdot\|w^l\partial_{\alpha_2}^{\beta_2} g\|_\nu
+\|w^l\partial_{\alpha_1}^{\beta_1}
g\|\cdot\|w^l\partial_{\alpha_2}^{\beta_2}
f\|_\nu\}\|w^l\partial_\alpha^\beta h\|_\nu\,.
\end{equation}
where $l\geq 0$, and summation is for $|\alpha|+|\beta|\leq N$ with
$\beta_1+\beta_2\leq\beta$ and $\alpha_2\leq \alpha$ componentwise.
\end{lem}
\begin{lem}\label{Soft-Est}
For the inverse power law with $-3 <\gamma <0$, for any $l \geq 0$,
there exist $C_{|\beta|}\,, C >0$ such that
\begin{equation}\label{linear-soft}
(w^{\{2l-2|\beta|\}|\gamma|}\partial^\beta_\alpha \mathcal{L} f\,,\partial^\beta_\alpha f) \geq
\frac{1}{2}\|w^{\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha f\|^2_\nu -C_{|\beta|}\|f\|^2_\nu\,,
\end{equation}
\begin{equation}\label{nonlinear-soft}
\begin{split}
(w^{\{2l-2|\beta|\}|\gamma|}\partial^\beta_\alpha\Gamma(f,g)\,,\partial^\beta_\alpha h) &\leq C\{\|w^{\{l-|\beta_1|\}|\gamma|}\partial^{\beta_1}_{\alpha_1} f\|\cdot\|w^{\{l-|\beta_2|\}|\gamma|}\partial^{\beta_2}_{\alpha_2} g\|_\nu\\
&+\|w^{\{l-|\beta_1|\}|\gamma|}\partial^{\beta_1}_{\alpha_1} g\|\cdot\|w^{\{l-|\beta_2|\}|\gamma|}\partial^{\beta_2}_{\alpha_2} f\|_\nu\}\\
&\times \|w^{\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha h\|_\nu\,,
\end{split}
\end{equation}
where the summation is taken over $|\alpha_1|+|\beta_1| \leq
|\alpha|+|\beta|\leq [\tfrac{N}{2}]+4$, and $\alpha_2 \leq \alpha$
and $\beta_2 \leq \beta$ componentwise.
\end{lem}

\begin{lem}\label{Landau-Est}
For the Landau kernel, for any $l \geq 0$, there exist $C_{|\beta|},
C >0$, such that
\begin{equation}\label{linear-landau}
(w^{2l-2|\beta|}\partial^\beta_\alpha \mathcal{L} f\,,\partial^\beta_\alpha f) \geq
\frac{1}{2}\|w^{l-|\beta|}\partial^\beta_\alpha f\|^2_\sigma
-C_{|\beta|}\|f\|^2_\sigma\,,
\end{equation}
\begin{equation}\label{nonlinear-landau}
\begin{split}
(w^{2l-2|\beta|}\partial^\beta_\alpha\Gamma(f,g)\,,\partial^\beta_\alpha h) &\leq C\{\|w^{l-|\beta_1|}\partial^{\beta_1}_{\alpha_1} f\|\cdot\|w^{l-|\beta_2|}\partial^{\beta_2}_{\alpha_2} g\|_\sigma\\
&+\|w^{l-|\beta_1|}\partial^{\beta_1}_{\alpha_1} g\|\cdot\|w^{l-|\beta_2|}\partial^{\beta_2}_{\alpha_2} f\|_\sigma\}\\
&\times \|w^{l-|\beta|}\partial^\beta_\alpha h\|_\sigma\,,
\end{split}
\end{equation}
where the summation is taken over $|\alpha|+|\beta|\leq N$, and
$\beta_1+\beta_2 \leq \beta$ and $\alpha_2 \leq \alpha$
componentwise.
\end{lem}

As a direct consequence of \eqref{nonlinear} in the above lemmas, we
can estimate the pure spatial derivatives for the nonlinear
collision operator $\Gamma$.

\begin{lem}
Let $\zeta(v)$ be a smooth function that decays exponentially, then
there is a given instant energy functional $\mathcal{E}_{N,0}(f)$
and $C_\zeta>0$, such that for summation over
$\alpha_1+\alpha_2=\alpha$, $|\alpha|\leq N$,
\begin{equation}\label{nonlinear2}
\begin{split}
(\partial_\alpha\Gamma(f\,,g)\,,\partial_\alpha h)&\leq\{\mathcal{E}^{1/2}_{N,0}(f)\|\partial_{\alpha_2}g\|_\nu+\mathcal{E}^{1/2}_{N,0}(g)\|\partial_{\alpha_2}f\|_\nu\}\|\partial_{\alpha_3}h\|_\nu\,,\\
\left\|\int\partial_\alpha\Gamma(f\,,g)\zeta\,dv\right\|&\leq
C_\zeta\{\mathcal{E}^{1/2}_{N,0}(f)\cdot\|\partial_{\alpha_2}g\|_\nu+\mathcal{E}^{1/2}_{N,0}(g)\cdot\|\partial_{\alpha_2}f\|_\nu\}\,.
\end{split}
\end{equation}
\end{lem}

\section{Derivation of Acoustic System}

In this section, we derive the acoustic system as the hydrodynamic
limit of solutions $f_\varepsilon $ to the rescaled Boltzmann equation
\eqref{P_B}. Since we have the uniform energy bound in $\varepsilon$
 by Theorem \ref{Acoustic},
there exists the unique limit $f$ of $f_\varepsilon $ in $\varepsilon$ and
we remark that due to higher order energy bound, all the limits in
the below are strongly convergent. First, by letting
$\varepsilon\rightarrow 0$ in \eqref{P_B}, one finds that
$\mathcal{L}f=0$. Thus $f$ can be written as follows:
\[
f = \{\rho+v\cdot
u+\left(\tfrac{|v|^2}{2}-\tfrac{3}{2}\right)\theta\}\sqrt{\mu},
\]
for $\rho,\,u,\,\theta$ are functions of $t,x$. In order to
determine the dynamics of $\rho,\,u,\,\theta$, project \eqref{P_B}
onto
$\{\sqrt{\mu},v\sqrt{\mu},(\tfrac{|v|^2}{2}-\tfrac{3}{2})\sqrt{\mu}\}$:
by collision invariants, first we get
\[
\langle \partial_t f_\varepsilon +v\cdot \nabla_x f_\varepsilon
,\;\{1,v,(\tfrac{|v|^2}{3}-1)\}\sqrt{\mu}\rangle=0
\]
and take the limit $\varepsilon\rightarrow 0$ to get
\[
\langle \partial_t f+v\cdot \nabla_x
f,\;\{1,v,(\tfrac{|v|^2}{3}-1)\}\sqrt{\mu}\rangle=0
\]
Since $f=\mathbf{P}f$, this is equivalent to
\begin{equation}\label{limit}
\begin{split}
&\partial_t\rho+\nabla_{\! x}\!\cdot\! u=0\\
&\partial_tu+\nabla_{\! x}(\rho+\theta)=0\\
&\partial_t\theta+\tfrac{2}{3}\nabla_{\! x}\!\cdot\! u=0
\end{split}
\end{equation}
Thus we have shown the following proposition on the mathematical
derivation of the acoustic system from the Boltzmann equation.

\begin{prop}\label{convergence}
Assume that $F_\varepsilon =\mu+\varepsilon\sqrt{\mu}f_\varepsilon $
 solves the rescaled Boltzmann equation \eqref{R_B} where $f_\varepsilon $
is obtained from Theorem \ref{Acoustic}. Then there exists the
hydrodynamic limit $f$ of $f_\varepsilon $ such that $f=\mathbf{P}f$, and
furthermore its macroscopic variables $\rho,\,u,\,\theta$ solve the
acoustic system \eqref{limit}.
\end{prop}

The acoustic system is a linear system and it is globally well-posed
in the Sobolev space.

\begin{lem}\label{regularity} The acoustic system \eqref{limit} is  globally well-posed in  $H^s(\mathbb{T}^3)$ space, for any $s\geq
0$. Moreover, we obtain the following estimates:
\begin{equation}\label{conser}
\frac{d}{dt}\{||\rho_1||^2_{H^s}+||u_1||^2_{H^s}+\tfrac{3}{2}||\theta_1||^2_{H^s}
\}=0
\end{equation}
\end{lem}

\begin{proof} The existence of solutions can be verified, for
instance by solving the ordinary differential equation after taking
Fourier transform in $x\in \mathbb{T}^3$. The energy estimates give rise to
the conservation of energy \eqref{conser}. The uniqueness is easily
deduced.
\end{proof}

\section{uniform spatial energy estimates}

In this section, we shall establish a uniform spatial energy
estimate for $f_\eps $,
 a solution to \eqref{P_B}:
\[
 \partial_t f_\varepsilon  +v\cdot\nabla_xf_\varepsilon
+\frac{1}{\varepsilon} \mathcal{L}f_\varepsilon =\Gamma(f_\varepsilon , f_\varepsilon )
\]
 For the convenience, we
rewrite the fluid part $\mathbf{P}f_\varepsilon $ as follows:
\[
\mathbf{P}f_\varepsilon =\{a^\varepsilon(t,x)+b^\varepsilon (t,x)\cdot
v+c^\varepsilon(t,x)|v|^2\}\sqrt{\mu}
\]
Our goal is to estimate $a^\varepsilon(t,x),\;b^\varepsilon
(t,x),\;\text{and }c^\varepsilon(t,x)$ in terms of
$(\mathbf{I-P})f_\varepsilon $.

\begin{lem}\label{positivity} Assume $f_\varepsilon $ is a solution
to \eqref{P_B} satisfying conservation of mass, momentum and energy:
\begin{equation}\label{conservation}
(f^{\varepsilon}(t),[1,v,|v|^2]\sqrt{\mu})=0.
\end{equation}
Then there exists $C_1>0$ such that
\begin{align}\label{PE}
\varepsilon\sum_{|\alpha|\leq N+1}\|\partial_\alpha\mathbf{P}f_\varepsilon
\|^2 \leq \varepsilon\frac{dG(t)}{dt}+\frac{C_1}{\varepsilon}
\sum_{|\alpha|\leq N+1}\|\partial_\alpha(\mathbf{I-P})f_\varepsilon
\|_\nu^2+ C_1\varepsilon \sum_{|\alpha|\leq
N}\|\partial_\alpha\Gamma(f_\varepsilon , f_\varepsilon )_{\|}\|^2
\end{align}
where $G(t)$ is defined as
\begin{equation}
\begin{split}
-&\sum_{|\alpha|\leq N}\int_{\mathbb{T}^3}\left(\langle(\mathbf{I-P})\partial_\alpha f_\varepsilon
\,,\zeta_{ij}\rangle\cdot\partial_j\partial_\alpha b^\varepsilon
-\langle(\mathbf{I-P})\partial_\alpha f_\varepsilon \,,\zeta_c\rangle\cdot\nabla_x\partial_\alpha c^\varepsilon\right)\,dx\\
&-\sum_{|\alpha|\leq N}\int_{\mathbb{T}^3}\left(\langle(\mathbf{I-P})\partial_\alpha f_\varepsilon
\,,\zeta\rangle\cdot\nabla_x\partial_\alpha a^\varepsilon\,-\partial_\alpha
b^\varepsilon\cdot\nabla_x\partial_\alpha a^\varepsilon\right)\,dx\,.
\end{split}
\end{equation}
Here $\zeta_{ij}(v)\,,\zeta_c(v)\,,\zeta_a(v)$ are some fixed linear
combinations of the basis
\[
[\sqrt{\mu}, v_i\sqrt{\mu}, v_iv_j\sqrt{\mu},v_i|v|^2\sqrt{\mu}]
\]
for $1\leq i\,,j\leq 3$, and $f_{\|}$ is the $L^2_v$ projection of
$f$ onto the subspace generated by the same basis. It is obvious
that $|G(t)|\leq C \mathcal{E}_{N,0}(t) $.
\end{lem}
\begin{proof}
The proof is similar to the one of Lemma 6.1 in \cite{G}. For the
clear presentation of this article, we provide the key ingredients
and estimates and point out the difference.
 From the conservation of mass, momentum, and energy
\eqref{conservation}, it follows that
\[
\int_{\mathbb{T}^3}a^\varepsilon(t,x)dx=
\int_{\mathbb{T}^3}b^\varepsilon(t,x)dx=
\int_{\mathbb{T}^3}c^\varepsilon(t,x)dx=0
\]
By Poincar$\acute{e}$ inequality, it suffices to estimate
\[
\nabla_x\partial_\alpha a^\varepsilon,\;\nabla_x\partial_\alpha
b^\varepsilon,\;\nabla_x\partial_\alpha c^\varepsilon,\;
\]
for $|\alpha|\leq N$. First, we use the local conservation laws:
Multiply $\sqrt{\mu}, v\sqrt{\mu},|v|^2\sqrt{\mu}$ with \eqref{P_B}
and integrate in $v\in \mathbb{R}^3$. By the collision invariants,
we obtain
\begin{equation}\label{hydro}
\begin{split}
&\partial_ta^\varepsilon=\frac{1}{2}\langle v\cdot
\nabla_x(\mathbf{I-P})f_\varepsilon ,|v|^2\sqrt{\mu}\rangle\\
&\partial_tc^\varepsilon+\frac{1}{3}\nabla_x\cdot
b^\varepsilon=-\frac{1}{6}\langle v\cdot
\nabla_x(\mathbf{I-P})f_\varepsilon ,|v|^2\sqrt{\mu}\rangle\\
&\partial_tb^\varepsilon+\nabla_xa^\varepsilon+5\nabla_xc^\varepsilon=-\langle
v\cdot \nabla_x(\mathbf{I-P})f_\varepsilon ,v\sqrt{\mu}\rangle
\end{split}
\end{equation}
The second ingredient of the proof is the macroscopic equations. By
plugging $f_\varepsilon =\mathbf{P}f_\varepsilon +(\mathbf{I-P}) f_\varepsilon $ into
\eqref{P_B}, we get
\[
\begin{split}
\{\partial_ta^\varepsilon+\partial_tb^\varepsilon \cdot
v+\partial_tc^\varepsilon |v|^2\}\sqrt{\mu}+ v\cdot
\{\nabla_xa^\varepsilon +\nabla_xb^\varepsilon \cdot
v+\nabla_xc^\varepsilon|v|^2\}\sqrt{\mu}\\
=-\{\partial_t+v\cdot\nabla_x\} (\mathbf{I-P})f_\varepsilon
-\frac{1}{\varepsilon}L(\mathbf{I-P})f_\varepsilon + \Gamma(f_\varepsilon ,f_\varepsilon
)
\end{split}
\]
Fix $t,x$, and compare the coefficients on both sides in front of
\[
[\sqrt{\mu}, v_i\sqrt{\mu}, v_iv_j\sqrt{\mu},v_i|v|^2\sqrt{\mu}].
\]
Then we get the following macroscopic equations as
\begin{align}
\partial_ic^\varepsilon&=l_c^\varepsilon+h_c^\varepsilon\\
\partial_tc^\varepsilon+ \partial_i b_i^\varepsilon&=
l_{i}^\varepsilon+h_{i}^\varepsilon\\
\partial_i b_j^\varepsilon+\partial_j b_i^\varepsilon
&=l_{ij}^\varepsilon+h_{ij}^\varepsilon,\; i\neq j\\
\partial_tb^\varepsilon_i+ \partial_i a^\varepsilon&=
l_{bi}^\varepsilon+h_{bi}^\varepsilon\\
\partial_ta^\varepsilon &= l_a^\varepsilon+h_a^\varepsilon
\end{align}
Here the linear parts $l_c^\varepsilon,l_{i}^\varepsilon,
l_{ij}^\varepsilon,l_{bi}^\varepsilon,l_{a}^\varepsilon$ are of the
form
\begin{equation}
\langle -\{\partial_t+v\cdot\nabla_x\} (\mathbf{I-P})f_\varepsilon
-\frac{1}{\varepsilon}\mathcal{L}(\mathbf{I-P})f_\varepsilon , \zeta
\rangle
\end{equation}
where $\zeta$ is a linear combination of the basis
\[
[\sqrt{\mu}, v_i\sqrt{\mu}, v_iv_j\sqrt{\mu},v_i|v|^2\sqrt{\mu}],
\]
and accordingly,
$h_{c}^\varepsilon,h_{i}^\varepsilon,h_{ij}^\varepsilon
h_{bi}^\varepsilon,h_{a}^\varepsilon$ are  defined as $\langle
\Gamma(f_\varepsilon ,f_\varepsilon ),\zeta\rangle$ with the same choices of
$\zeta$.

Following the proof of Lemma 6.1 in \cite{G}, we first deduce
\begin{align*}
\|\nabla\partial_{\alpha}b^{\varepsilon}\|^{2} &\leq-\frac{d}{dt}
\int_{\mathbb{T}^{3}}\langle(\mathbf{I-P})
\partial_{\alpha
}f^{\varepsilon},\zeta_{ij}\rangle\cdot\partial_j\partial_{\alpha
}b^{\varepsilon}dx\\
&\;\;+C\|\nabla_{x}\partial_{\alpha}(\mathbf{I-P})
f^{\varepsilon}\|_{\nu}\{\|\nabla\partial_{\alpha}a^{\varepsilon}\|
+\|\nabla\partial_{\alpha}c^{\varepsilon}\|\}\\
&\;\;+C\{\|
\partial_{\alpha}(\mathbf{I-P})
f^{\varepsilon}\|_{\nu}^2+\|\nabla_{x}
\partial_{\alpha}(\mathbf{I-P})
f^{\varepsilon}\|_{\nu}^2\}
\\
&\;\;+\frac{C}{\varepsilon}\{\|\nabla_{x}\partial_{\alpha}(\mathbf{I-P})
f^{\varepsilon}\|_{\nu}+\|\partial_{\alpha}(\mathbf{I-P})f^{\varepsilon
}\|_{\nu}\}\|\nabla\partial_{\alpha}b^{\varepsilon}\|
+C\|\partial_{\alpha
}h_{\|}^{\varepsilon}\|\cdot\|\nabla\partial_{\alpha}b^{\varepsilon}\|.
\end{align*}
Note that the difference from the estimate in \cite{G} so far is
that the scaling parameter
 $\varepsilon$ is absent in the $t$-derivative term due to the
acoustic scaling. Now multiply it by $\varepsilon$ and apply the
Cauchy-Schwarz inequality to get
\begin{align*}
\varepsilon\|\nabla\partial_{\alpha}b^{\varepsilon}\|^{2} &
\leq-\varepsilon \frac
{d}{dt}\int_{\mathbb{T}^{3}}\langle(\mathbf{I-P})\partial_{\alpha
}f^{\varepsilon},\zeta_{ij}\rangle\cdot\partial_{j}\partial_{\alpha
}b^{\varepsilon}dx
+\frac{\varepsilon^2}{2}\{\|\nabla\partial_{\alpha}a^{\varepsilon}\|^{2}
 +\|\nabla\partial_{\alpha}c^{\varepsilon}\|^{2}\}\\
&\;\;+\frac{C}{\varepsilon}\{\|\nabla_{x}\partial_{\alpha}(\mathbf{I-P})%
f^{\varepsilon}\|_{\nu}^{2}+\|\partial_{\alpha}(\mathbf{I-P})f^{\varepsilon
}\|_{\nu}^{2}\} +C\varepsilon\|\partial_{\alpha
}h_{\|}^{\varepsilon}\|^2+\frac{\varepsilon}{2}
\|\nabla\partial_{\alpha}b^{\varepsilon}\|^{2}.
\end{align*}
By the same token, we obtain the similar estimates on
$\nabla\partial_\alpha c^\varepsilon$ and $\nabla\partial_\alpha
a^\varepsilon$ as follows:
\begin{align*}
\varepsilon\|\nabla\partial_{\alpha}c^{\varepsilon}\|^{2}  &  \leq-
\varepsilon\frac{d}{dt}%
\int_{\mathbb{T}^{3}}\langle(\mathbf{I-P})\partial_{\alpha
}f^{\varepsilon},\zeta_{c}\rangle\cdot\nabla_{x}\partial_{\alpha
}c^{\varepsilon}dx+\frac{\varepsilon^2}{2}
\|\nabla\partial_{\alpha}b^{\varepsilon}\|^{2}\\
&\;\;+\frac{C}{\varepsilon}\{\|\nabla_{x}\partial_{\alpha}(\mathbf{I-P})%
f^{\varepsilon}\|_{\nu}^{2}+\|\partial_{\alpha}(\mathbf{I-P})f^{\varepsilon
}\|_{\nu}^{2}\} +C\varepsilon\|\partial_{\alpha
}h_{\|}^{\varepsilon}\|^2+\frac{\varepsilon}{2}
\|\nabla\partial_{\alpha}c^{\varepsilon}\|^{2},\\
\varepsilon\|\nabla\partial_{\alpha}a^{\varepsilon}\|^{2}  & \leq
-\varepsilon\frac{d}{dt}\{\int_{\mathbb{T}^{3}}\langle(\mathbf{I-P})%
\partial_{\alpha}f^{\varepsilon},\zeta\rangle\cdot\nabla_{x}\partial_{\alpha
}a^{\varepsilon}dx+\int_{\mathbb{T}^{3}}\partial_{\alpha
}b^{\varepsilon}\cdot\nabla_{x}\partial_{\alpha}a^{\varepsilon}dx\}+
\frac{\varepsilon^2}{2}
\|\nabla\partial_{\alpha}b^{\varepsilon}\|^{2}\\
&\;\;+\frac{C}{\varepsilon}\{\|\nabla_{x}\partial_{\alpha}(\mathbf{I-P})%
f^{\varepsilon}\|_{\nu}^{2}+\|\partial_{\alpha}(\mathbf{I-P})f^{\varepsilon
}\|_{\nu}^{2}\} +C\varepsilon\|\partial_{\alpha
}h_{\|}^{\varepsilon}\|^2+\frac{\varepsilon}{2}
\|\nabla\partial_{\alpha}a^{\varepsilon}\|^{2}.
\end{align*}
By absorbing the hydrodynamic terms in the right hand sides into the
left hand sides, we obtain the desired estimates \eqref{PE}.
\end{proof}

Next we perform the energy estimates of spatial derivatives.

\begin{lem}\label{energy} Assume that $f_\eps $ is a solution to
equation \eqref{P_B} and satisfies \eqref{conservation}; then there
exists a constant $C_1\geq 1$ such that the following energy
estimate is valid:
\begin{equation}\label{energy-est}
\begin{split}
\frac{d}{dt}\{C_1&\sum_{|\alpha|\leq N+1}\|\partial_\alpha f_\varepsilon
\|^2- \varepsilon\delta G(t)\}+\delta \sum_{|\alpha|\leq
N+1}\{\varepsilon\|
\partial_{\alpha}\mathbf{P}f_\varepsilon \|^{2}
+\frac{1}{\varepsilon} \|\partial_{\alpha}(\mathbf{I-P})f_\varepsilon
\|_{\nu}^{2}\}\\
&\leq 2C_1\sum_{|\alpha|\leq N+1}(\partial_\alpha\Gamma(f_\varepsilon ,
f_\varepsilon ),\partial_\alpha f_\varepsilon ) +\varepsilon
\delta\sum_{|\alpha|\leq N}\|\partial_\alpha\Gamma(f_\varepsilon ,
f_\varepsilon )_{\|}\|^2\\
&\leq C\{\mathcal{E}_{N,0}^{1/2}(f_\varepsilon ) +\mathcal{E}_{N,0}(f_\varepsilon
)\}\mathcal{D}_{N,0}(f_\varepsilon )
\end{split}
\end{equation}
\end{lem}
\begin{proof}
We take $\partial_\alpha$ of \eqref{P_B} and sum over $\alpha$ to get
\begin{align*}
\frac{1}{2}\frac{d}{dt}\|\partial_\alpha f_\varepsilon \|^2+
\frac{\delta}{\varepsilon}\|(\mathbf{I-P})\partial_\alpha f_\varepsilon
\|_\nu^2 \leq (\partial_\alpha\Gamma(f_\varepsilon , f_\varepsilon
),\partial_\alpha f_\varepsilon )\,.
\end{align*}
By Lemma \ref{positivity}, there is a constant $C_1\geq 1$ such that
\begin{equation}
\begin{split}
\frac{\delta}{2\varepsilon}\sum_{|\alpha|\leq N+1}&\|\partial_\alpha(\mathbf{I-P})f_\eps \|^2_\nu\\
&\geq \frac{\delta\varepsilon}{2C_1}\sum_{|\alpha|\leq
N+1}\|\partial_\alpha\mathbf{P}f_\eps \|^2-\frac{\delta\varepsilon}{2C_1}\frac{dG}{dt}-\frac{\delta\varepsilon}{2}
\sum_{|\alpha|\leq N}\|\partial_\alpha\Gamma(f_\eps ,f_\eps )_{\|}\|^2\,.
\end{split}
\end{equation}
Multiply by $C_1$ and collecting terms, we deduce the first
inequality in \eqref{energy-est}. By the nonlinear estimate in
\eqref{nonlinear2}, it is easy to derive that for $|\alpha|\leq N$
\begin{equation}\label{proj-est}
\|\partial_\alpha\Gamma(f_\varepsilon \,,f_\varepsilon )_{\|}\|^2\leq
C\mathcal{E}_{N,0}(f_\varepsilon )\mathcal{D}_{N,0}(f_\varepsilon )\,,
\end{equation}
and
\begin{equation}
\begin{split}
(\partial_\alpha\Gamma(f_\varepsilon \,,f_\varepsilon )\,,\partial_\alpha f_\varepsilon )&\leq C\mathcal{E}^{1/2}_{N,0}(f_\varepsilon )\|\varepsilon^{1/2}\partial_\alpha f_\varepsilon \|_\nu\|\varepsilon^{-1/2}\partial_\alpha(\mathbf{I-P})f_\eps \|_\nu\\
&\leq C\mathcal{E}^{1/2}_{N,0}(f_\varepsilon )\mathcal{D}_{N,0}(f_\varepsilon )\,.
\end{split}
\end{equation}
Thus, the second inequality in \eqref{nonlinear2} follows and this
finishes the proof of the lemma.
 \end{proof}

\section{The first order remainder}

In this section we finish the proof of Theorem \ref{Acoustic}. We
already established a pure spatial energy estimate for all collision
kernels in Lemma \ref{energy}. For general derivatives $\partial^\beta_\alpha$,
different collision kernels require different weight functions, we
treat separately in two cases: hard potentials then soft potentials
and Landau kernel.

\subsection{Proof of hard potential case of Theorem \ref{Acoustic}}

\begin{proof} First note that
for the hydrodynamic part $\mathbf{P}f_\varepsilon $,
\[
\|\partial_\alpha^\beta\mathbf{P}f_\varepsilon \|\leq
C\|\partial_\alpha\mathbf{P}f_\varepsilon \|
\]
which has been estimated in  Lemma \ref{energy}. In order to prove
Theorem \ref{Acoustic}, it remains to estimate the remaining
microscopic part $\partial_\alpha^\beta(\mathbf{I-P})f_\varepsilon $ for
$|\alpha|+ |\beta|\leq N$. We take $\partial^\beta_\alpha$ of equation \eqref{P_B}
and sum over $|\alpha|+ |\beta|\leq N$ to get
\begin{equation}\label{IP-equation}
\begin{split}
\partial_t\partial^\beta_\alpha&(\mathbf{I-P})f_\eps +v\cdot\nabla_x\partial^\beta_\alpha(\mathbf{I-P})f_\eps +\frac{1}{\varepsilon}\partial^\beta_\alpha \mathcal{L}(\mathbf{I-P})f_\eps \\
&+\left(\partial_t\partial^\beta_\alpha\mathbf{P}f_\eps +v\!\cdot\!\nabla\!_x\partial^\beta_\alpha\mathbf{P}f_\eps +\binom{\beta_1}{\beta}\partial_{\beta_1}v\!\cdot\!\nabla\!_x\partial^{\beta-\beta_1}_{\alpha}f_\varepsilon \right)\\
&=\partial^\beta_\alpha\Gamma(f_\varepsilon \,,f_\varepsilon )\,,
\end{split}
\end{equation}
where $|\beta_1|=1$. Taking the inner product with $w^{2l}\partial^\beta_\alpha(\mathbf{I-P})f_\eps $,
we get
\begin{equation}\label{energy-id}
\begin{split}
\frac{d}{dt}&\left\{\frac{1}{2}\|w^l\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2\right\}+\frac{1}{\varepsilon}(w^{2l}\partial^\beta_\alpha\mathcal{L}(\mathbf{I-P})f_\eps \,,\partial^\beta_\alpha(\mathbf{I-P})f_\eps )\\
&+\left(\partial_t\partial^\beta_\alpha\mathbf{P}f_\eps +v\!\cdot\!\nabla\!_x\partial^\beta_\alpha\mathbf{P}f_\eps +\binom{\beta_1}{\beta}\partial^{\beta_1}_\alphav\!\cdot\!\nabla\!_x\partial^{\beta-\beta_1}_{\alpha}f_\varepsilon
\,,w^{2l}\partial^\beta_\alpha(\mathbf{I-P})f_\eps
\right)\\
&\leq\left(w^{2l}\partial^\beta_\alpha\Gamma(f_\varepsilon \,,f_\varepsilon )\,,\partial^\beta_\alpha(\mathbf{I-P})f_\eps \right)\,.
\end{split}
\end{equation}
By the linear estimate \eqref{linear}, we have
\begin{equation}
\frac{1}{\varepsilon}(w^{2l}\partial^\beta_\alpha\mathcal{L}(\mathbf{I-P})f_\eps \,,\partial^\beta_\alpha(\mathbf{I-P})f_\eps )\geq
\frac{1}{2\varepsilon}\|w^l\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2_\nu-\frac{C}{\varepsilon}\|\partial_\alpha(\mathbf{I-P})f_\eps \|^2_\nu\,.
\end{equation}
From the local conservation laws \eqref{hydro} and the estimate
\eqref{proj-est},
\begin{equation}
\begin{split}
\|w^{2l}\partial_t\partial^\beta_\alpha\mathbf{P}f_\eps \|&\leq C\sum_{|\alpha|\leq N}(\|\partial_t\partial_\alpha a^\varepsilon\|+\|\partial_t\partial_\alpha b^\varepsilon\|+\|\partial_t\partial_\alpha c^\varepsilon\|)\\
&\leq C\left(\sum_{|\alpha|\leq N+1}\|\partial_\alpha f_\varepsilon \|_\nu+\sum_{|\alpha|\leq N}\|\partial_\alpha h^\varepsilon_{\|}\|\right)\\
&\leq C\left(\sum_{|\alpha|\leq N+1}\|\partial_\alpha f_\varepsilon
\|_\nu+\mathcal{E}^{1/2}_{N,0}(f_\varepsilon )\mathcal{D}^{1/2}_{N,0}(f_\varepsilon )\right)\,.
\end{split}
\end{equation}
We also have
\begin{equation}
\|w^{2l}v\!\cdot\!\nabla\!_x\partial^\beta_\alpha\mathbf{P}f_\eps \|\leq C\sum_{|\alpha|\leq N}\|\nabla_x\partial_\alpha\mathbf{P}f_\eps \|\leq
C\sum_{|\alpha|\leq N+1}\|\partial_\alpha\mathbf{P}f_\eps \|\,.
\end{equation}
Thus the first two inner products in the second line of
\eqref{energy-id} is bounded by
\begin{equation}
\frac{1}{8\varepsilon}\sum_{|\alpha|+|\beta|\leq N}\|\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2_\nu+C\left(\sum_{|\alpha|\leq
N+1}\|\partial_\alpha f_\varepsilon \|^2_\nu+\mathcal{E}^{1/2}_{N,0}(f_\varepsilon )\mathcal{D}_{N,0}(f_\varepsilon
)\right)\,.
\end{equation}
The last term in the second line of \eqref{energy-id} is bounded by
\begin{equation}
\begin{split}
C&|(\partial^{\beta_1}v\!\cdot\!\nabla\!_x\partial^{\beta-\beta_1}_\alpha(\mathbf{I-P})f_\eps \,,w^{2l}\partial^\beta_\alpha(\mathbf{I-P})f_\eps )|\\&+C|(\partial^{\beta_1}v\!\cdot\!\nabla\!_x\partial^{\beta-\beta_1}_\alpha\mathbf{P}f_\eps \,,
w^{2l}\partial^\beta_\alpha(\mathbf{I-P})f_\eps )|\\
&\leq C\|\nabla_x\partial^{\beta-\beta_1}_\alpha(\mathbf{I-P})f_\eps \|^2_\nu+\frac{1}{8\varepsilon}\|w^l\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2_\nu+C \varepsilon\|\partial_\alpha\mathbf{P}f_\eps \|^2\\
&\leq C \varepsilon\mathcal{D}_{N,l}(f_\varepsilon )+\frac{1}{8\varepsilon}\|w^l\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2_\nu+C
\varepsilon\|\partial_\alpha\mathbf{P}f_\eps \|^2\,,
\end{split}
\end{equation}
since $\nu(v)$ is bounded from below for hard potential.

Now we estimate the nonlinear term in \eqref{energy-id}. By the
nonlinear estimate in \eqref{nonlinear},
\begin{equation}
\begin{split}
(w^{2l}\partial^\beta_\alpha\Gamma(f_\varepsilon ,f_\varepsilon )\,,\partial^\beta_\alpha(\mathbf{I-P})f_\eps )&\leq C\mathcal{E}^{1/2}_{N,l}(f_\varepsilon )\|\varepsilon^{1/2}w^l\partial_\alpha f_\varepsilon \|_\nu\|\varepsilon^{-1/2}w^l\partial_\alpha(\mathbf{I-P})f_\eps \|_\nu\\
&\leq C\mathcal{E}^{1/2}_{N,l}(f_\varepsilon )\mathcal{D}_{N,l}(f_\varepsilon )\,.
\end{split}
\end{equation}
Using the coercivity of $\mathcal{L}$ \eqref{coer} and absorbing a
total of $\frac{1}{\varepsilon}\|w^l\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2_\nu$ from the right-hand
side, we have
\begin{equation}\label{IP-est}
\begin{split}
\sum_{|\alpha|+|\beta|\leq N}&\left(\frac{d}{dt}\left\{\frac{1}{2}\|w^l\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2\right\}+\frac{1}{4\varepsilon}\|w^l\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2_\nu\right)\\
&\leq C\sum_{aaN}\|\partial_\alpha f_\varepsilon \|^2_\nu+C\left(\mathcal{E}^{1/2}_{N,l}(f_\varepsilon
)+\varepsilon\right)\mathcal{D}_{N,l}(f_\varepsilon )\,.
\end{split}
\end{equation}

Multiplying \eqref{IP-est} by a factor and adding a large multiple
$K$ of \eqref{energy-est}, we have
\begin{equation}
\begin{split}
\frac{d}{dt}&(K\{C_1\sum_{|\alpha|\leq N+1}\|\partial_\alpha f_\varepsilon \|^2-\varepsilon\delta
G(t)\}+2\sum_{|\alpha|+|\beta|\leq N}\|w^l\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2)
+\mathcal{D}_{N,l}(f_\varepsilon )\\
&\leq C_K\left(\mathcal{E}^{1/2}_{N,l}(f_\varepsilon )+\mathcal{E}_{N,l}(f_\varepsilon
)+\varepsilon\right)\mathcal{D}_{N,l}(f_\varepsilon )\,.
\end{split}
\end{equation}
Notice that
\begin{equation}
\|w^l\partial^\beta_\alpha\mathbf{P}f_\eps \|^2\leq C\|\partial_\alpha\mathbf{P}f_\eps \|^2\leq C\|\partial_\alpha f_\varepsilon \|^2\,,
\end{equation}
and
\begin{equation}
G(t)\leq C\sum_{aaN}\|\partial_\alpha\mathbf{P}f_\eps \|(\|\mathbf{I-P}\partial_\alpha f_\varepsilon \|+\|\partial_\alpha \mathbf{P}f_\eps \|)\,.
\end{equation}
Thus we can redefine the instant energy by
\begin{equation}
\mathcal{E}_{N,l}(f_\varepsilon )=K\{C_1\sum_{|\alpha|\leq N+1}\|\partial_\alpha f_\varepsilon \|^2-\varepsilon\delta
G(t)\}+2\sum_{|\alpha|+|\beta|\leq N}\|w^l\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2
\end{equation}
for $\varepsilon$ sufficiently small. By a standard continuity argument, we
deduce our main estimate \eqref{en} by letting $\mathcal{E}_{N,l}(f_\varepsilon )$
be sufficiently small initially.
\end{proof}

\subsection{Proof of soft potential and Landau cases for Theorem \ref{Acoustic}}

We follow the same idea as in the hard potential case to establish
\eqref{en} for both soft potential and Landau kernels. First, for
soft potential cases, we take inner product of
$w^{2(l-|\beta|)|\gamma|}\partial^\beta_\alpha(\mathbf{I-P})f_\varepsilon$ with the equation
\eqref{IP-equation} and sum over $|\alpha|+|\beta| \leq N$ to get
\begin{equation}\label{energy-id-soft}
\begin{split}
\frac{d}{dt}&\left\{\frac{1}{2}\|w^{\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2\right\}+\frac{1}{\varepsilon}(w^{2\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha\mathcal{L}(\mathbf{I-P})f_\eps \,,\partial^\beta_\alpha(\mathbf{I-P})f_\eps )\\
&+\left(\partial_t\partial^\beta_\alpha\mathbf{P}f_\eps +v\!\cdot\!\nabla\!_x\partial^\beta_\alpha\mathbf{P}f_\eps +\binom{\beta_1}{\beta}\partial^{\beta_1}_\alphav\!\cdot\!\nabla\!_x\partial^{\beta-\beta_1}_{\alpha}f_\varepsilon
\,,w^{2\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha(\mathbf{I-P})f_\eps
\right)\\
&\leq\left(w^{2\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha\Gamma(f_\varepsilon \,,f_\varepsilon )\,,\partial^\beta_\alpha(\mathbf{I-P})f_\eps \right)\,,
\end{split}
\end{equation}
for $|\beta_1|=1$. By the linear estimate \eqref{linear-soft}, we
have
\begin{equation}
\begin{split}
&\frac{1}{\varepsilon}(w^{2\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha\mathcal{L}(\mathbf{I-P})f_\eps \,,\partial^\beta_\alpha(\mathbf{I-P})f_\eps )\\
&\geq
\frac{1}{2\varepsilon}\|w^{\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2_\nu-\frac{C}{\varepsilon}\|\partial_\alpha(\mathbf{I-P})f_\eps \|^2_\nu\,.
\end{split}
\end{equation}
From the local conservation laws \eqref{hydro}, we have
\begin{equation}
\|w^{2\{l-|\beta|\}|\gamma|}\partial_t\partial^\beta_\alpha\mathbf{P}f_\eps \| \leq
C\left(\sum_{|\alpha|\leq N+1}\|\partial_\alpha f_\varepsilon
\|_\nu+\mathcal{E}^{1/2}_{N,0}(f_\varepsilon )\mathcal{D}^{1/2}_{N,0}(f_\varepsilon )\right)\,.
\end{equation}
We also have
\begin{equation}
\|w^{2\{l-|\beta|\}|\gamma|}v\!\cdot\!\nabla\!_x\partial^\beta_\alpha\mathbf{P}f_\eps \|\leq
C\sum_{|\alpha|\leq N}\|\nabla_x\partial_\alpha\mathbf{P}f_\eps \|\leq C\sum_{|\alpha|\leq
N+1}\|\partial_\alpha\mathbf{P}f_\eps \|\,.
\end{equation}
Note that $\|\cdot\|_\nu$ is equivalent to $\|w^{\gamma/2}\cdot\|$,
the first two inner products in the second line of
\eqref{energy-id-soft} is bounded by
\begin{equation}
\frac{1}{8\varepsilon}\sum_{|\alpha|+|\beta|\leq N}\|w^{\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2_\nu+C\left(\sum_{|\alpha|\leq
N+1}\|\partial_\alpha f_\varepsilon \|^2_\nu+\mathcal{E}^{1/2}_{N,0}(f_\varepsilon )\mathcal{D}_{N,0}(f_\varepsilon
)\right)\,.
\end{equation}
The weight function $w^{|\beta|\gamma}$ is so designed to treat the
last term in the second line of \eqref{energy-id-soft}
\begin{equation}
\begin{split}
C&|(\partial^{\beta_1}v\!\cdot\!\nabla\!_x\partial^{\beta-\beta_1}_\alpha(\mathbf{I-P})f_\eps \,,w^{2\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha(\mathbf{I-P})f_\eps )|\\&+C|(\partial^{\beta_1}v\!\cdot\!\nabla\!_x\partial^{\beta-\beta_1}_\alpha\mathbf{P}f_\eps \,,
w^{2\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha(\mathbf{I-P})f_\eps )|\\
&\leq C\|w^{l+|\beta-\beta_1|\gamma}\nabla_x\partial^{\beta-\beta_1}_\alpha(\mathbf{I-P})f_\eps \|^2_\nu+\frac{1}{8\varepsilon}\|w^{\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2_\nu+C \varepsilon\|\partial_\alpha\mathbf{P}f_\eps \|^2\\
&\leq C \varepsilon\mathcal{D}_{N,l}(f_\varepsilon
)+\frac{1}{8\varepsilon}\|w^{\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2_\nu+C \varepsilon\|\partial_\alpha\mathbf{P}f_\eps \|^2\,.
\end{split}
\end{equation}
The nonlinear term in \eqref{energy-id-soft} is estimated by
\eqref{nonlinear-soft},
\begin{equation}
\begin{split}
&(w^{2\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha\Gamma(f_\varepsilon ,f_\varepsilon )\,,\partial^\beta_\alpha(\mathbf{I-P})f_\eps )\\
&\leq C\mathcal{E}^{1/2}_{N,l}(f_\varepsilon )\|\varepsilon^{1/2}w^{\{l-|\beta|\}|\gamma|}\partial_\alpha f_\varepsilon \|_\nu\|\varepsilon^{-1/2}w^{\{l-|\beta|\}|\gamma|}\partial_\alpha(\mathbf{I-P})f_\eps \|_\nu\\
&\leq C\mathcal{E}^{1/2}_{N,l}(f_\varepsilon )\mathcal{D}_{N,l}(f_\varepsilon )\,.
\end{split}
\end{equation}
The rest of the proof is similar to the hard potential case, the
nonlinear estimate \eqref{en} can be deduced by letting
\begin{equation}
\begin{split}
\mathcal{E}_{N,l}(f_\varepsilon )&=K\{C_1\sum_{|\alpha|\leq N+1}\|\partial_\alpha f_\varepsilon \|^2-\varepsilon\delta G(t)\}\\
&+2\sum_{|\alpha|+|\beta|\leq N}\|w^{\{l-|\beta|\}|\gamma|}\partial^\beta_\alpha(\mathbf{I-P})f_\eps \|^2\,.
\end{split}
\end{equation}
To establish the estimate \eqref{en} for the Landau case for which
the power of weight is $\gamma=-1$. We follow the same procedure as
in the soft potential case. Take the inner product with
$w^{2l-2|\beta|}\partial^\beta_\alpha(\mathbf{I-P})f_\eps $ for equation \eqref{IP-equation} to get
\eqref{energy-id-soft} with $\gamma=-1$. All the estimates for the
soft potential case can applied for the case $\gamma=-1$. So we omit
the details here.\\

\indent{\bf Acknowledgements.} This project was initiated by the
suggestion of Prof. Yan Guo when N. Jiang visited The Applied
Mathematics Division of Brown University in May 2007. The main part
of this work was carried out when J. Jang was a member of IAS
2007-2008 academic year. The authors would like to appreciate the
hospitality of these institutes. They also wish to thank Y. Guo for
many discussions and his long-term support.

redpajama_set_name: RedPajamaArXiv


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text:
Home Berne
Quick Berne Convention Overview
What You Didn't Know About the Berne Convention
The Berne Convention for the Protection
of Literary and Artistic Works is an international treaty that imposes certain
regulations and statutes that are accepted by its collective members, or
signatories. The Berne Convention was created to institute a cohesive
amalgamation of international copyright law for the purpose of convenience and
ease of interpretation between countries.
Before the Berne Convention, countries
had their own copyright processes and laws that would make the recognition of
other countries' copyright laws a confusing matter. The institution of an
international copyright law allows for the proper interpretation of general
copyright laws and legislation between countries so aspects such as copyright
infringement can be properly assessed and regulated.
The Berne Convention, as the international
agreement is commonly referred, derives its name from the city in which it was
first accepted as an international copyright law, Berne, Switzerland, in 1886.
The Berne Convention was first developed by a member of the Association
Litteraire et Artistique Internationale, Victor Hugo.
One the most notable
inclusions of provisions in the Berne Convention is the fact that under its
legislation, creative works and their copyrights are inferred upon their
creation and no proper registration or application for copyright is necessary.
Copyrights are inherited naturally upon a creative works production.
Once a work is termed as "fixed,"
or written and/or recorded in some physical form, the author of such work is
granted copyrights to that particular work, and any other creative works
stemming from the original. The copyright remains in the author's possession
until the copyright term expires, or the author explicitly gives up his/her
right to copyright ownership.
Thus, under this provision, nations under the
Berne Convention must recognize the copyrights of authors and their works from
foreign countries and consider them under domestic copyright law statutes as
having the same privileges and rights.
The Berne Convention became an
international copyright law that was influenced by the French concept of
"right of the author," as opposed to the United Kingdom's concept of
copyright, which was solely based on economic and monetary concerns. The
inherent development of the Berne Convention and its philosophy and structure
are evidence that there were different notions regarding copyrights that varied
from nation to nation, thus making the development of cohesive standards for
international copyright law deemed to be necessary.
Before the Berne Convention, any
particular published work could be created and be protected under its own
nation's copyright laws, but in another, the work could be copied and
distributed freely. Nations were not forced or required to observe foreign
copyright laws and would often refuse to adhere to them because they did not
apply under their own national copyright laws and standards. The Berne
Convention would allow for the integration of copyright laws so that all
nations' authors and their creative works or materials' copyrights could be
protected under one international copyright law, regardless if the respective
nations had their own particular system or process for copyright.
The Berne Convention was also influenced by the
Paris Convention, allowing for the integration of recognition of other types
intellectual property, as well as setting up a bureau to handle the
administrative responsibilities of the new statutes. By 1893, the Berne
Convention instituted the United International Bureaux for the Protection of
Intellectual Property, situated in Geneva.
The main purpose of this Bureau,
also known as BIRPI, was to simply apply and enforce the provisions of the Berne
Convention. BIRPI would eventually move its headquarters to Geneva in 1960,
which was also home to the United Nations. BIRPI would become the predecessor
to the World Intellectual Property Organization, or WIPO, which was created in
As per the structure and provisions granted in the
Berne Convention, the United States did become a member or signatory on March
1st, 1989, when the United States would pass the Berne Convention
Implementation Act of 1988. The reason that the U.S. refused to join the Berne
Convention previously was because copyright laws and the required registration
and mandatory notice of copyright were implemented into U.S. legislation.
Joining the convention would incur major revisions and changes to copyright
The inclusion of moral rights was also an
issue that prompted the refusal to join. However, the United States would
eventually realize the need for a governing international copyright law, and
even though it did have to amend and revise some legal statutes, the U.S. still
has its original system of copyright registration and overall statutes.
Currently, the Berne Convention is upheld by 164 countries around the world.
All You Need to Know About The Berne Convention Content
The Berne Convention for the Protection of Literary and Artistic Works of 1886 has as its main purpose of providing for unifying a collection of copyright laws to be adhered to at the international level. The provisions and guidelines provided by the Berne Convention has at its core a philosophy of simplicity. The Berne Convention is based upon three basic principles that contain certain guidelines outlining the basic means of copyright protection. The three basic principles are as follows:
All works created and published in one of the aligned countries with the Berne Convention shall grant the author and his/her perspective works or materials the same copyrights and protection in the other aligned nations under the Berne Convention.
In other words, countries must provide for the same copyrights and protections that its own citizens receive under law for their published works or materials to all foreign authors and their copyrighted works, as long as their countries of origin are part of the Berne Convention.
Copyrights and protection of the author under copyright laws are considered to be "automatic". Copyrights are to be considered inherent based upon the creation and publication of a work and are not subject to conditional or required registration or notification to acquire copyrights and protection.
The copyright recognition and protection administered by the Berne Convention is independent. In other words, the Berne Convention legislation is to apply regardless of a particular nation's own copyright laws and regulations.
However, in certain instances where a country applies a term of expiration of a copyright to their works or materials of an author, it must be noted that the Berne Convention has its own expiration terms.
If the country's expiration term for a copyright is longer than the original nation of origin of a certain work, protection and copyrights under the Berne Convention may be denied.
The copyright and protection of such a work will adhere to the expiration terms of its original place of origin or publication. This is often referred to as "the rule of shorter term." It is important to note that not all countries have accepted this regulation under the Berne Convention.
These three basic guidelines allow for a very basic and minimal implementation of legislation for the recognition of copyrights and the author and his/her created works or materials protection. Many countries, such as the United States, may have more regulatory or stricter statutes in their own legislation systems, but the recognition of foreign works in their own countries must adhere to the Berne Convention rules.
The United States proves to be a great example as to why international copyright legislation became necessary. The United States only allows for copyright infringement suits to be charged if the owner of a particular work had registered the work in question with the United States Copyright Office.
No copyright infringement suit would be recognized unless the work was properly registered. This legislation would prove to be a problem for works that originated or were first published in other countries and were not registered with the Copyright Office. The opportunity for copyright infringement to occur on such foreign works would not be held accountable or punishable by law.
Because of such legislation, the United States did not join the Berne Convention until 1989, but it still applies its original copyright registration process for national works and does not deem it necessary for international works to be properly registered in the United States in order to have the ability to bring up copyright infringement suits against violators or commissions of piracy.
The Berne Convention also institutes some basic copyright standards of protection to compliment the three basic principles. Firstly, copyright protection under this international copyright agreement will include works that can be considered of literary, artistic, or scientific origins, regardless of the means of reproduction or media in which they are produced. The Berne Convention also provides for a list of basic rights to be adhered to by all countries that are signatories to this international provision as it applies to international copyrighted works:
● The right to translate;
● The right to make certain adaptations or arrangements;
● The right to perform applicable works;
● The right to public recitation;
● The right to communicate the performance of applicable works (i.e: media reviews for concerts, plays, performances, etc.);
● The right to broadcast the performance of applicable works;
● The right to reproduction.
This provision may be subject to the nation or country allowing for permission for reproduction. Also, no actual authorization may be needed if the reproduction of a copyrighted work will not defile the original content or intention of the work and the owner of the copyrighted work will not be subject to defamation and will be properly compensated, if such compensation is in order (as in the case of the performing arts and their various modes of production).
The right to use a particular work as the basis of audiovisual purposes, and the right to reproduce, perform, or distributed that particular audiovisual work is also included in this Agreement.
Another aspect that the Berne Convention also incorporates is the concept of "moral rights." These rights are to give exclusive privilege to to the owner of a copyrighted work to denounce any kind of modification or derogatory action otherwise intended by the work itself or the author's personal will, as such modification may affect the author or copyright owner's reputation in a negative way.
The Berne Convention also applies a period of copyright protection and rights that has an expiration date. Generally, a work is protected under the Berne Convention provisions no less than fifty years after the author's death. If such a work is published anonymously or pseudonymously, the fifty year term is applied to the date of the work's first publication rather the author's death.
The exception for this particular instance is if a pseudonymously published work can provide for a positive identification of the actual author or copyright owner of such work, or if the author makes it known publicly that the work is published under a pseudonym. In that event, it reverts to the original limitation of fifty years after the death of the author.
For the case of audiovisual works, such as feature films and motion pictures, the fifty year period applies to the date of the work's release to the general public. If the work is not publicly released, then it applies to the date of its creation. For applied arts and photography, the term is shortened to twenty-five years from the date of creation of that particular work.
The Berne Convention also provides for the structure of its Council, which is made up of an Assembly and an Executive Committee. All countries and nations that are considered as signatories or adhere to the Berne Convention statutes are members of the Assembly.
The Executive Committee is elected by members of the union or Assembly, with the exception of Switzerland, which is a member by virtue of office. All countries and nations are eligible to join as signatories to the Berne
Uncover the Berne Convention
The Berne
Convention undertook the challenge of instituting an international copyright
law that all countries and nations could adhere to so that the violation of
copyrighted material and infringement at the transnational level could be
avoided. The Berne Convention modeled much of its structure according to the
Paris Convention for the Protection of Industrial Property of 1883.
creating a cohesive and simple structure for international copyright law, it
also allowed for integration of intellectual property, such as patents and
trademarks. In the architectural vein of the Paris Convention, the Berne
Convention itself decided to establish a bureau to be responsible for the
administrative processes to handle the legislation and enactment of the new
international copyright law.
In seeking
to create a bureau, the Paris Convention for the Protection of Literary and
Artistic Works and the Berne Convention for the Protection of Literary and
Artistic Works decided to combine their efforts and administrative factions,
and in 1893 the United International Bureaux for the Protection of Intellectual
Property was born, or BIRPI, which is the French acronym for the administrative
bureau. The consolidation of both administrative offices would coordinate and
allocate the necessary responsibilities to appropriately enact the Berne
Convention's legislation and international copyright law.
BIRPI was first established in the birth city of
the Convention itself, Berne, Switzerland. By 1960, the BIRPI was moved to
Geneva, which at the time was also the home of the United Nations. The move
would make it easier for BIRPI to consult with the various countries and
nations for the consolidation of a single conglomeration of international
copyright law for the purpose of effective copyright control and enforcement at
the international level.
However, BIRPI would only last seven years after its
move to Geneva because by 1967 BIRPI would change its name to the World
Intellectual Property Organization, or WIPO, which would become integrated as
an organization as part of the United Nations in 1974.
BIRPI still
exists today as the WIPO, which currently holds as its purpose "to
encourage creative activity, to promote the protection of intellectual property
throughout the world." In its current state, the WIPO boasts 184 countries
or member states. Almost all of the members of the United Nations are part of
the WIPO.
As of
October 1st, 2008, the Director-General of the WIPO is Francis Gurry. The last
BIRPI director was Georg Bodenhausen. Though the BIRPI became the WIPO, the
overall purpose of the organization as originally intended is still the
philosophy observed by its modern counterpart today.

redpajama_set_name: RedPajamaCommonCrawl