\label{lem:L2Green}
Let $(M,g)$ be a compact connected $2$-dimensional Riemannian manifold without boundary, and let $G$ be the symmetric mean-zero Green function of $-\Delta$.
It then holds that $G\in L^2(M\times M, dx\otimes dx)$, and so
\[
\sigma^2 \coloneqq \int_{M}\int_{M} G(x,y)^2\,dx\,dy <\infty.
\]

Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\mathrm{d}_g$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \ref{def:green}). We work with the normalized volume measure $dx=\mathrm{vol}(M)^{-1}\,\mathrm{d}\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\dots,x_n\in M$, we define the empirical measure:
\[
\mu_n \coloneqq \frac1n\sum_{i=1}^n \delta_{x_i}.
\]
Steinerberger \cite{Ste21} showed that for $d\ge 3$, the following Green--Wasserstein inequality holds:
\[
W_2(\mu_n,dx)\ \lesssim_{M}\ n^{-1/d} + \frac1n\left|\sum_{i\neq j} G(x_i,x_j)\right|^{1/2}.
\]
In \cite[Problem 53]{SteOpen}, he gives the following estimate:
\begin{equation}\label{eq:Steinerberger2D}
W_2\!\left(\frac1n\sum_{k=1}^n \delta_{x_k},\,dx\right)
\lesssim_M
\frac1n\left|\sum_{i\neq j} G(x_i,x_j)\right|^{1/2}
+
\begin{cases}
\sqrt{\frac{\log n}{n}} & \text{if } d=2,\\
n^{-1/d} & \text{if } d\ge 3.
\end{cases}
\end{equation}
In this note, we use
\[
\sum_{i\neq j}(\cdots)\ \coloneqq\ \sum_{\substack{1\le i,j\le n\\ i\neq j}}(\cdots)
\]
to denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i<j$. Steinerberger asked whether the factor $\sqrt{\log n}$ in \eqref{eq:Steinerberger2D} can be removed while keeping the same Green-energy term. The precise question is as follows.

\begin{problem}[Steinerberger {\cite[Problem 53]{SteOpen}}]\label{prob:Steinerberger}
In dimension $d=2$, can one replace the remainder $\sqrt{\frac{\log n}{n}}$ in \eqref{eq:Steinerberger2D}
by $\frac{1}{\sqrt n}$ while keeping the same off-diagonal Green term? In other terms, does there exist
$C_M>0$ such that for all $n\in\mathbb N$ and all $x_1,\dots,x_n\in M$,
\[
W_2\!\left(\frac1n\sum_{k=1}^n \delta_{x_k},\,dx\right)
\le C_M\left(\frac1{\sqrt n}
+\frac1n\left|\sum_{i\neq j} G(x_i,x_j)\right|^{1/2}\right)?
\]
\end{problem}
\noindent Our main result, Theorem \ref{thm:main}, gives a negative answer to Problem \ref{prob:Steinerberger}.

\begin{remark}[The diagonal]\label{rem:diagonal}
We note that the Green function $G$ is smooth on $(M\times M)\setminus \{(x,x):x\inM\}$ with a logarithmic singularity along the diagonal.
In a deterministic context, if $x_i=x_j$ for some $i\neq j$, then the term $G(x_i,x_j)$ is not finite. It is then natural to interpret the Green term in \eqref{eq:Steinerberger2D} and Problem \ref{prob:Steinerberger} as $+\infty$, in which case the inequality holds trivially.
In the random settings that we use in this note, collisions occur with probability $0$ because $dx$ is non-atomic.
\end{remark}

We now recall some definitions that will be useful in the main argument.
For Borel probability measures $\mu,\nu$ on $M$, let us denote by $\Gamma(\mu,\nu)$ the set of couplings of $\mu$ and $\nu$ on $M\times M$.
The quadratic Wasserstein distance is
\[
W_2(\mu,\nu)\coloneqq \left(\inf_{\gamma\in \Gamma(\mu,\nu)} \int_{M\times M} \mathrm{d}_g(x,y)^2\,\,\mathrm{d}\gamma(x,y)\right)^{1/2}.
\]
Given that $M$ is compact, it follows that $W_2(\mu,\nu)$ is always finite and satisfies $W_2(\mu,\nu)\le \mathrm{diam}(M)$. Let $\Delta$ be the Laplace--Beltrami operator on $(M,g)$, which is realized here as a self-adjoint operator on $L^2(M,dx)$ with domain $H^2(M)$. Again, since $M$ is compact and connected, $\ker(-\Delta)$ consists of the constant functions.

\begin{definition}[Mean-zero Green function]\label{def:green}
A mean-zero Green function for $-\Delta$ is a measurable function
\[
G:M\times M \to \mathbb{R}\cup\{+\infty\}
\]
with the following properties.
\begin{enumerate}
\item For each $x\in M$, $y\mapsto G(x,y)$ is locally integrable on $M$ and satisfies the normalization:
\begin{equation}\label{eq:GreenMeanZero}
\int_{M} G(x,y)\,dy=0.
\end{equation}
\item For each $f\in C^\infty(M)$ with $\int_{M} f\,dx=0$, the function
\[
u(x)\coloneqq \int_{M} G(x,y)\, f(y)\,dy
\]
is a weak solution of $-\Delta u = f$ with $\int_{M}u\,dx=0$.
\end{enumerate}
\end{definition}

Recall that the uniqueness and existence of the mean-zero Green function, as a distribution kernel, are guaranteed. We can also choose it to be symmetric (for example, see \cite{Aubin98} or \cite{Rosenberg97}). We now fix a symmetric mean-zero Green function $G$, so that $G(x,y)=G(y,x)$ holds.
By symmetry and \eqref{eq:GreenMeanZero}, we obtain:
\[
\int_{M}G(y,x)\,dy=0\qquad\text{for every }x\in M.
\]

\begin{remark}[Local singularity]\label{rem:logsing}
In dimension $2$, the following classical local expansion (in geodesic normal coordinates) is true:
\[
G(x,y)= -\frac{1}{2\pi}\log \mathrm{d}_g(x,y) + H(x,y).
\]
Note that $H$ extends continuously to the diagonal and is smooth off the diagonal.
In particular, we have $G(x,\cdot)\in L^1(M,dx)$ for each fixed $x$.
\end{remark}

Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\mathrm{d}_g$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \ref{def:green}). We work with the normalized volume measure $dx=\mathrm{vol}(M)^{-1}\,\mathrm{d}\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\dots,x_n\in M$, we define the empirical measure:
\[
\mu_n \coloneqq \frac1n\sum_{i=1}^n \delta_{x_i}.
\]
Steinerberger \cite{Ste21} showed that for $d\ge 3$, the following Green--Wasserstein inequality holds:
\[
W_2(\mu_n,dx)\ \lesssim_{M}\ n^{-1/d} + \frac1n\left|\sum_{i\neq j} G(x_i,x_j)\right|^{1/2}.
\]
In \cite[Problem 53]{SteOpen}, he gives the following estimate:
\begin{equation}\label{eq:Steinerberger2D}
W_2\!\left(\frac1n\sum_{k=1}^n \delta_{x_k},\,dx\right)
\lesssim_M
\frac1n\left|\sum_{i\neq j} G(x_i,x_j)\right|^{1/2}
+
\begin{cases}
\sqrt{\frac{\log n}{n}} & \text{if } d=2,\\
n^{-1/d} & \text{if } d\ge 3.
\end{cases}
\end{equation}
In this note, we use
\[
\sum_{i\neq j}(\cdots)\ \coloneqq\ \sum_{\substack{1\le i,j\le n\\ i\neq j}}(\cdots)
\]
to denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i<j$. Steinerberger asked whether the factor $\sqrt{\log n}$ in \eqref{eq:Steinerberger2D} can be removed while keeping the same Green-energy term. The precise question is as follows.

\begin{remark}[The diagonal]\label{rem:diagonal}
We note that the Green function $G$ is smooth on $(M\times M)\setminus \{(x,x):x\inM\}$ with a logarithmic singularity along the diagonal.
In a deterministic context, if $x_i=x_j$ for some $i\neq j$, then the term $G(x_i,x_j)$ is not finite. It is then natural to interpret the Green term in \eqref{eq:Steinerberger2D} and Problem \ref{prob:Steinerberger} as $+\infty$, in which case the inequality holds trivially.
In the random settings that we use in this note, collisions occur with probability $0$ because $dx$ is non-atomic.
\end{remark}

We now recall some definitions that will be useful in the main argument.
For Borel probability measures $\mu,\nu$ on $M$, let us denote by $\Gamma(\mu,\nu)$ the set of couplings of $\mu$ and $\nu$ on $M\times M$.
The quadratic Wasserstein distance is
\[
W_2(\mu,\nu)\coloneqq \left(\inf_{\gamma\in \Gamma(\mu,\nu)} \int_{M\times M} \mathrm{d}_g(x,y)^2\,\,\mathrm{d}\gamma(x,y)\right)^{1/2}.
\]
Given that $M$ is compact, it follows that $W_2(\mu,\nu)$ is always finite and satisfies $W_2(\mu,\nu)\le \mathrm{diam}(M)$. Let $\Delta$ be the Laplace--Beltrami operator on $(M,g)$, which is realized here as a self-adjoint operator on $L^2(M,dx)$ with domain $H^2(M)$. Again, since $M$ is compact and connected, $\ker(-\Delta)$ consists of the constant functions.

\begin{definition}[Mean-zero Green function]\label{def:green}
A mean-zero Green function for $-\Delta$ is a measurable function
\[
G:M\times M \to \mathbb{R}\cup\{+\infty\}
\]
with the following properties.
\begin{enumerate}
\item For each $x\in M$, $y\mapsto G(x,y)$ is locally integrable on $M$ and satisfies the normalization:
\begin{equation}\label{eq:GreenMeanZero}
\int_{M} G(x,y)\,dy=0.
\end{equation}
\item For each $f\in C^\infty(M)$ with $\int_{M} f\,dx=0$, the function
\[
u(x)\coloneqq \int_{M} G(x,y)\, f(y)\,dy
\]
is a weak solution of $-\Delta u = f$ with $\int_{M}u\,dx=0$.
\end{enumerate}
\end{definition}

Recall that the uniqueness and existence of the mean-zero Green function, as a distribution kernel, are guaranteed. We can also choose it to be symmetric (for example, see \cite{Aubin98} or \cite{Rosenberg97}). We now fix a symmetric mean-zero Green function $G$, so that $G(x,y)=G(y,x)$ holds.
By symmetry and \eqref{eq:GreenMeanZero}, we obtain:
\[
\int_{M}G(y,x)\,dy=0\qquad\text{for every }x\in M.
\]

\begin{remark}[Local singularity]\label{rem:logsing}
In dimension $2$, the following classical local expansion (in geodesic normal coordinates) is true:
\[
G(x,y)= -\frac{1}{2\pi}\log \mathrm{d}_g(x,y) + H(x,y).
\]
Note that $H$ extends continuously to the diagonal and is smooth off the diagonal.
In particular, we have $G(x,\cdot)\in L^1(M,dx)$ for each fixed $x$.
\end{remark}

\begin{definition}[Mean-zero Green function]\label{def:green}
A mean-zero Green function for $-\Delta$ is a measurable function
\[
G:\M\times \M \to \mathbb{R}\cup\{+\infty\}
\]
with the following properties.
\begin{enumerate}
\item For each $x\in \M$, $y\mapsto G(x,y)$ is locally integrable on $\M$ and satisfies the normalization:
\begin{equation}\label{eq:GreenMeanZero}
\int_{\M} G(x,y)\,dy=0.
\end{equation}
\item For each $f\in C^\infty(\M)$ with $\int_{\M} f\,dx=0$, the function
\[
u(x)\coloneqq \int_{\M} G(x,y)\, f(y)\,dy
\]
is a weak solution of $-\Delta u = f$ with $\int_{\M}u\,dx=0$.
\end{enumerate}
\end{definition}

\begin{equation}\label{eq:GreenMeanZero}
\int_{\M} G(x,y)\,dy=0.
\end{equation}

\begin{equation}\label{eq:Steinerberger2D}
\W\!\left(\frac1n\sum_{k=1}^n \delta_{x_k},\,dx\right)
\lesssim_\M
\frac1n\left|\sum_{i\neq j} G(x_i,x_j)\right|^{1/2}
+
\begin{cases}
\sqrt{\frac{\log n}{n}} & \text{if } d=2,\\
n^{-1/d} & \text{if } d\ge 3.
\end{cases}
\end{equation}

\begin{problem}[Steinerberger {\cite[Problem 53]{SteOpen}}]\label{prob:Steinerberger}
In dimension $d=2$, can one replace the remainder $\sqrt{\frac{\log n}{n}}$ in \eqref{eq:Steinerberger2D}
by $\frac{1}{\sqrt n}$ while keeping the same off-diagonal Green term? In other terms, does there exist
$C_\M>0$ such that for all $n\in\mathbb N$ and all $x_1,\dots,x_n\in \M$,
\[
\W\!\left(\frac1n\sum_{k=1}^n \delta_{x_k},\,dx\right)
\le C_\M\left(\frac1{\sqrt n}
+\frac1n\left|\sum_{i\neq j} G(x_i,x_j)\right|^{1/2}\right)?
\]
\end{problem}

Let $(\M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\dist$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \ref{def:green}). We work with the normalized volume measure $dx=\mathrm{vol}(\M)^{-1}\dd\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\dots,x_n\in \M$, we define the empirical measure:
\[
\mu_n \coloneqq \frac1n\sum_{i=1}^n \delta_{x_i}.
\]
Steinerberger \cite{Ste21} showed that for $d\ge 3$, the following Green--Wasserstein inequality holds:
\[
\W(\mu_n,dx)\ \lesssim_{\M}\ n^{-1/d} + \frac1n\left|\sum_{i\neq j} G(x_i,x_j)\right|^{1/2}.
\]
In \cite[Problem 53]{SteOpen}, he gives the following estimate:
\begin{equation}\label{eq:Steinerberger2D}
\W\!\left(\frac1n\sum_{k=1}^n \delta_{x_k},\,dx\right)
\lesssim_\M
\frac1n\left|\sum_{i\neq j} G(x_i,x_j)\right|^{1/2}
+
\begin{cases}
\sqrt{\frac{\log n}{n}} & \text{if } d=2,\\
n^{-1/d} & \text{if } d\ge 3.
\end{cases}
\end{equation}
In this note, we use
\[
\sum_{i\neq j}(\cdots)\ \coloneqq\ \sum_{\substack{1\le i,j\le n\\ i\neq j}}(\cdots)
\]
to denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i<j$. Steinerberger asked whether the factor $\sqrt{\log n}$ in \eqref{eq:Steinerberger2D} can be removed while keeping the same Green-energy term. The precise question is as follows.

\begin{remark}[The diagonal]\label{rem:diagonal}
We note that the Green function $G$ is smooth on $(\M\times \M)\setminus \{(x,x):x\in\M\}$ with a logarithmic singularity along the diagonal.
In a deterministic context, if $x_i=x_j$ for some $i\neq j$, then the term $G(x_i,x_j)$ is not finite. It is then natural to interpret the Green term in \eqref{eq:Steinerberger2D} and Problem \ref{prob:Steinerberger} as $+\infty$, in which case the inequality holds trivially.
In the random settings that we use in this note, collisions occur with probability $0$ because $dx$ is non-atomic.
\end{remark}

We now recall some definitions that will be useful in the main argument.
For Borel probability measures $\mu,\nu$ on $\M$, let us denote by $\Gamma(\mu,\nu)$ the set of couplings of $\mu$ and $\nu$ on $\M\times \M$.
The quadratic Wasserstein distance is
\[
\W(\mu,\nu)\coloneqq \left(\inf_{\gamma\in \Gamma(\mu,\nu)} \int_{\M\times \M} \dist(x,y)^2\,\dd\gamma(x,y)\right)^{1/2}.
\]
Given that $\M$ is compact, it follows that $\W(\mu,\nu)$ is always finite and satisfies $\W(\mu,\nu)\le \mathrm{diam}(\M)$. Let $\Delta$ be the Laplace--Beltrami operator on $(\M,g)$, which is realized here as a self-adjoint operator on $L^2(\M,dx)$ with domain $H^2(\M)$. Again, since $\M$ is compact and connected, $\ker(-\Delta)$ consists of the constant functions.

\begin{definition}[Mean-zero Green function]\label{def:green}
A mean-zero Green function for $-\Delta$ is a measurable function
\[
G:\M\times \M \to \mathbb{R}\cup\{+\infty\}
\]
with the following properties.
\begin{enumerate}
\item For each $x\in \M$, $y\mapsto G(x,y)$ is locally integrable on $\M$ and satisfies the normalization:
\begin{equation}\label{eq:GreenMeanZero}
\int_{\M} G(x,y)\,dy=0.
\end{equation}
\item For each $f\in C^\infty(\M)$ with $\int_{\M} f\,dx=0$, the function
\[
u(x)\coloneqq \int_{\M} G(x,y)\, f(y)\,dy
\]
is a weak solution of $-\Delta u = f$ with $\int_{\M}u\,dx=0$.
\end{enumerate}
\end{definition}

\begin{remark}[Local singularity]\label{rem:logsing}
In dimension $2$, the following classical local expansion (in geodesic normal coordinates) is true:
\[
G(x,y)= -\frac{1}{2\pi}\log \dist(x,y) + H(x,y).
\]
Note that $H$ extends continuously to the diagonal and is smooth off the diagonal.
In particular, we have $G(x,\cdot)\in L^1(\M,dx)$ for each fixed $x$.
\end{remark}

\begin{proof}
Recall that $\mathrm{vol}$ denotes the Riemannian volume measure on $(\M,g)$ and $dx = \mathrm{vol}(\M)^{-1}\dd\mathrm{vol}$. Let $\mathrm{inj}(\M)>0$ be the injectivity radius of $(\M,g)$ and set $r_0\coloneqq \mathrm{inj}(\M)/4$. We now apply standard results on the singularity structure of the Green function in dimension $2$ (for instance, see \cite{Aubin98}). In particular, the function
\[
H(x,y)\coloneqq G(x,y)+\frac{1}{2\pi}\log \dist(x,y),
\qquad x\neq y,
\]
extends continuously to the set $\{(x,y)\in \M\times\M: \dist(x,y)\le 2r_0\}$ (this also holds across the diagonal). Observe that this set is compact, so there exists $C_0>0$ such that
\[
|H(x,y)|\le C_0\qquad\text{whenever }\dist(x,y)\le 2r_0.
\]
As a result, for all $x\neq y$ with $\dist(x,y)<2r_0$, it follows that
\begin{equation}\label{eq:GreenLogBound}
|G(x,y)| \le \frac{1}{2\pi}|\log \dist(x,y)|+C_0 \le C\bigl(1+|\log \dist(x,y)|\bigr)
\end{equation}
for some constant $C>0$.

We now split $\M\times\M$ into the near-diagonal region
\[
D\coloneqq \{(x,y)\in\M\times\M: \dist(x,y)<r_0\}
\]
and its complement $D^c$. Notice that $D^c$ is compact and is always a positive distance away from the diagonal, so $G$ is smooth on $D^c$ and therefore bounded there. We obtain:
\[
\int_{D^c} G(x,y)^2\,dx\,dy<\infty.
\]
Let us now analyze $D$. By \eqref{eq:GreenLogBound}, we set
\[
I(x)\coloneqq \int_{B_{r_0}(x)} \bigl(1+|\log \dist(x,y)|\bigr)^2\,dy,
\]
so that
\[
\int_{D} G(x,y)^2\,dx\,dy
\le C^2\int_{\M} I(x)\,dx.
\]
We also fix $x\in\M$. Note that on the ball $B_{r_0}(x)$, we may use geodesic polar coordinates centered at $x$: $y=\exp_x(\rho\theta)$ with $\rho\in(0,r_0)$ and $\theta\in\mathbb{S}^1$. In these coordinates, the volume form has the following form:
\[
\dd\mathrm{vol}(y)=J_x(\rho,\theta)\,\dd\rho\,\dd\theta,
\qquad
J_x(\rho,\theta)=\rho\,a_x(\rho,\theta),
\]
where $a_x$ is smooth and satisfies $a_x(0,\theta)=1$.
In particular, by compactness, there exists $C_1>0$ with:
\begin{equation}\label{eq:JacobianBound}
J_x(\rho,\theta)\le C_1\rho\qquad\text{for all }x\in\M,\ \theta\in\mathbb{S}^1,\ \rho\in(0,r_0)
\end{equation}
(see, for instance, \cite{Chavel84}).
We now apply the normalization of $dx$ and obtain:
\begin{align*}
\int_{B_{r_0}(x)} \bigl(1+|\log \dist(x,y)|\bigr)^2\,dy
&=\frac{1}{\mathrm{vol}(\M)}\int_{\mathbb{S}^1}\!\!\int_0^{r_0} \bigl(1+|\log \rho|\bigr)^2 J_x(\rho,\theta)\,\dd\rho\,\dd\theta\\
&\le \frac{C_1}{\mathrm{vol}(\M)}\int_{\mathbb{S}^1}\!\!\int_0^{r_0} \bigl(1+|\log \rho|\bigr)^2 \rho\,\dd\rho\,\dd\theta.
\end{align*}
The right-hand side is finite since near $0$ the function $\rho\mapsto \rho(\log \rho)^2$ is integrable. We apply the substitution $t=-\log\rho$ (so $\rho=e^{-t}$ and $\rho\,\dd\rho=-e^{-2t}\,\dd t$) and get:
\[
\int_0^{r_0} \bigl(1+|\log \rho|\bigr)^2\rho\,\dd\rho
=\int_{\infty}^{-\log r_0} \bigl(1+|t|\bigr)^2(-e^{-2t})\,\dd t
=\int_{-\log r_0}^{\infty} \bigl(1+|t|\bigr)^2 e^{-2t}\,\dd t <\infty.
\]
It follows that $\int_{B_{r_0}(x)} (1+|\log \dist(x,y)|)^2\,dy$ is bounded uniformly in $x$.
If we integrate over $x$, we arrive at $\int_D G^2\,dx\,dy<\infty$. Finally, we combine the estimates on $D$ and $D^c$. This finishes the proof of $\sigma^2<\infty$.
\end{proof}

\begin{theorem}[The $\sqrt{\log n}$ remainder]\label{thm:main}
Let $G$ be the symmetric mean-zero Green function of $-\Delta$.
There does not exist a constant $C_{\M}>0$ such that for all $n\in\mathbb{N}$ and all $x_1,\dots,x_n\in \M$,
\begin{equation}\label{eq:NoUniversalO1sqrtN}
\W\!\left(\frac1n\sum_{i=1}^n\delta_{x_i},\, dx\right)
\le
C_{\M}\left(\frac{1}{\sqrt{n}}+\frac1n\left|\sum_{i\neq j} G(x_i,x_j)\right|^{1/2}\right).
\end{equation}
In particular, the $\sqrt{\log n}$ factor in the two-dimensional inequality \eqref{eq:Steinerberger2D} cannot be removed if the unrenormalized Green-energy term were to be preserved.
\end{theorem}

\begin{proof}
Recall that $\mathrm{vol}$ denotes the Riemannian volume measure on $(M,g)$ and $dx = \mathrm{vol}(M)^{-1}\,\mathrm{d}\mathrm{vol}$. Let $\mathrm{inj}(M)>0$ be the injectivity radius of $(M,g)$ and set $r_0\coloneqq \mathrm{inj}(M)/4$. We now apply standard results on the singularity structure of the Green function in dimension $2$ (for instance, see \cite{Aubin98}). In particular, the function
\[
H(x,y)\coloneqq G(x,y)+\frac{1}{2\pi}\log \mathrm{d}_g(x,y),
\qquad x\neq y,
\]
extends continuously to the set $\{(x,y)\in M\timesM: \mathrm{d}_g(x,y)\le 2r_0\}$ (this also holds across the diagonal). Observe that this set is compact, so there exists $C_0>0$ such that
\[
|H(x,y)|\le C_0\qquad\text{whenever }\mathrm{d}_g(x,y)\le 2r_0.
\]
As a result, for all $x\neq y$ with $\mathrm{d}_g(x,y)<2r_0$, it follows that
\begin{equation}\label{eq:GreenLogBound}
|G(x,y)| \le \frac{1}{2\pi}|\log \mathrm{d}_g(x,y)|+C_0 \le C\bigl(1+|\log \mathrm{d}_g(x,y)|\bigr)
\end{equation}
for some constant $C>0$.

We now split $M\timesM$ into the near-diagonal region
\[
D\coloneqq \{(x,y)\inM\timesM: \mathrm{d}_g(x,y)<r_0\}
\]
and its complement $D^c$. Notice that $D^c$ is compact and is always a positive distance away from the diagonal, so $G$ is smooth on $D^c$ and therefore bounded there. We obtain:
\[
\int_{D^c} G(x,y)^2\,dx\,dy<\infty.
\]
Let us now analyze $D$. By \eqref{eq:GreenLogBound}, we set
\[
I(x)\coloneqq \int_{B_{r_0}(x)} \bigl(1+|\log \mathrm{d}_g(x,y)|\bigr)^2\,dy,
\]
so that
\[
\int_{D} G(x,y)^2\,dx\,dy
\le C^2\int_{M} I(x)\,dx.
\]
We also fix $x\inM$. Note that on the ball $B_{r_0}(x)$, we may use geodesic polar coordinates centered at $x$: $y=\exp_x(\rho\theta)$ with $\rho\in(0,r_0)$ and $\theta\in\mathbb{S}^1$. In these coordinates, the volume form has the following form:
\[
\,\mathrm{d}\mathrm{vol}(y)=J_x(\rho,\theta)\,\,\mathrm{d}\rho\,\,\mathrm{d}\theta,
\qquad
J_x(\rho,\theta)=\rho\,a_x(\rho,\theta),
\]
where $a_x$ is smooth and satisfies $a_x(0,\theta)=1$.
In particular, by compactness, there exists $C_1>0$ with:
\begin{equation}\label{eq:JacobianBound}
J_x(\rho,\theta)\le C_1\rho\qquad\text{for all }x\inM,\ \theta\in\mathbb{S}^1,\ \rho\in(0,r_0)
\end{equation}
(see, for instance, \cite{Chavel84}).
We now apply the normalization of $dx$ and obtain:
\begin{align*}
\int_{B_{r_0}(x)} \bigl(1+|\log \mathrm{d}_g(x,y)|\bigr)^2\,dy
&=\frac{1}{\mathrm{vol}(M)}\int_{\mathbb{S}^1}\!\!\int_0^{r_0} \bigl(1+|\log \rho|\bigr)^2 J_x(\rho,\theta)\,\,\mathrm{d}\rho\,\,\mathrm{d}\theta\\
&\le \frac{C_1}{\mathrm{vol}(M)}\int_{\mathbb{S}^1}\!\!\int_0^{r_0} \bigl(1+|\log \rho|\bigr)^2 \rho\,\,\mathrm{d}\rho\,\,\mathrm{d}\theta.
\end{align*}
The right-hand side is finite since near $0$ the function $\rho\mapsto \rho(\log \rho)^2$ is integrable. We apply the substitution $t=-\log\rho$ (so $\rho=e^{-t}$ and $\rho\,\,\mathrm{d}\rho=-e^{-2t}\,\,\mathrm{d} t$) and get:
\[
\int_0^{r_0} \bigl(1+|\log \rho|\bigr)^2\rho\,\,\mathrm{d}\rho
=\int_{\infty}^{-\log r_0} \bigl(1+|t|\bigr)^2(-e^{-2t})\,\,\mathrm{d} t
=\int_{-\log r_0}^{\infty} \bigl(1+|t|\bigr)^2 e^{-2t}\,\,\mathrm{d} t <\infty.
\]
It follows that $\int_{B_{r_0}(x)} (1+|\log \mathrm{d}_g(x,y)|)^2\,dy$ is bounded uniformly in $x$.
If we integrate over $x$, we arrive at $\int_D G^2\,dx\,dy<\infty$. Finally, we combine the estimates on $D$ and $D^c$. This finishes the proof of $\sigma^2<\infty$.
\end{proof}

Using the 2D Green function singularity, write
\[
H(x,y)\coloneqq G(x,y)+\frac{1}{2\pi}\log \mathrm{d}_g(x,y)
\]
which “extends continuously” to a compact neighborhood of the diagonal, hence is bounded there. This gives the near-diagonal logarithmic bound
\[
|G(x,y)|\le C\bigl(1+|\log \mathrm{d}_g(x,y)|\bigr)\qquad (\mathrm{d}_g(x,y)<2r_0).
\]
Split \(M\times M\) into the near-diagonal region \(D=\{\mathrm{d}_g(x,y)<r_0\}\) and its complement \(D^c\). On \(D^c\), “\(G\) is smooth … and therefore bounded,” so \(\int_{D^c}G^2<\infty\).

On \(D\), use the bound above and reduce to showing finiteness of
\[
I(x)\coloneqq \int_{B_{r_0}(x)}\bigl(1+|\log \mathrm{d}_g(x,y)|\bigr)^2\,dy.
\]
In geodesic polar coordinates \(y=\exp_x(\rho\theta)\), the Jacobian satisfies \(J_x(\rho,\theta)=\rho a_x(\rho,\theta)\le C_1\rho\), hence
\[
I(x)\le \frac{C_1}{\mathrm{vol}(M)}\int_{\mathbb{S}^1}\!\!\int_0^{r_0}(1+|\log\rho|)^2\rho\,d\rho\,d\theta.
\]
The radial integral is finite because “near \(0\) the function \(\rho\mapsto \rho(\log\rho)^2\) is integrable,” shown via the substitution \(t=-\log\rho\) giving an integrand \((1+|t|)^2e^{-2t}\). This yields a uniform bound on \(I(x)\), so integrating over \(x\) gives \(\int_D G^2<\infty\). Combining \(D\) and \(D^c\) proves \(G\in L^2(M\times M)\) and hence \(\sigma^2<\infty\).

Using the 2D Green function singularity, write
\[
H(x,y)\coloneqq G(x,y)+\frac{1}{2\pi}\log \mathrm{d}_g(x,y)
\]
which extends continuously to a compact neighborhood of the diagonal, hence is bounded there. This gives the near-diagonal logarithmic bound
\[
|G(x,y)|\le C\bigl(1+|\log \mathrm{d}_g(x,y)|\bigr)\qquad (\mathrm{d}_g(x,y)<2r_0).
\]
Split \(M\times M\) into the near-diagonal region \(D=\{\mathrm{d}_g(x,y)<r_0\}\) and its complement \(D^c\). On \(D^c\), \(G\) is smooth and therefore bounded, so \(\int_{D^c}G^2<\infty\).

On \(D\), use the bound above and reduce to showing finiteness of
\[
I(x)\coloneqq \int_{B_{r_0}(x)}\bigl(1+|\log \mathrm{d}_g(x,y)|\bigr)^2\,dy.
\]
In geodesic polar coordinates \(y=\exp_x(\rho\theta)\), the Jacobian satisfies \(J_x(\rho,\theta)=\rho a_x(\rho,\theta)\le C_1\rho\), hence
\[
I(x)\le \frac{C_1}{\mathrm{vol}(M)}\int_{\mathbb{S}^1}\!\!\int_0^{r_0}(1+|\log\rho|)^2\rho\,d\rho\,d\theta.
\]
The radial integral is finite because near \(0\) the function \(\rho\mapsto \rho(\log\rho)^2\) is integrable; via the substitution \(t=-\log\rho\) the integrand becomes \((1+|t|)^2e^{-2t}\). This yields a uniform bound on \(I(x)\), so integrating over \(x\) gives \(\int_D G^2<\infty\). Combining \(D\) and \(D^c\) proves \(G\in L^2(M\times M)\) and hence \(\sigma^2<\infty\).

Question: Let $(M,g)$ be a compact connected $2$-dimensional Riemannian manifold without boundary, let $dx=\operatorname{vol}(M)^{-1}\,d\operatorname{vol}$ be the normalized volume measure, and let $G:M\times M\to \mathbb{R}\cup\{+\infty\}$ be a symmetric mean-zero Green function for $-\Delta$, meaning that $G(x,y)=G(y,x)$, for each fixed $x\in M$ the map $y\mapsto G(x,y)$ is locally integrable and satisfies $\int_M G(x,y)\,dy=0$, and for each $f\in C^\infty(M)$ with $\int_M f\,dx=0$, the function $u(x)=\int_M G(x,y)f(y)\,dy$ is a weak solution of $-\Delta u=f$ with $\int_M u\,dx=0$. Which statement holds for every such $(M,g)$ and $G$?

The Green function is square-integrable on $M\times M$ with respect to the product measure $dx\otimes dx$; equivalently, $G\in L^2(M\times M,dx\otimes dx)$ and therefore
\[
\sigma^2\coloneqq \int_M\int_M G(x,y)^2\,dx\,dy<\infty.
\]

Choices:
B.
The Green function is essentially bounded on $M\times M$ with respect to the product measure $dx\otimes dx$; equivalently, $G\in L^\infty(M\times M,dx\otimes dx)$ and therefore
\[
\int_M\int_M G(x,y)^2\,dx\,dy<\infty.
\]
C.
For each fixed $x\in M$, the section $y\mapsto G(x,y)$ belongs to $L^2(M,dy)$, and in particular
\[
\int_M G(x,y)^2\,dy<\infty\qquad\text{for every }x\in M.
\]
D.
The Green function is square-integrable off the diagonal but not on all of $M\times M$; more precisely, for every $\varepsilon>0$ one has
\[
G\in L^2\bigl(\{(x,y)\in M\times M:\ d_g(x,y)\ge \varepsilon\},dx\otimes dx\bigr),
\]
whereas
\[
\int_M\int_M G(x,y)^2\,dx\,dy=+\infty.
\]
E.
For each fixed $x\in M$, the map $y\mapsto G(x,y)$ belongs to $L^p(M,dy)$ for every $p>2$, and consequently
\[
G\in L^p(M\times M,dx\otimes dx)\qquad\text{for every }p>2.
\]

logarithmic singularity near the diagonal is bounded only after subtracting the log term, not in absolute size

joint $L^2(M\times M)$ conclusion weakened to fiberwise $L^2$ in the $y$-variable for each fixed $x$

near-diagonal integrability of $\rho(\log\rho)^2$

critical exponent from the logarithmic singularity: $L^2$ holds but no gain to all $p>2$

The stem gives only the setup and defining properties of the Green function; it does not reveal or strongly hint that square-integrability on M×M is the intended conclusion.

The item is not a bare restatement of a theorem in the stem. It asks the student to distinguish among several regularity/integrability claims (L1, L2, L∞, sectionwise bounds, divergence), so it involves competing conclusions.

A solver must reason about the 2D logarithmic singularity of the Green function and which integrability thresholds it satisfies, so some genuine mathematical reasoning is required. However, the pressure is weakened because choice C is also true, making the task less about uniquely generating the best conclusion.

B, D, and E are plausible and target common misconceptions about regularity and singularities. But C is a weaker true statement, so it is not a valid distractor under a single-answer format, which significantly harms the quality of the option set.

Conceptually strong and free of answer leakage, but the MCQ is flawed as a single-choice item because at least one non-key option (C) is also true. This ambiguity weakens both the distractor set and the assessment of reasoning.


Question: Let $(M,g)$ be a compact connected $2$-dimensional Riemannian manifold without boundary, let $dx=\operatorname{vol}(M)^{-1}\,d\operatorname{vol}$ be the normalized volume measure, and let $G:M\times M\to \mathbb{R}\cup\{+\infty\}$ be a symmetric mean-zero Green function for $-\Delta$, meaning that $G(x,y)=G(y,x)$, for each fixed $x\in M$ the map $y\mapsto G(x,y)$ is locally integrable and satisfies $\int_M G(x,y)\,dy=0$, and for each $f\in C^\infty(M)$ with $\int_M f\,dx=0$, the function $u(x)=\int_M G(x,y)f(y)\,dy$ is a weak solution of $-\Delta u=f$ with $\int_M u\,dx=0$. Which statement holds for every such $(M,g)$ and $G$?

Choices:
B.
The Green function is essentially bounded on $M\times M$ with respect to the product measure $dx\otimes dx$; equivalently, $G\in L^\infty(M\times M,dx\otimes dx)$ and therefore
\[
\int_M\int_M G(x,y)^2\,dx\,dy<\infty.
\]
C.
For each fixed $x\in M$, the section $y\mapsto G(x,y)$ belongs to $L^2(M,dy)$, and in particular
\[
\int_M G(x,y)^2\,dy<\infty\qquad\text{for every }x\in M.
\]
D.
The Green function is square-integrable off the diagonal but not on all of $M\times M$; more precisely, for every $\varepsilon>0$ one has
\[
G\in L^2\bigl(\{(x,y)\in M\times M:\ d_g(x,y)\ge \varepsilon\},dx\otimes dx\bigr),
\]
whereas
\[
\int_M\int_M G(x,y)^2\,dx\,dy=+\infty.
\]
E.
For each fixed $x\in M$, the map $y\mapsto G(x,y)$ belongs to $L^p(M,dy)$ for every $p>2$, and consequently
\[
G\in L^p(M\times M,dx\otimes dx)\qquad\text{for every }p>2.
\]

The stem does not state or strongly hint at the correct integrability conclusion; it only defines the Green function and asks about a nontrivial property.

This is not a restatement of the definition. It asks for a substantive consequence about global integrability of the 2D Green kernel, which goes beyond the setup.

The intended solution requires reasoning from the logarithmic diagonal singularity in dimension 2 to decide which integrability claims are valid. However, the item's ambiguity weakens the generative demand because more than one option appears true.

The distractors are not functioning well in a single-answer format: option C is a weaker true statement, and option E also appears true for a 2D logarithmic singularity (finite L^p for every finite p). This creates multiple correct answers and undermines validity.

Conceptually strong and non-tautological, but flawed as an MCQ because the answer set is not uniquely keyed: at least C, and likely E, are also true alongside A.


\label{main}
        Let $f_{\varepsilon}$  be of the form $f_{\varepsilon} = M + \varepsilon M g_{\varepsilon} $, and $\{f_{\varepsilon}\}$ is a family of solutions to \eqref{longtime_Boltzmann} with $\tau_\varepsilon=\varepsilon$ satisfying the assumptions \eqref{conservation_of_mass},\eqref{conservation_of_momentum}, \eqref{conservation_of_energy} and \eqref{entropy_inequality}.
        Let $(\rho_{\varepsilon}, u_{\varepsilon},\theta_{\varepsilon})$ be of the form $(\rho_{\varepsilon},u_{\varepsilon},\theta_{\varepsilon}) = (1+ \tilde{\rho}_{\varepsilon},\varepsilon \tilde{u}_{\varepsilon},1+ \varepsilon \tilde{\theta}_{\varepsilon})$, and $\{(\rho_{\varepsilon},u_{\varepsilon},\theta_{\varepsilon})\}$ is a family of solutions to \eqref{longtime_CNS}.
        Then we have, for $t \geqslant 0$,
        \begin{equation}\label{main_estimate}
            \frac{1}{{\varepsilon}^2} H(f_{\varepsilon}| M_{\varepsilon})(t)  \leqslant \frac{1}{\varepsilon^2}H(f^{\mathrm{in}}_{\varepsilon}| M^{\mathrm{in}}_{\varepsilon} ) + \mathcal{O}(\varepsilon),
        \end{equation}
        \begin{equation}\label{entropy_dissipation_bound}
           \int_0^t \int_{\mathbb{T}^3_x}\frac{1}{\varepsilon^4}D(f_\varepsilon)(s,x) - \frac{1}{2} \mu \sigma( \mathrm{u} ^b_\varepsilon): \sigma( \mathrm{u} ^b_\varepsilon) - \frac{5}{2}\kappa (\nabla_x \theta^b_{\varepsilon})^2\  dxds \lesssim \frac{1}{\varepsilon^2} H(f_{\varepsilon}^{\mathrm{in}}|M_{\varepsilon}^{\mathrm{in}}) + \mathcal{O}(\varepsilon),
         \end{equation}
         \begin{equation}\label{energy_momentum_flux_bound}
             \int_0^t \int_{\mathbb{T}^3_x} \frac{1}{2}\mu \sigma(\tilde{ \mathrm{u} }_{\varepsilon}- \mathrm{u} ^b_{\varepsilon}):\sigma(\tilde{ \mathrm{u} }_{\varepsilon}- \mathrm{u} ^b_{\varepsilon}) + \frac{5}{2}  \kappa(\nabla_x \tilde{\theta} -\nabla_x \theta^b_{\varepsilon})^2 \ dxds \lesssim \frac{1}{\varepsilon^2} H(f_{\varepsilon}^{\mathrm{in}}|M_{\varepsilon}^{\mathrm{in}}) + \mathcal{O}(\varepsilon),
        \end{equation}
        where
        \begin{equation}
            M_{\varepsilon}^{\mathrm{in}} = M(1+ \varepsilon \tilde\rho^{\mathrm{in}}, \varepsilon \tilde{u}^{\mathrm{in}},1 + \varepsilon \tilde{\theta}^{\mathrm{in}}).
        \end{equation}

\begin{equation}\label{INSF} \tag{INSF}
        \left\{
            \begin{aligned}
                &\partial_t \u + \u \mdot \nabla_x \u + \nabla_x p = \mu \Delta \u, \\
                &\partial_t \vartheta + \u \mdot \nabla_x \vartheta = \kappa \Delta \vartheta,\\
                &\nabla_x \cdot \u =0,
            \end{aligned}
        \right.
    \end{equation}

\begin{equation}\label{perturbed_Boltzmann}
    \left\{ \begin{matrix}
        \varepsilon^2 \partial_t g_{\varepsilon} + \varepsilon v\mdot \nabla_x g_{\varepsilon} + \mathcal{L}g_{\varepsilon} = \varepsilon Q(g_{\varepsilon},g_{\varepsilon}),\\
        g_{\varepsilon}(0,x,v) = g^{{in}}_{\varepsilon},
    \end{matrix}\right.
\end{equation}

\begin{equation}\label{g_moments}
    \begin{aligned}
        \rho^b_{\varepsilon} = \langle g_{\varepsilon} \rangle,\
        u^b_{\varepsilon} =\langle  vg_{\varepsilon} \rangle,\
        \theta^b_{\varepsilon} = \langle  \frac{|v|^2 -3}{3} g_{\varepsilon} \rangle.
    \end{aligned}
\end{equation}

\begin{equation}\label{short_CNSF} \tag{CNS}
    \left\{ \begin{aligned}
        &\partial_t \rho_{\varepsilon} + \nabla_x (\rho_{\varepsilon} u_{\varepsilon}) = 0,\\
        & \rho_{\varepsilon} (\partial_t + u_{\varepsilon} \mdot \nabla_x) u_{\varepsilon}  + \nabla_x ({\rho_{\varepsilon} \theta_{\varepsilon}}) = \varepsilon \nabla_x [ \mu_{\varepsilon}(\rho_{\varepsilon},\theta_{\varepsilon}) \sigma(u_{\varepsilon})],\\
        & \tfrac{3}{2} \rho_{\varepsilon} (\partial_t   + u_{\varepsilon} \mdot \nabla_x )\theta_{\varepsilon} + \rho_{\varepsilon} \theta_{\varepsilon}  \nabla_x \mdot u_{\varepsilon} = \varepsilon  \tfrac{1}{2} \mu_{\varepsilon}(\rho_{\varepsilon},\theta_{\varepsilon}) \sigma(u_{\varepsilon}):\sigma(u_{\varepsilon}) + \tfrac{5}{2} \varepsilon \nabla_x \cdot[\kappa(\rho_{\varepsilon},\theta_{\varepsilon}) \theta_{\varepsilon}],\\
        &(\rho_{\varepsilon}(0,x),\ u_{\varepsilon}(0,x), \ \theta_{\varepsilon}(0,x)) =  (\rho^{\mathrm{in}},\ u^{\mathrm{in}},\ \theta^{\mathrm{in}}).
    \end{aligned}\right.
\end{equation}

In this paper, we study, in the formal level, the asymptotic behavior of the following rescaled Boltzmann equation to the compressible Navier-Stokes system with small Mach number. To avoid the difficulty might be brought from the boundary, we consider the periodic domain $\mathbb{T}^3$.
\begin{equation}\label{longtime_Boltzmann} \tag{BE}
    \left\{\begin{aligned}
        &\tau_{\varepsilon} \partial_t f_{\varepsilon} + v \!\cdot\! \nabla_x f_{\varepsilon} =\frac{1}{\varepsilon} C(f_{\varepsilon},\ f_{\varepsilon}), \quad x\in \mathbb{T}^3, \quad v \in \mathbb{R}^3\\
        &f_{\varepsilon}(0,x) = f_{\varepsilon}^{\mathrm{in}} \geqslant 0,
    \end{aligned}\right.
\end{equation}
 where $f_{\varepsilon} = f_{\varepsilon}(t,x,v)$ denotes the number density of gas molecules at position $ x\in \mathbb{T}^3$ and time $t \geqslant 0$. Moreover, $\varepsilon$ denotes the Knudsen number which is the ratio of the mean free path and the macroscopic length. $\tau_{\varepsilon}$ denotes the Mach number which is the ratio of the bulk velocity to the sound speed.  The case $\tau_{\varepsilon} =1 $ is corresponding to the short time scale (or Euler time scale), and in this paper we shall focus on the case $\tau_{\varepsilon} = \varepsilon $ which is corresponding to the longer time scale (or Navier-Stokes scale).
The smaller the Knudsen number is, the behavior of the Boltzmann equation is closer to the fluid equations. The limiting process when $\varepsilon$ goes to zero is called hydrodynamic limit.

The operator $ C(f,g)$ is the collision operator which describe the binary elastic collision between particles and is defined as follows
\begin{equation}
    \begin{aligned}
        C( f , g)  = \frac{1}{2}\iint_{\mathbb{T}^3_x \times \mathbb{S}^2} B(v-v_1 , \sigma) \left( g^\prime_1  f^\prime + f^\prime_1  g^\prime -g_1 f -f_1 g   \right) d\sigma dv_1.
    \end{aligned}
\end{equation}
Here and in the sequel we use the notation
\begin{equation}
    f^{\prime}_{1} = f(t,x,v^{\prime}_1), \ f^{\prime} = f(t,x,v^{\prime}),\ f_{1} = f(t,x,v_1),
\end{equation}
and for $\sigma \in \mathbb{S}^2 $,
\begin{equation}
    v^{\prime} = \frac{v+ v_1}{2}+ \frac{|v-v_1|}{2} \sigma,  \ v^{\prime}_1 = \frac{v+ v_1}{2} - \frac{|v-v_1|}{2} \sigma.
\end{equation}
The function $B(v-v_1,\sigma)$, called cross-section, is assumed to depend only on $v-v_1$ and $\frac{v-v_1}{|v-v_1|} \cdot \sigma$.

It is well-known that the equilibrium of the Boltzmann collision operator $C$, i.e. the number density $\mathcal{M}$ such that $C(\mathcal{M}, \mathcal{M})=0$, have a specific form, which is called the Maxwellian distributions:
$$\mathcal{M}(\rho, u ,\theta) :=\frac{\rho}{(2\pi\theta)^{\frac{3}{2}}}\exp(-\frac{|v- \mathrm{u} |^2}{2\theta})\,,$$
where $(\rho, \mathrm{u} ,\theta)$ denotes the fluid variables: density, bulk velocity and temperature. When $(\rho,u,\theta)$ are functions of $(x,t)$, $\mathcal{M}(\rho,  \mathrm{u}  ,\theta)$ is called the local Maxwellians, while when $(\rho, \mathrm{u} ,\theta)$ are constant states, $\mathcal{M}(\rho,  \mathrm{u}  ,\theta)$ is called the global (or absolute) Maxwellians.

In different scales for time and fluctuations relative to Maxwellians, the fluid dynamics of Boltzmann equations are represented by different equations, such as compressible/incompressible Navier-Stokes or Euler equations. As formally derived in \cite{BGL1}, when $\tau_{\varepsilon} = \varepsilon $, this is the case where Mach number and Knudsen number both equals to $\varepsilon$ (then the Reynold number $\mathrm{Re}=O(1)$ from the von Kármán relation $\mathrm{Re}=\frac{\mathrm{Ma}}{\mathrm{Kn}}$ )
 and the incompressible Navier-Stokes limit can be expected. When considering small Mach number, it is natural to consider distributions  as perturbations about a given absolute Maxwellian such as $M = \mathcal{M}(1,0,1)$. More specifically, the solutions to the equation \eqref{longtime_Boltzmann} are sought in the form $f_{\varepsilon} = M + \varepsilon M g_{\varepsilon}$.
Then the equation \eqref{longtime_Boltzmann} can be rewritten as
\begin{equation}\label{perturbed_Boltzmann}
    \left\{ \begin{matrix}
        \varepsilon^2 \partial_t g_{\varepsilon} + \varepsilon v \!\cdot\! \nabla_x g_{\varepsilon} + \mathcal{L}g_{\varepsilon} = \varepsilon Q(g_{\varepsilon},g_{\varepsilon}),\\
        g_{\varepsilon}(0,x,v) = g^{{in}}_{\varepsilon},
    \end{matrix}\right.
\end{equation}
with the linearized Boltzmann operator
\begin{equation}
    \mathcal{L}(g) := - \frac{1}{M}(C(M,Mg) + C(Mg,M)),
\end{equation}
and
\begin{equation}
    \begin{aligned}
        &Q(f,g) := \frac{1}{M}C(Mf,Mg).\\
    \end{aligned}
\end{equation}
We shall use the following notations in the sequel:
    \begin{equation}
        \begin{aligned}
            &\langle f,g \rangle = \int_{\mathbb{R}^3_v} f g M\, \mathrm{d}  v, \ \ \
            \langle f \rangle = \int_{\mathbb{T}^3} f M \, \mathrm{d}  v,\\
            &\langle\! \langle  f  \rangle\! \rangle := \int_{\mathbb{R}^3_v} \int_{\mathbb{R}^3_v}  \int_{\mathbb{S}^2}f M M _1 b \, \mathrm{d}  v  \mathrm{d}  v_1  \mathrm{d} \sigma,
        \end{aligned}
    \end{equation}
    and the standard inner product $({a} \!\  , {b}) = \sum_{i=1}^5 a^ib^i$ in $\mathbb{R}^5$.

    It is well known that the linearized Boltzmann operator $\mathcal{L}$ is self-adjoint on $L^2(Mdv)$ with inner product $\langle f,g \rangle $.
    Moreover, the null space $\mathcal{N}$ of $\mathcal{L}$ is spanned by the set of collision invariants:
    \begin{equation}
        \begin{aligned}
            \mathcal{N} = \text{Span} \{ 1 , v, |v|^2\}.
        \end{aligned}
    \end{equation}
    We also let $\mathcal{N}^{\perp}$ to denote the orthogonal space of $\mathcal{N}$ with respect to the inner product $\langle \cdot, \cdot \rangle$ defined above.
    Furthermore, we use the notation $\Lpre g$ and $\Loth g$ to denote the projection of $g$ to $\mathcal{N}$ and $\mathcal{N}^{\perp}$ respectively.

    Formally, $g_{\varepsilon}$ should converge to some $g$ which lies in the kernel of $\mathcal{L}$. Specifically, $g$ should be of the form: $g = \rho + v \cdot  \mathrm{u}   +( \frac{1}{2} |v|^2 -\frac{3}{2})\theta$. Moreover,
    the velocity $u$ is divergence free and the density and the temperature fluctuations $\rho$ and $\theta$ satisfy the Boussinesq relation
    \begin{equation}
        \nabla_x \cdot  \mathrm{u}  =0; \ \ \rho + \theta = 0.
    \end{equation}
    Furthermore, $(\rho, \mathrm{u} , \theta)$ should satisfy the incompressible Navier-Stokes-Fourier system
    \begin{equation}\label{INSF} \tag{INSF}
        \left\{
            \begin{aligned}
                &\partial_t  \mathrm{u}  +  \mathrm{u}   \!\cdot\! \nabla_x  \mathrm{u}  + \nabla_x p = \mu \Delta  \mathrm{u} , \\
                &\partial_t \vartheta +  \mathrm{u}   \!\cdot\! \nabla_x \vartheta = \kappa \Delta \vartheta,\\
                &\nabla_x \cdot  \mathrm{u}  =0,
            \end{aligned}
        \right.
    \end{equation}
    where $\vartheta = \frac{3}{5} \theta - \frac{2}{5}\rho$. The viscosity coefficient $\mu$ and heat-conductivity coefficient $\kappa$ are defined by $\mu = \mu(1,1)$ and $\kappa = \kappa(1,1)$ as in \eqref{local_viscosity_and_heat_conductivity} by setting $(\rho,\theta) = (1,1)$. For more detailed derivation of the above incompressible Navier-Stokes system, see \cite{BGL1}.

    Various works have been contributed to justify this process since late 1970s. Among them, we shall emphasize the so-called BGL program started by Bardos, Golse, and Levermore since the late 1980s. They aimed to justify Leray's solutions to incompressible Navier-Stokes equations from Diperna-Lions renormalized solutions \cites{BGL1,BGL2}.
After that, Bardos, Golse, Levermore, Lions, Saint-Raymond, Masmoudi then made significant contributions (see e.g., \cites{BGL_2000,Golse_Levermore_2002,Lions_Masmoudi_2001}). Finally Golse and Saint-Raymond obtained the first complete convergence result without any additional compactness assumption in \cite{Golse_Raymond_2004} for cutoff Maxwellian collision kernel and in \cite{Golse_Raymond_2005} for hard cutoff potentials. These results were later extended to include soft potentials by Levermore and Masmoudi \cite{Levermore_Masmoudi_2010}, and to bounded domain by Jiang, Levermore, Masmoudi and Saint-Raymond (see e.g., \cites{Jiang_Masmoudi_2017, Jiang_Levermore_Masmoudi_2010,Masmoudi_Raymond_2003}).

Aside from the uss of renormalized solutions of Boltzmann equation to deduce hydrodynamic limits, numerous works have also been based on the classical solution of Boltzmann equation. The first work of this type is that of Bardos and Ukai \cite{Bardos_Ukai_1991}, in which they proved the global existence of classical solutions of \eqref{perturbed_Boltzmann} for cutoff hard potentials, and established estimates uniform in $\varepsilon$ for $0< \varepsilon<1$. Later Briant, Merino-Aceituno, and Mouhot \cites{Briant_2015,Briant_Merino-Aceituno_Mouhot_2019} used the semigroup approach to prove the incompressible Navier-Stokes limit on torus for cutoff kernels with hard potentials.
For a border class of collision kernels for both cutoff and non-cutoff cases, Jiang, Xu and Zhao proved the incompressible Navier-Stokes-Fourier limit in \cite{Jiang_Xu_Zhao_2018}, using the non-isotropic norm developed in \cites{AMUXY_2011,AMUXY_2012,AMUXY_2012_Anal_Appl} and equivalently in \cite{Gressman_Strain}.
There are also many works based on the Hilbert expansion in the context of classical solutions.
This approach was started from Caflisch and Nishida's work on compressible Euler limit \cites{Caflisch_1980,Nishida_1978}. Guo, Jiang, Jang also used this method combining with some nonlinear estimate to derive the  acoustic limit  \cites{Guo_Jang_Jiang_2009,Guo_Jang_Jiang_2010,Jang_Jiang_2009}. Afterwards, this approach was used for incompressible Navier-Stokes limit in \cites{Guo_CPAM_2006,Masi_Esposito_Lebowitz}. It was later extended to general initial data to include fast acoustic waves by Jiang and Xiong in \cite{Jiang_Xiong}.

In \cite{Golse_Levermore_2002}, Golse and Levermore made the following nice observation (in the {\em Concluding Remarks} of \cite{Golse_Levermore_2002}): they proposed the so-called {\em compressible
 Stokes system}, which is the linearization about a homogeneous state of the compressible Navier-Stokes system. In the short time scale (i.e. $\tau_\varepsilon=1$), as $\varepsilon\rightarrow 0$, the compressible Stokes system converges to the acoustic system. While in the longer time scale, (i.e. $\tau_\varepsilon=\varepsilon$), the limit is the incompressible Stokes equations. In this sense, the (compressible) acoustic system and the incompressible Stokes equations can be unified in one system, which is the compressible Stokes system.

Golse and Levermore's proposal is on the linear case. Motivated by their idea, in this paper, we start from the scaled general compressible Navier-Stokes, i.e. the diffusion terms are order $O(\varepsilon$. When the time scale $\tau_\varepsilon=1$, the limit will be the compressible Euler system. This is the inviscid limit, which is a well-known analytically hard problem. When the time scale $\tau_\varepsilon=\varepsilon$, the limit will be the incompressible Navier-Stokes equations, which has been well-studied, as mentioned above. The main novelty of this paper is that we propose a new relative entropy approach, which can be analytically proved, as least in the framework of classical solutions. For the clarity of the presentation, we mainly focus on the formal calculation. However, we clearly state the the approach how the rigorous analytical proof could be followed from our formal analysis. The main advantage of this approach is that we could obtain the convergence rate to the incompressible Navier-Stokes equations. This could not be obtained from the previous compactness arguments. We state the main ideas of our appraoch in the following.

When $\tau_{\varepsilon}=1$, it has been shown in the formal level (see e.g., \cite{BGL1}), using the Chapman-Enskog expansion, that the solution of \eqref{longtime_Boltzmann} can be approximated by $f_{\varepsilon} = M_{\varepsilon} (1 + \varepsilon g_{\varepsilon} + \varepsilon^2 w_{\varepsilon})$,
where $M_{\varepsilon} = \mathcal{M}(\rho_{\varepsilon}, u_{\varepsilon},\theta_{\varepsilon})$ and $(\rho_{\varepsilon}, u_{\varepsilon} ,\theta_{\varepsilon})$ satisfies the following compressible Navier-Stokes system with dissipation of the order $\varepsilon$:
\begin{equation}\label{short_CNSF} \tag{CNS}
    \left\{ \begin{aligned}
        &\partial_t \rho_{\varepsilon} + \nabla_x (\rho_{\varepsilon} u_{\varepsilon}) = 0,\\
        & \rho_{\varepsilon} (\partial_t + u_{\varepsilon}  \!\cdot\! \nabla_x) u_{\varepsilon}  + \nabla_x ({\rho_{\varepsilon} \theta_{\varepsilon}}) = \varepsilon \nabla_x [ \mu_{\varepsilon}(\rho_{\varepsilon},\theta_{\varepsilon}) \sigma(u_{\varepsilon})],\\
        & \tfrac{3}{2} \rho_{\varepsilon} (\partial_t   + u_{\varepsilon}  \!\cdot\! \nabla_x )\theta_{\varepsilon} + \rho_{\varepsilon} \theta_{\varepsilon}  \nabla_x  \!\cdot\! u_{\varepsilon} = \varepsilon  \tfrac{1}{2} \mu_{\varepsilon}(\rho_{\varepsilon},\theta_{\varepsilon}) \sigma(u_{\varepsilon}):\sigma(u_{\varepsilon}) + \tfrac{5}{2} \varepsilon \nabla_x \cdot[\kappa(\rho_{\varepsilon},\theta_{\varepsilon}) \theta_{\varepsilon}],\\
        &(\rho_{\varepsilon}(0,x),\ u_{\varepsilon}(0,x), \ \theta_{\varepsilon}(0,x)) =  (\rho^{\mathrm{in}},\ u^{\mathrm{in}},\ \theta^{\mathrm{in}}).
    \end{aligned}\right.
\end{equation}
Here $\sigma(u)= \nabla_x u + {\nabla_x u}^T  - \frac{2}{3}(\nabla_x  \!\cdot\! u) \mathrm{I}$ is the stress tensor,
    and the viscosity coefficient $\mu_{\varepsilon}(\rho,\theta)$ along with the heat-conductivity coefficient $\kappa_{\varepsilon}(\rho,\theta)$ are defined as follows
    \begin{equation}\label{local_viscosity_and_heat_conductivity}
        \begin{aligned}
            \mu(\rho,\theta) = \frac{1}{10} \int_{\mathbb{R}^3_v} A(V) :  \hat{A}(V) \mathcal{M} dv,\\
            \kappa(\rho,\theta) = \frac{2}{15} \int_{\mathbb{R}^3_v} B(V) \cdot \hat{B}(V) \mathcal{M} dv,
        \end{aligned}
    \end{equation}
    where
    \begin{equation}
        A(V) = V\otimes V - \frac{|V|^2}{3}I, \qquad B(V) = V(\frac{|V|^2}{2}- \frac{5}{2}),
    \end{equation}
    with $\displaystyle V= \frac{v-u}{\sqrt{\theta}}$.
    Moreover, $\hat{A} (V)$ and $\hat{B}(V)$ are the unique solutions in $(\text{Ker} \mathcal{L}_{\mathcal{M}}) ^{\perp}$ of the following equations respectively
    \begin{equation}
        \mathcal{L}_{\mathcal{M}} \hat{A} (V) = A(V), \quad  \mathcal{L}_{\mathcal{M}} \hat{B}(V) = B(V),
    \end{equation}
    with
        \begin{equation}
        \mathcal{L}_{\mathcal{M}}(g) := - \frac{1}{\mathcal{M}}(C(\mathcal{M},\mathcal{M}g) + C(\mathcal{M}g,\mathcal{M})).
    \end{equation}

    We note that the compressible Navier-Stokes system \eqref{short_CNSF} is not a limit of the Boltzmann equation as the Knudsen number $\varepsilon \rightarrow 0$ but a second order approximation in the Chapman-Enskog expansion.
    There are also a lot of results on this approximation, we refer to \cites{BGL1,Kawashima_Matsumura_Nishida_1979,Duan_Liu_2021,Liu_Yang_Zhao}  and the references within.

    When Mach number is small (when we set $\tau_\varepsilon = \varepsilon$ as will be assumed throughout), it is expected that the slightly compressible fluids are close to incompressible fluids.
    Then there is a natural relation between \eqref{short_CNSF} and \eqref{INSF} by studying the fluctuation of $(\rho_\varepsilon,u_\varepsilon,\theta_\varepsilon)$ around $(1,0,1)$. More specifically, one may consider the following  compressible Navier-Stokes system with small Mach number:
    \begin{equation} \label{longtime_CNS}\tag{$\text{CNS}_{\varepsilon}$}
        \left\{ \begin{aligned}
            &\varepsilon\partial_t \rho_{\varepsilon} + \nabla_x (\rho_{\varepsilon} u_{\varepsilon}) = 0,\\
            &\varepsilon\partial_t u_{\varepsilon} + u_{\varepsilon}  \!\cdot\! \nabla_x u_{\varepsilon}  + \frac{1}{\rho_{\varepsilon}} \nabla_x (\rho_{\varepsilon} \theta_{\varepsilon})  = \varepsilon \frac{1}{\rho_{\varepsilon}} \nabla_x [ \mu_{\varepsilon}(\rho_{\varepsilon},\theta_{\varepsilon}) \sigma(u_{\varepsilon})],\\
            &\varepsilon \partial_t \theta_{\varepsilon}  + u_{\varepsilon}  \!\cdot\! \nabla_x \theta_{\varepsilon} + \frac{2}{3} \theta_{\varepsilon}  \nabla_x  \!\cdot\! u_{\varepsilon} = \varepsilon  \frac{1}{\rho_{\varepsilon}} \frac{1}{3} \mu_{\varepsilon}(\rho_{\varepsilon},\theta_{\varepsilon}) \sigma(u_{\varepsilon}):\sigma(u_{\varepsilon}) +  \frac{1}{\rho_{\varepsilon}}\frac{5}{3} \varepsilon \nabla_x \cdot[\kappa(\rho_{\varepsilon},\theta_{\varepsilon}) \theta_{\varepsilon}],\\
            &(\rho_{\varepsilon}(0,x),\ u_{\varepsilon}(0,x), \ \theta_{\varepsilon}(0,x)) = (1+ \varepsilon \tilde{\rho}^{\mathrm{in}},\ \tilde{u}^{\mathrm{in}},\ 1+ \varepsilon \tilde{\theta}^{\mathrm{in}}).
        \end{aligned}\right.
    \end{equation}

    Then the fluctuations $(\tilde{\rho}_{\varepsilon}, \tilde{u}_{\varepsilon},\tilde{\theta}_{\varepsilon})$ of the solutions of \eqref{longtime_CNS}, which satisfying $ (\rho_{\varepsilon}, u_{\varepsilon},\theta_{\varepsilon} )= (1+ \varepsilon \tilde{\rho}_{\varepsilon}, \varepsilon \tilde{u}, 1+ \varepsilon \tilde{\theta}) $, should converge to the solutions of \eqref{INSF} in some strong or weak sense (depending on the initial data and the domain along with proper boundary conditions).
        This process is known as the low Mach number limit, and we refer to \cites{Klainerman_Majda_1981,Klainerman_Majda_1982,Danchin_AJM,Alazard_2006,Ebin,MS_ARMA,DGLM,Masmoudi_2022,CGHJ} for more information.
        We mention that when the domain is considered to be torus, and the initial data are ill-prepared(i.e. the initial data are not required to satisfy the incompressibility and Boussinesq relation), the convergence cannot be strong, due to the acoustic waves generated by initial compression, which will exist in the long time.

        This phenomenon shall also arise when considering the incompressible Navier-Stokes-Fourier limit of the Boltzmann equation, which prevents the strong convergence of $g_{\varepsilon}$ to $g= v \cdot u + \theta(\frac{3}{2} |v|^2 -\frac{5}{2}) $ if the kinetic part of initial data is well-prepared, but the fluid part of initial data is ill-prepared.
        Therefore, when considering the incompressible Navier-Stokes limit of the Boltzmann equation on torus with ill-prepared initial data, we need also to characterize the behavior of acoustic waves in the meantime.

        We deal with this problem by splitting the incompressible Navier-Stokes limit into two steps. The first step is the approximation of Boltzmann equation by compressible Navier-Stokes system with small Mach number, and the second step is the low Mach number limit of the compressible Navier-Stokes system.
        Their relationship is shown in the following diagram. The low Mach number limit process has been discussed in our former paper \cite{CGHJ}. In this paper we focus on the compressible Navier-Stokes approximation for the Boltzmann equation.
        \begin{figure}[H]
            \centering
            \begin{tikzpicture}[>=latex, node distance=2cm]
                \node [draw, rectangle]  (A) at (0,0) {Kinetic Level: };
                \node [draw, rectangle,text width=4cm, align=center] (A_1) at (5,0) {Boltzmann equation\\Kn=Ma=$\varepsilon$};
                \node [draw, rectangle] (B) at (0,-3) {Fluid Level:} ;
                \node [draw, rectangle, text width=3.5cm, align=center]  (C) at(12,-3) { Compressible \\ Navier-Stokes system\\  Ma = $\varepsilon$};
                \node [draw, rectangle, text width=3cm, align=center] (D)  at (5,-3) {Incompressible \\  Naiver-Stokes-Fourier system};

                 \draw[->] (A) -- (B);
                 \draw [->] (A_1) -- (C) node[midway, right,text width=4cm, align=center] {Hydrodynamic \\asymptotic\\ $\varepsilon \rightarrow 0$};
                 \draw [->] (A_1) -- (D) node[midway, left,text width=4cm, align=center] {Hydrodynamic \\limit\\ $\varepsilon \rightarrow 0$};
                 \draw [->] (C) -- (D)node[midway, above,text width=4cm, align=center] {Low Mach number}
                                        node[midway, below,text width=4cm, align=center] {limit $\varepsilon \rightarrow 0$};

            \end{tikzpicture}
            \caption{Relationship between Boltzmann equation, incompressible Navier-Stokes-Fourier system and compressible Navier-Stokes system.}
            \label{fig:tikz_example}
        \end{figure}

    As illustrated in \eqref{short_CNSF}, the compressible Navier-Stokes-Fourier system derived from Boltzmann equation is {\em not} a limit, since in this setting, the compressible Navier-Stokes-Fourier  system  depends on the Knudsen number itself. So it is not a {\em limit}, but an {\em asymptotic} problem.
    We shall use the relative entropy to describe the compressible Navier-Stokes approximation of the Boltzmann equation. The relative entropy $H(f|g)$ for $f>0$ and $g>0$
    is defined as
    \begin{equation}\label{RE}
        H(f|g) = \int_{\mathbb{T}^3_x}\int_{\mathbb{R}^3_v} f \log(\frac{f}{g}) - f + g \  dxdv.
    \end{equation}
    The key issue is that in this paper, the ``target'', i.e. $g$ in the relative entropy \eqref{RE} must depends on the Kundsen number $\varepsilon$. More specifically, we set $f$ in \eqref{RE} as the solutions to the scaled Boltzmann equation $f_\varepsilon$,  and $g$ as the local Maxwellian ($M_{\varepsilon}$ defined below) governed by the solutions to the scaled compressible NSF. We calculate the evolution of this scaled relative entropy $\frac{1}{{\varepsilon}^2} H(f_{\varepsilon}| M_{\varepsilon})(t)$ and prove its stability, i.e. the evolution of the scaled relative entropy in any time $t>0$ can be controlled by its initial data. More importantly, we characterize its dissipation rate. It is the first quantitative convergence result for the compressible Navier-Stokes-Fourier system in the sense of relative entropy. This approach avoids any expansion method, such as Chapman-Enskog or Hilbert expansions. The main disadvantage of the expansion methods is it can only construct {\em special} (in the format of expansions) solution of the original Boltzmann equation. On the other hand, the relative entropy method employed here is to directly measure the distance in the sense of relative entropy between {\em any} solutions of the scaled Boltzmann equation and those of compressible NSF. Furthermore, the asymptotic behavior is quantitatively characterized by relative entropy and the relative entropy dissipation is explicit. This is the main novelty of this paper.

    The idea of using the notion of relative entropy for this kind of problems comes from the notion of entropic convergence developed by Bardos, Golse and Levermore in \cite{BGL2}, and on the other hand from Yau's derivation of the hydrodynamic limit of Ginzburg-Landau lattice model \cite{Yau}.
    Latter, Saint-Raymond use relative entropy to deduce the incompressible Euler limit for well-prepared initial data in \cite{Saint-Raymond_2003} and for ill-prepared initial data in \cite{Saint-Raymond_2009}.

    We use the solution $(\rho_{\varepsilon},u_{\varepsilon},\theta_{\varepsilon}) = (1+ \varepsilon \tilde{\rho}_{\varepsilon}, \varepsilon \tilde {u}_{\varepsilon},1+ \varepsilon \tilde{\theta}_{\varepsilon})$ of \eqref{longtime_CNS} to construct a local Maxwellian $M_{\varepsilon} = \mathcal{M}(\rho_{\varepsilon},u_{\varepsilon},\theta_{\varepsilon}) = \mathcal{M} (1+ \varepsilon \tilde{\rho}_{\varepsilon}, \varepsilon \tilde {u}_{\varepsilon},1+ \varepsilon \tilde{\theta}_{\varepsilon}) $,
    and we would like to use the relative entropy $H(f_{\varepsilon}| M_{\varepsilon})$ to measure the difference between the solution of the rescaled Boltzmann equation $\eqref{longtime_Boltzmann}$ and the solution of the compressible Navier-Stokes system $\eqref{longtime_CNS}$.
    More specifically, we derive the time evolution of the relative entropy $\frac{d}{dt} \frac{1}{\varepsilon^2} H(f_{\varepsilon}| M_{\varepsilon})$, which leads to the following modulated entropy inequality
    \begin{equation}
        \begin{aligned}
            \tfrac{1}{{\varepsilon}^2} &H(f_{\varepsilon}| M_{\varepsilon})(t) +\int_0^t \int_{\mathbb{T}^3_x}\left\{\tfrac{1}{\varepsilon^4}D(f_\varepsilon)(s,x) - \tfrac{1}{2} \mu \sigma( \mathrm{u} ^b_\varepsilon): \sigma( \mathrm{u} ^b_\varepsilon) - \tfrac{5}{2}\kappa (\nabla_x \theta^b_{\varepsilon})^2\right\}\,   \mathrm{d}  x \mathrm{d}  s \\
            &+\int_0^t \int_{\mathbb{T}^3_x} \tfrac{1}{2}\mu \sigma(\tilde{\u_{\varepsilon}}- \mathrm{u} ^b_{\varepsilon}):\sigma(\tilde{ \mathrm{u} }_{\varepsilon}- \mathrm{u} ^b_{\varepsilon}) + \tfrac{5}{2}  \kappa(\nabla_x \tilde{\theta}_{\varepsilon} -\nabla_x \theta^b_{\varepsilon})^2 \ dxds\\
            &\lesssim \frac{1}{\varepsilon^2} H(f^{\mathrm{in}}_{\varepsilon}| M^{\mathrm{in}}_{\varepsilon}) + \mathcal{O}(\varepsilon).
        \end{aligned}
    \end{equation}
    Here and in the sequel $(\rho^b_{\varepsilon},  \mathrm{u} ^b_{\varepsilon},\theta^b_{\varepsilon})$ are moments of perturbation $g_{\varepsilon}$ as defined in \eqref{g_moments}.

    Under the assumption of vanishing initial relative entropy
    \begin{equation}
        \frac{1}{\varepsilon^2} H(f^{\mathrm{in}}_{\varepsilon}| M_{\varepsilon}^{\mathrm{in}}) \rightarrow 0  \ \ \text{as } \varepsilon \rightarrow 0,
    \end{equation}
    for each $t > 0$, we obtain the following asymptotics:
    \begin{enumerate}[label=\textbullet]
        \item Asymptotic of the entropy dissipation rate
            \begin{equation}
               \lim_{\varepsilon \rightarrow 0} \left[\int_0^t \int_{\mathbb{T}^3_x}\frac{1}{\varepsilon^4}D(f_\varepsilon)(s,x) - \frac{1}{2} \mu \sigma(u^b_{\varepsilon}): \sigma(u^b_{\varepsilon}) - \frac{5}{2}\kappa (\nabla_x \theta^b_{\varepsilon})^2\  dxds \right] = 0;
            \end{equation}
        \item Asymptotic of the momentum and energy flux
            \begin{equation}
                \lim_{\varepsilon \rightarrow 0} \left[ \int_0^t \int_{\mathbb{T}^3_x} \frac{1}{2}\mu \sigma(\tilde{ \mathrm{u} }_{\varepsilon}- \mathrm{u} ^b_{\varepsilon}):\sigma(\tilde{ \mathrm{u} }_{\varepsilon}- \mathrm{u} ^b_{\varepsilon}) + \frac{5}{2}  \kappa(\nabla_x \tilde{\theta}_{\varepsilon} -\nabla_x \theta^b_{\varepsilon})^2 \ dxds \right]=0;
            \end{equation}
        \item Asymptotic of the relative entropy
            \begin{equation}
                \lim_{\varepsilon \rightarrow 0}\frac{1}{{\varepsilon}^2} H(f_{\varepsilon}| M_{\varepsilon})(t)= 0.
            \end{equation}
    \end{enumerate}
   We state our main theorem as follows:

When Mach number is small (when we set $\tau_\varepsilon = \varepsilon$ as will be assumed throughout), it is expected that the slightly compressible fluids are close to incompressible fluids.
 Then there is a natural relation between \eqref{short_CNSF} and \eqref{INSF} by studying the fluctuation of $(\rho_\varepsilon,u_\varepsilon,\theta_\varepsilon)$ around $(1,0,1)$. More specifically, one may consider the following compressible Navier-Stokes system with small Mach number:
 \begin{equation} \label{longtime_CNS}\tag{$\text{CNS}_{\varepsilon}$}
 \left\{ \begin{aligned}
 &\varepsilon\partial_t \rho_{\varepsilon} + \nabla_x (\rho_{\varepsilon} u_{\varepsilon}) = 0,\\
 &\varepsilon\partial_t u_{\varepsilon} + u_{\varepsilon} \!\cdot\! \nabla_x u_{\varepsilon} + \frac{1}{\rho_{\varepsilon}} \nabla_x (\rho_{\varepsilon} \theta_{\varepsilon}) = \varepsilon \frac{1}{\rho_{\varepsilon}} \nabla_x [ \mu_{\varepsilon}(\rho_{\varepsilon},\theta_{\varepsilon}) \sigma(u_{\varepsilon})],\\
 &\varepsilon \partial_t \theta_{\varepsilon} + u_{\varepsilon} \!\cdot\! \nabla_x \theta_{\varepsilon} + \frac{2}{3} \theta_{\varepsilon} \nabla_x \!\cdot\! u_{\varepsilon} = \varepsilon \frac{1}{\rho_{\varepsilon}} \frac{1}{3} \mu_{\varepsilon}(\rho_{\varepsilon},\theta_{\varepsilon}) \sigma(u_{\varepsilon}):\sigma(u_{\varepsilon}) + \frac{1}{\rho_{\varepsilon}}\frac{5}{3} \varepsilon \nabla_x \cdot[\kappa(\rho_{\varepsilon},\theta_{\varepsilon}) \theta_{\varepsilon}],\\
 &(\rho_{\varepsilon}(0,x),\ u_{\varepsilon}(0,x), \ \theta_{\varepsilon}(0,x)) = (1+ \varepsilon \tilde{\rho}^{\mathrm{in}},\ \tilde{u}^{\mathrm{in}},\ 1+ \varepsilon \tilde{\theta}^{\mathrm{in}}).
 \end{aligned}\right.
 \end{equation}

We use the solution $(\rho_{\varepsilon},u_{\varepsilon},\theta_{\varepsilon}) = (1+ \varepsilon \tilde{\rho}_{\varepsilon}, \varepsilon \tilde {u}_{\varepsilon},1+ \varepsilon \tilde{\theta}_{\varepsilon})$ of \eqref{longtime_CNS} to construct a local Maxwellian $M_{\varepsilon} = \mathcal{M}(\rho_{\varepsilon},u_{\varepsilon},\theta_{\varepsilon}) = \mathcal{M} (1+ \varepsilon \tilde{\rho}_{\varepsilon}, \varepsilon \tilde {u}_{\varepsilon},1+ \varepsilon \tilde{\theta}_{\varepsilon}) $,
 and we would like to use the relative entropy $H(f_{\varepsilon}| M_{\varepsilon})$ to measure the difference between the solution of the rescaled Boltzmann equation $\eqref{longtime_Boltzmann}$ and the solution of the compressible Navier-Stokes system $\eqref{longtime_CNS}$.
 More specifically, we derive the time evolution of the relative entropy $\frac{d}{dt} \frac{1}{\varepsilon^2} H(f_{\varepsilon}| M_{\varepsilon})$, which leads to the following modulated entropy inequality
 \begin{equation}
 \begin{aligned}
 \tfrac{1}{{\varepsilon}^2} &H(f_{\varepsilon}| M_{\varepsilon})(t) +\int_0^t \int_{\mathbb{T}^3_x}\left\{\tfrac{1}{\varepsilon^4}D(f_\varepsilon)(s,x) - \tfrac{1}{2} \mu \sigma( \mathrm{u} ^b_\varepsilon): \sigma( \mathrm{u} ^b_\varepsilon) - \tfrac{5}{2}\kappa (\nabla_x \theta^b_{\varepsilon})^2\right\}\, \mathrm{d} x \mathrm{d} s \\
 &+\int_0^t \int_{\mathbb{T}^3_x} \tfrac{1}{2}\mu \sigma(\tilde{\u_{\varepsilon}}- \mathrm{u} ^b_{\varepsilon}):\sigma(\tilde{ \mathrm{u} }_{\varepsilon}- \mathrm{u} ^b_{\varepsilon}) + \tfrac{5}{2} \kappa(\nabla_x \tilde{\theta}_{\varepsilon} -\nabla_x \theta^b_{\varepsilon})^2 \ dxds\\
 &\lesssim \frac{1}{\varepsilon^2} H(f^{\mathrm{in}}_{\varepsilon}| M^{\mathrm{in}}_{\varepsilon}) + \mathcal{O}(\varepsilon).
 \end{aligned}
 \end{equation}
 Here and in the sequel $(\rho^b_{\varepsilon}, \mathrm{u} ^b_{\varepsilon},\theta^b_{\varepsilon})$ are moments of perturbation $g_{\varepsilon}$ as defined in \eqref{g_moments}.

Under the assumption of vanishing initial relative entropy
 \begin{equation}
 \frac{1}{\varepsilon^2} H(f^{\mathrm{in}}_{\varepsilon}| M_{\varepsilon}^{\mathrm{in}}) \rightarrow 0 \ \ \text{as } \varepsilon \rightarrow 0,
 \end{equation}
 for each $t > 0$, we obtain the following asymptotics:
 \begin{enumerate}[label=\textbullet]
 \item Asymptotic of the entropy dissipation rate
 \begin{equation}
 \lim_{\varepsilon \rightarrow 0} \left[\int_0^t \int_{\mathbb{T}^3_x}\frac{1}{\varepsilon^4}D(f_\varepsilon)(s,x) - \frac{1}{2} \mu \sigma(u^b_{\varepsilon}): \sigma(u^b_{\varepsilon}) - \frac{5}{2}\kappa (\nabla_x \theta^b_{\varepsilon})^2\ dxds \right] = 0;
 \end{equation}
 \item Asymptotic of the momentum and energy flux
 \begin{equation}
 \lim_{\varepsilon \rightarrow 0} \left[ \int_0^t \int_{\mathbb{T}^3_x} \frac{1}{2}\mu \sigma(\tilde{ \mathrm{u} }_{\varepsilon}- \mathrm{u} ^b_{\varepsilon}):\sigma(\tilde{ \mathrm{u} }_{\varepsilon}- \mathrm{u} ^b_{\varepsilon}) + \frac{5}{2} \kappa(\nabla_x \tilde{\theta}_{\varepsilon} -\nabla_x \theta^b_{\varepsilon})^2 \ dxds \right]=0;
 \end{equation}
 \item Asymptotic of the relative entropy
 \begin{equation}
 \lim_{\varepsilon \rightarrow 0}\frac{1}{{\varepsilon}^2} H(f_{\varepsilon}| M_{\varepsilon})(t)= 0.
 \end{equation}
 \end{enumerate}
 We state our main theorem as follows:

\begin{equation}\label{temp_relative_entropy}
 \begin{aligned}
 &\frac{1}{{\varepsilon}^2} H(f_{\varepsilon}| M_{\varepsilon})(t) +\frac{1}{\varepsilon^4}\int_0^t \int_{\mathbb{T}^3_x} D(f_\varepsilon)(s,x)\ ds dx + \tilde{R} \\
 &+ \int^t_0 \int_{\mathbb{T}^3_x} \nabla_x [ \mu(\rho,\theta) \sigma(\tilde{u})] \frac{1}{\rho} \frac{u^b-\tilde{u}}{\theta} - \frac{1}{\theta} \mu \sigma(u^b):\nabla_x \tilde{u} \ dx ds \\
 & + \int^t_0 \int_{\mathbb{T}^3_x} \frac{1}{\rho}\left(\frac{5}{2} \nabla_x [\kappa(\rho,\theta) \nabla_x \tilde{\theta}]\right) \frac{1}{\theta^2} (\theta^b -\tilde{\theta}) -\frac{1}{\theta^2}\frac{5}{2}\kappa \nabla_x \theta^b \cdot \nabla_x \tilde{\theta} \ dx ds \\
 &+ \int^t_0 \int_{\mathbb{T}^3_x} \frac{1}{\theta} \left[ (\tilde{u}-u^b)\otimes(\tilde{u}-u^b) - \frac{1}{3} (\tilde{u} - u^b)^2 I \right] : \nabla_x \tilde{u} \ dx ds \\
 &+ \int^t_0 \int_{\mathbb{T}^3_x} \left[\frac{5}{2} \frac{1}{\theta^2} (\tilde{u}-u^b)(\tilde{\theta}-\theta^b) \right] \cdot \nabla_x \tilde{\theta} \ dx ds \\
 &\leqslant \frac{1}{\varepsilon^2} H(f^{\mathrm{in}}_{\varepsilon}| M^{\mathrm{in}}_{\varepsilon} ),
 \end{aligned}
 \end{equation}
where
\begin{equation}
 \begin{aligned}
 \tilde{R} = R_1 + R_2 + R_3 + R_4 + \int^t_0 \int_{\mathbb{T}^3_x} R_A : \frac{\nabla_x \tilde{u}}{\theta} + R_B \cdot \frac{\nabla_x \tilde{\theta}}{\theta^2} \ dxds.
 \end{aligned}
\end{equation}
And we denote
\begin{align}
 & \text{\Rmnum{1}}=\int^t_0 \int_{\mathbb{T}^3_x} \nabla_x [ \mu(\rho,\theta) \sigma(\tilde{u})] \frac{1}{\rho} \frac{u^b-\tilde{u}}{\theta} - \frac{1}{\theta} \mu \sigma(u^b):\nabla_x \tilde{u} \ dx ds \label{viscous_term},\\
 & \text{\Rmnum{2}}= \int^t_0 \int_{\mathbb{T}^3_x} \frac{1}{\rho}\left(\frac{5}{2} \nabla_x [\kappa(\rho,\theta) \nabla_x \tilde{\theta}]\right) \frac{1}{\theta^2} (\theta^b -\tilde{\theta}) -\frac{1}{\theta^2}\frac{5}{2}\kappa \nabla_x \theta^b \cdot \nabla_x \tilde{\theta} \ dx ds \label{heat_conduct_term} ,\\
 & \text{\Rmnum{3}}= \int^t_0 \int_{\mathbb{T}^3_x} \frac{1}{\theta} \left[ (\tilde{u}-u^b)\otimes(\tilde{u}-u^b) - \frac{1}{3} (\tilde{u} - u^b)^2 I \right] : \nabla_x \tilde{u} \ dx ds \label{u_convection_term}, \\
 & \text{\Rmnum{4}}= \int^t_0 \int_{\mathbb{T}^3_x} \left[\frac{5}{2} \frac{1}{\theta^2} (\tilde{u}-u^b)(\tilde{\theta}-\theta^b) \right] \cdot \nabla_x \tilde{\theta} \ dx ds \label{theta_convection_term} .\\
 \end{align}
For the term \Rmnum{1}, by symmetry of $\sigma(\tilde{u})$ and integration by parts, we have
\begin{equation}\label{integration_by_part_u}
 \begin{aligned}
 \int^t_0 \int_{\mathbb{T}^3_x} &\nabla_x \mdot [ \mu(\rho,\theta) \sigma(\tilde{u})] \mdot \frac{1}{\rho} \frac{u^b-\tilde{u}}{\theta} - \frac{1}{\theta} \mu \sigma(u^b):\nabla_x \tilde{u} \ dx ds\\
 &= \int^t_0 \int_{\mathbb{T}^3_x}\frac{1}{2} \mu \sigma(\tilde{u}-u^b):\sigma(\tilde{u}-u^b) - \frac{1}{2} \mu \sigma(u^b):\sigma(u^b) \ dxds + R_5,
 \end{aligned}
\end{equation}
where
\begin{equation}
 \begin{aligned}
 R_5&= \int^t_0 \int_{\mathbb{T}^3_x} \nabla_x \mdot [ \mu(\rho,\theta) \sigma(\tilde{u})] \mdot (\frac{-\varepsilon\tilde{\rho} -\varepsilon \tilde{\theta} -\varepsilon^2 \tilde{\rho}\tilde{\theta}}{\rho \theta})(u^b-\tilde{u})+ (\frac{\varepsilon\theta}{\theta})\mu \sigma(u^b):\nabla_x \tilde{u} \ dxds\\
 &+\int^t_0 \int_{\mathbb{T}^3_x} \frac{1}{2} \left( \mu(\rho,\theta) - \mu \right)\sigma(\tilde{u}):\sigma(\tilde{u}-u^b) \ dxds.
 \end{aligned}
\end{equation}
For the term II, same reasoning shows that
\begin{equation}\label{integration_by_part_theta}
 \begin{aligned}
 \int^t_0 \int_{\mathbb{T}^3_x} \frac{1}{\rho}\left(\frac{5}{2} \nabla_x \mdot [\kappa(\rho,\theta) \nabla_x \tilde{\theta}]\right) \frac{1}{\theta^2} (\theta^b -\tilde{\theta}) -\frac{1}{\theta^2}\frac{5}{2}\kappa \nabla_x \theta^b \cdot \nabla_x \tilde{\theta} \ dx ds \\
 = \int^t_0 \int_{\mathbb{T}^3_x} \frac{5}{2} \kappa(\nabla_x \tilde{\theta} -\nabla_x \theta^b)^2 -\frac{5}{2}\kappa( \nabla_x \theta^b )^2 \ dxds + R_6,
 \end{aligned}
\end{equation}
where
\begin{equation}
 \begin{aligned}
 R_6&= \int^t_0 \int_{\mathbb{T}^3_x} (\frac{1}{\rho \theta^2}-1 ) \frac{5}{2} \nabla_x \mdot [\kappa(\rho,\theta) \nabla_x \tilde{\theta} ](\theta^b -\tilde{\theta}) + (1-\frac{1}{\theta^2})\kappa \nabla_x \theta^b \cdot \nabla_x \tilde{\theta}\ dxds\\
 &+ \int^t_0 \int_{\mathbb{T}^3_x} \frac{5}{2} (\kappa - \kappa(\rho,\theta))\nabla_x \tilde{\theta}\cdot (\nabla_x \tilde{\theta}-\nabla_x \theta^b) \ dxds.
 \end{aligned}
\end{equation}

\subsection{Conclusion} \label{SS_Conclusion}
We now complete the proof of Theorem \ref{main}.
Note that
\begin{equation}\label{quadratic_term}
 \int_0^t \int_{\mathbb{T}^3_x} \frac{1}{2}\mu \sigma(\tilde{u}-u^b):\sigma(\tilde{u}-u^b) + \frac{5}{2} \kappa(\nabla_x \tilde{\theta} -\nabla_x \theta^b)^2 \ dxds \geqslant 0.
\end{equation}
Combining with \eqref{mid_result} and \eqref{dissipation_control_final_result} we have
\begin{equation}
 \begin{aligned}
 \frac{1}{{\varepsilon}^2} &H(f_{\varepsilon}| M_{\varepsilon})(t) \\
 &\lesssim \frac{1}{\varepsilon^2} H(f^{\mathrm{in}}_{\varepsilon}| M^{\mathrm{in}}_{\varepsilon}) + \int^t_0 \|(\nabla_x \tilde{u},\nabla_x \tilde{\theta})\|_{L^{\infty}(dx)} \frac{1}{\varepsilon^2} H(f_{\varepsilon}|M_{\varepsilon}) ds \\
 & +\tilde{R}+ R_5+R_6+R_7 -R_{11}-R_{12} - 2R_{13}.
 \end{aligned}
\end{equation}
All the remainder terms can be shown to tend to zero as $\varepsilon$ tends to zero by using that $(\rho^b_{\varepsilon}, u^b_{\varepsilon}, \theta^b_{\varepsilon})$, $(\tilde{\rho}_{\varepsilon},\tilde{u}_{\varepsilon},\tilde{\theta}_{\varepsilon})$, $g_{\varepsilon}$ and $\frac{1}{\varepsilon} \Loth g_{\varepsilon}$
are separately uniformly bounded in some proper functional spaces, and Grönwall's inequality then implies that
\begin{equation}
 \begin{aligned}
 \frac{1}{{\varepsilon}^2} H(f_{\varepsilon}| M_{\varepsilon})(t) \leqslant
 \left( \frac{1}{\varepsilon^2} H(f^{\mathrm{in}}_{\varepsilon}| M^{\mathrm{in}}_{\varepsilon})+ \mathcal{O}(\varepsilon)\right) \exp \left( C \int_0^{\infty} \| (\nabla_x \tilde{u}, \nabla_x \tilde{\theta}) \|_{L^{\infty}(dx)} dt \right).
 \end{aligned}
\end{equation}
Hence,
\begin{equation}\label{entropy_control}
 \begin{aligned}
 \frac{1}{{\varepsilon}^2} H(f_{\varepsilon}| M_{\varepsilon})(t) \lesssim
 \frac{1}{\varepsilon^2} H(f^{\mathrm{in}}_{\varepsilon}| M^{\mathrm{in}}_{\varepsilon})+ \mathcal{O}(\varepsilon).
 \end{aligned}
\end{equation}
This estimate together with \eqref{mid_result}, \eqref{quadratic_term} and the fact $H(f|g) \geqslant 0$ leads to
\begin{equation}\label{entropy_dissipation_bound_temp}
 \int_0^t \int_{\mathbb{T}^3_x}\frac{1}{\varepsilon^4}D(f_\varepsilon)(s,x) - \frac{1}{2} \mu \sigma(u^b): \sigma(u^b) - \frac{5}{2}\kappa (\nabla_x \theta^b)^2\ dxds \lesssim \frac{1}{\varepsilon^2} H(f_{\varepsilon}^{\mathrm{in}}|M_{\varepsilon}^{\mathrm{in}}) + \mathcal{O}(\varepsilon).
 \end{equation}
 Using \eqref{mid_result}, \eqref{entropy_control}, \eqref{entropy_dissipation_bound_temp} and \eqref{dissipation_control_final_result},we then have
 \begin{equation}
 \int_0^t \int_{\mathbb{T}^3_x} \frac{1}{2}\mu \sigma(\tilde{u}-u^b):\sigma(\tilde{u}-u^b) + \frac{5}{2} \kappa(\nabla_x \tilde{\theta} -\nabla_x \theta^b)^2 \ dxds \lesssim \frac{1}{\varepsilon^2} H(f_{\varepsilon}^{\mathrm{in}}|M_{\varepsilon}^{\mathrm{in}}) + \mathcal{O}(\varepsilon),
\end{equation}
which completes the proof of Theorem \ref{main}.

\begin{remark}
  We remark that although the above theorem is in the formal level, its proof is indeed provide a clear way to the later rigorous proof. If we work in the framework of the classical solutions, all the corresponding existence results are available, for the Boltzmann equations, compressible and incompressible Navier-Stokes equations, at least near the constant states. Because the scalings themselves are near the constant states, so in this sense, the formal analysis is not far from the rigorous proof.

    In fact, the estimates \eqref{main_estimate}, \eqref{entropy_dissipation_bound} and \eqref{energy_momentum_flux_bound} hold for $0 \leqslant t \leqslant T^*$, where $T^*$ is determined by the minimal lifespan of the solutions to \eqref{longtime_Boltzmann} and \eqref{longtime_CNS}. In particular, if  for all $ \varepsilon > 0$ both $f_\varepsilon$ and $(\rho_\varepsilon, u_\varepsilon, \theta_\varepsilon)$ exist globally in time, then these estimates also hold globally. Furthermore, the validity of the $\mathcal{O}(\varepsilon)$ residual term relies on uniform in $\varepsilon$ estimates for the solutions. As we are working within a perturbation framework, such existence of global solutions of \eqref{longtime_CNS} are provided in \cites{Danchin_Inventiones,Chen-Gui-Jiang}, while the required uniform bounds for $(\rho_\varepsilon, u_{\varepsilon}, \theta_{\varepsilon})$ can be found in \cite{CGHJ}. Moreover, the global existence and uniform in $\varepsilon$ estimates for the solution $f_{\varepsilon}$ of \eqref{longtime_Boltzmann} can be found in \cite{Jiang_Xu_Zhao_2018}.
\end{remark}
    In the next section, we shall prove, at a formal level, the above estimate by calculating the evolution of the relative entropy. In section \ref{SS_REC}, we show that the unsigned terms will be controlled by the relative entropy itself. In section \ref{SS_EDC}  we show that the negative terms will be controlled by the entropy dissipation.
    And in section \ref{SS_Conclusion} we conclude the estimate \eqref{main_estimate}, \eqref{entropy_dissipation_bound}, \eqref{energy_momentum_flux_bound} by Grönwall's inequality.

To prove Theorem~\ref{main} ("at a formal level"), the paper proceeds as follows:
\begin{itemize}
\item First, it proves the estimate by "calculating the evolution of the relative entropy."
\item In Section~\ref{SS_REC}, it shows that the "unsigned terms will be controlled by the relative entropy itself."
\item In Section~\ref{SS_EDC}, it shows that the "negative terms will be controlled by the entropy dissipation."
\item Finally, in Section~\ref{SS_Conclusion}, it concludes \eqref{main_estimate}, \eqref{entropy_dissipation_bound}, \eqref{energy_momentum_flux_bound} "by Gr\"onwall's inequality."
\end{itemize}
It is also remarked that the argument is "in the formal level" but "provide[s] a clear way to the later rigorous proof," and that the validity of the $\mathcal{O}(\varepsilon)$ residual term "relies on uniform in $\varepsilon$ estimates for the solutions" (with cited sources providing global existence and uniform bounds in the perturbative framework).

To prove the main theorem ("at a formal level"), the paper proceeds as follows:
\begin{itemize}
\item First, it proves the estimate by "calculating the evolution of the relative entropy."
\item Next, in the part labeled
\begin{aligned}
        R_6&= \int^t_0 \int_{\mathbb{T}^3_x} (\frac{1}{\rho \theta^2}-1 ) \frac{5}{2} \nabla_x \mdot [\kappa(\rho,\theta) \nabla_x \tilde{\theta} ](\theta^b -\tilde{\theta}) + (1-\frac{1}{\theta^2})\kappa \nabla_x \theta^b \cdot \nabla_x \tilde{\theta}\ dxds\\
         &+ \int^t_0 \int_{\mathbb{T}^3_x} \frac{5}{2} (\kappa - \kappa(\rho,\theta))\nabla_x \tilde{\theta}\cdot (\nabla_x \tilde{\theta}-\nabla_x \theta^b) \ dxds.
    \end{aligned}
\end{equation}

\subsection{Relative entropy control}\label{SS_REC}
The terms \Rmnum{3} and \Rmnum{4} are unsigned, however they can be controlled by the quadratic of $(\rho^b-\tilde{\rho}^{\varepsilon}, u^b - \tilde{u}^{\varepsilon},\theta^b - \tilde{\theta}^{\varepsilon})$ which  can be controlled by relative entropy $\frac{1}{\varepsilon^2} H(f_{\varepsilon}|M_{\varepsilon})$.
In fact, we have the following two lemmas.
\begin{lemma} \label{same_moments}
    \begin{equation}
        \begin{aligned}
            H(f_{\varepsilon}|M_{\varepsilon}) = H(f_{\varepsilon}| M_{f_\varepsilon}) + H(M_{f_{\varepsilon}}|M_{\varepsilon}),
        \end{aligned}
\end{itemize}
(so that the "unsigned terms will be controlled by the relative entropy itself").
\begin{itemize}
\item Then, in the part labeled
\begin{aligned}
        \frac{1}{{\varepsilon}^2} &H(f_{\varepsilon}| M_{\varepsilon})(t) +\int_0^t \int_{\mathbb{T}^3_x}\frac{1}{\varepsilon^4}D(f_\varepsilon)(s,x) - \frac{1}{2} \mu \sigma(u^b): \sigma(u^b) - \frac{5}{2}\kappa (\nabla_x \theta^b)^2\  dxds \\
        &+\int_0^t \int_{\mathbb{T}^3_x} \frac{1}{2}\mu \sigma(\tilde{u}-u^b):\sigma(\tilde{u}-u^b) + \frac{5}{2}  \kappa(\nabla_x \tilde{\theta} -\nabla_x \theta^b)^2 \ dxds\\
        &\lesssim \frac{1}{\varepsilon^2} H(f^{\mathrm{in}}_{\varepsilon}| M^{\mathrm{in}}_{\varepsilon}) + \int^t_0 \|(\nabla_x \tilde{u},\nabla_x \tilde{\theta})\|_{L^{\infty}(dx)} \frac{1}{\varepsilon^2} H(f_{\varepsilon}|M_{\varepsilon}) ds +\tilde{R} + R_5+R_6+R_7.
    \end{aligned}
\end{equation}

\subsection{Entropy dissipation control}\label{SS_EDC}
 We now deal with the term
\begin{equation}\label{dissipation_control}
    \begin{aligned}
        \int_0^t \int_{\mathbb{T}^3_x}\frac{1}{\varepsilon^4}D(f_\varepsilon)(s,x) - \frac{1}{2} \mu \sigma(u^b): \sigma(u^b) - \frac{5}{2}\kappa (\nabla_x \theta^b)^2\  dxds ,
    \end{aligned}
\end{itemize}
(it shows that the "negative terms will be controlled by the entropy dissipation").
\begin{itemize}
\item Finally, in the part labeled
\begin{aligned}
        R_{12}= \int_0^t \int_{\mathbb{T}^3_x} \left( \langle v \mdot \nabla_x \Loth g_{\varepsilon} , \hat{A}\rangle + \langle \varepsilon \partial_t g_{\varepsilon} , \hat{A}\rangle\right) :\sigma(u^b) \ dxds ,\\
        R_{13}=\int_0^t \int_{\mathbb{T}^3_x}  \left(\langle v \mdot \nabla_x \Loth g_{\varepsilon} , \hat{B}\rangle + \langle \varepsilon \partial_t g_{\varepsilon} , \hat{B}\rangle\right) \cdot \nabla_x \theta^b \ dxds.
    \end{aligned}
\end{equation}

\subsection{Conclusion} \label{SS_Conclusion}
We now complete the proof of Theorem \ref{main}.
Note that
\begin{equation}\label{quadratic_term}
    \int_0^t \int_{\mathbb{T}^3_x} \frac{1}{2}\mu \sigma(\tilde{u}-u^b):\sigma(\tilde{u}-u^b) + \frac{5}{2}  \kappa(\nabla_x \tilde{\theta} -\nabla_x \theta^b)^2 \ dxds \geqslant 0.
\end{equation}
Combining with \eqref{mid_result} and \eqref{dissipation_control_final_result} we have
\begin{equation}
    \begin{aligned}
        \frac{1}{{\varepsilon}^2} &H(f_{\varepsilon}| M_{\varepsilon})(t)  \\
        &\lesssim \frac{1}{\varepsilon^2} H(f^{\mathrm{in}}_{\varepsilon}| M^{\mathrm{in}}_{\varepsilon}) + \int^t_0 \|(\nabla_x \tilde{u},\nabla_x \tilde{\theta})\|_{L^{\infty}(dx)} \frac{1}{\varepsilon^2} H(f_{\varepsilon}|M_{\varepsilon}) ds \\
        & +\tilde{R}+ R_5+R_6+R_7 -R_{11}-R_{12} - 2R_{13}.
    \end{aligned}
\end{equation}

it concludes the bounds
\begin{equation}\label{main_estimate}
            \frac{1}{{\varepsilon}^2} H(f_{\varepsilon}| M_{\varepsilon})(t)  \leqslant \frac{1}{\varepsilon^2}H(f^{\mathrm{in}}_{\varepsilon}| M^{\mathrm{in}}_{\varepsilon} ) + \mathcal{O}(\varepsilon),
        \end{equation}
\begin{equation}\label{entropy_dissipation_bound}
           \int_0^t \int_{\mathbb{T}^3_x}\frac{1}{\varepsilon^4}D(f_\varepsilon)(s,x) - \frac{1}{2} \mu \sigma(\u^b_\varepsilon): \sigma(\u^b_\varepsilon) - \frac{5}{2}\kappa (\nabla_x \theta^b_{\varepsilon})^2\  dxds \lesssim \frac{1}{\varepsilon^2} H(f_{\varepsilon}^{\mathrm{in}}|M_{\varepsilon}^{\mathrm{in}}) + \mathcal{O}(\varepsilon),
         \end{equation}
\begin{equation}\label{energy_momentum_flux_bound}
             \int_0^t \int_{\mathbb{T}^3_x} \frac{1}{2}\mu \sigma(\tilde{\u}_{\varepsilon}-\u^b_{\varepsilon}):\sigma(\tilde{\u}_{\varepsilon}-\u^b_{\varepsilon}) + \frac{5}{2}  \kappa(\nabla_x \tilde{\theta} -\nabla_x \theta^b_{\varepsilon})^2 \ dxds \lesssim \frac{1}{\varepsilon^2} H(f_{\varepsilon}^{\mathrm{in}}|M_{\varepsilon}^{\mathrm{in}}) + \mathcal{O}(\varepsilon),
        \end{equation}
"by Gr\"onwall's inequality."
\end{itemize}
It is also remarked that the argument is "in the formal level" but "provide[s] a clear way to the later rigorous proof," and that the validity of the $\mathcal{O}(\varepsilon)$ residual term "relies on uniform in $\varepsilon$ estimates for the solutions" (with cited sources providing global existence and uniform bounds in the perturbative framework).

\label{main}
        Let $f_{\varepsilon}$  be of the form $f_{\varepsilon} = M + \varepsilon M g_{\varepsilon} $, and $\{f_{\varepsilon}\}$ is a family of solutions to
        \begin{equation}\label{longtime_Boltzmann} \tag{BE}
    \left\{\begin{aligned}
        &\tau_{\varepsilon} \partial_t f_{\varepsilon} + v\mdot \nabla_x f_{\varepsilon} =\frac{1}{\varepsilon} C(f_{\varepsilon},\ f_{\varepsilon}), \quad x\in \mathbb{T}^3, \quad v \in \mathbb{R}^3\\
        &f_{\varepsilon}(0,x) = f_{\varepsilon}^{\mathrm{in}} \geqslant 0,
    \end{aligned}\right.
\end{equation}
        with $\tau_\varepsilon=\varepsilon$ satisfying the assumptions
        \begin{align}
            &\partial_t \int_{\mathbb{R}^3_v} f_{\varepsilon} dv + \nabla_x \cdot  \int_{\mathbb{R}^3_v} f_{\varepsilon}dv = 0, \label{conservation_of_mass} \\
            &\partial_t \int_{\mathbb{R}^3_v} vf_{\varepsilon} dv + \nabla_x \cdot \int_{\mathbb{R}^3_v} vf_{\varepsilon}dv =0,\label{conservation_of_momentum}\\
            &\partial_t \int_{\mathbb{R}^3_v} |v|^2 f_{\varepsilon} dv + \nabla_x \cdot \int_{\mathbb{R}^3_v} |v|^2f_{\varepsilon}dv=0.\label{conservation_of_energy}\\
        \end{align}
        and
        \begin{equation}\label{entropy_inequality}
        H(f_{\varepsilon}| M)(t) + \frac{1}{\varepsilon^2}\int_0^t \int_{\mathbb{T}^3_x} D(f_\varepsilon)(s,x) \ ds dx \leqslant H(f^{\mathrm{in}}_{\varepsilon}| M ),
    \end{equation}
        Let $(\rho_{\varepsilon}, u_{\varepsilon},\theta_{\varepsilon})$ be of the form $(\rho_{\varepsilon},u_{\varepsilon},\theta_{\varepsilon}) = (1+ \tilde{\rho}_{\varepsilon},\varepsilon \tilde{u}_{\varepsilon},1+ \varepsilon \tilde{\theta}_{\varepsilon})$, and $\{(\rho_{\varepsilon},u_{\varepsilon},\theta_{\varepsilon})\}$ is a family of solutions to
        \begin{equation} \label{longtime_CNS}\tag{$\text{CNS}_{\varepsilon}$}
        \left\{ \begin{aligned}
            &\varepsilon\partial_t \rho_{\varepsilon} + \nabla_x (\rho_{\varepsilon} u_{\varepsilon}) = 0,\\
            &\varepsilon\partial_t u_{\varepsilon} + u_{\varepsilon} \mdot \nabla_x u_{\varepsilon}  + \frac{1}{\rho_{\varepsilon}} \nabla_x (\rho_{\varepsilon} \theta_{\varepsilon})  = \varepsilon \frac{1}{\rho_{\varepsilon}} \nabla_x [ \mu_{\varepsilon}(\rho_{\varepsilon},\theta_{\varepsilon}) \sigma(u_{\varepsilon})],\\
            &\varepsilon \partial_t \theta_{\varepsilon}  + u_{\varepsilon} \mdot \nabla_x \theta_{\varepsilon} + \frac{2}{3} \theta_{\varepsilon}  \nabla_x \mdot u_{\varepsilon} = \varepsilon  \frac{1}{\rho_{\varepsilon}} \frac{1}{3} \mu_{\varepsilon}(\rho_{\varepsilon},\theta_{\varepsilon}) \sigma(u_{\varepsilon}):\sigma(u_{\varepsilon}) +  \frac{1}{\rho_{\varepsilon}}\frac{5}{3} \varepsilon \nabla_x \cdot[\kappa(\rho_{\varepsilon},\theta_{\varepsilon}) \theta_{\varepsilon}],\\
            &(\rho_{\varepsilon}(0,x),\ u_{\varepsilon}(0,x), \ \theta_{\varepsilon}(0,x)) = (1+ \varepsilon \tilde{\rho}^{\mathrm{in}},\ \tilde{u}^{\mathrm{in}},\ 1+ \varepsilon \tilde{\theta}^{\mathrm{in}}).
        \end{aligned}\right.
    \end{equation}
        Then we have, for $t \geqslant 0$,
        \begin{equation}\label{main_estimate}
            \frac{1}{{\varepsilon}^2} H(f_{\varepsilon}| M_{\varepsilon})(t)  \leqslant \frac{1}{\varepsilon^2}H(f^{\mathrm{in}}_{\varepsilon}| M^{\mathrm{in}}_{\varepsilon} ) + \mathcal{O}(\varepsilon),
        \end{equation}
        \begin{equation}\label{entropy_dissipation_bound}
           \int_0^t \int_{\mathbb{T}^3_x}\frac{1}{\varepsilon^4}D(f_\varepsilon)(s,x) - \frac{1}{2} \mu \sigma( \mathrm{u} ^b_\varepsilon): \sigma( \mathrm{u} ^b_\varepsilon) - \frac{5}{2}\kappa (\nabla_x \theta^b_{\varepsilon})^2\  dxds \lesssim \frac{1}{\varepsilon^2} H(f_{\varepsilon}^{\mathrm{in}}|M_{\varepsilon}^{\mathrm{in}}) + \mathcal{O}(\varepsilon),
         \end{equation}
         \begin{equation}\label{energy_momentum_flux_bound}
             \int_0^t \int_{\mathbb{T}^3_x} \frac{1}{2}\mu \sigma(\tilde{ \mathrm{u} }_{\varepsilon}- \mathrm{u} ^b_{\varepsilon}):\sigma(\tilde{ \mathrm{u} }_{\varepsilon}- \mathrm{u} ^b_{\varepsilon}) + \frac{5}{2}  \kappa(\nabla_x \tilde{\theta} -\nabla_x \theta^b_{\varepsilon})^2 \ dxds \lesssim \frac{1}{\varepsilon^2} H(f_{\varepsilon}^{\mathrm{in}}|M_{\varepsilon}^{\mathrm{in}}) + \mathcal{O}(\varepsilon),
        \end{equation}
        where
        \begin{equation}
            M_{\varepsilon}^{\mathrm{in}} = M(1+ \varepsilon \tilde\rho^{\mathrm{in}}, \varepsilon \tilde{u}^{\mathrm{in}},1 + \varepsilon \tilde{\theta}^{\mathrm{in}}).
        \end{equation}

Question:
Let $M$ be the reference Maxwellian, and suppose $f_{\varepsilon}=M+\varepsilon M g_{\varepsilon}$ is a family of nonnegative solutions of the rescaled Boltzmann equation on $\mathbb T^3_x\times\mathbb R^3_v$ with $\tau_{\varepsilon}=\varepsilon$,
\[
\varepsilon\,\partial_t f_{\varepsilon}+v\cdot\nabla_x f_{\varepsilon}=\frac1\varepsilon C(f_{\varepsilon},f_{\varepsilon}),\qquad f_{\varepsilon}(0,x)=f_{\varepsilon}^{\mathrm{in}},
\]
which satisfy the local conservation laws of mass, momentum, and energy, as well as the entropy inequality
\[
H(f_{\varepsilon}\mid M)(t)+\frac1{\varepsilon^2}\int_0^t\!\int_{\mathbb T^3_x} D(f_{\varepsilon})(s,x)\,ds\,dx\le H(f_{\varepsilon}^{\mathrm{in}}\mid M),
\]
where $H(\cdot\mid\cdot)$ is the relative entropy and $D(f_{\varepsilon})$ is the Boltzmann entropy dissipation. Suppose also that $(\rho_{\varepsilon},u_{\varepsilon},\theta_{\varepsilon})=(1+\tilde\rho_{\varepsilon},\varepsilon\tilde u_{\varepsilon},1+\varepsilon\tilde\theta_{\varepsilon})$ is a family of solutions of
\[
\left\{\begin{aligned}
&\varepsilon\partial_t \rho_{\varepsilon}+\nabla_x(\rho_{\varepsilon}u_{\varepsilon})=0,\\
&\varepsilon\partial_t u_{\varepsilon}+u_{\varepsilon}\cdot\nabla_xu_{\varepsilon}+\frac1{\rho_{\varepsilon}}\nabla_x(\rho_{\varepsilon}\theta_{\varepsilon})=\varepsilon\frac1{\rho_{\varepsilon}}\nabla_x[\mu_{\varepsilon}(\rho_{\varepsilon},\theta_{\varepsilon})\sigma(u_{\varepsilon})],\\
&\varepsilon\partial_t\theta_{\varepsilon}+u_{\varepsilon}\cdot\nabla_x\theta_{\varepsilon}+\frac23\theta_{\varepsilon}\nabla_x\cdot u_{\varepsilon}=\varepsilon\frac1{\rho_{\varepsilon}}\frac13\mu_{\varepsilon}(\rho_{\varepsilon},\theta_{\varepsilon})\sigma(u_{\varepsilon}):\sigma(u_{\varepsilon})+\frac1{\rho_{\varepsilon}}\frac53\varepsilon\nabla_x\cdot[\kappa(\rho_{\varepsilon},\theta_{\varepsilon})\theta_{\varepsilon}],\\
&(\rho_{\varepsilon}(0,x),u_{\varepsilon}(0,x),\theta_{\varepsilon}(0,x))=(1+\varepsilon\tilde\rho^{\mathrm{in}},\tilde u^{\mathrm{in}},1+\varepsilon\tilde\theta^{\mathrm{in}}),
\end{aligned}\right.
\]
and define the local Maxwellians
\[
M_{\varepsilon}=\mathcal M(\rho_{\varepsilon},u_{\varepsilon},\theta_{\varepsilon}),\qquad M_{\varepsilon}^{\mathrm{in}}=\mathcal M(1+\varepsilon\tilde\rho^{\mathrm{in}},\varepsilon\tilde u^{\mathrm{in}},1+\varepsilon\tilde\theta^{\mathrm{in}}).
\]
Let $(\rho^b_{\varepsilon},u^b_{\varepsilon},\theta^b_{\varepsilon})$ denote the density, momentum, and temperature moments associated with the perturbation $g_{\varepsilon}$. Which statement holds for every such pair of families and every $t\ge 0$?

For every $t\ge 0$, all three estimates hold:
\[
\frac1{\varepsilon^2}H(f_{\varepsilon}\mid M_{\varepsilon})(t)\le \frac1{\varepsilon^2}H(f_{\varepsilon}^{\mathrm{in}}\mid M_{\varepsilon}^{\mathrm{in}})+\mathcal O(\varepsilon),
\]
\[
\int_0^t\!\int_{\mathbb T^3_x}\left(\frac1{\varepsilon^4}D(f_{\varepsilon})(s,x)-\frac12\mu\,\sigma(u^b_{\varepsilon}):\sigma(u^b_{\varepsilon})-\frac52\kappa\,(\nabla_x\theta^b_{\varepsilon})^2\right)dx\,ds\lesssim \frac1{\varepsilon^2}H(f_{\varepsilon}^{\mathrm{in}}\mid M_{\varepsilon}^{\mathrm{in}})+\mathcal O(\varepsilon),
\]
\[
\int_0^t\!\int_{\mathbb T^3_x}\left(\frac12\mu\,\sigma(\tilde u_{\varepsilon}-u^b_{\varepsilon}):\sigma(\tilde u_{\varepsilon}-u^b_{\varepsilon})+\frac52\kappa\,(\nabla_x\tilde\theta_{\varepsilon}-\nabla_x\theta^b_{\varepsilon})^2\right)dx\,ds\lesssim \frac1{\varepsilon^2}H(f_{\varepsilon}^{\mathrm{in}}\mid M_{\varepsilon}^{\mathrm{in}})+\mathcal O(\varepsilon).
\]

Choices:
B.
For every $t\ge 0$, all three estimates hold with the same right-hand side but with the dissipation term scaled only by $\frac1{\varepsilon^2}$, namely
\[
\frac1{\varepsilon^2}H(f_{\varepsilon}\mid M_{\varepsilon})(t)\le \frac1{\varepsilon^2}H(f_{\varepsilon}^{\mathrm{in}}\mid M_{\varepsilon}^{\mathrm{in}})+\mathcal O(\varepsilon),
\]
\[
\int_0^t\!\int_{\mathbb T^3_x}\left(\frac1{\varepsilon^2}D(f_{\varepsilon})(s,x)-\frac12\mu\,\sigma(u^b_{\varepsilon}):\sigma(u^b_{\varepsilon})-\frac52\kappa\,(\nabla_x\theta^b_{\varepsilon})^2\right)dx\,ds\lesssim \frac1{\varepsilon^2}H(f_{\varepsilon}^{\mathrm{in}}\mid M_{\varepsilon}^{\mathrm{in}})+\mathcal O(\varepsilon),
\]
\[
\int_0^t\!\int_{\mathbb T^3_x}\left(\frac12\mu\,\sigma(\tilde u_{\varepsilon}-u^b_{\varepsilon}):\sigma(\tilde u_{\varepsilon}-u^b_{\varepsilon})+\frac52\kappa\,(\nabla_x\tilde\theta_{\varepsilon}-\nabla_x\theta^b_{\varepsilon})^2\right)dx\,ds\lesssim \frac1{\varepsilon^2}H(f_{\varepsilon}^{\mathrm{in}}\mid M_{\varepsilon}^{\mathrm{in}})+\mathcal O(\varepsilon).
\]
C.
For every $t\ge 0$, the relative entropy estimate holds:
\[
\frac1{\varepsilon^2}H(f_{\varepsilon}\mid M_{\varepsilon})(t)\le \frac1{\varepsilon^2}H(f_{\varepsilon}^{\mathrm{in}}\mid M_{\varepsilon}^{\mathrm{in}})+\mathcal O(\varepsilon).
\]
D.
For every $t\ge 0$, all three estimates hold with the same left-hand sides as above, but the error term can be taken uniformly negligible in the sense that
\[
\frac1{\varepsilon^2}H(f_{\varepsilon}\mid M_{\varepsilon})(t)\le \frac1{\varepsilon^2}H(f_{\varepsilon}^{\mathrm{in}}\mid M_{\varepsilon}^{\mathrm{in}}),
\]
\[
\int_0^t\!\int_{\mathbb T^3_x}\left(\frac1{\varepsilon^4}D(f_{\varepsilon})(s,x)-\frac12\mu\,\sigma(u^b_{\varepsilon}):\sigma(u^b_{\varepsilon})-\frac52\kappa\,(\nabla_x\theta^b_{\varepsilon})^2\right)dx\,ds\lesssim \frac1{\varepsilon^2}H(f_{\varepsilon}^{\mathrm{in}}\mid M_{\varepsilon}^{\mathrm{in}}),
\]
\[
\int_0^t\!\int_{\mathbb T^3_x}\left(\frac12\mu\,\sigma(\tilde u_{\varepsilon}-u^b_{\varepsilon}):\sigma(\tilde u_{\varepsilon}-u^b_{\varepsilon})+\frac52\kappa\,(\nabla_x\tilde\theta_{\varepsilon}-\nabla_x\theta^b_{\varepsilon})^2\right)dx\,ds\lesssim \frac1{\varepsilon^2}H(f_{\varepsilon}^{\mathrm{in}}\mid M_{\varepsilon}^{\mathrm{in}}).
\]
E.
For every $t\ge 0$, all three estimates hold after replacing the initial modulated entropy by the entropy relative to the fixed reference Maxwellian $M$, namely
\[
\frac1{\varepsilon^2}H(f_{\varepsilon}\mid M_{\varepsilon})(t)\le \frac1{\varepsilon^2}H(f_{\varepsilon}^{\mathrm{in}}\mid M)+\mathcal O(\varepsilon),
\]
\[
\int_0^t\!\int_{\mathbb T^3_x}\left(\frac1{\varepsilon^4}D(f_{\varepsilon})(s,x)-\frac12\mu\,\sigma(u^b_{\varepsilon}):\sigma(u^b_{\varepsilon})-\frac52\kappa\,(\nabla_x\theta^b_{\varepsilon})^2\right)dx\,ds\lesssim \frac1{\varepsilon^2}H(f_{\varepsilon}^{\mathrm{in}}\mid M)+\mathcal O(\varepsilon),
\]
\[
\int_0^t\!\int_{\mathbb T^3_x}\left(\frac12\mu\,\sigma(\tilde u_{\varepsilon}-u^b_{\varepsilon}):\sigma(\tilde u_{\varepsilon}-u^b_{\varepsilon})+\frac52\kappa\,(\nabla_x\tilde\theta_{\varepsilon}-\nabla_x\theta^b_{\varepsilon})^2\right)dx\,ds\lesssim \frac1{\varepsilon^2}H(f_{\varepsilon}^{\mathrm{in}}\mid M)+\mathcal O(\varepsilon).
\]

The stem does not explicitly reveal the correct choice. It gives the hypotheses and asks for the valid estimate, but the exact conclusion is not stated in the prompt itself.

This is essentially a theorem-recall item: the stem lists the assumptions of a specific result and asks for its conclusion. The correct option is a near-verbatim statement of that conclusion.

Some reasoning is needed to distinguish subtle variants involving the dissipation scaling, the reference Maxwellian, and the error term. However, the task is mostly recognition of the exact theorem statement rather than genuine generation of a conclusion.

The distractors are technically plausible and target realistic failure modes: incorrect epsilon scaling, replacing the sharp statement by a weaker true one, confusing the reference Maxwellian, and altering the remainder size.

Technically strong distractors, but the item is largely a direct theorem-restatement question rather than a high-quality generative reasoning MCQ.