\label{thMain}
There exist constants $\delta, \eps > 0$ and a sequence of finite groups $(G_n)_{n \geq 1}$ with $|G_n| \to \infty$
with bounded-size generating sets $X_n, Y_n \subseteq G_n$ such that
\begin{enumerate}[\normalfont(a)]
    \item $\op{Cay}(G_n,X_n)$ are $\eps$-expander graphs, and
    \item $\op{diam}( \op{Cay}(G_n,Y_n) ) \geq \exp(\delta \sqrt{\log|G_n|})$.
\end{enumerate}

Let $G$ be an infinite group generated by a finite set $X$.
Several important properties of the Cayley graph $\op{Cay}(G,X)$ do not depend on the choice of $X$,
and are actually group properties:
the growth type, the number of ends, hyperbolicity, amenability (via the F{\o}lner condition), and property (T).
The key observation behind these facts is that, for any other finite subset $Y$ of $G$,
there exists a constant $c$ such that every element in $Y$ can be written as the product of at most $c$ elements in $X$.

The situation is different for a sequence of finite groups.
Answering a question of Lubotzky--Weiss~\cite{LW93}, Alon, Lubotzky, and Wigderson~\cite{ALW01}
constructed finite groups whose Cayley graphs are expanders with respect to one bounded-size generating set but not with respect to another.
Specifically, they showed that the groups $C_2^{p+1} \rtimes \PSL_2(p)$ have this property.
In the seminal paper~\cite{Kas07}, Kassabov showed that the symmetric groups also have this property,
and subsequently Kassabov, Lubotzky, and Nikolov~\cite{KLN06} proved that almost all nonabelian finite simple groups are expanders with respect to suitable bounded-size generating sets.

However, all of the groups listed above likely have worst-case polylogarithmic diameter.
In the case of the finite simple groups, this is the content of the famous conjecture of Babai~\cite{BS92}*{Conjecture~1.7}.
This observation prompted Pyber and Szab\'o to ask the following question:

\begin{question}[Pyber \& Szab\'o~\cite{PS13}*{Question~23}]
 \label{quesPS}
 Let $(G_n)_{n \geq 1}$ be a sequence of finite groups and suppose that
 $\op{Cay}(G_n,X_n)$ are expander graphs for some sets $(X_n)_{n \geq 1}$ of bounded cardinality.
 Is it true that for every sequence of generating sets $(Y_n)_{n \geq 1}$ we have
 $$ \op{diam}( \op{Cay}(G_n,Y_n) ) \> \le \> C (\log |G_n|)^C $$
 for some constant $C$?
\end{question}

A similar question was asked by the second author at the 2024 Oberwolfach meeting \emph{Growth and Expansion in Groups} \cite{BD24}*{p.~1028, Question 5}:
there it was asked whether it is enough to assume that $\op{diam}(\op{Cay}(G_n, X_n))$ is polylogarithmic, i.e., whether polylogarithmic diameter is a group property.

In this note we answer both questions negatively. In fact we show that the diameter of $G$ with respect to the second generating set can be as large as $\exp(c \sqrt{\log |G|})$.

\begin{abstract}
 We study how the spectral gap and diameter of Cayley graphs depend strongly on the choice of generating set.
 We answer a question of Pyber and Szab\'o (2013)
 by exhibiting a sequence of finite groups $G_n$ with $|G_n| \to \infty$ admitting bounded generating sets $X_n,Y_n$ such that $\operatorname{Cay}(G_n,X_n)$ is an expander
 while $\operatorname{Cay}(G_n,Y_n)$ has super-polylogarithmic diameter.
 The construction uses the semidirect product $G_n = C_p^{n-1} \rtimes S_n$ with $p$ exponentially large in $n$,
 and the analysis reduces to bounding some exponential sums of permutational type.
\end{abstract}

\begin{question}[Pyber \& Szab\'o~\cite{PS13}*{Question~23}]
 \label{quesPS}
 Let $(G_n)_{n \geq 1}$ be a sequence of finite groups and suppose that
 $\Cay(G_n,X_n)$ are expander graphs for some sets $(X_n)_{n \geq 1}$ of bounded cardinality.
 Is it true that for every sequence of generating sets $(Y_n)_{n \geq 1}$ we have
 $$ \diam( \Cay(G_n,Y_n) ) \> \le \> C (\log |G_n|)^C $$
 for some constant $C$?
\end{question}

In this note we answer both questions negatively. In fact we show that the diameter of $G$ with respect to the second generating set can be as large as $\exp(c \sqrt{\log |G|})$.

By contrast having polynomially large diameter \emph{is} a group property,
and such groups are called \emph{almost flat}: see \cite{BT16}*{Theorem~4.1, Corollary~4.16}.
In particular, if (a) holds then $\diam(\Cay(G_n, Y_n)) \le |G_n|^{o(1)}$,
and it is an interesting open question to find the optimal bound in (b).

\begin{theorem} \label{thExpSum}
 There exist constants $\delta, \eps >0$ such that the following holds.
 For every integer $n \ge 1$ and prime $p \le e^{\delta n}$,
 there exists $v \in V_0$ such that, for all nonconstant $w \in V$,
 \[
 \left| \frac1{n!} \sum_{\sigma \in S_n} e_p\br{\gen{v, w^\sigma}} \right|
 \> \le \> 1 - \eps.
 \]
\end{theorem}

\begin{lemma}
 \label{lem:kaz-basics}
 Let $G$ be a finite group and let $S, T \subseteq G$.
 \begin{enumerate}
 \item $\kaz(G, S) \le \kaz(G, T)$ if $S \subseteq T$;
 \item $\kaz(G, S) \le 2$;
 \item $\kaz(G, G) \ge \sqrt 2$;
 \item $\kaz(G, S) \ge \frac1n \kaz(G, S^n)$.
 \end{enumerate}
\end{lemma}
\begin{proof}
 (1)--(2) are clear. For (3)--(4) see \cite{Kas07}*{Propositions~1.3--1.4}.
\end{proof}

\begin{lemma} \label{lemSean1}
 Let $\pi \colon G \to U(V)$ and let $\xi \in V$ be $\eps$-almost $S$-invariant.
 Assume $\kaz(G, S) > 0$ and let $\eps' = \eps / \kaz(G, S)$.
 Then
 \begin{enumerate}[(i)]
 \item there is a $G$-invariant vector $\xi_1 \in V$ such that $\|\xi - \xi_1\| < \eps'\|\xi\|$;
 \item $\xi$ is $2\eps'$-almost $G$-invariant.
 \end{enumerate}
\end{lemma}
\begin{proof}
 \emph{(i)}
 Orthogonally decompose $\xi = \xi_1 + \xi_2$ where $\xi_1 \in V^G$ and $\xi_2 \in (V^G)^\perp$.
 Since $((V^G)^\perp)^G = 0$, there is some $s \in S$ such that
 \[
 \kaz(G, S) \|\xi_2\| \le \|\pi(s) \xi_2 - \xi_2\| = \|\pi(s) \xi - \xi\| < \eps \|\xi\|.
 \]
 Hence $\|\xi_2\| < \eps' \|\xi\|$.

We now prove Theorem~\ref{thMain}(a) assuming Theorem~\ref{thExpSum} (which will be proved in the next section).
Let $V= \F_p^n$ and let $V_0 < V$ be the deleted permutation module of dimension $n-1$.
Let $G = V_0 \rtimes S_n$ and $v \in V_0$.
By Kassabov~\cite{Kas07} there is a bounded-size generating set $T \subseteq S_n$ such that $\kaz(S_n, T)$ is bounded away from zero.
If $X = \{v\} \cup T$,
then by Proposition~\ref{prop:semidirect} and Lemma~\ref{lemKazSG} we have
\begin{align*}
 \kaz(G, X)
 &\ge \frac{\sqrt 2}{48} \kaz(V_0, v^{S_n}) \kaz(S_n, T)
 \\
 &\ge \frac{1}{24} \gap(V_0, v^{S_n})^{1/2} \kaz(S_n, T).
 \end{align*}
We now focus on $\gap(V_0, v^{S_n})$.
Observe that $\Cay(V,v^{S_n})$ is the disjoint union of $p$ graphs all isomorphic to $\Cay(V_0,v^{S_n})$.
It follows that the eigenvalues of the adjacency operator of $\Cay(V_0, v^{S_n})$ are the same as for
$\Cay(V, v^{S_n})$, the only difference being their multiplicities.
Since $V$ is abelian, its regular representation is isomorphic to a direct sum of one-dimensional representations
indexed by the irreducible complex characters.
We can identify these characters with vectors $w \in V$, using the formula
\[
 \chi_w(u) = e_p(\gen{u, w}) \qquad (u,w \in V).
\]
(Recall that $e_p(x) := \exp(2 \pi i x / p)$.)
It follows that the eigenvalues of $\Cay(V,v^{S_n})$ and so of $\Cay(V_0,v^{S_n})$ are given by
\[
 \frac1{2|S_n|} \sum_{\sigma \in S_n} \br{\chi_w(v^\sigma) + \chi_w(-v^{\sigma})} \hspace{1cm} (w \in V) .
\]
These are precisely the real values $\{ \Re[\lambda_{v,w}] \}_{w \in V}$, where
\[
 \lambda_{v,w} \> := \>
 \frac1{n!} \sum_{\sigma \in S_n} e_p (\gen{v, w^\sigma}) .
\]
The trivial eigenvalues are associated to the constant vectors, and so
$\gap(V_0,v^{S_n}) = 1 - \max_{w \in V \setminus \gen \one} \Re[\lambda_{v,w}]$.
 Since $\Re[\lambda_{v,w}] \le |\lambda_{v,w}|$, the reduction follows.

\begin{theorem} \label{thExpSum}
    There exist constants $\delta, \eps >0$ such that the following holds.
    For every integer $n \ge 1$ and prime $p \le e^{\delta n}$,
    there exists $v \in V_0$ such that, for all nonconstant $w \in V$,
    \[
       \left| \frac1{n!} \sum_{\sigma \in S_n} e_p\br{\gen{v, w^\sigma}} \right|
        \> \le \> 1 - \eps.
    \]
\end{theorem}

\begin{theorem} \label{thMain}
There exist constants $\delta, \eps > 0$ and a sequence of finite groups $(G_n)_{n \geq 1}$ with $|G_n| \to \infty$
with bounded-size generating sets $X_n, Y_n \subseteq G_n$ such that
\begin{enumerate}[\normalfont(a)]
    \item $\Cay(G_n,X_n)$ are $\eps$-expander graphs, and
    \item $\diam( \Cay(G_n,Y_n) ) \geq \exp(\delta \sqrt{\log|G_n|})$.
\end{enumerate}
\end{theorem}

By contrast having polynomially large diameter \emph{is} a group property,
and such groups are called \emph{almost flat}: see \cite{BT16}*{Theorem~4.1, Corollary~4.16}.
In particular, if (a) holds then $\op{diam}(\op{Cay}(G_n, Y_n)) \le |G_n|^{o(1)}$,
and it is an interesting open question to find the optimal bound in (b).

We now describe the construction.
Notably, there are elements in common with both Alon--Lubotzky--Wigderson and Kassabov.
From now on we suppress the subscript $n$, which will be clear from context.
The group is just
\[G \> = \> V_0 \rtimes S_n,\]
where $V_0 < V = \F_p^n$ is the deleted permutation module of dimension $n-1$,
and $p \sim e^{cn}$ is a large prime.
Note that
\[
    \> |G| \> = \> p^{n-1} n! \> = \> \exp\br{cn^2 + O(n \log n)} .
\]
The two generating sets $X$ and $Y$ both have the form $\{v\} \cup T$ where $v \in V_0$ and $T \subset S_n$.
For $Y$ we make the unimaginative choice
\[
    v = (1, -1, 0, \dots, 0), \qquad T = \{(1, 2), (1, 2, \dots, n) \}.
\]
Clearly $\op{diam}(\op{Cay}(G, Y)) \ge \floor{pn/2} \ge p \sim e^{cn}$, so (b) holds.

The nontrivial part of the proof is the demonstration that there is a choice of $v$ and $T$ such that $\op{Cay}(G, X)$ has a uniform spectral gap.
For $T$ we take a bounded-size expanding generating set for $S_n$ (which exists by the result of Kassabov).
By an analysis similar to that in \cite{ALW01},
the problem of finding $v \in V_0$ reduces to the following ``permutational exponential sum'' estimate.

For $v, w \in V = \F_p^n$, let $\gen {v,w} = \sum_{i=1}^n v_i w_i$.
Let $\mathbf{1} = (1, \dots, 1)$ denote the all-one vector, so that $V_0 = \mathbf{1}^\perp$,
and let $e_p(x) := \exp(2 \pi i x / p)$.

\begin{theorem} \label{thExpSum}
    There exist constants $\delta, \eps >0$ such that the following holds.
    For every integer $n \ge 1$ and prime $p \le e^{\delta n}$,
    there exists $v \in V_0$ such that, for all nonconstant $w \in V$,
    \[
       \left| \frac1{n!} \sum_{\sigma \in S_n} e_p\br{\langle{v, w^\sigma}\rangle} \right|
        \> \le \> 1 - \eps.
    \]
\end{theorem}

In fact we show that a random $v \in V_0$ works.
When $p$ is bounded, this can be proved by a straightforward modification of the analysis in \cite{ALW01}.
The main novelty in our result is that $p$ is allowed to be enormous:
in technical terms, the reason we can take $p$ exponential in $n$ is that we do not need a union bound over all vectors,
but only over support-one vectors, and we can then use a deterministic switching argument based on the Cauchy--Schwarz inequality to deal with all other vectors.

\begin{remark}
    We have chosen to present the simplest argument that answers Question~\ref{quesPS}, but
    a more elaborate analysis shows that Theorems~\ref{thMain} and \ref{thExpSum} are true for any constant $\delta > 0$ and some suitable $\eps = \eps(\delta) > 0$.
    Moreover, $\eps(\delta)$ can be taken to be a continuous monotonic function such that $\eps(\delta) \to 1$ as $\delta \to 0$.
    On the other hand, a standard ``rectification'' argument in additive combinatorics (see \cite{BLR98}*{Section~3} for example) shows that necessarily $\eps(\delta) \to 0$ as $\delta \to \infty$ in Theorem~\ref{thExpSum}.
\end{remark}

We “describe the construction” of the groups in Theorem~\ref{thMain} as follows. “The group is just”
\[G = V_0 \rtimes S_n,\]
where $V_0<\F_p^n$ is the deleted permutation module and “$p\sim e^{cn}$ is a large prime,” so $|G|=p^{n-1}n!=\exp(cn^2+O(n\log n))$. The generating sets “both have the form $\{v\}\cup T$ where $v\in V_0$ and $T\subset S_n$.”

For the large-diameter generating set $Y$, one takes
\[v=(1,-1,0,\dots,0),\qquad T=\{(1,2),(1,2,\dots,n)\},\]
and then “clearly $\operatorname{diam}(\operatorname{Cay}(G,Y))\ge \lfloor pn/2\rfloor\ge p\sim e^{cn}$, so (b) holds.”

“The nontrivial part of the proof” is to choose $v$ and $T$ so that $\operatorname{Cay}(G,X)$ has “a uniform spectral gap.” For $T$ one uses “a bounded-size expanding generating set for $S_n$ (which exists by the result of Kassabov).” Then, “by an analysis similar to that in \cite{ALW01}, the problem of finding $v\in V_0$ reduces to” the exponential-sum estimate stated as Theorem~\ref{thExpSum}. The text adds that “in fact we show that a random $v\in V_0$ works.” The key point enabling $p$ to be “enormous” (exponential in $n$) is that “we do not need a union bound over all vectors, but only over support-one vectors, and we can then use a deterministic switching argument based on the Cauchy--Schwarz inequality to deal with all other vectors.”

We “describe the construction” of the groups in establishing the main theorem as follows. “The group is just”
\[G = V_0 \rtimes S_n,\]
where $V_0<\F_p^n$ is the deleted permutation module and “$p\sim e^{cn}$ is a large prime,” so $|G|=p^{n-1}n!=\exp(cn^2+O(n\log n))$. The generating sets “both have the form $\{v\}\cup T$ where $v\in V_0$ and $T\subset S_n$.”

For the large-diameter generating set $Y$, one takes
\[v=(1,-1,0,\dots,0),\qquad T=\{(1,2),(1,2,\dots,n)\},\]
and then “clearly $\operatorname{diam}(\operatorname{Cay}(G,Y))\ge \lfloor pn/2\rfloor\ge p\sim e^{cn}$, so (b) holds.”

“The nontrivial part of the proof” is to choose $v$ and $T$ so that $\operatorname{Cay}(G,X)$ has “a uniform spectral gap.” For $T$ one uses “a bounded-size expanding generating set for $S_n$ (which exists by the result of Kassabov).” Then, “by an analysis similar to that in A. Alon, Y. Roichman, and N. Linial, \emph{The spectral gap of random Cayley graphs} (2001), the problem of finding $v\in V_0$ reduces to” the following theorem.

\begin{theorem} \label{thExpSum}
    There exist constants $\delta, \eps >0$ such that the following holds.
    For every integer $n \ge 1$ and prime $p \le e^{\delta n}$,
    there exists $v \in V_0$ such that, for all nonconstant $w \in V$,
    \[
       \left| \frac1{n!} \sum_{\sigma \in S_n} e_p\br{\gen{v, w^\sigma}} \right|
        \> \le \> 1 - \eps.
    \]
\end{theorem}

The text adds that “in fact we show that a random $v\in V_0$ works.” The key point enabling $p$ to be “enormous” (exponential in $n$) is that “we do not need a union bound over all vectors, but only over support-one vectors, and we can then use a deterministic switching argument based on the Cauchy--Schwarz inequality to deal with all other vectors.”

Question: Which statement holds? Here, for a finite group $G$ and a generating set $S \subseteq G$, $\operatorname{Cay}(G,S)$ denotes the Cayley graph of $G$ with respect to $S$, $\operatorname{diam}(\operatorname{Cay}(G,S))$ is its graph diameter, and saying that $\operatorname{Cay}(G,S)$ is an $\varepsilon$-expander means that the family has expansion uniformly bounded below by $\varepsilon>0$. Also, “bounded-size generating sets” means that the cardinalities of the generating sets are uniformly bounded independently of $n$.

There exist constants $\delta,\varepsilon>0$ and a sequence of finite groups $(G_n)_{n\ge 1}$ with $|G_n|\to\infty$, together with generating sets $X_n,Y_n\subseteq G_n$ of uniformly bounded size, such that for every $n\ge 1$, the Cayley graph $\operatorname{Cay}(G_n,X_n)$ is an $\varepsilon$-expander graph and $$\operatorname{diam}(\operatorname{Cay}(G_n,Y_n))\ge \exp\!\bigl(\delta\sqrt{\log|G_n|}\bigr).$$

Choices:
B. There exist constants $\delta,\varepsilon>0$ and a sequence of finite groups $(G_n)_{n\ge 1}$ with $|G_n|\to\infty$, together with generating sets $X_n,Y_n\subseteq G_n$ of uniformly bounded size, such that for every $n\ge 1$, both Cayley graphs $\operatorname{Cay}(G_n,X_n)$ and $\operatorname{Cay}(G_n,Y_n)$ are $\varepsilon$-expander graphs and $$\operatorname{diam}(\operatorname{Cay}(G_n,Y_n))\ge \exp\!\bigl(\delta\sqrt{\log|G_n|}\bigr).$$
C. There exist constants $\delta,\varepsilon>0$ and a sequence of finite groups $(G_n)_{n\ge 1}$ with $|G_n|\to\infty$, together with generating sets $X_n,Y_n\subseteq G_n$ of uniformly bounded size, such that for every $n\ge 1$, the Cayley graph $\operatorname{Cay}(G_n,X_n)$ is an $\varepsilon$-expander graph and $$\operatorname{diam}(\operatorname{Cay}(G_n,Y_n))\ge (\log|G_n|)^{\delta}.$$
D. There exist constants $\delta,\varepsilon>0$ such that for every sequence of finite groups $(G_n)_{n\ge 1}$ with $|G_n|\to\infty$ and every choice of uniformly bounded-size generating sets $X_n,Y_n\subseteq G_n$, if the Cayley graphs $\operatorname{Cay}(G_n,X_n)$ are $\varepsilon$-expander graphs, then one has $$\operatorname{diam}(\operatorname{Cay}(G_n,Y_n))\ge \exp\!\bigl(\delta\sqrt{\log|G_n|}\bigr)$$ for every $n\ge 1$.
E. There exist constants $\delta,\varepsilon>0$ and a sequence of finite groups $(G_n)_{n\ge 1}$ with $|G_n|\to\infty$, together with generating sets $X_n,Y_n\subseteq G_n$ of uniformly bounded size, such that for every $n\ge 1$, the Cayley graph $\operatorname{Cay}(G_n,X_n)$ is an $\varepsilon$-expander graph and $$\operatorname{diam}(\operatorname{Cay}(G_n,Y_n))\ge \exp\!\bigl(\delta\log|G_n|\bigr).$$

separation between expanding set and large-diameter set

existential choice of groups and generating sets

size of the diameter lower bound relative to $|G_n|$

The stem only supplies definitions and notation. It does not explicitly or implicitly signal which of the closely related existence statements is correct.

The item is not a bare restatement of a theorem in the stem. The choices differ in quantifiers, strength of the diameter bound, and whether both generating sets must expand, so the student must discriminate among genuinely competing conclusions.

The question requires some reasoning about logical strength and asymptotic bounds, but it mainly tests precise recognition/recall of a theorem statement rather than generating a conclusion from given premises.

The distractors are mathematically plausible and target common errors: strengthening the conclusion too far, weakening it, changing existential to universal quantifiers, or incorrectly imposing expansion on both generating sets.

A strong MCQ with no answer leakage and high-quality distractors. Its main limitation is that it leans more toward theorem recognition than fully generative mathematical reasoning.


\begin{definition}\label{defcomNS}
    For a given $T>0$, we call $(\rho, u)$ a weak solution on $[0, T]$ to the constant viscosity system \eqref{eq:comNS}\eqref{con:initial} if:
    \begin{itemize}
        \item The momentum equations \eqref{eq:comNS} hold in $\mathcal{D}^{\prime}((0, T) \times \Omega)$ with the regularity:
        \begin{equation}\label{con:defcomNS}
            \rho^\gamma, \rho|u|^2 \in L^{\infty}\left(0, T ; L^1(\Omega)\right), \quad u \in L^2\left(0, T ; H_0^1(\Omega)\right);
        \end{equation}

        \item The initial conditions \eqref{con:initial} hold in $\mathcal{D}^{\prime}(\Omega)$;

        \item $(\rho, u)$ is a renormalized solution of the continuity equation in the sense of \cite{diperna1989ordinary};

        \item The energy inequality holds for almost every $t \in [0,T]$:
        $$
        \begin{aligned}
            \int_{\Omega}\left(\frac{1}{2} \rho|u|^2+\frac{\rho^\gamma}{\gamma-1}\right) \mathrm{d} x & +\int_0^t \int_{\Omega}\left(\mu|\nabla u|^2+(\mu+\lambda)(\operatorname{div} u)^2\right) \mathrm{d} x \mathrm{d} s \\
            & \leq \int_{\Omega}\left(\frac{1}{2} \rho_0\left|u_0\right|^2+\frac{\rho_0^\gamma}{\gamma-1}\right) \mathrm{d} x.
        \end{aligned}
        $$
    \end{itemize}
\end{definition}

The compressible Navier-Stokes equations are a fundamental model in fluid dynamics, describing the motion of a viscous compressible fluid. In this paper, we mainly study the energy conservation of the following system in a bounded domain $\Omega \subset \mathbb{R}^N$ ($N = 2, 3$):\begin{equation}\label{eq:comNS}
    \begin{split}
        (\rho u)_t + \text{div}(\rho u \otimes u) - \mu \Delta u - (\mu+\lambda) \nabla \text{div} \, u + \nabla P &= 0,  \\
        \rho_t + \text{div}(\rho u) &= 0,
    \end{split}
\end{equation}
with initial data
\begin{equation}
    \rho|_{t=0} = \rho_0(x), \quad \rho u|_{t=0} = \rho_0 u_0, \label{con:initial}
\end{equation}
where $P = \rho^\gamma$, $\gamma > 1$, is the pressure, $\rho$ is the density, $u$ is the velocity, and the viscosity coefficients satisfy $\mu > 0$, $2\mu + N\lambda \geq 0$. Here we define $u_0 = 0$ on the set $\{x|\rho_0(x) = 0\}$. However, the weak solutions of these equations do not, in general, satisfy the energy equalities, i.e.
\begin{equation}
    \begin{aligned}
    	&\int_{\Omega}\left(\frac{1}{2} \rho_0\left|u_0\right|^2+\frac{\rho_0^\gamma}{\gamma-1}\right) \mathrm{d} x-\int_{\Omega}\left(\frac{1}{2} \rho|u|^2+\frac{\rho^\gamma}{\gamma-1}\right) \mathrm{d} x\\
    	=&\int_0^T \int_{\Omega}\left( \mu|\nabla u|^2+(\mu+\lambda)|\operatorname{div} u|^2 \right)\mathrm{d} x \mathrm{d} t. \label{eq:energy_comNS}
    \end{aligned}
\end{equation}
This phenomenon, along with incompressible cases, has drawn attention and sparked interest in exploring the relationship between energy conservation and the regularity of weak solutions \cite{buckmaster2016dissipative,constantin1994onsager,feireisl2017regularity,isett2018proof}.

In fact, this subject originated from the famous Onsager’s conjecture for incompressible Euler flows \cite{onsager1949statistical}. This conjecture states that 1/3 should be the threshold of Hölder continuous exponent for energy conservation. Consequently, \cite{constantin1994onsager,eyink1994energy,cheskidov2008energy} obtained the energy equality with Besov regularity greater than 1/3 and \cite{buckmaster2015anomalous,buckmaster2016dissipative,isett2018proof} constructed the non-conservation solutions with Hölder continuity smaller than 1/3. These significant advances almost reached the threshold.

In the context of the Navier–Stokes equations, the pioneering studies were done by Lions \cite{lions1960regularite} and Serrin \cite{serrininitial}. Lions proved that a weak solution $u$ to a incompressible fluids conserves its energy provided $u \in L^{4}_{t,x}$ and Serrin gave a dimension-dependent condition
\begin{equation}\label{con:serrin}
    u \in L^{p}_{t}L^{q}_{x}\quad\mbox{for}\quad \frac{2}{p}+\frac{N}{q}\leq 1,\quad q > N.
\end{equation}
Later, Shinbrot removed the dimensional dependence in \cite{shinbrot1974energy} and proved the same conclusion if
\begin{equation}\label{con:shinbrot}
    \frac{2}{p}+\frac{2}{q}\leq 1,\quad q\geq 4,
\end{equation}
which improved the previous work. Meanwhile, for $3\leq q<4$, the following criteria
\begin{equation}\label{con:shinbrot2}
    \frac{1}{p}+\frac{3}{q}\leq 1
\end{equation}
could also yield energy conservation due to the embedding relationship \cite{da2020shinbrot}. Furthermore, a range of new types of conditions have been obtained recently, see \cite{drivas2019onsager,cheskidov2020energy,leslie2018conditions}.

As for the compressible systems, corresponding results are more recent and the analysis becomes inherently more complex. Within the framework of Onsager's theory \cite{onsager1949statistical}, Drivas and Eyink \cite{drivas2018onsager1,eyink2018cascades} adapted methods from the incompressible fluids \cite{constantin1994onsager,duchon2000inertial,eyink1994energy} to the compressible Euler equations and obtain necessary conditions for the energy dissipative anomalies of turbulent solutions. Meanwhile, sufficient conditions for energy conservation in terms of Besov regularity for the compressible Euler fluids with a $C^{\alpha}$ ($1<\alpha\leq2$) pressure law $p(\rho)$ were later established by Feireisl et al. \cite{feireisl2017regularity} and Akramov et al. \cite{akramov2020energy}.

Regarding the energy equality of compressible Navier-Stokes \eqref{eq:comNS}, the pioneering breakthrough was made by Yu \cite{yu2017energy}. He proved the equalities \eqref{eq:energy_comNS} and \eqref{eq:energy_comNSdeg} with the conditions that the weak solution $(\rho,u)$ satisfies
\begin{equation}\label{con:Yu}
    u \in L^{p}_{t}(0,T;L^{q}_{x}(\mathbb{T}^{N}))\quad\mbox{for}\quad \frac{1}{p}+\frac{1}{q}\leq \frac{5}{12},\quad q\geq 6
\end{equation}
and
\begin{equation}\label{con:genhao rou}
    \sqrt{\rho} \in L^\infty (0, T; H^1(\mathbb{T}^{N})).
\end{equation}
Later, Chen et al. \cite{chen2020energy} treated the  boundary effect to investigate energy conservation on a more general bounded domain with no-slip boundary condition, when
\begin{equation}\label{con:Chen et al.}
    u \in L^{p}_{t}(0,T;L^{q}_{x}(\Omega))\quad\mbox{for}\quad p\geq4,\quad q\geq 6,
\end{equation}
which is just the end point of \eqref{con:Yu}. Particularly, Ye et al. \cite{ye2022energy} have reached $L^{4}L^{4}$ criteria for the compressible Navier-Stokes equations, while they provided that $\nabla\sqrt{\rho}\in L^{4}L^{4}$ instead of $L^{\infty}L^{2}$. However, the $L^{\infty}H^{1}$ regularity of $\sqrt{\rho}$ is of independent significance, as the Bresch-Desjardins entropy for \eqref{eq:comNSdeg} implies such an estimate \cite{vasseur2016existence}. In conclusion, a gap persists between these results and the Shinbrot criteria, due to the complexity of the compressible system.

\begin{figure}[htbp]
  \centering
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        xlabel={$1/p$},
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  \caption{The shaded region illustrates the exponent ranges covered by Theorem \ref{theoremcomNS}.The dashed and grid subregions represent new ranges of exponents not obtained before in the literature, where the grid area only corresponds to bounded domain with no-slip boundary and the dashed area corresponds to both bounded domain and torus.}
  \label{fig:embedding-diagram}
\end{figure}

In fact, temporal derivative term $\partial_{t}(\rho u)$ is the main challenges in filling this gap. It requests a spatiotemporal mollification of the equation, not a pure spatial mollification of the incompressible system. Then, new errors associated to temporal mollification have emerged. In the proofs of \cite{chen2020energy,ye2022energy,yu2017energy}, the energy flux from $\partial_{t}(\rho u)$ was handled by a commutator estimate providing the integrability of $\partial_{t}\rho$. Particularly, it is the mass equation which allows us to 'transform' the spatial regularity of $u$ into the temporal regularity of $\rho$. However, this strategy meets some restriction. One need to assume that the velocity $u$ complies with the Serrin type criteria at least, not the Lions or Shinbrot type criteria.

\subsection{Methodology}

The present paper aims to bridge the gap between known research (\eqref{con:Yu} and \eqref{con:Chen et al.}) and the Shinbrot type criteria (\eqref{con:shinbrot} and \eqref{con:shinbrot2}) on a periodic domain $\mathbb{T}^{d}$ or a bounded domain with $C^{1}$ boundary. For this purpose, we shall reorganize the framework and introduce a new mollification and estimate. These approaches allow us to deal with the energy flux caused by the nonlinear terms $(\rho u)_t$ under lower integrability conditions. In the following paragraphs, we briefly describe the ideas of our method.

\emph{Framework and mollification.} The proof framework developed in this paper consists of two main steps: first, establishing the local energy equality, and then extending it to a global version. In the first step, we derive the local energy equality using the specific test function $(\varphi u_{\varepsilon}^{\varepsilon})_{\varepsilon}^{\varepsilon}$, introduced in Section 2, and passing to the limit as $\varepsilon \to 0$. By structuring the mollifier in this manner, we can separate errors present in different directions. In particular, we establish the vanishing of the temporal error via our weak-type temporal commutator estimate, which improves upon the classical treatment. Meanwhile, the other terms, such as $\text{div}(\rho u \otimes u)$, are rigorously re-examined in this step to ensure convergence under the weaker assumptions. In the second step, the result is extended to the whole domain by setting $\varphi=\psi_{\tau}\phi_{\delta}$ (with $\varphi=\psi_{\tau}$ for the torus $\mathbb{T}^{d}$) and letting $\tau, \delta \to 0$. The spatial cut-off function $\phi_{\delta}$ is constructed by the approaches used in \cite{chen2020energy} to control the regularity of the solution near the boundary. Particularly, the presence of a solid boundary complicates the dissipative mechanisms, as the behavior of the solution near the wall can differ significantly from its behavior in the interior. For further discussion on boundary effects in energy conservation contexts, we refer the reader to the related literature \cite{bardos2019onsager,bardos2019extension,drivas2018onsager}.

\emph{Weak-type commutator estimate.} Our main challenge is to justify the convergence of the commutator
$$
\partial_{t}(\rho u^{\varepsilon})_{\varepsilon}-\partial_{t}(\rho u_{\varepsilon}^{\varepsilon})
$$
under the Shinbrot-type condition on velocity. Compared with the previous work, these criteria entail a corresponding loss of regularity of $\partial_{t}\rho$, which is the key condition of the commutator estimate used in them. Since one usually employs the continuity equation:
$$
\partial_{t}\rho = - \text{div}(\rho u) = - (\rho \text{div} u + \nabla\rho\cdot u),
$$
to obtain integrability of $\partial_{t}\rho$, it may be difficult to derive the necessary $L_{t}^{p}L_{x}^{q}$ estimate of $\partial_{t}\rho$ under the assumption of the present paper. To overcome this obstacle, we prove Lemma \ref{commutator}, in which the convergence is established directly without establishing any $L^{p}$-type commutator estimate. Lemma \ref{commutator}, together with the viscous properties of the fluid, allows us to show that the error term vanishes even when $\partial_{t}\rho$ is only a distribution, which is ensured by the mass conservation.

\subsection{Main results}

Our results consist of the energy conservation criteria of two kinds of compressible Navier--Stokes systems: constant viscosities and degenerate viscosity. Here, we present our main results regarding the energy conservation for weak solutions to the equations with constant viscosities, whereas the remaining content, pertaining to degenerate viscosity cases, is outlined in the final section (Section \ref{sec:deg}). Throughout the present paper, let $\Omega$ be either the torus $\mathbb{T}^{N}$ ($N=2,3$) or an open, bounded domain in $\mathbb{R}^N$ with $C^1$ boundary $\partial \Omega$ with $u=0$ on $\partial\Omega$.

We first present the result for the Navier-Stokes equations \eqref{eq:comNS} with constant viscosity, where the presence of vacuum is allowed.

In the context of the Navier–Stokes equations, the pioneering studies were done by Lions \cite{lions1960regularite} and Serrin \cite{serrininitial}. Lions proved that a weak solution $u$ to a incompressible fluids conserves its energy provided $u \in L^{4}_{t,x}$ and Serrin gave a dimension-dependent condition
\begin{equation}\label{con:serrin}
 u \in L^{p}_{t}L^{q}_{x}\quad\mbox{for}\quad \frac{2}{p}+\frac{N}{q}\leq 1,\quad q > N.
\end{equation}
Later, Shinbrot removed the dimensional dependence in \cite{shinbrot1974energy} and proved the same conclusion if
\begin{equation}\label{con:shinbrot}
 \frac{2}{p}+\frac{2}{q}\leq 1,\quad q\geq 4,
\end{equation}
which improved the previous work. Meanwhile, for $3\leq q<4$, the following criteria
\begin{equation}\label{con:shinbrot2}
 \frac{1}{p}+\frac{3}{q}\leq 1
\end{equation}
could also yield energy conservation due to the embedding relationship \cite{da2020shinbrot}. Furthermore, a range of new types of conditions have been obtained recently, see \cite{drivas2019onsager,cheskidov2020energy,leslie2018conditions}.

Regarding the energy equality of compressible Navier-Stokes \eqref{eq:comNS}, the pioneering breakthrough was made by Yu \cite{yu2017energy}. He proved the equalities \eqref{eq:energy_comNS} and \eqref{eq:energy_comNSdeg} with the conditions that the weak solution $(\rho,u)$ satisfies
\begin{equation}\label{con:Yu}
 u \in L^{p}_{t}(0,T;L^{q}_{x}(\mathbb{T}^{N}))\quad\mbox{for}\quad \frac{1}{p}+\frac{1}{q}\leq \frac{5}{12},\quad q\geq 6
\end{equation}
and
\begin{equation}\label{con:genhao rou}
 \sqrt{\rho} \in L^\infty (0, T; H^1(\mathbb{T}^{N})).
\end{equation}
Later, Chen et al. \cite{chen2020energy} treated the boundary effect to investigate energy conservation on a more general bounded domain with no-slip boundary condition, when
\begin{equation}\label{con:Chen et al.}
 u \in L^{p}_{t}(0,T;L^{q}_{x}(\Omega))\quad\mbox{for}\quad p\geq4,\quad q\geq 6,
\end{equation}
which is just the end point of \eqref{con:Yu}. Particularly, Ye et al. \cite{ye2022energy} have reached $L^{4}L^{4}$ criteria for the compressible Navier-Stokes equations, while they provided that $\nabla\sqrt{\rho}\in L^{4}L^{4}$ instead of $L^{\infty}L^{2}$. However, the $L^{\infty}H^{1}$ regularity of $\sqrt{\rho}$ is of independent significance, as the Bresch-Desjardins entropy for \eqref{eq:comNSdeg} implies such an estimate \cite{vasseur2016existence}. In conclusion, a gap persists between these results and the Shinbrot criteria, due to the complexity of the compressible system.

\emph{Weak-type commutator estimate.} Our main challenge is to justify the convergence of the commutator
$$
\partial_{t}(\rho u^{\varepsilon})_{\varepsilon}-\partial_{t}(\rho u_{\varepsilon}^{\varepsilon})
$$
under the Shinbrot-type condition on velocity. Compared with the previous work, these criteria entail a corresponding loss of regularity of $\partial_{t}\rho$, which is the key condition of the commutator estimate used in them. Since one usually employs the continuity equation:
$$
\partial_{t}\rho = - \text{div}(\rho u) = - (\rho \text{div} u + \nabla\rho\cdot u),
$$
to obtain integrability of $\partial_{t}\rho$, it may be difficult to derive the necessary $L_{t}^{p}L_{x}^{q}$ estimate of $\partial_{t}\rho$ under the assumption of the present paper. To overcome this obstacle, we prove Lemma \ref{commutator}, in which the convergence is established directly without establishing any $L^{p}$-type commutator estimate. Lemma \ref{commutator}, together with the viscous properties of the fluid, allows us to show that the error term vanishes even when $\partial_{t}\rho$ is only a distribution, which is ensured by the mass conservation.

Our results consist of the energy conservation criteria of two kinds of compressible Navier--Stokes systems: constant viscosities and degenerate viscosity. Here, we present our main results regarding the energy conservation for weak solutions to the equations with constant viscosities, whereas the remaining content, pertaining to degenerate viscosity cases, is outlined in the final section (Section \ref{sec:deg}). Throughout the present paper, let $\Omega$ be either the torus $\mathbb{T}^{N}$ ($N=2,3$) or an open, bounded domain in $\mathbb{R}^N$ with $C^1$ boundary $\partial \Omega$ with $u=0$ on $\partial\Omega$.

We first present the result for the Navier-Stokes equations \eqref{eq:comNS} with constant viscosity, where the presence of vacuum is allowed.

Regarding the energy equality of compressible Navier-Stokes \eqref{eq:comNS}, the pioneering breakthrough was made by Yu \cite{yu2017energy}. He proved the equalities \eqref{eq:energy_comNS} and \eqref{eq:energy_comNSdeg} with the conditions that the weak solution $(\rho,u)$ satisfies
\begin{equation}\label{con:Yu}
 u \in L^{p}_{t}(0,T;L^{q}_{x}(\mathbb{T}^{N}))\quad\mbox{for}\quad \frac{1}{p}+\frac{1}{q}\leq \frac{5}{12},\quad q\geq 6
\end{equation}
and
\begin{equation}\label{con:genhao rou}
 \sqrt{\rho} \in L^\infty (0, T; H^1(\mathbb{T}^{N})).
\end{equation}
Later, Chen et al. \cite{chen2020energy} treated the boundary effect to investigate energy conservation on a more general bounded domain with no-slip boundary condition, when
\begin{equation}\label{con:Chen et al.}
 u \in L^{p}_{t}(0,T;L^{q}_{x}(\Omega))\quad\mbox{for}\quad p\geq4,\quad q\geq 6,
\end{equation}
which is just the end point of \eqref{con:Yu}. Particularly, Ye et al. \cite{ye2022energy} have reached $L^{4}L^{4}$ criteria for the compressible Navier-Stokes equations, while they provided that $\nabla\sqrt{\rho}\in L^{4}L^{4}$ instead of $L^{\infty}L^{2}$. However, the $L^{\infty}H^{1}$ regularity of $\sqrt{\rho}$ is of independent significance, as the Bresch-Desjardins entropy for \eqref{eq:comNSdeg} implies such an estimate \cite{vasseur2016existence}. In conclusion, a gap persists between these results and the Shinbrot criteria, due to the complexity of the compressible system.

We first present the result for the Navier-Stokes equations \eqref{eq:comNS} with constant viscosity, where the presence of vacuum is allowed.

The remainder of this paper is organized as follows. Section 2 introduces the necessary preliminaries, including definitions of weak solutions and key technical lemmas. Section 3 focuses on establishing the local energy conservation law in the sense of distributions using a mollification argument. In Section 4, we extend this local result to the global energy balance on the domain $\Omega$, completing the proof of Theorem \ref{theoremcomNS}. Finally, we apply our methods on the compressible Navier-Stokes equations with degenerate viscosity in the last Section 5.

It is easy to see, for any $\alpha \geq \frac{1}{2}$,
\begin{equation*}
 \partial_{t}(\rho^{\alpha}) = -\alpha \rho^{\alpha} \mathrm{div} u - 2\alpha \rho^{\alpha - \frac{1}{2}} u \cdot \nabla \sqrt{\rho},
\end{equation*}
which, together with \eqref{con:defcomNS} and \eqref{condi1-th2}, implies
\begin{equation*}
 \rho^{\alpha} \in L^{\infty}(0, T; W^{1,\frac{3}{2}}(\Omega)), \quad \partial_{t}(\rho^{\alpha}) \in L^{2}(0, T; L^{\frac{12}{11}}(\Omega)).
\end{equation*}
It follows that, by the Lemma \ref{aubinlions},
\begin{equation*}
 \rho^{\alpha} \in C([0, T]; L^{r}(\Omega)), \quad r < \frac{3N}{2N-3},
\end{equation*}
so we can use $u_{0} \in L^{\frac{6N}{6-N}}$ to deduce that
\begin{equation}\label{continuity_sqrt_rho_u}
 (\sqrt{\rho }u) (t) \rightarrow (\sqrt{\rho }u) (0) \quad \text{strongly in } L^2(\Omega) \text{ as } t \to 0^+.
\end{equation}
Following the similar manner of proof in \cite{yu2017energy}, we can complete the proof of Theorem \ref{theoremcomNS}.

\section{The energy conservation for the compressible Navier-Stokes equations with degenerate viscosity}\label{sec:deg}
Next, we consider the compressible Navier-Stokes equations with degenerate viscosity, where the density is strictly bounded away from vacuum. The systems are stated as follows:
 \begin{equation}\label{eq:comNSdeg}
 \begin{split}
 (\rho u)_t + \text{div}(\rho u \otimes u) - 2\nu \text{div}(\rho \mathbb{D} u) + \nabla P &= 0, \\
 \rho_t + \text{div}(\rho u) &= 0,
 \end{split}
 \end{equation}
 where $\mathbb{D} u = \frac{1}{2}(\nabla u + \nabla^T u)$ is the strain tensor and the viscosity coefficients satisfy $\nu > 0$. Then, we present the definition and the energy conservation criteria of the corresponding weak solutions.
 \begin{definition}\label{defcomNSdeg}
 The pair $(\rho, u)$ is called a global weak solution to the degenerate viscosity system \eqref{eq:comNSdeg} with initial data \eqref{con:initial} if, for any $t \in[0, T]$:
 \begin{itemize}
 \item The momentum equations \eqref{eq:comNSdeg} hold in $\mathcal{D}^{\prime}((0, T) \times \Omega)$ satisfying:
 $$
 \begin{gathered}
 \rho \geq 0, \quad \rho \in L^{\infty}\left(0, T ; L^\gamma(\Omega)\right),\quad \nabla \sqrt{\rho} \in L^{\infty}\left(0, T ; L^\frac{3}{2}(\Omega)\right)\\
 \sqrt{\rho} u \in L^{\infty}\left(0, T ; L^2(\Omega)\right), \quad \sqrt{\rho} \nabla u \in L^2\left(0, T ; L^2(\Omega)\right);
 \end{gathered}
 $$

\item The energy inequality holds for almost every $t \in [0,T]$:
 \begin{equation}\label{eq:energy_comNSdeg}
 \int_{\Omega}\left(\frac{1}{2} \rho|u|^2+\frac{\rho^\gamma}{\gamma-1}\right) \mathrm{d} x+\int_0^t \int_{\Omega} \rho|\mathbb{D} u|^2 \mathrm{d} x \mathrm{d} s \leq \int_{\Omega}\left(\frac{1}{2} \rho_0\left|u_0\right|^2+\frac{\rho_0^\gamma}{\gamma-1}\right) \mathrm{d} x.
 \end{equation}
 \end{itemize}
\end{definition}
\begin{remark}
 The condition $\nabla\sqrt{\rho}\in L^{\infty}L^{\frac{3}{2}}$ is reasonable in this context, as it is consistent with the known existence theory for weak solutions. Specifically, for the degenerate compressible Navier–Stokes equations, global existence results \cite{vasseur2016existence} ensure that $\nabla \rho^{\frac{\gamma}{2}}\in L^{2}L^{2}$ and $\nabla\sqrt{\rho}\in L^{\infty}L^{2}$.
\end{remark}

\begin{theorem}\label{theoremcomNSdeg}
 Let $(\rho, u)$ be a weak solution of \eqref{eq:comNSdeg} in the sense of Definition \ref{defcomNSdeg}. Assume that the density satisfies
 \begin{equation}\label{condi1-th2}
 0 < \underline{\rho} \leq \rho(t, x)\leq \bar{\rho} < \infty.
 \end{equation}
 and the initial condition satisfies
 \begin{equation}\label{condi2-th2}
 \sqrt{\rho_0} u_0 \in L^{\frac{4N}{N+2}}(\Omega).
 \end{equation}
 If the velocity satisfies
 \begin{equation}\label{con:shinbrot_cond2}
 u \in L^p\left(0, T ; L^q(\Omega)\right) \quad \text{with} \quad \begin{cases}
 \frac{1}{p}+\frac{3}{q} \leq 1,\quad\mbox{if} \quad 3\leq q < 4,\\
 \frac{2}{p}+\frac{2}{q} \leq 1,\quad\mbox{if} \quad 4\leq q \leq\infty,
 \end{cases}
 \end{equation}
 then the energy equality \eqref{eq:energy_comNSdeg} holds for any $t \in[0, T]$.
\end{theorem}
\begin{remark}
 Different from Theorem \ref{theoremcomNS}, we have obtained the energy conservation under the assumption $u\in L^{2}L^{\infty}$, which is one of the endpoints.
\end{remark}
Finally, let us provide the proof of Theorem \ref{theoremcomNSdeg}. We can modify the proof in Section 3 and 4 slightly to arrive the case $(p,q)=(4,4)$. Note that, for any weak solution ($\rho, u$), condition \eqref{condi1-th2} implies that

\begin{remark}
    Our result improves the known regularity criteria in the literature and closes a notable gap. We relax the stronger Serrin-type integrability required in prior works (e.g., \eqref{con:Yu} and \eqref{con:Chen et al.}) via a novel weak-type temporal commutator estimate (Lemma \ref{commutator}), which handles the nonlinear terms without demanding strong regularity of $\partial_t\rho$. The extended range of admissible exponents is visualized in {\rm Figure 1}. Particularly, we provide that $\nabla\sqrt{\rho}\in L^{\infty}L^{2}$ instead of $L^{\infty}L^{2}$, which is also weaker than \cite{yu2017energy,chen2020energy}.
\end{remark}
\begin{remark}
     It is worth noting that, due to the non‑separability of the space $L^{2}(0,T;L^{\infty}(\Omega))$, our method does not currently apply to the endpoint $(p,q)=(2,\infty)$; this remains for future investigation.
\end{remark}
\begin{remark}
    The approaches employed to validate this theorem are equally applicable to the compressible Navier-Stokes equations with degenerate viscosity, a fact we will demonstrate by proving Theorem \ref{theoremcomNSdeg} in Section 5.
\end{remark}

The remainder of this paper is organized as follows. Section 2 introduces the necessary preliminaries, including definitions of weak solutions and key technical lemmas. Section 3 focuses on establishing the local energy conservation law in the sense of distributions using a mollification argument. In Section 4, we extend this local result to the global energy balance on the domain $\Omega$, completing the proof of Theorem \ref{theoremcomNS}. Finally, we apply our methods on the compressible Navier-Stokes equations with degenerate viscosity in the last Section 5.

A proof outline is given after Theorem~\ref{theoremcomNS}: Section~3 "focuses on establishing the local energy conservation law in the sense of distributions using a mollification argument." Then, in Section~4, they "extend this local result to the global energy balance on the domain $\Omega$, completing the proof of Theorem~\ref{theoremcomNS}." A key ingredient enabling the relaxed integrability assumptions is "a novel weak-type temporal commutator estimate (Lemma \ref{commutator}), which handles the nonlinear terms without demanding strong regularity of $\partial_t\rho$."

A proof outline is given after
\begin{theorem}\label{theoremcomNS}
    Let $(\rho, u)$ be a weak solution of \eqref{eq:comNS} in the sense of Definition \ref{defcomNS}. Assume that the density satisfies
    \begin{equation}\label{con:thcomNSrhoover0}
         0 \leq \rho(t,x) \leq \bar{\rho} < \infty, \quad \text{and} \quad \nabla \sqrt{\rho} \in L^{\infty}\left(0, T ; L^{\frac{3}{2}}(\Omega)\right),
    \end{equation}
    and the initial velocity satisfies
    \begin{equation}\label{con:initial_1}
        u_0 \in L^{\frac{6N}{6-N}}(\Omega).
    \end{equation}
    If the velocity satisfies
    \begin{equation}\label{con:shinbrot_cond}
        u \in L^p\left(0, T ; L^q(\Omega)\right) \quad \text{with} \quad \begin{cases}
            \frac{1}{p}+\frac{3}{q} \leq 1, \quad\mbox{if} \quad 3\leq q < 4,\\
            \frac{2}{p}+\frac{2}{q} \leq 1, \quad\mbox{if} \quad 4\leq q <\infty,
        \end{cases}
    \end{equation}
    then the energy equality \eqref{eq:energy_comNS} holds for any $t \in[0, T]$.
\end{theorem}
Next, they focus on establishing the local energy conservation law in the sense of distributions using a mollification argument. Then, they extend this local result to the global energy balance on the domain $\Omega$, thereby completing the proof of the main theorem. A key ingredient enabling the relaxed integrability assumptions is the following commutator estimate.

\begin{lemma}\label{commutator}
Let $1 \leq \bar{p}, \bar{q}, p_1, q_1, p_2, q_2 \leq \infty$, with $\frac{1}{\bar{p}}+\frac{1}{p_1}+\frac{1}{p_2}=1$ and $\frac{1}{\bar{q}}+\frac{1}{q_1}+\frac{1}{q_2}=1$. Let $f\in L^{p_1}\left(0, T ; L^{q_1}(\Omega)\right), \partial_{t} f \in L^{p_1}\left(0, T ; W^{-1,q_1}(\Omega)\right), g \in L^{p_2}\left(0, T ; W^{1,q_2}(\Omega)\right), \varphi \in L^{\bar{p}}\left(0, T ; W_{0}^{1,\bar{q}}(\Omega)\right)$. Then, there holds
$$
\int_0^T \int_{\Omega}\varphi\left[\partial_t\left(f g\right)_{\varepsilon}-\partial_t(f g_{\varepsilon})\right]\mathrm{d} x \mathrm{d} t \rightarrow 0,
$$
as $\varepsilon \rightarrow 0$ if $p_2, q_2,\bar{p},\bar{q}<\infty$.
\end{lemma}
This lemma handles the nonlinear terms without demanding strong regularity of $\partial_t\rho$.

\label{theoremcomNS}
    Let $(\rho, u)$ be a weak solution of \begin{equation}\label{eq:comNS}
    \begin{split}
        (\rho u)_t + \text{div}(\rho u \otimes u) - \mu \Delta u - (\mu+\lambda) \nabla \text{div} \, u + \nabla P &= 0,  \\
        \rho_t + \text{div}(\rho u) &= 0,
    \end{split}
\end{equation} in the sense of the following definition. \begin{definition}\label{defcomNS}
    For a given $T>0$, we call $(\rho, u)$ a weak solution on $[0, T]$ to the constant viscosity system \eqref{eq:comNS}\eqref{con:initial} if:
    \begin{itemize}
        \item The momentum equations \eqref{eq:comNS} hold in $\mathcal{D}^{\prime}((0, T) \times \Omega)$ with the regularity:
        \begin{equation}\label{con:defcomNS}
            \rho^\gamma, \rho|u|^2 \in L^{\infty}\left(0, T ; L^1(\Omega)\right), \quad u \in L^2\left(0, T ; H_0^1(\Omega)\right);
        \end{equation}

        \item The initial conditions \eqref{con:initial} hold in $\mathcal{D}^{\prime}(\Omega)$;

        \item $(\rho, u)$ is a renormalized solution of the continuity equation in the sense of DiPerna--Lions, ``Ordinary differential equations, transport theory and Sobolev spaces'' (1989);

        \item The energy inequality holds for almost every $t \in [0,T]$:
        $$
        \begin{aligned}
            \int_{\Omega}\left(\frac{1}{2} \rho|u|^2+\frac{\rho^\gamma}{\gamma-1}\right) \mathrm{d} x & +\int_0^t \int_{\Omega}\left(\mu|\nabla u|^2+(\mu+\lambda)(\operatorname{div} u)^2\right) \mathrm{d} x \mathrm{d} s \\
            & \leq \int_{\Omega}\left(\frac{1}{2} \rho_0\left|u_0\right|^2+\frac{\rho_0^\gamma}{\gamma-1}\right) \mathrm{d} x.
        \end{aligned}
        $$
    \end{itemize}
\end{definition}
    Assume that the density satisfies
    \begin{equation}\label{con:thcomNSrhoover0}
         0 \leq \rho(t,x) \leq \bar{\rho} < \infty, \quad \text{and} \quad \nabla \sqrt{\rho} \in L^{\infty}\left(0, T ; L^{\frac{3}{2}}(\Omega)\right),
    \end{equation}
    and the initial velocity satisfies
    \begin{equation}\label{con:initial_1}
        u_0 \in L^{\frac{6N}{6-N}}(\Omega).
    \end{equation}
    If the velocity satisfies
    \begin{equation}\label{con:shinbrot_cond}
        u \in L^p\left(0, T ; L^q(\Omega)\right) \quad \text{with} \quad \begin{cases}
            \frac{1}{p}+\frac{3}{q} \leq 1, \quad\mbox{if} \quad 3\leq q < 4,\\
            \frac{2}{p}+\frac{2}{q} \leq 1, \quad\mbox{if} \quad 4\leq q <\infty,
        \end{cases}
    \end{equation}
    then the energy equality
    \begin{aligned}
    	&\int_{\Omega}\left(\frac{1}{2} \rho_0\left|u_0\right|^2+\frac{\rho_0^\gamma}{\gamma-1}\right) \mathrm{d} x-\int_{\Omega}\left(\frac{1}{2} \rho|u|^2+\frac{\rho^\gamma}{\gamma-1}\right) \mathrm{d} x\\
    	=&\int_0^T \int_{\Omega}\left( \mu|\nabla u|^2+(\mu+\lambda)|\operatorname{div} u|^2 \right)\mathrm{d} x \mathrm{d} t. \label{eq:energy_comNS}
    \end{aligned} holds for any $t \in[0, T]$.

Question:
Let \(\Omega\) be either the torus \(\mathbb T^N\) with \(N=2,3\), or a bounded open \(C^1\) domain in \(\mathbb R^N\) (with no-slip boundary condition when \(\partial\Omega\neq\varnothing\)). Consider a weak solution \((\rho,u)\) on \([0,T]\) of the constant-viscosity compressible Navier--Stokes system
\[
(\rho u)_t+\operatorname{div}(\rho u\otimes u)-\mu\Delta u-(\mu+\lambda)\nabla\operatorname{div}u+\nabla P=0,
\qquad
\rho_t+\operatorname{div}(\rho u)=0,
\]
with initial data \((\rho_0,u_0)\), in the following sense: the momentum equation holds in \(\mathcal D'((0,T)\times\Omega)\);
\(\rho^\gamma,\rho|u|^2\in L^\infty(0,T;L^1(\Omega))\); \(u\in L^2(0,T;H_0^1(\Omega))\); the initial conditions hold in \(\mathcal D'(\Omega)\); \((\rho,u)\) is a renormalized solution of the continuity equation in the DiPerna--Lions sense; and for almost every \(t\in[0,T]\),
\[
\int_{\Omega}\left(\tfrac12\rho|u|^2+\frac{\rho^\gamma}{\gamma-1}\right)dx
+\int_0^t\!\int_{\Omega}\bigl(\mu|\nabla u|^2+(\mu+\lambda)(\operatorname{div}u)^2\bigr)\,dx\,ds
\le
\int_{\Omega}\left(\tfrac12\rho_0|u_0|^2+\frac{\rho_0^\gamma}{\gamma-1}\right)dx.
\]
Assume in addition that
\[
0\le \rho(t,x)\le \bar\rho<\infty,
\qquad
\nabla\sqrt\rho\in L^\infty(0,T;L^{3/2}(\Omega)),
\qquad
u_0\in L^{\frac{6N}{6-N}}(\Omega),
\]
and that
\[
u\in L^p(0,T;L^q(\Omega))
\quad\text{with}\quad
\begin{cases}
\frac1p+\frac3q\le 1, & 3\le q<4,\\[2mm]
\frac2p+\frac2q\le 1, & 4\le q<\infty.
\end{cases}
\]
Which statement holds for every such weak solution?

For every \(t\in[0,T]\), the energy inequality becomes an equality:
\[
\int_{\Omega}\left(\tfrac12\rho_0|u_0|^2+\frac{\rho_0^\gamma}{\gamma-1}\right)dx
-
\int_{\Omega}\left(\tfrac12\rho(t)|u(t)|^2+\frac{\rho(t)^\gamma}{\gamma-1}\right)dx
=
\int_0^t\!\int_{\Omega}\bigl(\mu|\nabla u|^2+(\mu+\lambda)|\operatorname{div}u|^2\bigr)\,dx\,ds.
\]

Choices:
B.
For every \(t\in[0,T]\), the energy inequality becomes an equality:
\[
\int_{\Omega}\left(\tfrac12\rho_0|u_0|^2+\frac{\rho_0^\gamma}{\gamma-1}\right)dx
-
\int_{\Omega}\left(\tfrac12\rho(t)|u(t)|^2+\frac{\rho(t)^\gamma}{\gamma-1}\right)dx
=
\int_0^t\!\int_{\Omega}\bigl(\mu|\nabla u|^2+(\mu+\lambda)|\operatorname{div}u|^2\bigr)\,dx\,ds,
\]
provided the velocity satisfies
\[
u\in L^p(0,T;L^q(\Omega))
\quad\text{with}\quad
\begin{cases}
\frac1p+\frac3q\le 1, & 3\le q\le 4,\\[2mm]
\frac2p+\frac2q\le 1, & 4< q<\infty.
\end{cases}
\]
C.
The weak solution satisfies the original energy inequality for almost every \(t\in[0,T]\):
\[
\int_{\Omega}\left(\tfrac12\rho(t)|u(t)|^2+\frac{\rho(t)^\gamma}{\gamma-1}\right)dx
+
\int_0^t\!\int_{\Omega}\bigl(\mu|\nabla u|^2+(\mu+\lambda)(\operatorname{div}u)^2\bigr)\,dx\,ds
\le
\int_{\Omega}\left(\tfrac12\rho_0|u_0|^2+\frac{\rho_0^\gamma}{\gamma-1}\right)dx.
\]
D.
For every \(t\in[0,T]\), the energy inequality becomes an equality under the same assumptions, and in fact the conclusion remains valid if the Shinbrot-type condition is replaced by the endpoint-extended condition
\[
u\in L^p(0,T;L^q(\Omega))
\quad\text{with}\quad
\begin{cases}
\frac1p+\frac3q\le 1, & 3\le q<4,\\[2mm]
\frac2p+\frac2q\le 1, & 4\le q\le \infty.
\end{cases}
\]
E.
For every such weak solution, the same energy equality holds for every \(t\in[0,T]\) even if one drops the assumption that \((\rho,u)\) is a renormalized solution of the continuity equation in the DiPerna--Lions sense, retaining only that the continuity equation holds in \(\mathcal D'((0,T)\times\Omega)\) together with
\[
0\le \rho\le \bar\rho,
\qquad
\nabla\sqrt\rho\in L^\infty(0,T;L^{3/2}(\Omega)),
\qquad
u\in L^p(0,T;L^q(\Omega))
\]
and the stated Shinbrot-type conditions on \((p,q)\).

finite-exponent requirement excluding q=\infty in the commutator/local energy argument

renormalized continuity equation needed to pass from local mollified identities to global energy balance

The stem does not explicitly state the conclusion, and the correct option is not lexically signaled. It presents hypotheses only, so there is no direct answer leakage.

This is very close to a theorem-statement recall item: the stem lists the full hypotheses and asks for the theorem's conclusion. The subtle alternative formulations prevent it from being a pure tautology, but it is still a near-restatement.

Some reasoning is needed to distinguish 'for every t' from 'for a.e. t' and to detect endpoint/assumption tampering. However, the item mainly tests exact recall of a known result rather than substantial generative mathematical reasoning.

Several distractors are plausible theorem distortions (endpoint shift, weakened time conclusion, altered density regularity). But the set is weakened by logical ambiguity: option C is a weaker true statement implied by A, and option D is also potentially read as a weaker true statement under the given assumptions.

A technically sophisticated but theorem-recall-heavy MCQ. It avoids direct leakage, but it is close to restating the source result, and the distractor set is only moderately strong because at least one distractor is weaker-true or logically ambiguous.


Question:
Let \(\Omega\) be either the torus \(\mathbb T^N\) with \(N=2,3\), or a bounded open \(C^1\) domain in \(\mathbb R^N\) (with no-slip boundary condition when \(\partial\Omega\neq\varnothing\)). Consider a weak solution \((\rho,u)\) on \([0,T]\) of the constant-viscosity compressible Navier--Stokes system
\[
(\rho u)_t+\operatorname{div}(\rho u\otimes u)-\mu\Delta u-(\mu+\lambda)\nabla\operatorname{div}u+\nabla P=0,
\qquad
\rho_t+\operatorname{div}(\rho u)=0,
\]
with initial data \((\rho_0,u_0)\), in the following sense: the momentum equation holds in \(\mathcal D'((0,T)\times\Omega)\);
\(\rho^\gamma,\rho|u|^2\in L^\infty(0,T;L^1(\Omega))\); \(u\in L^2(0,T;H_0^1(\Omega))\); the initial conditions hold in \(\mathcal D'(\Omega)\); \((\rho,u)\) is a renormalized solution of the continuity equation in the DiPerna--Lions sense; and for almost every \(t\in[0,T]\),
\[
\int_{\Omega}\left(\tfrac12\rho|u|^2+\frac{\rho^\gamma}{\gamma-1}\right)dx
+\int_0^t\!\int_{\Omega}\bigl(\mu|\nabla u|^2+(\mu+\lambda)(\operatorname{div}u)^2\bigr)\,dx\,ds
\le
\int_{\Omega}\left(\tfrac12\rho_0|u_0|^2+\frac{\rho_0^\gamma}{\gamma-1}\right)dx.
\]
Assume in addition that
\[
0\le \rho(t,x)\le \bar\rho<\infty,
\qquad
\nabla\sqrt\rho\in L^\infty(0,T;L^{3/2}(\Omega)),
\qquad
u_0\in L^{\frac{6N}{6-N}}(\Omega),
\]
and that
\[
u\in L^p(0,T;L^q(\Omega))
\quad\text{with}\quad
\begin{cases}
\frac1p+\frac3q\le 1, & 3\le q<4,\\[2mm]
\frac2p+\frac2q\le 1, & 4\le q<\infty.
\end{cases}
\]
Which statement holds for every such weak solution?

Choices:
B.
For every \(t\in[0,T]\), the energy inequality becomes an equality:
\[
\int_{\Omega}\left(\tfrac12\rho_0|u_0|^2+\frac{\rho_0^\gamma}{\gamma-1}\right)dx
-
\int_{\Omega}\left(\tfrac12\rho(t)|u(t)|^2+\frac{\rho(t)^\gamma}{\gamma-1}\right)dx
=
\int_0^t\!\int_{\Omega}\bigl(\mu|\nabla u|^2+(\mu+\lambda)|\operatorname{div}u|^2\bigr)\,dx\,ds,
\]
provided the velocity satisfies
\[
u\in L^p(0,T;L^q(\Omega))
\quad\text{with}\quad
\begin{cases}
\frac1p+\frac3q\le 1, & 3\le q\le 4,\\[2mm]
\frac2p+\frac2q\le 1, & 4< q<\infty.
\end{cases}
\]
C.
The weak solution satisfies the original energy inequality for almost every \(t\in[0,T]\):
\[
\int_{\Omega}\left(\tfrac12\rho(t)|u(t)|^2+\frac{\rho(t)^\gamma}{\gamma-1}\right)dx
+
\int_0^t\!\int_{\Omega}\bigl(\mu|\nabla u|^2+(\mu+\lambda)(\operatorname{div}u)^2\bigr)\,dx\,ds
\le
\int_{\Omega}\left(\tfrac12\rho_0|u_0|^2+\frac{\rho_0^\gamma}{\gamma-1}\right)dx.
\]
D.
For every \(t\in[0,T]\), the energy inequality becomes an equality under the same assumptions, and in fact the conclusion remains valid if the Shinbrot-type condition is replaced by the endpoint-extended condition
\[
u\in L^p(0,T;L^q(\Omega))
\quad\text{with}\quad
\begin{cases}
\frac1p+\frac3q\le 1, & 3\le q<4,\\[2mm]
\frac2p+\frac2q\le 1, & 4\le q\le \infty.
\end{cases}
\]
E.
For every such weak solution, the same energy equality holds for every \(t\in[0,T]\) even if one drops the assumption that \((\rho,u)\) is a renormalized solution of the continuity equation in the DiPerna--Lions sense, retaining only that the continuity equation holds in \(\mathcal D'((0,T)\times\Omega)\) together with
\[
0\le \rho\le \bar\rho,
\qquad
\nabla\sqrt\rho\in L^\infty(0,T;L^{3/2}(\Omega)),
\qquad
u\in L^p(0,T;L^q(\Omega))
\]
and the stated Shinbrot-type conditions on \((p,q)\).

The stem does not explicitly state the upgraded conclusion of full energy equality for every t. It gives the hypotheses and the baseline energy inequality, but the exact correct conclusion is not leaked outright.

This is very close to a theorem-recall item: the stem lists essentially the full hypotheses and asks for the theorem’s conclusion. However, the alternatives do introduce nearby competing formulations, so it is not a pure verbatim restatement.

There is some reasoning pressure in checking the endpoint conditions at q=4, the role of renormalization, and whether equality is strengthened from the assumed inequality. But the task is still mostly recognition of the exact theorem statement rather than substantial mathematical generation.

B, D, and E are plausible theorem-neighbor distractors based on common endpoint/generalization mistakes. But C is problematic because it is explicitly true from the stem itself (a weaker true statement), so the distractor set is not cleanly single-answer.

Moderate-quality theorem-recognition MCQ: no major answer leakage, but it is close to direct theorem recall and includes a weaker true option that harms single-answer clarity and reduces generative reasoning.


\begin{theo} \label{maintheore}
Let $U \subset \mathbb{R}^d$ be a (possibly unbounded) open set with $d \geq 2$. Assume that $\mathbf{G} \in L^2_{loc}(U, \mathbb{R}^d)$ and $c \in L^1_{loc}(U)$. Let $A = (a_{ij})_{1 \leq i,j \leq d}$ be a symmetric matrix of functions such that $a_{ij} \in VMO_{loc}(U)$ for all $1 \leq i,j \leq d$ (see Definition~\ref{defnvmo}), and suppose that ${\rm div} A \in L^2_{loc}(U, \mathbb{R}^d)$ (see Definition~\ref{defndivmat}). Also assume that $A$ is locally uniformly strictly elliptic and bounded on $U$, i.e. for every open ball $V$ with $\overline{V} \subset U$, there exist constants $\lambda_V, M_V > 0$ such that
\begin{equation} \label{locellst}
\lambda_V \| \xi \|^2 \leq \langle A(x) \xi, \xi \rangle, \quad \;\; \max_{1 \leq i,j \leq d} |a_{ij}(x)| \leq M_V
\quad \text{for a.e. } x \in V \text{ and all } \xi \in \mathbb{R}^d.
\end{equation}
Let $\tilde{\mathbf{F}} \in L^2_{loc}(U, \mathbb{R}^d)$ and $\tilde{f} \in L^1_{loc}(U)$. Suppose that $h \in L^{\infty}_{loc}(U)$ satisfies
\[
\int_U \left({\rm trace}(A \nabla^2 \varphi) + \langle \mathbf{G}, \nabla \varphi \rangle + c \varphi \right) h \, dx
= \int_U \tilde{f} \varphi \, dx + \int_U \langle \tilde{\mathbf{F}}, \nabla \varphi \rangle \, dx \quad \text{for all } \varphi \in C_0^{\infty}(U).
\]
Then $h \in H^{1,2}_{loc}(U)$.
\end{theo}

\label{intro}
The Fokker-Planck equation is fundamentally connected to stochastic analysis, particularly to the theory of stochastic differential equations (SDEs). Given a stochastic process $(X_t)_{t \geq 0}$ defined on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$ and a standard Brownian motion $(W_t)_{t \geq 0}$, consider the following SDE:
\[
X_t = x + \int_0^t \sigma(X_s) \, dW_s + \int_0^t \mathbf{G}(X_s) \, ds, \quad 0 \leq t < \infty, \quad \mathbb{P}\text{-a.s.}
\]
where $x \in \mathbb{R}^d$, $\sigma$ is a $d \times d$ matrix of functions, and $\mathbf{G}$ is a vector field that is locally integrable.
For any $f \in C_0^\infty(\mathbb{R}^d)$ and $t > 0$, applying It\^{o}'s formula (\cite[Chapter IV, Theorem 3.3]{R99}) yields
\begin{align*}
f(X_t) - f(X_0) &= \int_0^t \nabla f(X_s)  \, dX_s + \frac{1}{2} \sum_{i,j=1}^d \int_0^t \frac{\partial^2 f}{\partial x_i \partial x_j}(X_s) \, d\langle X^i, X^j \rangle_s \\
&= \int_0^t \nabla f(X_s) \sigma(X_s)\, dW_s + \int_0^t Lf(X_s) \, ds, \quad \text{ $\mathbb{P}$-a.s.},
\end{align*}
where the operator $L$ is given by
$$
Lf = {\rm trace}(A \nabla^2 f) + \langle \mathbf{G}, \nabla f \rangle, \quad \text{with } A = \frac12 \sigma \sigma^T.
$$
Taking expectations on both sides of the above identity, we obtain
\begin{equation}\label{fokkeriden}
\mathbb{E}[f(X_t)] - f(x) = \int_0^t \mathbb{E}[L f(X_s)]\,ds \quad \text{for all } f \in C_0^\infty(\mathbb{R}^d) \text{ and } t > 0,
\end{equation}
where $\mathbb{E}$ is the expectation with respect to $\mathbb{P}$. Let $p(t, x, \cdot)$ denote the probability density function of $X_t$ under the probability measure $\mathbb{P}$ (such a density exists under quite general assumptions and see \cite{L25ai}). Then, equation \eqref{fokkeriden} is equivalent to the following integral identity:
\begin{equation}\label{fokkerpllan}
\int_{\mathbb{R}^d} f(y)\, p(t, x, y)\,dy - f(x)
= \int_0^t \int_{\mathbb{R}^d} Lf(y)\, p(s, x, y)\,dy\,ds \quad \text{for all } f \in C_0^\infty(\mathbb{R}^d) \text{ and } t > 0.
\end{equation}
The family of measures $\left(p(t, x, y)\,dy\right)_{t \geq 0}$ is referred to as the solution to the Cauchy problem for the Fokker-Planck equation associated with the operator $L$ and the initial distribution $\delta_x$, the Dirac delta measure at $x$ (cf. \cite[Definition 6.1, Proposition 6.1.2]{BKRS15}).
We refer the reader to the introduction of \cite{L25ai} for a more detailed discussion of the Fokker-Planck equation. In fact, under appropriate ergodicity assumptions, one can show the existence of a probability measure $\nu$ on $\mathcal{B}(\mathbb{R}^d)$ such that
\begin{equation} \label{weakconver}
\lim_{t \to \infty} p(t, x, y)\,dy = \nu(dy) \quad \text{weakly on } \mathbb{R}^d,
\end{equation}
(see \cite[Theorem 1.1(iii)]{L25ai}).
Under suitable regularity assumptions on the coefficients $A$ and the drift vector field $\mathbf{G}$, it follows from \eqref{fokkerpllan} and \eqref{weakconver} that $L^* \nu=0$ on $\mathbb{R}^d$, i.e.
\begin{equation} \label{stafokkerp}
\int_{\mathbb{R}^d} Lf \, d\nu = 0  \quad \text{for all } f \in C_0^\infty(\mathbb{R}^d).
\end{equation}
A Borel measure $\nu$ satisfying \eqref{stafokkerp} is called a solution to the stationary Fokker-Planck equation associated with the operator $L$. In stochastic analysis, $\nu$ is also called an infinitesimally invariant measure for $(L, C_0^{\infty}(\mathbb{R}^d))$. In this paper, we investigate the regularity of the density of $\nu$. A systematic study of stationary Fokker-Planck equations was carried out in \cite{BKRS15, BKR01, BKR09, BRS12, BS17, B18}. In particular, if the diffusion coefficients satisfy $a_{ij} \in H^{1,p}_{loc}(\mathbb{R}^d)$ for all $1 \leq i,j \leq d$ with $p \in (d, \infty)$, $A = (a_{ij})_{1 \leq i,j \leq d}$ is locally uniformly strictly elliptic and bounded on $\mathbb{R}^d$ (i.e. \eqref{locellst} holds for $U=\mathbb{R}^d$), $\mathbf{G} \in L^p_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ and \eqref{stafokkerp} holds, then according to \cite[Corollary 1.6.9]{BKRS15}, there exists a function $\rho \in H^{1,p}_{loc}(\mathbb{R}^d) \cap C(\mathbb{R}^d)$ such that  $\nu = \rho \,dx$.\\
Stationary Fokker-Planck equations can be viewed as a class of partial differential equations in which the second-order derivatives act on test functions. The regularity theory for their solutions has developed through successive advancements, beginning with the classical Weyl lemma \cite{W40}, and subsequently extended in works such as \cite{S73, F77, BS17}. For further details, we refer the reader to \cite{L25bv} and the references therein. A recent result (\cite[Corollary 1.2]{L25bv}) concerning the regularity of solutions to stationary Fokker-Planck equations generalizes \cite[Corollary 1.6.9]{BKRS15} under the assumption that (cf. Definitions \ref{defndivmat}, \ref{defnvmo})
${\rm div} A \in L^p_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ with $p \in (d, \infty)$, $a_{ij} \in VMO_{loc}(\mathbb{R}^d)$ for all $1 \leq i,j \leq d$, and that the matrix $A = (a_{ij})_{1 \leq i,j \leq d}$ is locally uniformly strictly elliptic and bounded on $\mathbb{R}^d$. If, in addition, $\mathbf{G} \in L^p_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ and \eqref{stafokkerp} holds, then it has been shown in \cite[Corollary 1.2]{L25bv} that there exists a function $\rho \in H^{1,p}_{loc}(\mathbb{R}^d) \cap C(\mathbb{R}^d)$ such that  $\nu = \rho\,dx$.\\
However, the assumption $\mathbf{G} \in L_{loc}^p(\mathbb{R}^d, \mathbb{R}^d)$ may still be regarded as rather restrictive. It is therefore natural to consider relaxing this condition to the case $\mathbf{G} \in L^2_{loc}(\mathbb{R}^d, \mathbb{R}^d)$. In fact, as shown in \cite{F81, St99}, a stochastic process associated with the operator $L$ can be constructed even when the drift is merely locally $L^2$-integrable with respect to a reference measure. This observation motivates the extension of the results in \cite[Corollary 1.6.9]{BKRS15} and \cite[Corollary 1.2]{L25bv} to this more general setting.
In particular, when the diffusion matrix is the identity, the assumption $\mathbf{G} \in L^2_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ has been considered in \cite[Theorem 1(ii)]{BKR97}, where corresponding regularity results for the density of the solution were established. \\
In our setting, however, we require the a priori assumption that the solution $\nu$ admits a density belonging to $L_{loc}^\infty(U)$.

Fokker-Planck equation. In fact, under appropriate ergodicity assumptions, one can show the existence of a probability measure $\nu$ on $\mathcal{B}(\mathbb{R}^d)$ such that
\begin{equation} \label{weakconver}
\lim_{t \to \infty} p(t, x, y)\,dy = \nu(dy) \quad \text{weakly on } \mathbb{R}^d,
\end{equation}
(see \cite[Theorem 1.1(iii)]{L25ai}).
Under suitable regularity assumptions on the coefficients $A$ and the drift vector field $\mathbf{G}$, it follows from \eqref{fokkerpllan} and \eqref{weakconver} that $L^* \nu=0$ on $\mathbb{R}^d$, i.e.
\begin{equation} \label{stafokkerp}
\int_{\mathbb{R}^d} Lf \, d\nu = 0 \quad \text{for all } f \in C_0^\infty(\mathbb{R}^d).
\end{equation}
A Borel measure $\nu$ satisfying \eqref{stafokkerp} is called a solution to the stationary Fokker-Planck equation associated with the operator $L$. In stochastic analysis, $\nu$ is also called an infinitesimally invariant measure for $(L, C_0^{\infty}(\mathbb{R}^d))$. In this paper, we investigate the regularity of the density of $\nu$. A systematic study of stationary Fokker-Planck equations was carried out in \cite{BKRS15, BKR01, BKR09, BRS12, BS17, B18}. In particular, if the diffusion coefficients satisfy $a_{ij} \in H^{1,p}_{loc}(\mathbb{R}^d)$ for all $1 \leq i,j \leq d$ with $p \in (d, \infty)$, $A = (a_{ij})_{1 \leq i,j \leq d}$ is locally uniformly strictly elliptic and bounded on $\mathbb{R}^d$ (i.e. \eqref{locellst} holds for $U=\mathbb{R}^d$), $\mathbf{G} \in L^p_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ and \eqref{stafokkerp} holds, then according to \cite[Corollary 1.6.9]{BKRS15}, there exists a function $\rho \in H^{1,p}_{loc}(\mathbb{R}^d) \cap C(\mathbb{R}^d)$ such that $\nu = \rho \,dx$.\\
Stationary Fokker-Planck equations can be viewed as a class of partial differential equations in which the second-order derivatives act on test functions. The regularity theory for their solutions has developed through successive advancements, beginning with the classical Weyl lemma \cite{W40}, and subsequently extended in works such as \cite{S73, F77, BS17}. For further details, we refer the reader to \cite{L25bv} and the references therein. A recent result (\cite[Corollary 1.2]{L25bv}) concerning the regularity of solutions to stationary Fokker-Planck equations generalizes \cite[Corollary 1.6.9]{BKRS15} under the assumption that (cf. Definitions \ref{defndivmat}, \ref{defnvmo})
${\rm div} A \in L^p_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ with $p \in (d, \infty)$, $a_{ij} \in VMO_{loc}(\mathbb{R}^d)$ for all $1 \leq i,j \leq d$, and that the matrix $A = (a_{ij})_{1 \leq i,j \leq d}$ is locally uniformly strictly elliptic and bounded on $\mathbb{R}^d$. If, in addition, $\mathbf{G} \in L^p_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ and \eqref{stafokkerp} holds, then it has been shown in \cite[Corollary 1.2]{L25bv} that there exists a function $\rho \in H^{1,p}_{loc}(\mathbb{R}^d) \cap C(\mathbb{R}^d)$ such that $\nu = \rho\,dx$.\\
However, the assumption $\mathbf{G} \in L_{loc}^p(\mathbb{R}^d, \mathbb{R}^d)$ may still be regarded as rather restrictive. It is therefore natural to consider relaxing this condition to the case $\mathbf{G} \in L^2_{loc}(\mathbb{R}^d, \mathbb{R}^d)$. In fact, as shown in \cite{F81, St99}, a stochastic process associated with the operator $L$ can be constructed even when the drift is merely locally $L^2$-integrable with respect to a reference measure. This observation motivates the extension of the results in \cite[Corollary 1.6.9]{BKRS15} and \cite[Corollary 1.2]{L25bv} to this more general setting.
In particular, when the diffusion matrix is the identity, the assumption $\mathbf{G} \in L^2_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ has been considered in \cite[Theorem 1(ii)]{BKR97}, where corresponding regularity results for the density of the solution were established. \\
In our setting, however, we require the a priori assumption that the solution $\nu$ admits a density belonging to $L_{loc}^\infty(U)$.

\begin{defn} \label{defndivmat}
Let $U$ be an open subset of $\mathbb{R}^d$, $\mathbf{E} = (e_1, \dots, e_d) \in L^1_{loc}(U, \mathbb{R}^d)$ and $B = (b_{ij})_{1 \leq i,j \leq d}$ be a (possibly non-symmetric) matrix of functions with $b_{ij} \in L^1_{loc}(U)$ for all $1 \leq i,j \leq d$. Let $\mathbf{F} \in L^1_{loc}(U, \mathbb{R}^d)$ and $f \in L^1_{loc}(U)$.
\begin{itemize}
\item[(i)] We say that ${\rm div} B = \mathbf{E}$ if
\[
\int_U \sum_{i,j=1}^d b_{ij} \, \partial_i \phi_j \, dx = - \int_U \sum_{j=1}^d e_j \, \phi_j \, dx \quad \text{for all } (\phi_1, \dots, \phi_d) \in C_0^\infty(U)^d.
\]
\item[(ii)] We say that ${\rm div} \mathbf{F} = f$ if
\[
\int_U \langle \mathbf{F}, \nabla \varphi \rangle \, dx = - \int_U f \varphi \, dx \quad \text{for all } \varphi \in C_0^\infty(U).
\]
\end{itemize}
\end{defn}

\begin{defn} \label{defnvmo}
Let $\omega$ be a positive continuous function on $[0,\infty)$ with $\omega(0) = 0$. The space $VMO_\omega$ consists of all functions $g \in L^1_{loc}(\mathbb{R}^d)$ such that
\[
\sup_{z \in \mathbb{R}^d,\, r < R} \, r^{-2d} \int_{B_r(z)} \int_{B_r(z)} |g(x) - g(y)| \, dx \, dy \leq \omega(R) \quad \text{for all } R \in (0,\infty).
\]
Let $B$ be an open ball in $\mathbb{R}^d$. We define $VMO(B)$ as the set of all functions $f \in L^1(B)$ for which there exists an extension $\tilde{f} \in VMO_{\omega}$  (for some modulus $\omega$) such that $\tilde{f}|_B = f$. For an open set $U \subset \mathbb{R}^d$, the space $VMO_{loc}(U)$ is defined as the collection of all functions $f \in L^1_{loc}(U)$ such that $f \in VMO(B)$ for every open ball $B \subset \mathbb{R}^d$ with $\overline{B} \subset U$.
\end{defn}

\begin{equation} \label{locellst}
\lambda_V \| \xi \|^2 \leq \langle A(x) \xi, \xi \rangle, \quad \;\; \max_{1 \leq i,j \leq d} |a_{ij}(x)| \leq M_V
\quad \text{for a.e. } x \in V \text{ and all } \xi \in \mathbb{R}^d.
\end{equation}

\begin{equation} \label{stafokkerp}
\int_{\mathbb{R}^d} Lf \, d\nu = 0  \quad \text{for all } f \in C_0^\infty(\mathbb{R}^d).
\end{equation}

\noindent
{Keywords: Stationary Fokker-Planck equations, Elliptic regularity, Resolvent, Weak maximum principle, Singular drifts, Invariant measures
}

\section{Application to the uniqueness results} \label{sec4}
The local regularity result established in the main theorem enables us to convert solutions of the stationary Fokker–Planck equations into weak solutions of a divergence form equation. Based on this, our main theorem can be applied to derive a uniqueness result for solutions to the stationary Fokker–Planck equations.
\begin{theo} \label{applicationth}
Let $A = (a_{ij})_{1 \leq i,j \leq d}$ be a symmetric matrix of functions whose components satisfy $a_{ij} \in VMO_{loc}(\mathbb{R}^d)$ for all $1 \leq i,j \leq d$. Assume that the condition \eqref{ellipticit} holds for some strictly positive constants $\lambda$ and $M$. Let $C = (c_{ij})_{1 \leq i,j \leq d}$ be an anti-symmetric matrix of functions satisfying $c_{ij} = -c_{ji} \in L^{\infty}(\mathbb{R}^d)$ for all $1 \leq i,j \leq d$. Assume that ${\rm div}(A+C) \in L^2_{loc}(\mathbb{R}^d, \mathbb{R}^d)$. Let $\rho \in H^{1,2}_{loc}(\mathbb{R}^d) \cap L^{\infty}(\mathbb{R}^d)$ be such that for some constant $\theta>0$, $\rho(x) > \theta$ for a.e. $x \in \mathbb{R}^d$. Then, the following statements hold:
\begin{itemize}
\item[(i)]
\begin{align*}
 \int_{\mathbb{R}^d} \bigg( {\rm trace}(A \nabla^2 f) + \left\langle {\rm div}(A+C) + \frac{1}{\rho}(A+C^T) \nabla \rho, \nabla f \right\rangle \bigg)\, \rho\, dx=0 \quad \text{ for all $f \in C_0^{\infty}(\mathbb{R}^d)$.}
\end{align*}
\item[(ii)]
If $h \in L^{\infty}_{loc}(\mathbb{R}^d)$ with $h \geq 0$ in $\mathbb{R}^d$ satisfies
\[
\int_{\mathbb{R}^d} \bigg( {\rm trace}(A \nabla^2 f) + \left\langle {\rm div}(A+C) + \frac{1}{\rho}(A+C^T) \nabla \rho, \nabla f \right\rangle \bigg)\, h\, dx = 0 \quad \text{for all } f \in C_0^{\infty}(\mathbb{R}^d),
\]
then there exists a constant $c \geq 0$ such that $h = c \rho$ in $\mathbb{R}^d$.
\item[(iii)]
If $h \in L^{\infty}(\mathbb{R}^d)$ satisfies
\[
\int_{\mathbb{R}^d} \bigg( {\rm trace}(A \nabla^2 f) + \left\langle {\rm div}(A+C) + \frac{1}{\rho}(A+C^T) \nabla \rho, \nabla f \right\rangle \bigg)\, h\, dx = 0 \quad \text{for all } f \in C_0^{\infty}(\mathbb{R}^d),
\]
then there exists a constant $c \in \mathbb{R}$ such that $h = c \rho$ in $\mathbb{R}^d$.
\end{itemize}
\end{theo}
\begin{proof}
(i) Let $f \in C_0^{\infty}(\mathbb{R}^d)$.
Then, by \cite[Proposition 5.7]{L25bv},
\begin{equation} \label{divtracrep}
 \frac{1}{\rho} {\rm div} \Big( \rho(A+C) \nabla f \Big) = {\rm trace}(A \nabla^2 f) + \left\langle {\rm div}(A+C) + \frac{1}{\rho}(A+C^T) \nabla \rho, \nabla f \right\rangle.
\end{equation}
Take an open ball $V$ so that ${\rm supp}(f) \subset V$. Let $\chi \in C_0^{\infty}(\mathbb{R}^d)$ with $\chi \equiv 1$ on $\overline{V}$. Then,
\begin{align*}
&\int_{\mathbb{R}^d} \bigg( {\rm trace}(A \nabla^2 f) + \left\langle {\rm div}(A+C) + \frac{1}{\rho}(A+C^T) \nabla \rho, \nabla f \right\rangle \bigg)\, \rho\, dx \\
&=\int_{\mathbb{R}^d} {\rm div} \Big( \rho(A+C) \nabla f \Big)\,dx = \int_{\mathbb{R}^d} {\rm div} \Big( \rho(A+C) \nabla f \Big) \chi \,dx \\
& = -\int_{V} \langle \rho(A+C) \nabla f, \nabla \chi \rangle\,dx=0.
\end{align*}
(ii) First, by Theorem \ref{maintheore}, $h \in H^{1,2}_{loc}(\mathbb{R}^d)$.
Let $\hat{h}:=\frac{h}{\rho}$. Then, $\hat{h} \in H^{1,2}_{loc}(\mathbb{R}^d)$ with $\hat{h} \geq 0$ in $\mathbb{R}^d$ and it follows from \eqref{divtracrep} that
\begin{align*}
0=&\int_{\mathbb{R}^d} \bigg( {\rm trace}(A \nabla^2 f) + \left\langle {\rm div}(A+C) + \frac{1}{\rho}(A+C^T) \nabla \rho, \nabla f \right\rangle \bigg)\, h\, dx \\
& = \int_{\mathbb{R}^d} {\rm div} \Big( \rho(A+C) \nabla f \Big) \cdot \hat{h}\,dx = -\int_{\mathbb{R}^d} \langle \rho (A+C^T) \nabla \hat{h}, \nabla f \rangle \,dx \quad \text{ for all $f \in C_0^{\infty}(\mathbb{R}^d)$}.
\end{align*}
By the Liouville-type theorem \cite[Theorem 9.11(i)]{F07}, $\hat{h}$ is constant, so that the assertion follows. \\ \\
(iii) As in (ii), $h \in H^{1,2}_{loc}(\mathbb{R}^d)$. Moreover, $\hat{h}:=\frac{h}{\rho} \in H^{1,2}_{loc}(\mathbb{R}^d) \cap L^{\infty}(\mathbb{R}^d)$ satisfies
$$
 \int_{\mathbb{R}^d} \langle \rho (A+C^T) \nabla \hat{h}, \nabla f \rangle \,dx=0 \quad \text{ for all $f \in C_0^{\infty}(\mathbb{R}^d)$}.
$$
Using the Liouville-type theorem \cite[Theorem 9.11(ii)]{F07}, $\hat{h}$ is constant, so that the assertion follows.
\end{proof}
\centerline{}
\noindent
To present a concrete example, we consider the following function:
Let $\varepsilon \in (0, 10^{-2026})$ be fixed. Define
\begin{equation} \label{defnpsi}
\Psi(t) :=
\begin{cases}
1, & t \in (-\infty, -1), \\[6pt]
1 + \displaystyle\int_{-1}^{t} |s|^{-\frac{1}{2+\varepsilon}}\,ds, & t \in [-1,1], \\[6pt]
1 + \displaystyle\int_{-1}^{1} |s|^{-\frac{1}{2+\varepsilon}}\,ds, & t \in (1, \infty).
\end{cases}
\end{equation}
Then $\Psi \in C(\mathbb{R}) \cap L^{\infty}(\mathbb{R})$, and its derivative $\Psi' \in \bigcap_{p \in (1,\; 2+\varepsilon)} L^p(\mathbb{R})$ is given by
\[
\Psi'(t) :=
\begin{cases}
0, & t \in (-\infty, -1), \\[6pt]
|t|^{-\frac{1}{2+\varepsilon}}, & t \in (-1,0) \cup (0,1), \\[6pt]
0, & t \in (1, \infty).
\end{cases}
\]
In particular, $\Psi' \notin L_{loc}^{2+\varepsilon}(\mathbb{R})$, and
\[
1 \leq \Psi(t) \leq 1 + \int_{-1}^{1} |s|^{-\frac{1}{2+\varepsilon}}\,ds \quad \text{for all } t \in \mathbb{R}.
\]

\begin{exam} \label{ourexam1}
Let $d \geq 2$ and $\varepsilon \in (0,10^{-2026})$ be fixed. Let $\Psi$ be the function defined in \eqref{defnpsi}. Define
$$
\rho(x_1, \ldots, x_d) = \prod_{i=1}^d \Psi(x_i), \quad x =(x_1, \ldots, x_d) \in \mathbb{R}^d.
$$
Then $\rho \in C(\mathbb{R}^d) \cap L^{\infty}(\mathbb{R}^d) \cap H^{1,2}_{loc}(\mathbb{R}^d)$, and $\nabla \rho \notin L^{2+\varepsilon}_{loc}(\mathbb{R}^d)$. Moreover,
$$
1 \leq \rho(x) \leq \left( 1 + \int_{-1}^{1} |s|^{-\frac{1}{2+\varepsilon}}\,ds \right)^d \quad \text{for all } x \in \mathbb{R}^d.
$$
Let $A = I$, $\mathbf{G} = \frac{1}{\rho} \nabla \rho$. Then
\begin{equation} \label{infinitesinvar}
\int_{\mathbb{R}^d} \left( {\rm trace}(A \nabla^2 f) + \langle \mathbf{G}, \nabla f \rangle \right) \rho\,dx = 0 \quad \text{for all } f \in C_0^{\infty}(\mathbb{R}^d).
\end{equation}
Assume that $h \in L^{\infty}_{loc}(\mathbb{R}^d)$ with $h \geq 0$ in $\mathbb{R}^d$ satisfies
\begin{equation} \label{infininvah}
\int_{\mathbb{R}^d} \left({\rm trace}(A \nabla^2 f) + \langle \mathbf{G}, \nabla f \rangle \right) h \,dx = 0 \quad \text{for all } f \in C_0^{\infty}(\mathbb{R}^d).
\end{equation}
Then it follows from Theorem \ref{applicationth}(ii) that $h$ is a constant multiple of $\rho$. If $h \in L^{\infty}(\mathbb{R}^d)$ satisfies \eqref{infininvah}, then it follows from Theorem \ref{applicationth}(iii) that $h$ is also a constant multiple of $\rho$.
\end{exam}

The main idea of this work is to construct a constant $\lambda_0 > 0$ and a sub-Markovian resolvent $\left(R_{\alpha}\right)_{\alpha \geq \lambda_0}$ associated with the principal elliptic operator $L^A = {\rm trace}(A \nabla^2)$, and to show that the function $h$ can be represented as the local weak $H^{1,2}$-limit of the family $(\alpha R_{\alpha} h)_{\alpha \geq \lambda_0}$ as $\alpha \rightarrow \infty$. This idea is inspired by \cite[Theorem 2.1]{St99}. In that result, however, the diffusion coefficients $a_{ij}$ are assumed to be locally Hölder continuous for all $1 \leq i,j \leq d$, and the density of the solution measure $\nu$ is assumed to belong to $L^{\infty}(\mathbb{R}^d)$. By contrast, in the present work, we establish resolvent estimates under the weaker assumptions that the coefficients belong to $VMO_{loc}(\mathbb{R}^d)$ and that ${\rm div} A \in L^2_{loc}(\mathbb{R}^d, \mathbb{R}^d)$, which enable us to derive the main result, Theorem \ref{maintheore}.\\
It is worth noting that determining the extent to which the assumptions in our main theorem can be further relaxed remains an open and challenging problem. In particular, it is unclear whether the assumptions that $a_{ij} \in VMO_{loc}(U)$ for all $1 \leq i,j \leq d$ and ${\rm div} A \in L^2_{loc}(U, \mathbb{R}^d)$ can be weakened. Similarly, whether the a priori condition $h \in L^{\infty}_{loc}(U)$ can be replaced by the weaker assumption $h \in L^2_{loc}(U)$ remains open. Furthermore, one may ask whether the result could still hold under the sole assumptions that the solution measure $\nu$ is locally finite and that $Lf \in L^1_{loc}(U, \nu)$ for every $f \in C_0^{\infty}(U)$. Understanding the regularity of solutions to such stationary Fokker-Planck equations is expected to play a crucial role in the study of invariant measures associated with stochastic differential equations having singular coefficients. \\
This paper is organized as follows. Section \ref{sec2} introduces the basic notations and definitions used throughout this paper. Section \ref{sec3} presents the resolvent approaches to elliptic regularity and provides the proof of the main result, Theorem \ref{maintheore}, at the end of the section. Section \ref{sec4} discusses applications to uniqueness results based on the main result and presents some concrete examples.

To derive Theorem~\ref{maintheore}, the paper’s stated strategy is to (i) “construct a constant $\lambda_0>0$ and a sub-Markovian resolvent $(R_\alpha)_{\alpha\ge \lambda_0}$ associated with the principal elliptic operator $L^A={\rm trace}(A\nabla^2)$”, and then (ii) “show that the function $h$ can be represented as the local weak $H^{1,2}$-limit of the family $(\alpha R_\alpha h)_{\alpha\ge \lambda_0}$ as $\alpha\to\infty$.” The introduction emphasizes that the proof relies on “resolvent estimates under the weaker assumptions that the coefficients belong to $VMO_{loc}(\mathbb{R}^d)$ and that ${\rm div}A\in L^2_{loc}(\mathbb{R}^d,\mathbb{R}^d)$, which enable us to derive the main result, Theorem~\ref{maintheore}.”

To prove the main theorem, the paper’s stated strategy is to (i) “construct a constant $\lambda_0>0$ and a sub-Markovian resolvent $(R_\alpha)_{\alpha\ge \lambda_0}$ associated with the principal elliptic operator $L^A={\rm trace}(A\nabla^2)$”, and then (ii) “show that the function $h$ can be represented as the local weak $H^{1,2}$-limit of the family $(\alpha R_\alpha h)_{\alpha\ge \lambda_0}$ as $\alpha\to\infty$.” The introduction emphasizes that the proof relies on “resolvent estimates under the weaker assumptions that the coefficients belong to $VMO_{loc}(\mathbb{R}^d)$ and that ${\rm div}A\in L^2_{loc}(\mathbb{R}^d,\mathbb{R}^d)$, which enable us to derive the main result.”

\label{maintheore}
Let $U \subset \mathbb{R}^d$ be a (possibly unbounded) open set with $d \geq 2$. Assume that $\mathbf{G} \in L^2_{loc}(U, \mathbb{R}^d)$ and $c \in L^1_{loc}(U)$. Let $A = (a_{ij})_{1 \leq i,j \leq d}$ be a symmetric matrix of functions such that $a_{ij} \in VMO_{loc}(U)$ for all $1 \leq i,j \leq d$, where $VMO_{loc}(U)$ is defined as follows.

\begin{defn} \label{defnvmo}
Let $\omega$ be a positive continuous function on $[0,\infty)$ with $\omega(0) = 0$. The space $VMO_\omega$ consists of all functions $g \in L^1_{loc}(\mathbb{R}^d)$ such that
\[
\sup_{z \in \mathbb{R}^d,\, r < R} \, r^{-2d} \int_{B_r(z)} \int_{B_r(z)} |g(x) - g(y)| \, dx \, dy \leq \omega(R) \quad \text{for all } R \in (0,\infty).
\]
Let $B$ be an open ball in $\mathbb{R}^d$. We define $VMO(B)$ as the set of all functions $f \in L^1(B)$ for which there exists an extension $\tilde{f} \in VMO_{\omega}$  (for some modulus $\omega$) such that $\tilde{f}|_B = f$. For an open set $U \subset \mathbb{R}^d$, the space $VMO_{loc}(U)$ is defined as the collection of all functions $f \in L^1_{loc}(U)$ such that $f \in VMO(B)$ for every open ball $B \subset \mathbb{R}^d$ with $\overline{B} \subset U$.
\end{defn}

Assume moreover that ${\rm div} A \in L^2_{loc}(U, \mathbb{R}^d)$, where the meaning of ${\rm div}$ for a matrix field is as follows.

\begin{defn} \label{defndivmat}
Let $U$ be an open subset of $\mathbb{R}^d$, $\mathbf{E} = (e_1, \dots, e_d) \in L^1_{loc}(U, \mathbb{R}^d)$ and $B = (b_{ij})_{1 \leq i,j \leq d}$ be a (possibly non-symmetric) matrix of functions with $b_{ij} \in L^1_{loc}(U)$ for all $1 \leq i,j \leq d$. Let $\mathbf{F} \in L^1_{loc}(U, \mathbb{R}^d)$ and $f \in L^1_{loc}(U)$.
\begin{itemize}
\item[(i)] We say that ${\rm div} B = \mathbf{E}$ if
\[
\int_U \sum_{i,j=1}^d b_{ij} \, \partial_i \phi_j \, dx = - \int_U \sum_{j=1}^d e_j \, \phi_j \, dx \quad \text{for all } (\phi_1, \dots, \phi_d) \in C_0^\infty(U)^d.
\]
\item[(ii)] We say that ${\rm div} \mathbf{F} = f$ if
\[
\int_U \langle \mathbf{F}, \nabla \varphi \rangle \, dx = - \int_U f \varphi \, dx \quad \text{for all } \varphi \in C_0^\infty(U).
\]
\end{itemize}
\end{defn}

Also assume that $A$ is locally uniformly strictly elliptic and bounded on $U$, i.e. for every open ball $V$ with $\overline{V} \subset U$, there exist constants $\lambda_V, M_V > 0$ such that
\begin{equation} \label{locellst}
\lambda_V \| \xi \|^2 \leq \langle A(x) \xi, \xi \rangle, \quad \;\; \max_{1 \leq i,j \leq d} |a_{ij}(x)| \leq M_V
\quad \text{for a.e. } x \in V \text{ and all } \xi \in \mathbb{R}^d.
\end{equation}
Let $\tilde{\mathbf{F}} \in L^2_{loc}(U, \mathbb{R}^d)$ and $\tilde{f} \in L^1_{loc}(U)$. Suppose that $h \in L^{\infty}_{loc}(U)$ satisfies
\[
\int_U \left({\rm trace}(A \nabla^2 \varphi) + \langle \mathbf{G}, \nabla \varphi \rangle + c \varphi \right) h \, dx
= \int_U \tilde{f} \varphi \, dx + \int_U \langle \tilde{\mathbf{F}}, \nabla \varphi \rangle \, dx \quad \text{for all } \varphi \in C_0^{\infty}(U).
\]
Then, in establishing the main theorem, $h \in H^{1,2}_{loc}(U).

Question:
Let U be a possibly unbounded open subset of R^d with d >= 2. Assume G ∈ L^2_loc(U, R^d), c ∈ L^1_loc(U), \tilde{F} ∈ L^2_loc(U, R^d), and \tilde{f} ∈ L^1_loc(U). Let A = (a_ij) be a symmetric matrix field on U such that each a_ij belongs to VMO_loc(U), meaning: for every open ball B with closure contained in U, the restriction a_ij|_B extends to some function g on R^d for which there exists a positive continuous modulus ω with ω(0) = 0 such that sup_{z in R^d, r < R} r^{-2d} \int_{B_r(z)} \int_{B_r(z)} |g(x) - g(y)| dx dy <= ω(R) for all R > 0. Assume also that div A ∈ L^2_loc(U, R^d) in the distributional sense, i.e. there exists E = (E_j) with \int_U \sum_{i,j=1}^d a_ij \partial_i \phi_j dx = -\int_U \sum_{j=1}^d E_j \phi_j dx for all (\phi_1, ..., \phi_d) ∈ C_0^∞(U)^d, and that A is locally uniformly strictly elliptic and bounded: for every open ball V with closure contained in U, there exist constants λ_V, M_V > 0 such that λ_V |ξ|^2 <= <A(x)ξ, ξ> and max_{i,j} |a_ij(x)| <= M_V for almost every x ∈ V and all ξ ∈ R^d. Suppose h ∈ L^∞_loc(U) satisfies, for every test function φ ∈ C_0^∞(U),
\int_U (trace(A ∇^2 φ) + <G, ∇φ> + c φ) h dx = \int_U \tilde{f} φ dx + \int_U <\tilde{F}, ∇φ> dx,
where trace(A ∇^2 φ) = \sum_{i,j=1}^d a_ij \partial_{ij} φ. Which statement holds for every such h?

Every such h belongs to H^{1,2}_loc(U); equivalently, h has locally square-integrable weak first derivatives on U.

Choices:
B. Every such h belongs to H^{1,p}_{loc}(U) for some p>2 depending only on d and the local ellipticity and VMO data of A; in particular, h has locally p-integrable weak first derivatives on U.
C. Every such h belongs to W^{1,1}_{loc}(U); in particular, h has locally integrable weak first derivatives on U.
D. Every such h belongs to H^{1,2}_{loc}(U) provided, for each open ball V with \overline{V}\subset U, the ellipticity and boundedness constants \lambda_V,M_V can be chosen independently of V; equivalently, the conclusion holds under a uniform strict ellipticity and boundedness condition on all of U.
E. Every such h belongs to H^{1,2}_{loc}(U) as soon as A is bounded and locally uniformly strictly elliptic and each a_{ij}\in L^{\infty}_{loc}(U), even without assuming either a_{ij}\in VMO_{loc}(U) or {\rm div}\,A\in L^2_{loc}(U,\mathbb{R}^d).

local versus global uniformity of ellipticity/boundedness

use of VMO and div A in resolvent estimates

The stem does not explicitly state the conclusion that h is in H^{1,2}_{loc}(U). Although the setup strongly signals a specific regularity theorem, the correct answer is not leaked verbatim in the stem.

This is essentially a direct theorem-recall item: the stem lists the full hypotheses of a regularity result and option A states the theorem’s conclusion almost verbatim.

There is some reasoning pressure because the student must distinguish the exact guaranteed regularity from stronger, weaker, or hypothesis-altered alternatives. But the task is still mainly recognition/application of a known theorem rather than genuine derivation.

The distractors are mathematically plausible and target common failure modes: overclaiming higher integrability, settling for a weaker true statement, confusing local with global uniformity, or dropping essential VMO/div A assumptions.

A mathematically well-constructed theorem-application MCQ with strong distractors, but it is too close to a direct restatement of the underlying theorem to strongly test generative reasoning.