# Analytical validation of the helicity conservation for the compressible
Euler equations

Yanqing Wang111 College of Mathematics and Information Science, Zhengzhou
University of Light Industry, Zhengzhou, Henan 450002, P. R. China Email:
<EMAIL_ADDRESS>Wei Wei222School of Mathematics and Center for
Nonlinear Studies, Northwest University, Xi’an, Shaanxi 710127, P. R. China
Email<EMAIL_ADDRESS>and Yulin Ye333Corresponding author. School of
Mathematics and Statistics, Henan University, Kaifeng, 475004, P. R. China.
Email<EMAIL_ADDRESS>
###### Abstract

In [25], Moffatt introduced the concept of helicity in an inviscid fluid and
examined the helicity preservation of smooth solution to barotropic
compressible flow. In this paper, it is shown that the weak solutions of the
above system in Onsager type spaces $\dot{B}^{1/3}_{p,c(\mathbb{N})}$
guarantee the conservation of the helicity. The parallel results of
homogeneous incompressible Euler equations and the surface quasi-geostrophic
equation are also obtained.

MSC(2020): 35Q30, 35Q35, 76D03, 76D05
Keywords: compressible Euler equations; incompressible Euler equations;
helicity conservation; surface quasi-geostrophic equation

##  1 Introduction

The Euler equations describing incompressible inviscid fluids with constant
density in $\Omega\times(0,\,T)$ read

$\left\\{\begin{aligned} &v_{t}+v\cdot\nabla v+\nabla\Pi=0,\\\
&\text{div}\,v=0,\\\ &v|_{t=0}=v_{0}(x),\end{aligned}\right.$ (1.1)

where the unknown vector $v$ represents the flow velocity field, and the
scalar function $\Pi=-\Delta^{-1}\partial_{i}\partial_{j}(v_{i}v_{j})$ stands
for the pressure. The initial velocity $v_{0}$ satisfies
$\text{div}\,v_{0}=0$. Let $\Omega$ be the whole space $\mathbb{R}^{d}$ or the
periodic domain $\mathbb{T}^{d}$, where $d\geq 2$ is the spatial dimension. We
denote $\omega=\text{curl\,}v$ by the vorticity of the flow velocity in
(LABEL:Euler), whose equations read

$\omega_{t}+v\cdot\nabla\omega-\omega\cdot\nabla
v=0,~{}~{}\text{div\,}\omega=0.$ (1.2)

It is well known that there exist two physically conserved quantities for the
regular solutions of the Euler system (LABEL:Euler):

  * •

energy conservation

$\int_{\Omega}|v(x,t)|^{2}dx=\int_{\Omega}|v(x,0)|^{2}dx.$ (1.3)

  * •

helicity preservation

$\int_{\Omega}\omega(x,t)\cdot v(x,t)dx=\int_{\Omega}\omega(x,0)\cdot
v(x,0)dx.$ (1.4)

Besides the above two conserved laws of the Euler equations (LABEL:Euler), one
can find others in [24, p.24]. In a seminal paper [27], Onsager conjectured
that weak solutions of the Euler equation with Hölder continuity exponent
$\alpha>\frac{1}{3}$ do conserve energy and that turbulent or anomalous
dissipation occurs when $\alpha\leq\frac{1}{3}$. The first attempt to the
energy conservation issue for weak solutions of the Euler equations
(LABEL:Euler) was given by Eyink [18] in a stronger Hölder space
$C_{*}^{\alpha}$ with $\alpha>\frac{1}{3}$, which is included in $C^{\alpha}$.
Subsequently Constantin-E-Titi [11] successfully solved the positive part of
the Onsager’s conjecture and proved that the energy is conserved if a weak
solution $v$ is in the Besov space
$L^{3}(0,T;B^{\alpha}_{3,\infty}(\mathbb{T}^{3}))$ with $\alpha>1/3$. Later
on, by deriving a local energy equation which contains a term $D(v)$
representing the dissipation or production of energy caused by the lack of
smoothness of solution, Duchon and Robert gave a weaker condition on the
solutions to conserve energy in [17], that is, if $v$ satisfies
$\int|v(t,x+\xi)-v(t,x)|^{3}dx\leq C(t)|\xi|\sigma(|\xi|)$, where $C(t)\in
L^{1}(0,T)$ and $\sigma(a)\rightarrow 0$ as $a\rightarrow 0$, then energy is
conserved. Recently, Cheskidov-Constantin-Friedlander-Shvydkoy [8] refined the
above spaces to the critical space $L^{3}(0,T;B^{1/3}_{3,c(\mathbb{N})})$,
where $B^{1/3}_{3,c(\mathbb{N})}=\\{v\in
B^{1/3}_{3,\infty}:\lim_{q\rightarrow\infty}2^{q}\|\Delta_{q}v\|^{3}_{L^{3}}=0\\}$
and $\Delta_{q}$ represents a smooth restriction of $v$ into Fourier modes of
order $2^{q}$. In the follow-up paper, Shvydkoy [28, 29] stated that this
condition of energy conservation can be equivalent to

$\lim_{z\rightarrow
0}\frac{\int_{0}^{T}\int_{\mathbb{R}^{d}}|v(x+z,t)-v(x,t)|^{3}dxdt}{|z|}=0.$

It is worth remarking that $B^{1/3}_{3,q}$ for $1\leq q<\infty$ are included
in $B^{1/3}_{3,c(\mathbb{N})}$, which is called as the Onsager-critical space.
In this direction, the recent progress can be found in [20]. The proof of the
other part of Onsager’s conjecture in three-dimensional space is given by
Isett [21], which adopted the approach developed by De Lellis and Szekelyhidi
[13, 14, 12, 15].

Next, we turn our attention to the helicity conservation (1.4) in the Euler
equations. The concept of the helicity in an inviscid fluid was introduced by
Moffatt in [25]. As pointed in [25, 26], helicity is important at a
fundamental level in relation to flow kinematics because it admits topological
interpretation in relation to the linkage or linkages of vortex lines of the
flow. Indeed, Moffatt [25] examined that the helicity of smooth solutions to
the following compressible Euler equations (LABEL:CEuler) is invariant in
time,

$\left\\{\begin{aligned} &\rho_{t}+\nabla\cdot(\rho v)=0,\\\ &(\rho
v)_{t}+\mathrm{div}\,(\rho v\otimes v)+\nabla\pi(\rho)=0.\end{aligned}\right.$
(1.5)

The proof in [25] relies on the transport formula involving moving region. To
the best of our knowledge, there has been little literature concerning the
helicity preservation of weak solutions to the compressible Euler equations
(LABEL:CEuler). In this direction, most previous works focus on the
homogeneous incompressible Euler equations (LABEL:Euler). In particular, the
study of helicity conservation for the incompressible Euler equations
(LABEL:Euler) was originated from Chae’s interesting works [5, 4]. In [4],
Chae considered the preservation of the helicity of weak solutions to the
Euler equations via $\omega$ in spaces
$L^{3}(0,T;B^{\alpha}_{\frac{9}{5},\infty})$ with $\alpha>\frac{1}{3}$.
Subsequently, it is shown that $v\in
L^{r_{1}}(0,T;\dot{B}^{\alpha}_{\frac{9}{2},q})$ and $\omega\in
L^{r_{2}}(0,T;\in\dot{B}^{\alpha}_{\frac{9}{5},q})$, with
$\alpha>\frac{1}{3}$, $q\in[2,\infty]$, $r_{1}\in[2,\infty]$,
$r_{2}\in[1,\infty]$ and $\frac{2}{r_{1}}+\frac{1}{r_{2}}=1$, ensure the
helicity conservation of weak solutions in [5]. After that, Cheskidov-
Constantin-Friedlander-Shvydkoy [8] established the helicity conservation
class based on the velocity $v$ in Onsager critical space

$L^{3}(0,T;B^{\frac{2}{3}}_{3,c(\mathbb{N})}).$ (1.6)

Very recently, De Rosa [16] proved that the helicity is a constant provided
that $v\in L^{2r}(0,T;W^{\theta,2p})$ and $\omega\in
L^{\kappa}(0,T;W^{\alpha,q})$ with
$\frac{1}{p}+\frac{1}{q}=\frac{1}{r}+\frac{1}{\kappa}$ and $2\theta+\alpha\geq
1$. Notice that there holds the embedding relation
$W^{\theta,p}\hookrightarrow B^{\theta}_{p,c(\mathbb{N})}$, which can be found
in [16, 8]. The first objective of this paper is to address the question how
much regularity is needed for a weak solution of the compressible Euler
equations (LABEL:CEuler) to conserve the helicity. To this end, we shall need
to give the equation for the vorticity $\omega$ from (LABEL:CEuler). Our
observation is that for any smooth solutions $(\rho,v)$ of compressible Euler
equations (LABEL:CEuler) with $0<c_{1}\leq\rho\leq c_{2}<\infty$, dividing the
both sides of the momentum equation $\eqref{CEuler}_{2}$ by $\rho$, we can
reformulate the system (LABEL:CEuler) as

$\left\\{\begin{aligned} &\rho_{t}+\nabla\cdot(\rho v)=0,\\\
&v_{t}+v\cdot\nabla v+\nabla\Pi(\rho)=0,\end{aligned}\right.$ (1.7)

where $\Pi(\rho)=\int_{1}^{\rho}\frac{\pi^{\prime}(s)}{s}ds$. Then the
momentum equation in (LABEL:rCEuler1) is the same as that in the
incompressible case. However, this is not valid for weak solutions $(\rho,v)$
even with $0<c_{1}\leq\rho\leq c_{2}<\infty$. To use the equations
(LABEL:rCEuler1), we shall apply the techniques due to Feireisl, Gwiazda,
Swierczewska-Gwiazda and Wiedemann in [19] to show that the weak solutions of
compressible Euler equations (LABEL:CEuler) in Onsager type spaces
$B^{1/3}_{p,c(\mathbb{N})}$ are also the weak solutions of equations
(LABEL:rCEuler1). Now we state our first main result on the helicity
conservation of weak solutions for the compressible Euler equations
(LABEL:rCEuler1) as follows.

######  Theorem 1.1.

Let $(\rho,v)$ be a solution of (LABEL:CEuler) in the sense of distributions
and $\text{div\,}v,\text{curl\,}v\in C([0,T];L^{\frac{2d}{d+1}}(\Omega))$.
Assume that there exist two positive constants $c_{1}$ and $c_{2}$ such that

$\displaystyle 0<c_{1}\leq\rho\leq c_{2}<\infty,~{}~{}\rho\in
L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,c(\mathbb{N})}),~{}\rho v\in
L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,\infty}),$ (1.8) $\displaystyle v\in
L^{3}(0,T;B^{\frac{1}{3}}_{3,c(\mathbb{N})}),\pi\in C^{2}[c_{1},c_{2}],$

and one of the following four conditions is satisfied

  1. (1)

$\omega\in L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,\infty});$

  2. (2)

$\omega\in L^{3}(0,T;L^{3}(\Omega));$

  3. (3)

$v\in L^{\frac{p}{p-2}}(0,T;L^{\frac{q}{q-2}}(\Omega)),\omega\in
L^{p}(0,T;L^{q}(\Omega)),$ with $2<p,q<\infty$;

  4. (4)

$\text{div\,}v,\omega\in L^{3}(0,T;L^{\frac{9}{4}}(\Omega)),$ with $d=3$.

Then the helicity of the compressible Euler equation (LABEL:CEuler) is
conserved, that is, for any $0<t<T$,

$\int_{\Omega}\omega(x,t)\cdot v(x,t)dx=\int_{\Omega}\omega(x,0)\cdot
v(x,0)dx.$

######  Remark 1.1.

The helicity conservation of weak solutions for the compressible Euler
equations (LABEL:CEuler) in Onsager’s critical space
$B^{\theta}_{p,c(\mathbb{N})}$ and $B^{\theta}_{p,\infty}$ in terms of the
velocity and the vorticity is studied in Theorem 1.2 and Corollary 1.3. This
is consistent with Moffatt’s work [25]. The regularity condition that
$\text{div\,}v,\text{curl\,}v\in C([0,T];L^{\frac{2d}{d+1}}(\Omega))$ is
necessary to ensure that the helicity is well defined.

######  Remark 1.2.

The constraint that $\rho v\in L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,\infty})$
can be removed, if the condition that $\rho\in
L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,c(\mathbb{N})})$ is replaced by $\rho\in
L^{\infty}(0,T;\dot{B}^{\frac{1}{3}}_{\infty,c(\mathbb{N})})$ due to Lemma
2.5.

######  Remark 1.3.

This theorem is valid for the compressible isentropic Euler system. Indeed,
for the isentropic pressure law $\pi(\rho)=\kappa\rho^{\gamma}$ with
$\gamma>1$ and $\kappa=\frac{(\gamma-1)^{2}}{4\gamma},$ the pressure condition
that $\pi\in C^{2}[c_{1},c_{2}]$ is obviously satisfied if
$0<c_{1}\leq\rho\leq c_{2}<\infty$.

######  Remark 1.4.

The sufficient conditions (3) in this theorem and (4)-(5) in Theorem 1.2 below
are partially motivated by the recent works [31] on energy balance of the
Navier-Stokes equations based on the combination of the velocity and its
gradient.

######  Remark 1.5.

Since the following embedding relations in $\mathbb{R}^{3}$ is valid

$W^{1,\frac{9}{4}}\hookrightarrow H^{\frac{7}{6}}\hookrightarrow
B^{\frac{2}{3}}_{3,c(\mathbb{N})},$

the fourth criterion of this theorem is also in Onsager’s critical space. It
is an interesting question to show the velocity $v$ in Onsager’s critical
space $L^{3}(0,T;B^{\frac{2}{3}}_{3,c(\mathbb{N})})$ keeping the helicity
conservation for the compressible Euler equations.

In what follows, we formulate our second main result involving helicity
conservation for the incompressible Euler equations (LABEL:Euler).

######  Theorem 1.2.

Let $v$ be a weak solution of incompressible Euler equations (LABEL:Euler) in
the sense of Definition 2.2 and $\omega\in
C([0,T];L^{\frac{2d}{d+1}}(\Omega))$. Then the helicity conservation (1.4) is
valid provided that one of the following five conditions is satisfied

  1. (1)

$v\in L^{k}(0,T;\dot{B}^{\alpha}_{p,c(\mathbb{N})}),\omega\in
L^{\ell}(0,T;\dot{B}^{\beta}_{q,\infty}),$ with
$\frac{2}{k}+\frac{1}{\ell}=1,\frac{2}{p}+\frac{1}{q}=1,2\alpha+\beta\geq 1;$

  2. (2)

$v\in L^{k}(0,T;\dot{B}^{\alpha}_{p,\infty}),\omega\in
L^{\ell}(0,T;\dot{B}^{\beta}_{q,c(\mathbb{N})}),$ with
$\frac{2}{k}+\frac{1}{\ell}=1,\frac{2}{p}+\frac{1}{q}=1,2\alpha+\beta\geq 1;$

  3. (3)

$\omega\in L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{\frac{3d}{d+2},c(\mathbb{N})});$

  4. (4)

$v\in L^{\frac{p}{p-2}}(0,T;L^{\frac{q}{q-2}}(\Omega)),\omega\in
L^{p}(0,T;L^{q}(\Omega)),$ with $2<p,q<\infty$;

  5. (5)

$v\in
L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\Omega)),\text{curl}\,\omega\in
L^{p}(0,T;L^{q}(\Omega)),$ with $1\leq p\leq\infty,1\leq q<\infty$.

######  Remark 1.6.

Theorem 1.2 is an improvement of corresponding results in [16, 5, 4].

######  Remark 1.7.

An approach to (1.6) in homogeneous Besov spaces is presented in the proof of
Theorem 1.2.

Recently, when dimension $d=3$, the authors in [23] provide a proof of energy
conservation criteria for the Euler equations in terms of the vorticity

$\omega\in L^{3}(0,T;L^{\frac{9}{5}}(\Omega)),$ (1.9)

which is also deduced from Cheskidov-Constantin-Friedlander-Shvydkoy’s
classical condition that $L^{3}(0,T;B^{1/3}_{3,c(\mathbb{N})})$ (see also
related work [9]). In the spirit of [23], we have

######  Corollary 1.3.

Let $v$ be a weak solution of the 3D incompressible Euler equations
(LABEL:Euler) in the sense of Definition 2.2 and $\omega\in
C([0,T];L^{\frac{3}{2}}(\Omega))$. Then the helicity conservation (1.4) is
valid provided that one of the following two conditions is satisfied

  1. (1)

$\omega\in L^{3}(0,T;L^{\frac{9}{4}}(\Omega))$;

  2. (2)

$\text{curl\,}\omega\in L^{3}(0,T;L^{\frac{9}{7}}(\Omega))$.

######  Remark 1.8.

From (1.9) and the condition (1) in Corollary 1.3, we see that there may exist
a weak solution of the Euler equations that keep the energy rather than the
helicity. This is also previously pointed out by Chae in [4].

######  Remark 1.9.

In dimension two, the criterion (1) in Corollary 1.3 can be replaced by a new
condition that $\omega\in L^{3}(0,T;L^{2^{+}}(\Omega))$.

The starting point of (1) in Theorem 1.1 and (1)-(2) in Theorem 1.2 is the
study of the cross-helicity in the ideal MHD equations. Precisely, for any
vector functions $A,B$, there hold the identities below

$\displaystyle\nabla(A\cdot B)=A\cdot\nabla B+B\cdot\nabla
A+A\times\text{curl}B+B\times\text{curl}A,$ $\displaystyle\text{curl}(A\times
B)=A\text{div}B-B\text{div}A+B\cdot\nabla A-A\cdot\nabla B,$

which together with the condition that $\mathrm{div}\,\omega=0$ lead to

$\displaystyle v\cdot\nabla v=\frac{1}{2}\nabla|v|^{2}+\omega\times v,$ (1.10)
$\displaystyle\text{curl}(\omega\times
v)=\omega\text{div}v+v\cdot\nabla\omega-\omega\cdot\nabla v.$

Then we can use (1.10) to reformulate the compressible Euler system
(LABEL:rCEuler1) and its vorticity equations as

$\left\\{\begin{aligned} &v_{t}+\omega\times
v+\nabla(\Pi(\rho)+\frac{1}{2}|v|^{2})=0,\\\
&\omega_{t}-\text{curl}(v\times\omega)=0,\\\
&\text{div\,}\omega=0,\end{aligned}\right.$ (1.11)

which is very closed to the inviscid ideal Magnetohydrodynamics (MHD)
equations

$\left\\{\begin{aligned} &v_{t}+\omega\times v-b\cdot\nabla
b+\nabla(p+\frac{1}{2}|v|^{2}+\frac{1}{2}|b|^{2})=0,\\\
&b_{t}-\text{curl}(v\times b)=0,\\\
&\mathrm{div}\,v=\mathrm{div}\,b=0,\end{aligned}\right.$

where $b$ describes the magnetic field. For this ideal MHD system, the cross-
helicity $\int_{\Omega}v(x,t)\cdot h(x,t)dx$ is conserved for the smooth
solutions. To the knowledge of the authors, Yu [34] obtained cross-helicity
conservation criterion of weak solutions for ideal MHD equations based on the
condition that $v\in L^{3}(0,T;B^{\alpha_{1}}_{3,\infty})$ and $b\in
L^{3}(0,T;B^{\alpha_{2}}_{3,c(\mathbb{N})})$ with $\alpha_{1}+2\alpha_{2}\geq
1$ and $\alpha_{2}\geq\frac{1}{3}$ (see also [22, 30, 3]). Inspired by Yu’s
work [34], we consider the helicity conservation of weak solutions to the
incompressible Euler equations (LABEL:Euler) in spaces
$B^{\theta}_{p,c(\mathbb{N})}$ and $B^{\theta}_{p,\infty}$. It should be
pointed out that this coincides with the appearance of helicity conservation
of weak solutions to the compressible Euler equations (LABEL:CEuler) motivated
by the cross-helicity conservation in the MHD system (see [25, 26]). The proof
of (1) in Theorem 1.1 and (1)-(2) in Theorem 1.2 is very closed to that in [8,
34]. The key point is our new observation that functions in the Onsager type
spaces $\dot{B}^{1/3}_{p,c(\mathbb{N})}$ can satisfy the corresponding
Constantin-E-Titi type commutator estimates in physical spaces (see Lemma
2.6). It seems that the argument presented here is quite general and can be
applied in other fluid equations. Particularly, we shall show that our
approach here can be applied in the 2D surface quasi-geostrophic (SQG)
equation below,

$\left\\{\begin{aligned}
&\theta_{t}+v\cdot\nabla\theta=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{in
}~{}(0,T)\times\Omega,\\\
&v=\nabla^{\perp}\Psi=(-\frac{\partial\Psi}{\partial_{x_{2}}},\frac{\partial\Psi}{\partial_{x_{1}}}),~{}\theta=-(-\Delta)^{\frac{1}{2}}\Psi,~{}~{}~{}~{}~{}\text{in
}~{}[0,T)\times\Omega,\\\
&\theta|_{t=0}=\theta_{0},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{in
}~{}\Omega,\end{aligned}\right.$ (1.12)

where the unknown scalar function
$\theta(x,t)\colon\mathbb{R}^{2}\times[0,\infty)\to\mathbb{R}$ stands for the
temperature, $v$ is the velocity field, and
$\Psi=-\int_{\Omega}\frac{\theta(y,t)}{|x-y|}dy$ is the stream function. Here
we assume that $\Omega$ is the whole space $\mathbb{R}^{2}$ or torus
$\mathbb{T}^{2}$ with periodic boundary conditions. The surface quasi-
geostrophic equation arises in geophysical fluids and shares many striking
similarities with the 3D Euler equations (see [10] and references therein).
Zhou [35] studied the following general helicity

$\int_{\Omega}\theta\partial_{i}\theta dx,i=1,2,$

of weak solutions for the 2D surface quasi-geostrophic equation (LABEL:qg). It
is shown that if $\nabla\theta\in C([0,T];L^{\frac{4}{3}})\cap
L^{3}(0,T;B^{\alpha}_{\frac{3}{2},\infty})$ for $\alpha>1/3$, then the general
helicity of weak solutions for 2-D surface quasi-geostrophic equation is
conserved in [35]. In the present paper, we are going to give some new
sufficient conditions to guarantee the conservation of the helicity for weak
solutions to the 2-D surface quasi-geostrophic equation in Onsager-critical
spaces. Our third main result can be stated as follows:

######  Theorem 1.4.

Let $\theta$ be a weak solution to the 2D surface quasi-geostrophic equation
(LABEL:qg) in the sense of Definition 2.3 and $\nabla\theta\in
C([0,T];L^{\frac{4}{3}}(\Omega))$. Then the helicity conservation

$\int_{\Omega}\theta(x,t)\partial_{i}\theta(x,t)dx=\int_{\Omega}\theta_{0}(x)\partial_{i}\theta_{0}(x)dx,i=1,2,$
(1.13)

is valid provided that $\nabla\theta\in
L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{\frac{3}{2},c(\mathbb{N})})\cap
C([0,T];L^{\frac{4}{3}}(\Omega)).$

######  Remark 1.10.

According to the Bernstein inequality, one may replace the condition that
$\nabla\theta\in L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{\frac{3}{2},c(\mathbb{N})})$
by $\theta\in L^{3}(0,T;\dot{B}^{\frac{4}{3}}_{\frac{3}{2},c(\mathbb{N})})$ in
this theorem.

The rest of this paper is organized as follows. In Section 2, we present some
notations and auxiliary lemmas which will be used in the present paper.
Particularly, in the spirit of [34], we shall show that the functions in
Onsager type spaces $\dot{B}^{\frac{1}{3}}_{p,c(\mathbb{N})}$ mean the
Constantin-E-Titi type commutator estimates in physical spaces, which plays an
important role in the follow-up study. Section 3 and Section 4 are devoted to
the proof of helicity conservation of weak solutions to the compressible Euler
equations and the homogeneous incompressible Euler equations, respectively.
The helicity conservation of weak solutions to the 2-D surface quasi-
geostrophic equation is considered in Section 5.

##  2 Notations and some auxiliary lemmas

First, we introduce some notations used in this paper. For $p\in[1,\,\infty]$,
the notation $L^{p}(0,\,T;X)$ stands for the set of measurable functions $f$
on the interval $(0,\,T)$ with values in $X$ and $\|f\|_{X}$ belonging to
$L^{p}(0,\,T)$. The classical Sobolev space $W^{k,p}(\Omega)$ is equipped with
the norm
$\|f\|_{W^{k,p}(\Omega)}=\sum\limits_{|\alpha|=0}^{k}\|D^{\alpha}f\|_{L^{p}(\Omega)}$.
$\mathcal{S}$ represents the Schwartz class of rapidly decreasing functions,
$\mathcal{S}^{\prime}$ the space of tempered distributions,
$\mathcal{S}^{\prime}/\mathcal{P}$ the quotient space of tempered
distributions which modulo polynomials. We use $\mathcal{F}f$ or $\widehat{f}$
to denote the Fourier transform of a tempered distribution $f$. $a\approx b$
means that $C^{-1}b\leq a\leq Cb$ for some constant $C>1$. For simplicity, we
write

$\int_{0}^{t}\int_{\Omega}f(t,x)dxds=\int_{0}^{t}\int f\
~{}~{}\text{and}~{}~{}\|f\|_{L^{p}(0,T;X(\Omega))}=\|f\|_{L^{p}(0,T;X)}.$

To define Besov spaces, we need the following dyadic unity partition (see e.g.
[1]). Choose two nonnegative radial functions $\varrho$, $\varphi\in
C^{\infty}(\mathbb{R}^{d})$ supported respectively in the ball
$\\{\xi\in\mathbb{R}^{d}:|\xi|\leq\frac{3}{4}\\}$ and the shell
$\\{\xi\in\mathbb{R}^{n}:\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\\}$ such that

$\varrho(\xi)+\sum_{j\geq
0}\varphi(2^{-j}\xi)=1,\quad\forall\xi\in\mathbb{R}^{d};\qquad\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1,\quad\forall\xi\neq
0.$

Then for every $\xi\in\mathbb{R}^{d},$
$\varphi(\xi)=\varrho(\xi/2)-\varrho(\xi)$. Denote $h=\mathcal{F}^{-1}\varphi$
and $\tilde{h}=\mathcal{F}^{-1}\varrho$, then nonhomogeneous dyadic blocks
$\Delta_{j}$ are defined by

$\Delta_{j}u:=0~{}~{}\text{if}~{}~{}j\leq-2,~{}~{}\Delta_{-1}u:=\varrho(D)u=\int_{\mathbb{R}^{d}}\tilde{h}(y)u(x-y)dy,$
$\text{and}~{}~{}\Delta_{j}u:=\varphi\left(2^{-j}D\right)u=2^{jd}\int_{\mathbb{R}^{d}}h(2^{j}y)u(x-y)dy~{}~{}\text{if}~{}~{}j\geq
0.$

The nonhomogeneous low-frequency cut-off operator $S_{j}$ is defined by

$S_{j}u:=\sum_{k\leq j-1}\Delta_{k}u.$

The homogeneous dyadic blocks $\dot{\Delta}_{j}$ and homogeneous low-frequency
cut-off operators $\dot{S}_{j}$ are defined for every $j\in\mathbb{Z}$ by

$\dot{\Delta}_{j}u:=\varphi(2^{-j}D)u=2^{jd}\int_{\mathbb{R}^{d}}h(2^{j}y)u(x-y)dy,$
$\text{ and
}~{}~{}\dot{S}_{j}u:=\varrho(2^{-j}D)u=2^{jd}\int_{\mathbb{R}^{d}}\tilde{h}(2^{j}y)u(x-y)dy.$

Then for $-\infty<s<\infty$ and $1\leq p,q\leq\infty,$ the homogeneous Besov
semi-norm $\|f\|_{\dot{B}^{s}_{p,q}}$ of
$f\in\mathcal{S}^{\prime}/\mathcal{P}$ is given by

$\displaystyle\left\|f\right\|_{\dot{B}^{s}_{p,q}}:=\left\\{\begin{array}[]{lll}\left(\sum_{j\in\mathbb{Z}}2^{jqs}\left\|\dot{\Delta}_{j}f\right\|_{L^{p}}^{q}\right)^{1/q},~{}~{}\text{if}\
q\in[1,\infty),\\\
\sup_{j\in\mathbb{Z}}2^{js}\left\|\dot{\Delta}_{j}f\right\|_{L^{p}},~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{if}~{}q=\infty.\end{array}\right.$

Moreover, for $s\in\mathbb{R}$ and $1\leq p,q\leq\infty$, we define the
inhomogeneous Besov norm $\left\|f\right\|_{B^{s}_{p,q}}$ of
$f\in\mathcal{S}^{{}^{\prime}}$ as

$\left\|f\right\|_{B^{s}_{p,q}}=\left\|f\right\|_{{L^{p}}}+\left\|f\right\|_{\dot{B}^{s}_{p,q}}.$

Motivated by [8], we denote $\dot{B}^{\alpha}_{p,c(\mathbb{N})}$ with
$\alpha\in\mathbb{R}$ and $1\leq p\leq\infty$ as the class of all tempered
distributions $f$ satisfying

$\left\|f\right\|_{\dot{B}^{\alpha}_{p,\infty}}<\infty~{}\text{and}~{}\lim_{j\rightarrow\infty}2^{j\alpha}\left\|\dot{\Delta}_{j}f\right\|_{L^{p}}=0.$
(2.1)

It is clear that the Besov spaces $\dot{B}^{\alpha}_{p,q}$ are included in
$\dot{B}^{\alpha}_{p,c(\mathbb{N})}$ for any $1\leq q<\infty$. For more
background on harmonic analysis in the context of fluids, we refer the readers
to [6].

Furthermore, we let $\eta_{\varepsilon}:\mathbb{R}^{d}\rightarrow\mathbb{R}$
be a standard mollifier, i.e. $\eta(x)=C_{0}e^{-\frac{1}{1-|x|^{2}}}$ for
$|x|<1$ and $\eta(x)=0$ for $|x|\geq 1$, where $C_{0}$ is a constant such that
$\int_{\mathbb{R}^{d}}\eta(x)dx=1$. For $\varepsilon>0$, the rescaled
mollifier
$\eta_{\varepsilon}(x)=\frac{1}{\varepsilon^{d}}\eta(\frac{x}{\varepsilon})$,
and for any function $f\in L^{1}_{loc}(\mathbb{R}^{d})$, its mollified version
is defined by

$f^{\varepsilon}(x)=(f*\eta_{\varepsilon})(x)=\int_{\mathbb{R}^{d}}f(x-y)\eta_{\varepsilon}(y)dy,\
\ x\in\mathbb{R}^{d}.$

Subsequently, we collect some Lemmas which will be used in the present paper.

######  Lemma 2.1 (Bernstein inequality).

For any $b\geq a\geq 1$, there holds that

$\left\|\dot{\Delta}_{j}f\right\|_{L^{b}}\leq
2^{jd(\frac{1}{a}-\frac{1}{b})}\left\|\dot{\Delta}_{j}f\right\|_{L^{a}}.$
(2.2)

The following lemma gives a characterization of Besov spaces in terms of
finite differences in the same spirit of [28, 34, 17].

######  Lemma 2.2.

Let $\alpha\in(0,1)$ and $p\in[1,\infty]$, then for any
$f\in\mathcal{S}^{{}^{\prime}}_{h}$, there holds

$f\in{\dot{B}^{\alpha}_{p,\infty}}\Longleftrightarrow
ess\sup_{|y|>0}\frac{\|f(\cdot-y)-f(\cdot)\|_{L^{p}}}{|y|^{\alpha}}<\infty;$
(2.3)
$\lim_{j\rightarrow\infty}2^{j\alpha}\left\|\dot{\Delta}_{j}f\right\|_{L^{p}}=0\Longleftrightarrow\lim\limits_{|y|\rightarrow
0}\frac{\|f(\cdot-y)-f(\cdot)\|_{L^{p}}}{|y|^{\alpha}}=0.$ (2.4)

###### Proof.

(1) “LHS$\Rightarrow$ RHS” in (2.3). First, by the mean value theorem, the
Bernstein inequality and the Minkowski inequality, we have for any tempered
distribution $f$ in $\mathbb{R}^{d}$,

$\|f(\cdot-y)-f(\cdot)\|_{L^{p}}\leq C\left(\sum_{j\leq
N}2^{j}|y|\|\dot{\Delta}_{j}f\|_{L^{p}}+\sum_{j>N}\|\dot{\Delta}_{j}f\|_{L^{p}}\right).$
(2.5)

Before going further, just as [8], we set the following localized kernel

$K(j)=\left\\{\begin{aligned}
&2^{j\alpha},~{}~{}~{}~{}~{}~{}~{}~{}\text{if}~{}j\leq 0,\\\
&2^{-(1-\alpha)j},~{}\text{if}~{}j>0,\end{aligned}\right.$

and denote $d_{j}=2^{j\alpha}\|\dot{\Delta}_{j}f\|_{L^{p}}$.

As a consequence, using the mean value theorem, the Bernstein inequality and
the Minkowski inequality again, we rewrite (2.5) as

$\displaystyle\|f(\cdot-y)-f(\cdot)\|_{L^{p}}$ $\displaystyle\leq$
$\displaystyle C\left(2^{N(1-\alpha)}|y|\sum_{j\leq
N}2^{-(N-j)(1-\alpha)}2^{j\alpha}\|\dot{\Delta}_{j}f\|_{L^{p}}+2^{-\alpha
N}\sum_{j>N}2^{(N-j)\alpha}2^{j\alpha}\|\dot{\Delta}_{j}f\|_{L^{p}}\right)$
$\displaystyle\leq$ $\displaystyle C\left(2^{N(1-\alpha)}|y|+2^{-\alpha
N}\right)(K\ast d_{j})(N).$

Thanks to the localized kernel $K(\cdot)\in l^{1}$, by choosing $N$ such that

$2^{N(1-\alpha)}|y|\approx 2^{-N\alpha},$

we arrive at

$\|f(\cdot-y)-f(\cdot)\|_{L^{p}}\leq C|y|^{\alpha}(K\ast d_{j})(N)\leq
C|y|^{\alpha}\sup_{j\in\mathbb{Z}}d_{j}.$ (2.6)

This implies that
$ess\sup_{|y|>0}\frac{\|f(\cdot-y)-f(\cdot)\|_{L^{p}}}{|y|^{\alpha}}\leq
C\|f\|_{\dot{B}^{\alpha}_{p,\infty}}$.

Now we will prove the reverse inequality “LHS $\Longleftarrow$ RHS” in (2.3).
As the mean value of the function $h$ is $0$, we can write

$\displaystyle\dot{\Delta}_{j}f(x)$ $\displaystyle=2^{jd}\int
h(2^{j}y)f(x-y)dy$ (2.7) $\displaystyle=2^{jd}\int
h(2^{j}y)\left(f(x-y)-f(x)\right)dy,$

which together with Minkowski inequality yields that

$\displaystyle 2^{j\alpha}\|\dot{\Delta}_{j}f\|_{L^{p}}$ $\displaystyle\leq
2^{jd}\int 2^{j\alpha}|h(2^{j}y)|\|f(\cdot-y)-f(\cdot)\|_{L^{p}}dy$ (2.8)
$\displaystyle\leq
ess\sup_{y\in\mathbb{R}^{d}}\frac{\|f(\cdot-y)-f(\cdot)\|_{L^{p}}}{|y|^{\alpha}}\int
2^{j(\alpha+d)}|y|^{\alpha}|h(2^{j}y)|dy$ $\displaystyle\leq
C\,ess\sup_{y\in\mathbb{R}^{d}}\frac{\|f(\cdot-y)-f(\cdot)\|_{L^{p}}}{|y|^{\alpha}}.$

Then we conclude the result (2.3).

(2) The proof of (2.4) is similar to (2.3). Due to (2.6) and (2.8), we
immediately obtain the result (2.4) after taking the limits. We complete the
proof of this lemma. ∎

As a corollary of Lemma 2.2, we have the following properties.

######  Corollary 2.3.

Let $0<\alpha,\beta<1$ and $1\leq q_{1},q_{2}\leq\infty$. Assume that
$f\in\dot{B}^{\alpha}_{q_{1},\infty}$ and
$g\in\dot{B}^{\beta}_{q_{2},c(\mathbb{N})}$, then we have

$\displaystyle f\in\dot{B}^{\alpha}_{q_{1},\infty}\Leftrightarrow
ess\sup_{|y|>0}\frac{\|f(\cdot-y)-f(\cdot)\|_{L^{q_{1}}}}{|y|^{\alpha}}<\infty,$
(2.9) $\displaystyle g\in\dot{B}^{\beta}_{q_{2},c(\mathbb{N})}\Leftrightarrow
ess\sup_{|y|>0}\frac{\|g(\cdot-y)-g(\cdot)\|_{L^{q_{2}}}}{|y|^{\beta}}<\infty~{}\text{and}~{}\lim\limits_{|y|\rightarrow
0}\frac{\|g(\cdot-y)-g(\cdot)\|_{L^{q_{2}}}}{|y|^{\beta}}=0.$

Moreover, there hold

$\displaystyle\|f(\cdot-y)-f(\cdot)\|_{L^{q_{1}}}=\text{O}(|y|^{\alpha})\left\|f\right\|_{\dot{B}^{\alpha}_{q_{1},\infty}},$
(2.10)
$\displaystyle\|g(\cdot-y)-g(\cdot)\|_{L^{q_{2}}}=\text{o}(|y|^{\beta})\left\|g\right\|_{\dot{B}^{\beta}_{q_{2},c(\mathbb{N})}}.$

Combining the properties of mollifier with Corollary 2.3, we derive the
following Lemma.

######  Lemma 2.4.

Let $\alpha,\beta\in(0,1)$, $p,q\in[1,\infty]$, and $k\in\mathbb{N}^{+}$.
Assume that $f\in L^{p}(0,T;\dot{B}^{\alpha}_{q,\infty})$, $g\in
L^{p}(0,T;\dot{B}^{\beta}_{q,c(\mathbb{N})})$, then there hold

  1. (1)

$\|f^{\varepsilon}-f\|_{L^{p}(0,T;L^{q})}\leq\text{O}(\varepsilon^{\alpha})\|f\|_{L^{p}(0,T;\dot{B}^{\alpha}_{q,\infty})}$;

  2. (2)

$\|\nabla^{k}f^{\varepsilon}\|_{L^{p}(0,T;L^{q})}\leq\text{O}(\varepsilon^{\alpha-k})\|f\|_{L^{p}(0,T;\dot{B}^{\alpha}_{q,\infty})}$;

  3. (3)

$\|g^{\varepsilon}-g\|_{L^{p}(0,T;L^{q})}\leq\text{o}(\varepsilon^{\beta})\|g\|_{L^{p}(0,T;\dot{B}^{\beta}_{q,c(\mathbb{N})})}$;

  4. (4)

$\|\nabla^{k}g^{\varepsilon}\|_{L^{p}(0,T;L^{q})}\leq\text{o}(\varepsilon^{\beta-k})\|g\|_{L^{p}(0,T;\dot{B}^{\beta}_{q,c(\mathbb{N})})}.$

###### Proof.

(1) Since $\int_{\mathbb{R}^{d}}\eta_{\varepsilon}(y)dy=1$, we deduce from
direct calculations that

$\displaystyle f^{\varepsilon}(x)-f(x)=$ $\displaystyle\int
f(x-y)\eta_{\varepsilon}(y)dy-f(x)\int\eta_{\varepsilon}(y)dy$
$\displaystyle=$ $\displaystyle\int[f(x-y)-f(x)]\eta_{\varepsilon}(y)dy.$

According to the Minkowski inequality and Corollary 2.3, we see that

$\displaystyle\|f^{\varepsilon}-f\|_{L^{p}(0,T;L^{q})}\leq$ $\displaystyle
C\int_{B(0,\varepsilon)}\|f(x-y)-f(x)\|_{L^{p}(0,T;L^{q})}\eta_{\varepsilon}(y)dy$
$\displaystyle\leq$
$\displaystyle\int_{B(0,\varepsilon)}O(|y|^{\alpha})\|f\|_{L^{p}(0,T;\dot{B}^{\alpha}_{q,\infty})}\eta_{\varepsilon}(y)dy$
$\displaystyle\leq$
$\displaystyle\int_{B(0,\varepsilon)}O(\varepsilon^{\alpha})\|f\|_{L^{p}(0,T;\dot{B}^{\alpha}_{q,\infty})}\eta_{\varepsilon}(y)dy$
$\displaystyle\leq$ $\displaystyle
O(\varepsilon^{\alpha})\|f\|_{L^{p}(0,T;\dot{B}^{\alpha}_{q,\infty})},$

which concludes (1).

(2) Some straightforward computations yield that

$\displaystyle\nabla^{k}f^{\varepsilon}(x)=\nabla^{k}(f\ast\eta_{\varepsilon})(x)=$
$\displaystyle\int_{B(0,\varepsilon)}\nabla^{k}_{x}f(x-y)\eta_{\varepsilon}(y)dy$
$\displaystyle=$
$\displaystyle(-1)^{k}\int_{B(0,\varepsilon)}\nabla^{k}_{y}f(x-y)\eta_{\varepsilon}(y)dy$
$\displaystyle=$
$\displaystyle{\varepsilon}^{-k}\int_{B(0,\varepsilon)}f(x-y)\nabla^{k}\eta_{\varepsilon}(y)dy.$

Using the fact $\int_{B(0,\varepsilon)}\nabla^{k}\eta(y)dy=0$, we infer that

${\varepsilon}^{-k}\int\nabla^{k}\eta_{\varepsilon}(y)dy=0.$ (2.11)

Hence, we arrive at

$\nabla
f^{\varepsilon}(x)={\varepsilon}^{-k}\int_{B(0,\varepsilon)}[f(x-y)-f(x)]\nabla^{k}\eta_{\varepsilon}(y)dy.$

Then making use of the Minkowski inequality and Corollary 2.3 once again, we
know that

$\displaystyle\|\nabla^{k}f^{\varepsilon}\|_{L^{p}(0,T;L^{q})}\leq$
$\displaystyle
C{\varepsilon}^{-k}\int_{B(0,\varepsilon)}\|f(x-y)-f(x)\|_{L^{p}(0,T;L^{q})}|\nabla^{k}\eta_{\varepsilon}(y)|dy$
$\displaystyle\leq$
$\displaystyle{\varepsilon}^{-k}\int_{B(0,\varepsilon)}O(|y|^{\alpha})\|f\|_{L^{p}(0,T;\dot{B}^{\alpha}_{q,\infty})}|\nabla^{k}\eta_{\varepsilon}(y)|dy$
$\displaystyle\leq$
$\displaystyle{\varepsilon}^{-k}\int_{B(0,\varepsilon)}O(\varepsilon^{\alpha})\|f\|_{L^{p}(0,T;\dot{B}^{\alpha}_{q,\infty})}|\nabla^{k}\eta_{\varepsilon}(y)|dy$
$\displaystyle\leq$ $\displaystyle
O(\varepsilon^{\alpha-k})\|f\|_{L^{p}(0,T;\dot{B}^{\alpha}_{q,\infty})}.$

This verifies the second part of this lemma. Exactly as the above derivation,
we can finish the proof of the rest part of this lemma. ∎

######  Lemma 2.5.

Let $\alpha\in(0,1)$ and $p\in[1,\infty]$, then there hold

$\|fg\|_{{\dot{B}^{\alpha}_{p,\infty}}}\leq
C\left(\|f\|_{L^{\infty}}\|g\|_{{\dot{B}^{\alpha}_{p,\infty}}}+\|f\|_{{\dot{B}^{\alpha}_{\infty,\infty}}}\|g\|_{L^{p}}\right),~{}\text{for
any}~{}f\in B^{\alpha}_{\infty,\infty},g\in B^{\alpha}_{p,\infty},$ (2.12)

and

$\|fg\|_{{\dot{B}^{\alpha}_{p,c(\mathbb{N})}}}\leq
C\left(\|f\|_{L^{\infty}}\|g\|_{{\dot{B}^{\alpha}_{p,c(\mathbb{N})}}}+\|f\|_{{\dot{B}^{\alpha}_{\infty,c(\mathbb{N})}}}\|g\|_{L^{p}}\right),\text{for
any}~{}f\in B^{\alpha}_{\infty,c(\mathbb{N})},g\in
B^{\alpha}_{p,c(\mathbb{N})}.$ (2.13)

###### Proof.

Using the triangle inequality ,we have

$\displaystyle|f(x-y)g(x-y)-f(x)g(x)|$ (2.14) $\displaystyle\leq$
$\displaystyle{|f(x-y)\Big{(}g(x-y)-g(x)\Big{)}|+|\Big{(}f(x-y)-f(x)\Big{)}g(x)|},$

which together with the Minkowski inequality implies that

$\displaystyle\frac{\|f(\cdot-y)g(\cdot-y)-fg(\cdot)\|_{L^{p}}}{|y|^{\alpha}}$
(2.15) $\displaystyle\leq$ $\displaystyle
C\left(\frac{\|f(\cdot-y)\Big{(}g(\cdot-y)-g(\cdot)\Big{)}\|_{L^{p}}}{|y|^{\alpha}}+\frac{\|\Big{(}f(\cdot-y)-f(\cdot)\Big{)}g(\cdot)\|_{L^{p}}}{|y|^{\alpha}}\right)$
$\displaystyle\leq$ $\displaystyle
C\left(\|f\|_{L^{\infty}}\frac{\|g(\cdot-y)-g(\cdot)\|_{L^{p}}}{|y|^{\alpha}}+\frac{\|f(\cdot-y)-f(\cdot)\|_{L^{\infty}}}{|y|^{\alpha}}\|g\|_{L^{p}}\right)$
$\displaystyle\leq$ $\displaystyle
C\left(\|f\|_{L^{\infty}}\|g\|_{{\dot{B}^{\alpha}_{p,\infty}}}+\|f\|_{{\dot{B}^{\alpha}_{\infty,\infty}}}\|g\|_{L^{p}}\right),$

where we have used the Corollary 2.3. Moreover, taking the limits on (2.15)
and using the Corollary 2.3 again, we have

$\displaystyle\lim\limits_{|y|\rightarrow
0}\frac{\|f(\cdot-y)g(\cdot-y)-fg(\cdot)\|_{L^{p}}}{|y|^{\alpha}}$ (2.16)
$\displaystyle\leq C\left(\|f\|_{L^{\infty}}\lim\limits_{|y|\rightarrow
0}\frac{\|g(\cdot-y)-g(\cdot)\|_{L^{p}}}{|y|^{\alpha}}+\lim\limits_{|y|\rightarrow
0}\frac{\|f(\cdot-y)-f(\cdot)\|_{L^{\infty}}}{|y|^{\alpha}}\|g\|_{L^{p}}\right)$
$\displaystyle\leq
C\left(\|f\|_{L^{\infty}}\|g\|_{{\dot{B}^{\alpha}_{p,c(\mathbb{N})}}}+\|f\|_{{\dot{B}^{\alpha}_{\infty,c(\mathbb{N})}}}\|g\|_{L^{p}}\right).$

∎

Next, we will state a new Constantin-E-Titi type commutator estimate as
follows.

######  Lemma 2.6.

Assume that $0<\alpha,\beta<1$, $1\leq p,q,p_{1},p_{2}\leq\infty$ and
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$. Then, there holds

$\displaystyle\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q})}\leq\text{o}(\varepsilon^{\alpha+\beta}),$
(2.17)

provided that one of the following three conditions is satisfied,

  1. (1)

$f\in L^{p_{1}}(0,T;\dot{B}^{\alpha}_{q_{1},c(\mathbb{N})})$, $g\in
L^{p_{2}}(0,T;\dot{B}^{\beta}_{q_{2},\infty})$, $1\leq q_{1},q_{2}\leq\infty$,
$\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$;

  2. (2)

$\nabla f\in L^{p_{1}}(0,T;\dot{B}^{\alpha}_{q_{1},c(\mathbb{N})})$, $\nabla
g\in L^{p_{2}}(0,T;\dot{B}^{\beta}_{q_{2},\infty})$,
$\frac{2}{d}+\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$, $1\leq
q_{1},q_{2}<d$;

  3. (3)

$f\in L^{p_{1}}(0,T;\dot{B}^{\alpha}_{q_{1},c(\mathbb{N})})$, $\nabla g\in
L^{p_{2}}(0,T;\dot{B}^{\beta}_{q_{2},\infty})$,
$\frac{1}{d}+\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$, $1\leq q_{2}<d$,
$1\leq q_{1}\leq\infty$.

######  Remark 2.1.

The estimate that
$\int_{0}^{T}\int|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}||\nabla
h|dxdt$ frequently appears in the study of energy (helicity) conservation of
fluid equations (see [11, 32, 33, 7, 16, 31]). Therefore, it seems that this
lemma is very helpful in this research of weak solutions in critical Besov
spaces.

######  Remark 2.2.

We would like to point out that $o(\cdot)$ should be replaced by $O(\cdot)$ if
the space $\dot{B}^{\alpha}_{q_{1},c(\mathbb{N})}$ in the condition of this
lemma is replaced by the space $\dot{B}^{\alpha}_{q_{1},\infty}$.

######  Remark 2.3.

It is worth remarking that $\nabla$ may be replaced by curl for incompressible
flow if $1<q_{1},q_{2}<d$.

######  Remark 2.4.

The results of Lemma 2.2-2.6 still hold for the nonhomogeneous Besov spaces.

###### Proof.

First, we recall the following identity observed by Constantin-E-Titi in [11]
that

$\displaystyle(fg)^{\varepsilon}(x)-f^{\varepsilon}g^{\varepsilon}(x)$
$\displaystyle=$
$\displaystyle\int_{\Omega}\eta_{\varepsilon}(y)[f(x-y)-f(x)][g(x-y)-g(x)]dy-(f-f^{\varepsilon})(g-g^{\varepsilon})(x).$

(1) Using the Hölder’s inequality and Minkowski inequality, we obtain

$\displaystyle\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q})}$
$\displaystyle\leq$ $\displaystyle
C\int_{|y|\leq\varepsilon}\eta_{\varepsilon}(y)\|f(\cdot-y)-f(\cdot)\|_{L^{p_{1}}(0,T;L^{q_{1}})}\|g(\cdot-y)-g(\cdot)\|_{L^{p_{2}}(0,T;L^{q_{2}})}dy$
$\displaystyle+C\|f-f^{\varepsilon}\|_{L^{p_{1}}(0,T;L^{q_{1}})}\|g-g^{\varepsilon}\|_{L^{p_{2}}(0,T;L^{q_{2}})}$
$\displaystyle\leq$ $\displaystyle
o(\varepsilon^{\alpha+\beta})\|f\|_{L^{p_{1}}(0,T;\dot{B}^{\alpha}_{q_{1},c(\mathbb{N})})}\|g\|_{L^{p_{2}}(0,T;\dot{B}^{\beta}_{q_{2},\infty})},$

where Corollary 2.3 and Lemma 2.4 are used.

(2) Let $\frac{1}{q}=\frac{1}{q_{1}^{\natural}}+\frac{1}{q_{2}^{\natural}}$
and $\frac{1}{q_{i}^{\natural}}=\frac{1}{q_{i}}-\frac{1}{d}$, $i=1,2.$ A
slight variant of the above proof implies that

$\displaystyle\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q})}$
$\displaystyle\leq$ $\displaystyle
C\int_{|y|\leq\varepsilon}\eta_{\varepsilon}(y)\|f(\cdot-y)-f(\cdot)\|_{L^{p_{1}}(0,T;L^{q_{1}^{\natural}})}\|g(\cdot-y)-g(\cdot)\|_{L^{p_{2}}(0,T;L^{q_{2}^{\natural}})}dy$
$\displaystyle+C\|f-f^{\varepsilon}\|_{L^{p_{1}}(0,T;L^{q_{1}^{\natural}})}\|g-g^{\varepsilon}\|_{L^{p_{2}}(0,T;L^{q_{2}^{\natural}})}$
$\displaystyle\leq$ $\displaystyle
C\int_{|y|\leq\varepsilon}\eta_{\varepsilon}(y)\|\nabla f(\cdot-y)-\nabla
f(\cdot)\|_{L^{p_{1}}(0,T;L^{q_{1}})}\|\nabla g(\cdot-y)-\nabla
g(\cdot)\|_{L^{p_{2}}(0,T;L^{q_{2}})}dy$ $\displaystyle+C\|\nabla f-\nabla
f^{\varepsilon}\|_{L^{p_{1}}(0,T;L^{q_{1}})}\|\nabla g-\nabla
g^{\varepsilon}\|_{L^{p_{2}}(0,T;L^{q_{2}})}$ $\displaystyle\leq$
$\displaystyle o(\varepsilon^{\alpha+\beta})\|\nabla
f\|_{L^{p_{1}}(0,T;\dot{B}^{\alpha}_{q_{1},c(\mathbb{N})})}\|\nabla
g\|_{L^{p_{2}}(0,T;\dot{B}^{\beta}_{q_{2},\infty})},$

where we have used the Hölder inequality and Sobolev embedding theorem.

(3) Arguing in the same manner as above, we know that

$\displaystyle\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q})}$
$\displaystyle\leq$ $\displaystyle
C\int_{|y|\leq\varepsilon}\eta_{\varepsilon}(y)\|f(\cdot-y)-f(\cdot)\|_{L^{p_{1}}(0,T;L^{q_{1}})}\|g(\cdot-y)-g(\cdot)\|_{L^{p_{2}}(0,T;L^{q_{2}^{\natural}})}dy$
$\displaystyle+C\|f-f^{\varepsilon}\|_{L^{p_{1}}(0,T;L^{q_{1}})}\|g-g^{\varepsilon}\|_{L^{p_{2}}(0,T;L^{q_{2}^{\natural}})}$
$\displaystyle\leq$ $\displaystyle
C\int_{|y|\leq\varepsilon}\eta_{\varepsilon}(y)\|f(\cdot-y)-f(\cdot)\|_{L^{p_{1}}(0,T;L^{q_{1}})}\|\nabla
g(\cdot-y)-\nabla g(\cdot)\|_{L^{p_{2}}(0,T;L^{q_{2}})}dy$
$\displaystyle+C\|f-f^{\varepsilon}\|_{L^{p_{1}}(0,T;L^{q_{1}})}\|\nabla
g-\nabla g^{\varepsilon}\|_{L^{p_{2}}(0,T;L^{q_{2}})}$ $\displaystyle\leq$
$\displaystyle o(\varepsilon^{\alpha+\beta})\|\nabla
f\|_{L^{p_{1}}(0,T;\dot{B}^{\alpha}_{q_{1},c(\mathbb{N})})}\|\nabla
g\|_{L^{p_{2}}(0,T;\dot{B}^{\beta}_{q_{2},\infty})},$

where the indices are required to satisfy
$\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}^{\natural}}$ and
$\frac{1}{q_{2}^{\natural}}=\frac{1}{q_{2}}-\frac{1}{d}$ with $1\leq q_{2}<d$.

Then the proof of this lemma is completed. ∎

######  Lemma 2.7.

Let $p,q,p_{1},q_{1},p_{2},q_{2}\in[1,\infty)$ with
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}},\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$.
Assume that $f\in L^{p_{1}}(0,T;L^{q_{1}}(\Omega))$ and $g\in
L^{p_{2}}(0,T;L^{q_{2}}(\Omega))$. Then, as $\varepsilon\rightarrow 0$, there
hold

$\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\Omega))}\rightarrow
0,$ (2.18)

and

$\|(f\times g)^{\varepsilon}-f^{\varepsilon}\times
g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\Omega))}\rightarrow 0.$ (2.19)

###### Proof.

First, it follows from the triangle inequality that

$\displaystyle\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q})}$
$\displaystyle\leq$ $\displaystyle
C\left(\|(fg)^{\varepsilon}-(fg)\|_{L^{p}(0,T;L^{q})}+\|fg-f^{\varepsilon}g\|_{L^{p}(0,T;L^{q})}+\|f^{\varepsilon}g-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q})}\right)$
$\displaystyle\leq$ $\displaystyle
C\Big{(}\|(fg)^{\varepsilon}-fg\|_{L^{p}(0,T;L^{q})}+\|f-f^{\varepsilon}\|_{L^{p_{1}}(0,T;L^{q_{1}})}\|g\|_{L^{p_{2}}(0,T;L^{q_{2}})}$
$\displaystyle\ \ \ \ \
+\|f^{\varepsilon}\|_{L^{p_{1}}(0,T;L^{q_{1}})}\|g-g^{\varepsilon}\|_{L^{p_{2}}(0,T;L^{q_{2}})}\Big{)},$

which together with the properties of the standard mollifiers yields (2.18).

Furthermore, to conclude (2.19), we only consider the case that the spatial
dimension $d=3$, since the proof for the case $d=2$ is similar. Without loss
of generality, we assume that $f=(f_{1},f_{2},f_{3})$ and
$g=(g_{1},g_{2},g_{3})$. Then by a direct computation, we have

$\displaystyle\|(f\times g)^{\varepsilon}-f^{\varepsilon}\times
g^{\varepsilon}\|_{L^{p}(0,T;L^{q})}=\Big{\|}\left(\left|\begin{matrix}\overrightarrow{i}&\overrightarrow{j}&\overrightarrow{k}\\\
f_{1}&f_{2}&f_{3}\\\ g_{1}&g_{2}&g_{3}\\\
\end{matrix}\right|\right)^{\varepsilon}-\left|\begin{matrix}\overrightarrow{i}&\overrightarrow{j}&\overrightarrow{k}\\\
f_{1}^{\varepsilon}&f_{2}^{\varepsilon}&f_{3}^{\varepsilon}\\\
g_{1}^{\varepsilon}&g_{2}^{\varepsilon}&g_{3}^{\varepsilon}\\\
\end{matrix}\right|\Big{\|}_{L^{p}(0,T;L^{q})}$ (2.20) $\displaystyle=$
$\displaystyle\Big{\|}\left((f_{2}g_{3}-f_{3}g_{2})\overrightarrow{i}-(f_{1}g_{3}-f_{3}g_{1})\overrightarrow{j}+(f_{1}g_{2}-f_{2}g_{1})\overrightarrow{k}\right)^{\varepsilon}$
$\displaystyle~{}~{}~{}~{}-\left((f_{2}^{\varepsilon}g_{3}^{\varepsilon}-f_{3}^{\varepsilon}g_{2}^{\varepsilon})\overrightarrow{i}-(f_{1}^{\varepsilon}g_{3}^{\varepsilon}-f_{3}^{\varepsilon}g_{1}^{\varepsilon})\overrightarrow{j}+(f_{1}^{\varepsilon}g_{2}^{\varepsilon}-f_{2}^{\varepsilon}g_{1}^{\varepsilon})\overrightarrow{k}\right)\Big{\|}_{L^{p}(0,T;L^{q})}$
$\displaystyle=$
$\displaystyle\Big{\|}\left[\Big{(}(f_{2}g_{3})^{\varepsilon}-f_{2}^{\varepsilon}g_{3}^{\varepsilon}\Big{)}-\Big{(}(f_{3}g_{2})^{\varepsilon}-f_{3}^{\varepsilon}g_{2}^{\varepsilon}\Big{)}\right]\overrightarrow{i}-\left[\Big{(}(f_{1}g_{3})^{\varepsilon}-f_{1}^{\varepsilon}g_{3}^{\varepsilon}\Big{)}-\Big{(}(f_{3}g_{1})^{\varepsilon}-f_{3}^{\varepsilon}g_{1}^{\varepsilon}\Big{)}\right]\overrightarrow{j}$
$\displaystyle+\left[\Big{(}(f_{1}g_{2})^{\varepsilon}-f_{1}^{\varepsilon}g_{2}^{\varepsilon}\Big{)}-\Big{(}(f_{2}g_{1})^{\varepsilon}-f_{2}^{\varepsilon}g_{1}^{\varepsilon}\Big{)}\right]\overrightarrow{k}\Big{\|}_{L^{p}(0,T;L^{q})},$

which together with the triangle inequality and (2.18) leads to (2.19). ∎

For the convenience of readers, we end this section by presenting the
definition of weak solutions to the compressible Euler equations
(LABEL:CEuler), incompressible Euler equations (LABEL:Euler) and the surface
quasi-geostrophic equation (LABEL:qg), respectively.

######  Definition 2.1.

A pair ($\rho,v$) is called a weak solution to the compressible Euler
equations (LABEL:CEuler), if ($\rho,v$) satisfies

  1. (i)

for any test function $\phi\in C_{0}^{\infty}((0,T)\times\Omega)$, there holds

$\int^{T}_{0}\int_{\Omega}\rho(x,t)\partial_{t}\phi(x,t)+\rho(x,t)v(x,t)\nabla\phi(x,t)dxdt=0.$

  2. (ii)

for any test vector field $\varphi\in C_{0}^{\infty}((0,T)\times\Omega)$,
there holds

$\displaystyle\int_{0}^{T}\int_{\Omega}\rho(x,t)v(x,t)\partial_{t}\varphi(x,t)$
$\displaystyle+(\rho(x,t)v(x,t)\otimes v(x,t))\nabla\varphi(x,t)$
$\displaystyle+$ $\displaystyle\pi(\rho)(x,t)\text{div}\varphi(x,t)dxdt=0.$

  3. (iii)

the energy inequality holds

$\displaystyle\mathcal{E}(t)\leq\mathcal{E}(0),$ (2.21)

where
$\mathcal{E}(t)=\int_{\Omega}\left(\frac{1}{2}\rho|v|^{2}+\kappa\frac{\rho^{\gamma}}{\gamma-1}\right)dx$.

######  Definition 2.2.

A vector field $v\in C_{\text{weak}}([0,T];L^{2}(\Omega))$ is called a weak
solution of the Euler equations (LABEL:Euler) with initial data $v_{0}\in
L^{2}(\Omega)$, if $v$ satisfies

  1. (i)

for any divergence-free test function $\varphi\in
C_{0}^{\infty}((0,T)\times\Omega)$, there holds

$\int_{0}^{T}\int_{\Omega}v(x,t)\partial_{t}\varphi(x,t)+v(x,t)\otimes
v(x,t)\cdot\nabla\varphi(x,t)dxdt=0.$

  2. (ii)

$v$ is weakly divergence-free, namely, for every scalar test function $\psi\in
C_{0}^{\infty}((0,T)\times\Omega)$,

$\int_{0}^{T}\int_{\Omega}\mathrm{div}\,v\cdot\psi dxdt=0.$

######  Definition 2.3.

A vector field $\theta\in C([0,T];L^{2}(\Omega))$ is called a weak solution of
the 2-D quasi-geostrophic equation (LABEL:qg) with initial data $\theta_{0}\in
L^{2}(\Omega)$, if there hold

$\int_{\Omega}[\theta(x,T)\varphi(x,T)-\theta(x,0)\varphi(x,0)]dx=\int_{0}^{T}\int_{\Omega}\theta(x,t)(\partial_{t}\varphi(x,t)+v\cdot\nabla\varphi)dxdt,$
(2.22)

and

$v(x,t)=-\nabla^{\perp}\int_{\Omega}\frac{\theta(y,t)}{|x-y|}dy$ (2.23)

for any test function $\varphi\in C_{0}^{\infty}([0,T];C^{\infty}(\Omega))$,
where $\nabla^{\perp}$ in (2.23) is in the sense of distributions.

##  3  Helicity conservation for the compressible Euler equations

This section contains the proof of helicity conservation of weak solutions in
Onsager type spaces $\dot{B}^{\frac{1}{3}}_{p,c(\mathbb{N})}$ for the
compressible Euler equations.

######  Proposition 3.1.

Let $(\rho,v)$ be a weak solution of compressible Euler equations
(LABEL:CEuler) in the sense of Definition 2.1, and
$\text{div\,}v,\text{curl\,}v\in C([0,T];L^{\frac{2d}{d+1}}(\Omega))$. If
$(\rho,v)$ additionally satisfies

$\displaystyle 0<c_{1}\leq\rho\leq c_{2}<\infty,~{}\rho\in
L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,c(\mathbb{N})}),~{}\rho v\in
L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,\infty}),$ $\displaystyle v\in
L^{3}(0,T;B^{\frac{1}{3}}_{3,c(\mathbb{N})}),$

then $(\rho,v)$ is also a weak solution of system (LABEL:rCEuler1) in the
sense of distributions.

###### Proof.

First, we mollify the momentum equation $\eqref{CEuler}_{2}$ in space (with
the kernel and notation as in Section 2):

$\displaystyle(\rho v)^{\varepsilon}_{t}+\text{div}(\rho v\otimes
v)^{\varepsilon}+\nabla\pi(\rho)^{\varepsilon}=0.$ (3.1)

Then, in virtue of appropriate commutators, the above equation can be
rewritten as

$\displaystyle(\rho^{\varepsilon}v^{\varepsilon})_{t}$
$\displaystyle+\text{div}((\rho v)^{\varepsilon}\otimes
v^{\varepsilon})+\nabla\pi(\rho^{\varepsilon})$ (3.2) $\displaystyle=-[(\rho
v)^{\varepsilon}-(\rho^{\varepsilon}v^{\varepsilon})]_{t}-\text{div}[(\rho
v\otimes v)^{\varepsilon}-(\rho v)^{\varepsilon}\otimes
v^{\varepsilon}]-\nabla[\pi(\rho^{\varepsilon})-\pi(\rho)^{\varepsilon}].$

Taking into account of mollified version of the continuity equation

$\rho^{\varepsilon}_{t}+\text{div}(\rho v)^{\varepsilon}=0,$ (3.3)

a direct computation shows

$\displaystyle\rho^{\varepsilon}v^{\varepsilon}_{t}+\rho^{\varepsilon}v^{\varepsilon}\cdot\nabla
v^{\varepsilon}+\nabla\pi(\rho^{\varepsilon})=$ $\displaystyle-[(\rho
v)^{\varepsilon}-\rho^{\varepsilon}v^{\varepsilon}]\cdot\nabla
v^{\varepsilon}-[(\rho
v)^{\varepsilon}-(\rho^{\varepsilon}v^{\varepsilon})]_{t}$ (3.4)
$\displaystyle-\text{div}[(\rho v\otimes v)^{\varepsilon}-(\rho
v)^{\varepsilon}\otimes
v^{\varepsilon}]-\nabla[\pi(\rho^{\varepsilon})-\pi(\rho)^{\varepsilon}].$

Since $0<c_{1}\leq\rho\leq c_{2}<\infty$, setting
$\Pi(\rho^{\varepsilon})=\int_{1}^{\rho^{\varepsilon}}\frac{\pi^{\prime}(s)}{s}ds$
and dividing both sides of equation (3.4) by $\rho^{\varepsilon}$, we arrive
at

$\displaystyle v^{\varepsilon}_{t}+\omega^{\varepsilon}\times
v^{\varepsilon}+\nabla\left(\Pi(\rho^{\varepsilon})+\frac{1}{2}|v^{\varepsilon}|^{2}\right)$
(3.5) $\displaystyle=$ $\displaystyle-\frac{1}{\rho^{\varepsilon}}[(\rho
v)^{\varepsilon}-\rho^{\varepsilon}v^{\varepsilon}]\cdot\nabla
v^{\varepsilon}-\frac{1}{\rho^{\varepsilon}}[(\rho
v)^{\varepsilon}-(\rho^{\varepsilon}v^{\varepsilon})]_{t}$
$\displaystyle-\frac{1}{\rho^{\varepsilon}}\text{div}[(\rho v\otimes
v)^{\varepsilon}-(\rho v)^{\varepsilon}\otimes
v^{\varepsilon}]-\frac{1}{\rho^{\varepsilon}}\nabla[\pi(\rho^{\varepsilon})-\pi(\rho)^{\varepsilon}],$

where we have used the identity that $v^{\varepsilon}\cdot\nabla
v^{\varepsilon}=\frac{1}{2}\nabla|v^{\varepsilon}|^{2}+\omega^{\varepsilon}\times
v^{\varepsilon}$ with $\omega=\text{curl\,}v$.

Let $\varphi(x,t)\in C_{0}^{\infty}((0,T)\times\Omega)$ be a test function.
Multiplication with $\varphi$ and then integration yield that

$\displaystyle\int_{0}^{t}\int_{\Omega}$
$\displaystyle\left(v^{\varepsilon}_{\tau}+\omega^{\varepsilon}\times
v^{\varepsilon}+\nabla\left(\Pi(\rho^{\varepsilon})+\frac{1}{2}|v^{\varepsilon}|^{2}\right)\right)\varphi
dxd\tau$ (3.6) $\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{\Omega}\Big{(}-\frac{1}{\rho^{\varepsilon}}[(\rho
v)^{\varepsilon}-\rho^{\varepsilon}v^{\varepsilon}]\cdot\nabla
v^{\varepsilon}-\frac{1}{\rho^{\varepsilon}}[(\rho
v)^{\varepsilon}-(\rho^{\varepsilon}v^{\varepsilon})]_{\tau}$
$\displaystyle-\frac{1}{\rho^{\varepsilon}}\text{div}[(\rho v\otimes
v)^{\varepsilon}-(\rho v)^{\varepsilon}\otimes
v^{\varepsilon}]-\frac{1}{\rho^{\varepsilon}}\nabla[\pi(\rho^{\varepsilon})-\pi(\rho)^{\varepsilon}]\Big{)}\varphi
dxd\tau$ $\displaystyle=$ $\displaystyle I_{1}+I_{2}+I_{3}+I_{4}.$

Next we will show that the four terms $I_{1}-I_{4}$ on the RHS of (3.6) tend
to zero as $\varepsilon\rightarrow 0$.

Indeed, note that

$\rho\in L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,c(\mathbb{N})}),\rho v\in
L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,\infty}),v\in
L^{3}(0,T;{B}^{\frac{1}{3}}_{3,c(\mathbb{N})}).$

Using Corollary 2.3, Lemma 2.4 and the Hölder inequality, the first term
$I_{1}$ on the RHS of (3.6) can be handled as

$\displaystyle|I_{1}|$ $\displaystyle\leq C\|(\rho
v)^{\varepsilon}-\rho^{\varepsilon}v^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\Omega))}\|\nabla
v^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\Omega))}\|\varphi\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}$
(3.7) $\displaystyle\leq
o(1)\|\rho\|_{L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,c(\mathbb{N})})}\|v\|_{L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,c(\mathbb{N})})}^{2}$
$\displaystyle\leq o(1).$

For second term $I_{2}$, using integration by parts, conbination the Corollary
2.3 with Lemma 2.4, we have

$\displaystyle|I_{2}|$ $\displaystyle=\left|\int_{0}^{t}\int_{\Omega}[(\rho
v)^{\varepsilon}-(\rho^{\varepsilon}v^{\varepsilon})]\left(-\frac{\partial_{\tau}\rho^{\varepsilon}}{(\rho^{\varepsilon})^{2}}\varphi+\frac{1}{\rho^{\varepsilon}}\partial_{\tau}\varphi\right)dxd\tau\right|$
(3.8) $\displaystyle=\left|\int_{0}^{t}\int_{\Omega}[(\rho
v)^{\varepsilon}-(\rho^{\varepsilon}v^{\varepsilon})]\left(\frac{\text{div}(\rho
v)^{\varepsilon}}{(\rho^{\varepsilon})^{2}}\varphi+\frac{1}{\rho^{\varepsilon}}\partial_{\tau}\varphi\right)dxd\tau\right|$
$\displaystyle\leq C\|(\rho
v)^{\varepsilon}-\rho^{\varepsilon}v^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\Omega))}\left(\|\nabla(\rho
v)^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\Omega))}+\|\partial_{\tau}\varphi\|_{L^{3}(0,T;L^{3}(\Omega))}\right)$
$\displaystyle\leq
o(\varepsilon^{\frac{2}{3}})\|\rho\|_{L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,c(\mathbb{N})})}\|v\|_{L^{3}(0,T;{\dot{B}^{\frac{1}{3}}_{3,c(\mathbb{N})}})}\big{(}O(\varepsilon^{\frac{1}{3}-1})\|\rho
v\|_{L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,\infty})}+1\big{)}$ $\displaystyle\leq
o(1)+o(\varepsilon^{\frac{2}{3}}).$

The third term $I_{3}$ can be estimated similarly as $I_{2}$:

$\displaystyle\begin{aligned} |I_{3}|&=\left|\int_{0}^{t}\int_{\Omega}[(\rho
v\otimes v)^{\varepsilon}-(\rho v)^{\varepsilon}\otimes
v^{\varepsilon}]\left(-\frac{\nabla\rho^{\varepsilon}}{(\rho^{\varepsilon})^{2}}\varphi+\frac{\nabla\varphi}{\rho^{\varepsilon}}\right)dxd\tau\right|\\\
&\leq C\|(\rho v\otimes v)^{\varepsilon}-(\rho v)^{\varepsilon}\otimes
v^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\Omega))}\left(\|\nabla\rho^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\Omega))}+\|\nabla\varphi\|_{L^{3}(0,T;L^{3}(\Omega))}\right)\\\
&\leq o(\varepsilon^{\frac{2}{3}})\|\rho
v\|_{L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,\infty})}\|v\|_{L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,c(\mathbb{N})})}\left(o(\varepsilon^{\frac{1}{3}-1})\|\rho\|_{L^{3}({\dot{B}^{\frac{1}{3}}_{3,c(\mathbb{N})}})}+1\right)\\\
&\leq o(1)+o(\varepsilon^{\frac{2}{3}}).\end{aligned}$ (3.9)

For the last term $I_{4}$, notice that if $\pi(s)\in C^{2}([a,b])$, then for
any $s,s_{0}\in[a,b]$ there holds

$|\pi(s)-\pi(s_{0})-\pi^{{}^{\prime}}(s_{0})(s-s_{0})|\leq C(s-s_{0})^{2},$

where the constant $C$ can be independent of $s$ and $s_{0}$. This yields that

$|\pi(\rho^{\varepsilon}(t,x))-\pi(\rho(t,x))-\pi^{{}^{\prime}}(\rho(t,x))(\rho^{\varepsilon}(t,x)-\rho(t,x))|\leq
C|\rho^{\varepsilon}(t,x)-\rho(t,x)|^{2}.$ (3.10)

Similarly, we also have

$|\pi(\rho(t,y))-\pi(\rho(t,x))-\pi^{{}^{\prime}}(\rho(t,x))(\rho(t,y)-\rho(t,x))|\leq
C|\rho(t,y)-\rho(t,x)|^{2}.$ (3.11)

Applying convolution with respect to $y$ to the last inequality and invoking
Jensen’s inequality, we get

$\displaystyle|\pi^{\varepsilon}(\rho(t,x))-\pi(\rho(t,x))-\pi^{{}^{\prime}}(\rho(t,x))(\rho^{\varepsilon}(t,x)-\rho(t,x))|\leq
C(\rho(t,\cdot)-\rho(t,x))^{2}*\eta_{\varepsilon}(x),$ (3.12)

which together with (3.10) gives

$\displaystyle|\pi(\rho^{\varepsilon}(t,x))-\pi^{\varepsilon}(\rho(t,x))|$
$\displaystyle\leq
C|\rho^{\varepsilon}(t,x)-\rho(t,x)|^{2}+C(\rho(t,\cdot)-\rho(t,x))^{2}*\eta_{\varepsilon}(x).$
(3.13)

Then it follows from integration by parts that $I_{4}$ can be handled as

$\displaystyle|I_{4}|=$
$\displaystyle\left|\int_{0}^{t}\int_{\Omega}[\pi(\rho^{\varepsilon})-\pi^{\varepsilon}(\rho)]\left(\frac{1}{\rho^{\varepsilon}}\text{div}\varphi-\varphi\cdot\frac{\nabla\rho^{\varepsilon}}{(\rho^{\varepsilon})^{2}}\right)dxd\tau\right|$
(3.14) $\displaystyle\leq$ $\displaystyle
C\|\rho^{\varepsilon}(t,x)-\rho(t,x)|^{2}+(\rho(t,\cdot)-\rho(t,x))^{2}*\eta_{\varepsilon}(x)\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\Omega))}$
$\displaystyle\times\left(\|\nabla\varphi\|_{L^{3}(0,T;L^{3}(\Omega))}+\|\nabla\rho^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\Omega))}\right)$
$\displaystyle\leq$ $\displaystyle
o(\varepsilon^{\frac{2}{3}})\|\rho\|_{L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,c(\mathbb{N})})}^{2}\left(1+o(\varepsilon^{\frac{1}{3}-1})\|\rho\|_{L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,c(\mathbb{N})})}\right)$
$\displaystyle\leq$ $\displaystyle o(\varepsilon^{\frac{2}{3}})+o(1).$

Hence, letting $\varepsilon\rightarrow 0$ and using the Lebesgue’s dominated
convergence theorem for LHS of (3.6), we have

$\int_{0}^{t}\int_{\Omega}\left(v_{\tau}+\omega\times
v+\nabla\left(\Pi(\rho)+\frac{1}{2}|v|^{2}\right)\right)\varphi dxd\tau=0,$
(3.15)

which means that $(\rho,v)$ is also a weak solution of equations

$\left\\{\begin{aligned} &\rho_{t}+\nabla\cdot(\rho v)=0,\\\
&v_{t}+\omega\times
v+\nabla\left(\Pi(\rho)+\frac{1}{2}|v|^{2}\right)=0.\end{aligned}\right.$
(3.16)

This concludes the proof of this proposition. ∎

###### Proof of Theorem 1.1.

Let $(\rho,v)$ be a weak solution to the compressible Euler equations
(LABEL:CEuler). Combining (1.8) and Proposition 3.1, we know that $(\rho,v)$
is also a weak solution to the system (3.16). Then one can mollify the
equation (3.16) in space direction to deduce that

$\left\\{\begin{aligned} &v^{\varepsilon}_{t}+(\omega\times
v)^{\varepsilon}+\nabla(\Pi(\rho)+\frac{1}{2}|v|^{2})^{\varepsilon}=0,\\\
&\omega^{\varepsilon}_{t}-\text{curl}(v\times\omega)^{\varepsilon}=0,\\\
&\text{div}\omega^{\varepsilon}=0.\end{aligned}\right.$

Consequently, by a straightforward computation, we have

$\displaystyle\frac{d}{dt}\int_{\Omega}(v^{\varepsilon}\cdot\omega^{\varepsilon})dx=$
$\displaystyle\int_{\Omega}v_{t}^{\varepsilon}\cdot\omega^{\varepsilon}+\omega_{t}^{\varepsilon}\cdot
v^{\varepsilon}dx$ (3.17) $\displaystyle=$
$\displaystyle\int_{\Omega}-\left((\omega\times
v)^{\varepsilon}+\nabla(\Pi(\rho)+\frac{1}{2}|v|^{2})^{\varepsilon}\right)\cdot\omega^{\varepsilon}+\left(\text{curl}\left(v\times\omega\right)\right)^{\varepsilon}\cdot
v^{\varepsilon}dx$ $\displaystyle=$
$\displaystyle\int_{\Omega}-\left(\omega\times
v\right)^{\varepsilon}\cdot\omega^{\varepsilon}+\left(v\times\omega\right)^{\varepsilon}\cdot\omega^{\varepsilon}dx$
$\displaystyle=$ $\displaystyle\int_{\Omega}-2\Big{(}(\omega\times
v)^{\varepsilon}-\omega^{\varepsilon}\times
v^{\varepsilon}\Big{)}\cdot\omega^{\varepsilon}-2(\omega^{\varepsilon}\times
v^{\varepsilon})\cdot\omega^{\varepsilon}dx$ $\displaystyle=$
$\displaystyle\int_{\Omega}-2\Big{(}(\omega\times
v)^{\varepsilon}-\omega^{\varepsilon}\times
v^{\varepsilon}\Big{)}\cdot\omega^{\varepsilon}dx$ $\displaystyle=$
$\displaystyle I,$

where we have used the identity that $\omega^{\varepsilon}\times
v^{\varepsilon}\cdot
v^{\varepsilon}=\omega^{\varepsilon}\times\omega^{\varepsilon}\cdot
v^{\varepsilon}=0$.

Next we will show that $I$ tends to zero as $\varepsilon\rightarrow 0$.

(1) The Hölder inequality ensures that

$\displaystyle|I|\leq$ $\displaystyle C\|(\omega\times
v)^{\varepsilon}-\omega^{\varepsilon}\times
v^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\Omega))}\|\omega^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\Omega))}$
(3.18) $\displaystyle\leq$ $\displaystyle C\|(\omega\times
v)^{\varepsilon}-\omega^{\varepsilon}\times
v^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\Omega))}\|\nabla
v^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\Omega))}.$

By $v\in L^{3}(0,T;{B}^{\frac{1}{3}}_{3,c(\mathbb{N})})$, $\omega\in
L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{3,\infty})$ and Lemma 2.6, we deduce that

$\|(\omega\times v)^{\varepsilon}-\omega^{\varepsilon}\times
v^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\Omega))}\leq
o(\varepsilon^{\frac{2}{3}}).$

Combining $v\in L^{3}(0,T;{B}^{\frac{1}{3}}_{3,c(\mathbb{N})})$ and (2) in
Lemma 2.4, we observe that

$\|\nabla v^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\Omega))}\leq
o(\varepsilon^{-\frac{2}{3}}).$

Hence there holds

$|I|\leq o(1).$

(2) By the Hölder inequality and Lemma 2.7, we get

$|I|\leq C\|(\omega\times v)^{\varepsilon}-\omega^{\varepsilon}\times
v^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\Omega))}\|\omega^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\Omega))}\rightarrow
0,$ (3.19)

due to the condition that $v\in
L^{3}(0,T;B^{\frac{1}{3}}_{3,c(\mathbb{N})}(\Omega))$ and $\omega\in
L^{3}(0,T;L^{3}(\Omega))$.

(3) It follows from the Hölder inequality that

$\displaystyle|I|\leq$ $\displaystyle C\|(\omega\times
v)^{\varepsilon}-\omega^{\varepsilon}\times
v^{\varepsilon}\|_{L^{\frac{p}{p-1}}(0,T;L^{\frac{q}{q-1}})}\|\omega^{\varepsilon}\|_{L^{p}(0,T;L^{q})}$
(3.20)

and

$\|\omega\times v\|_{L^{\frac{p}{p-1}}(0,T;L^{\frac{q}{q-1}}(\Omega))}\leq
C\|v\|_{L^{\frac{p}{p-2}}(0,T;L^{\frac{q}{q-2}}(\Omega))}\|\omega\|_{L^{p}(0,T;L^{q}(\Omega))}.$

Thus, by using the properties of the standard mollifiers, we know that

$\limsup_{\varepsilon\rightarrow 0}|I|=0.$

(4) Recall the identical equation

$-\Delta A=\text{curl\,}\text{curl\,}A-\nabla\text{div\,}A.$

The classical elliptic estimate yields that

$\|\nabla A\|_{L^{p}(\Omega)}\leq
C(\|\text{curl\,}A\|_{L^{p}(\Omega)}+\|\text{div\,}A\|_{L^{p}(\Omega)}).$

Therefore, there holds

$\|\nabla v\|_{L^{\frac{9}{4}}(\Omega)}\leq
C(\|\text{curl\,}v\|_{L^{\frac{9}{4}}(\Omega)}+\|\text{div\,}v\|_{L^{\frac{9}{4}}(\Omega)}).$

When dimemsion $d=3$, the Sobolev inequality implies that

$\|v\|_{L^{9}(\Omega)}\leq C\|\nabla v\|_{L^{\frac{9}{4}}(\Omega)}.$

Integrating this estimate in time direction, we derive

$\|v\|_{L^{3}(0,T;L^{9}(\Omega))}\leq C\|\nabla
v\|_{L^{3}(0,T;L^{\frac{9}{4}}(\Omega))}.$

Finally, we get $v\in L^{3}(0,T;L^{9}(\Omega))$ and $\text{div\,}u,\omega\in
L^{3}(0,T;L^{\frac{9}{4}}(\Omega))$. From the previous case, we complete the
proof of this case. This concludes Theorem 1.1. ∎

##  4  Helicity conservation for the homogeneous incompressible Euler
equations

This section is devoted to the proof of helicity conservation of weak
solutions in Onsager type spaces $\dot{B}^{\frac{1}{3}}_{p,c(\mathbb{N})}$ for
the homogeneous incompressible Euler equations.

###### Proof of Theorem 1.2.

Let $v$ be a weak solution to the homogeneous incompressible Euler equations
(LABEL:Euler). Then one can mollify the equation (1.11) in space direction to
deduce that

$\left\\{\begin{aligned} &v^{\varepsilon}_{t}+(\omega\times
v)^{\varepsilon}+\nabla(\Pi+\frac{1}{2}|v|^{2})^{\varepsilon}=0,\\\
&\omega^{\varepsilon}_{t}-\text{curl}(v\times\omega)^{\varepsilon}=0,\\\
&\text{div}\omega^{\varepsilon}=0.\end{aligned}\right.$

Consequently, by a straightforward computation, we have

$\displaystyle\frac{d}{dt}\int_{\Omega}(v^{\varepsilon}\cdot\omega^{\varepsilon})dx=$
$\displaystyle\int_{\Omega}v_{t}^{\varepsilon}\cdot\omega^{\varepsilon}+\omega_{t}^{\varepsilon}\cdot
v^{\varepsilon}dx$ (4.1) $\displaystyle=$
$\displaystyle\int_{\Omega}-\left((\omega\times
v)^{\varepsilon}+\nabla(\Pi+\frac{1}{2}|v|^{2})^{\varepsilon}\right)\cdot\omega^{\varepsilon}+\left(\text{curl}\left(v\times\omega\right)\right)^{\varepsilon}\cdot
v^{\varepsilon}dx$ $\displaystyle=$
$\displaystyle\int_{\Omega}-\left(\omega\times
v\right)^{\varepsilon}\cdot\omega^{\varepsilon}+\left(v\times\omega\right)^{\varepsilon}\cdot\omega^{\varepsilon}dx$
$\displaystyle=$ $\displaystyle\int_{\Omega}-2\Big{(}(\omega\times
v)^{\varepsilon}-\omega^{\varepsilon}\times
v^{\varepsilon}\Big{)}\cdot\omega^{\varepsilon}-2(\omega^{\varepsilon}\times
v^{\varepsilon})\cdot\omega^{\varepsilon}dx$ $\displaystyle=$
$\displaystyle\int_{\Omega}-2\Big{(}(\omega\times
v)^{\varepsilon}-\omega^{\varepsilon}\times
v^{\varepsilon}\Big{)}\cdot\omega^{\varepsilon}dx,$

where integration by parts and the fact that
$(v^{\varepsilon}\times\omega^{\varepsilon})\cdot\omega^{\varepsilon}=v^{\varepsilon}\cdot(\omega^{\varepsilon}\times\omega^{\varepsilon})=0$
were utilized.

From $\eqref{identity}_{1}$ and the divergence-free condition that div$\,v=0$,
we infer that

$\omega\times v=v\cdot\nabla v-\frac{1}{2}\nabla|v|^{2}=\text{div}(v\otimes
v)-\frac{1}{2}\nabla|v|^{2},$

which entails that

$(\omega\times v)^{\varepsilon}=(v\cdot\nabla
v)^{\varepsilon}-\frac{1}{2}\nabla(|v|^{2})^{\varepsilon}=\text{div}(v\otimes
v)^{\varepsilon}-\frac{1}{2}\nabla(|v|^{2})^{\varepsilon}$

and

$\omega^{\varepsilon}\times v^{\varepsilon}=v^{\varepsilon}\cdot\nabla
v^{\varepsilon}-\frac{1}{2}\nabla|v^{\varepsilon}|^{2}=\text{div}(v^{\varepsilon}\otimes
v^{\varepsilon})-\frac{1}{2}\nabla|v^{\varepsilon}|^{2}.$

Plugging the latter two equations into (4.1), we use integration by parts and
observe that

$\displaystyle\frac{d}{dt}\int_{\Omega}v^{\varepsilon}\cdot\omega^{\varepsilon}dx=$
$\displaystyle-2\int_{\Omega}\text{div}\big{(}(v\otimes
v)^{\varepsilon}-(v^{\varepsilon}\otimes
v^{\varepsilon})\big{)}\cdot\omega^{\varepsilon}+\frac{1}{2}\nabla\left(|v^{\varepsilon}|^{2}-(|v|^{2})^{\varepsilon}\right)\cdot\omega^{\varepsilon}dx$
$\displaystyle=$ $\displaystyle 2\int_{\Omega}\big{(}(v\otimes
v)^{\varepsilon}-v^{\varepsilon}\otimes
v^{\varepsilon}\big{)}\nabla\omega^{\varepsilon}dx.$

Then integrating the latter relation in time over $(0,t)$, we see that

$\int_{\Omega}v^{\varepsilon}(x,t)\cdot\omega^{\varepsilon}(x,t)dx-\int_{\Omega}v^{\varepsilon}(x,0)\cdot\omega^{\varepsilon}(x,0)dx=2\int_{0}^{t}\int_{\Omega}\big{(}v^{\varepsilon}\otimes
v^{\varepsilon}-(v\otimes
v)^{\varepsilon}\big{)}\nabla\omega^{\varepsilon}dxds.$ (4.2)

Now we are in a position to apply Lemma 2.6 to complete the proof of Theorem
1.2.

(1) Note that $v\in L^{k}(0,T;\dot{B}^{\alpha}_{p,c(\mathbb{N})})$, we apply
Lemma 2.6 to obtain

$\|(v\otimes v)^{\varepsilon}-v^{\varepsilon}\otimes
v^{\varepsilon}\|_{L^{\frac{k}{2}}(0,T;L^{\frac{p}{2}}(\Omega))}\leq
o(\varepsilon^{2\alpha}).$

Since $\omega\in L^{\ell}(0,T;\dot{B}^{\beta}_{q,\infty}),$ we adopt Lemma 2.4
to get

$\|\nabla\omega^{\varepsilon}\|_{L^{\ell}(0,T;L^{q}(\Omega))}\leq
O(\varepsilon^{\beta-1}).$

In light of the Hölder inequality, we see that

$\displaystyle\left|\int_{0}^{t}\int_{\Omega}\big{(}(v\otimes
v)^{\varepsilon}-v^{\varepsilon}\otimes
v^{\varepsilon}\big{)}\nabla\omega^{\varepsilon}dxds\right|$
$\displaystyle\leq$ $\displaystyle C\|(v\otimes
v)^{\varepsilon}-v^{\varepsilon}\otimes
v^{\varepsilon}\|_{L^{\frac{k}{2}}(0,T;L^{\frac{p}{2}}(\Omega))}\|\nabla\omega^{\varepsilon}\|_{L^{\ell}(0,T;L^{q}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
o(\varepsilon^{2\alpha})O(\varepsilon^{\beta-1})$ $\displaystyle\leq$
$\displaystyle o(\varepsilon^{\beta+2\alpha-1}).$

Then letting $\varepsilon\rightarrow 0$ in (4.2), this completes the proof of
the first part.

(2) In the same manner as above, making use of Remark 2.2, we have

$\displaystyle\left|\int_{0}^{t}\int_{\Omega}\big{(}(v\otimes
v)^{\varepsilon}-v^{\varepsilon}\otimes
v^{\varepsilon}\big{)}\nabla\omega^{\varepsilon}dxds\right|$
$\displaystyle\leq$ $\displaystyle C\|(v\otimes
v)^{\varepsilon}-v^{\varepsilon}\otimes
v^{\varepsilon}\|_{L^{\frac{k}{2}}(0,T;L^{\frac{p}{2}}(\Omega))}\|\nabla\omega^{\varepsilon}\|_{L^{\ell}(0,T;L^{q}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
O(\varepsilon^{2\alpha})o(\varepsilon^{\beta-1})$ $\displaystyle\leq$
$\displaystyle o(\varepsilon^{\beta+2\alpha-1}).$

Then letting $\varepsilon\rightarrow 0$, this together with (4.2) yields the
second part of Theorem 1.2.

(3) Choose $q_{1}=q_{2}=\frac{3d}{d+2}$. Then there holds
$\frac{2}{d}+\frac{2d-2}{3d}=\frac{d+2}{3d}+\frac{d+2}{3d}$. Hence, we apply
(2) in Lemma 2.6 and Remark 2.3 to obtain

$\|(v\otimes v)^{\varepsilon}-v^{\varepsilon}\otimes
v^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3d}{2d-2}}(\Omega))}\leq
o(\varepsilon^{\frac{2}{3}}),$

where we have used the condition that $\omega\in
L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{\frac{3d}{d+2},c(\mathbb{N})}).$ Following
the same path as above, we get

$\displaystyle\left|\int_{0}^{t}\int_{\Omega}\big{(}(v\otimes
v)^{\varepsilon}-v^{\varepsilon}\otimes
v^{\varepsilon}\big{)}\nabla\omega^{\varepsilon}dxds\right|$
$\displaystyle\leq$ $\displaystyle C\|(v\otimes
v)^{\varepsilon}-v^{\varepsilon}\otimes
v^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3d}{2d-2}}(\Omega))}\|\nabla\omega^{\varepsilon}\|_{L^{3}(0,T;L^{\frac{3d}{d+2}}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
O(\varepsilon^{\frac{2}{3}})o(\varepsilon^{-\frac{2}{3}})$ $\displaystyle\leq$
$\displaystyle o(1),$

which together with (4.2) concludes the third part of this theorem.

(An alternative approach to (3) and (1.6)):
Next we provide an alternative approach to proving this part. According to the
boundedness of Riesz Transform in homogeneous Besov spaces and the condition
that $\omega\in
L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{\frac{3d}{d+2},c(\mathbb{N})}),$ we have
$\nabla v\in L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{\frac{3d}{d+2},c(\mathbb{N})}).$
The Bernstein inequality further helps us to get $v\in
L^{3}(0,T;\dot{B}^{\frac{4}{3}}_{\frac{3d}{d+2},c(\mathbb{N})}).$ Recall that
$\dot{B}^{\frac{4}{3}}_{\frac{3d}{d+2},c(\mathbb{N})}\hookrightarrow\dot{B}^{\frac{2}{3}}_{3,c(\mathbb{N})}$,
we may achieve the desired proof of (3) from (1.6).
Subsequently, to make the paper more readable and more self-contained, we also
present the proof of (1.6). Notice that

$\|\nabla\omega^{\varepsilon}\|_{L^{q}(\Omega)}\leq
C\|\nabla^{2}v^{\varepsilon}\|_{L^{q}(\Omega)}~{}\text{for}~{}1<q<\infty,$

which together with the fact that $v\in
L^{3}(0,T;B^{\frac{2}{3}}_{3,c(\mathbb{N})}(\mathbb{R}^{3}))$ and Lemma 2.4
means that

$\|\nabla\omega^{\varepsilon}\|_{L^{3}(\Omega)}\leq
C\|\nabla^{2}v^{\varepsilon}\|_{L^{3}(\Omega)}\leq
o(\varepsilon^{-\frac{4}{3}}).$

Using the Hölder inequality and (1) in Lemma 2.6, we arrive at

$\displaystyle\left|\int_{0}^{t}\int_{\Omega}\big{(}(v\otimes
v)^{\varepsilon}-v^{\varepsilon}\otimes
v^{\varepsilon}\big{)}\nabla\omega^{\varepsilon}dxds\right|$
$\displaystyle\leq$ $\displaystyle C\|(v\otimes
v)^{\varepsilon}-v^{\varepsilon}\otimes
v^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\Omega))}\|\nabla\omega^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
o(\varepsilon^{\frac{4}{3}})o(\varepsilon^{-\frac{4}{3}})$ $\displaystyle\leq$
$\displaystyle o(1).$

This together with (4.2) gives (1.6).

(4) Following the same path of derivation of (2) in Theorem 1.2, Lemma 2.7
gives the proof of this part.

(5) With (4.2) and div$\,\omega=0$ in hand, a slight variant of the proof of
(2) in Theorem 1.2 provides the proof of this part. ∎

##  5  Helicity conservation for 2-D surface quasi-geostrophic equations

In this section, we are concerned with the helicity conservation of weak
solutions to 2-D surface quasi-geostrophic equations.

###### Proof of Theorem 1.4.

First, due to the divergence-free condition of velocity $v(x,t)$, we rewrite
the equation (LABEL:qg) as

$\left\\{\begin{aligned} &\theta_{t}+\mathrm{div}\,(v\otimes\theta)=0,\\\
&v(x,t)=(-\mathcal{R}_{2}\theta,\mathcal{R}_{1}\theta),\\\
&\theta|_{t=0}=\theta_{0},\end{aligned}\right.$ (5.1)

where $\mathcal{R}_{i}$ represents the Riesz transform for $i=1,2$. Next, we
regularize the equation (LABEL:qg1) to obtain

$\theta^{\varepsilon}_{t}+\partial_{j}(v_{j}\theta)^{\varepsilon}=0,$

and

$\partial_{i}\theta^{\varepsilon}_{t}+\partial_{j}(\partial_{i}v_{j}\theta)^{\varepsilon}+\partial_{j}(v_{j}\partial_{i}\theta)^{\varepsilon}=0.$

Thus, it follows from a direct computation that for $i=1,2$,

$\displaystyle\frac{d}{dt}\int_{\Omega}\theta^{\varepsilon}\partial_{i}\theta^{\varepsilon}dx$
$\displaystyle=\int_{\Omega}\theta^{\varepsilon}\partial_{i}\partial_{t}\theta^{\varepsilon}dx+\int_{\Omega}\partial_{t}\theta^{\varepsilon}\partial_{i}\theta^{\varepsilon}dx$
(5.2)
$\displaystyle=-\int_{\Omega}\theta^{\varepsilon}[\partial_{j}(\partial_{i}v_{j}\theta)^{\varepsilon}+\partial_{j}(v_{j}\partial_{i}\theta)^{\varepsilon}]dx-\int_{\Omega}\partial_{j}(v_{j}\theta)^{\varepsilon}\partial_{i}\theta^{\varepsilon}dx.$

Using the divergence-free condition of $v(x,t)$ again, we have the fact that
$\int\partial_{j}(\partial_{i}v_{j}^{\varepsilon}\theta^{\varepsilon})\theta^{\varepsilon}dx=0$.
Then we can reformulate (5.2) as

$\displaystyle\frac{d}{dt}\int_{\Omega}\theta^{\varepsilon}\partial_{i}\theta^{\varepsilon}dx=$
$\displaystyle-\int_{\Omega}\theta^{\varepsilon}\big{[}\partial_{j}(\partial_{i}v_{j}\theta)^{\varepsilon}-\partial_{j}(\partial_{i}v_{j}^{\varepsilon}\theta^{\varepsilon})\big{]}dx-\int_{\Omega}\theta^{\varepsilon}\big{[}\partial_{j}(v_{j}\partial_{i}\theta)^{\varepsilon}-\partial_{j}(v_{j}^{\varepsilon}\partial_{i}\theta^{\varepsilon})\big{]}dx$
$\displaystyle-\int_{\Omega}\theta^{\varepsilon}\partial_{j}(v_{j}^{\varepsilon}\partial_{i}\theta^{\varepsilon})dx-\int_{\Omega}\partial_{j}(v_{j}\theta)^{\varepsilon}\partial_{i}\theta^{\varepsilon}dx$
$\displaystyle=$
$\displaystyle\int_{\Omega}\partial_{j}\theta^{\varepsilon}\big{[}(\partial_{i}v_{j}\theta)^{\varepsilon}-(\partial_{i}v_{j}^{\varepsilon}\theta^{\varepsilon})\big{]}dx+\int_{\Omega}\big{[}(v_{j}\partial_{i}\theta)^{\varepsilon}-(v_{j}^{\varepsilon}\partial_{i}\theta^{\varepsilon})\big{]}\partial_{j}\theta^{\varepsilon}dx$
$\displaystyle-\int_{\Omega}\theta^{\varepsilon}v_{j}^{\varepsilon}\partial_{i}\partial_{j}\theta^{\varepsilon}dx+\int_{\Omega}(v_{j}\theta)^{\varepsilon}\partial_{j}\partial_{i}\theta^{\varepsilon}dx$
$\displaystyle=$
$\displaystyle\int_{\Omega}\partial_{j}\theta^{\varepsilon}\big{[}(\partial_{i}v_{j}\theta)^{\varepsilon}-(\partial_{i}v_{j}^{\varepsilon}\theta^{\varepsilon})\big{]}dx+\int_{\Omega}\big{[}(v_{j}\partial_{i}\theta)^{\varepsilon}-(v_{j}^{\varepsilon}\partial_{i}\theta^{\varepsilon})\big{]}\partial_{j}\theta^{\varepsilon}dx$
$\displaystyle+\int_{\Omega}\big{[}(v_{j}\theta)^{\varepsilon}-\theta^{\varepsilon}v_{j}^{\varepsilon}\big{]}\partial_{i}\partial_{j}\theta^{\varepsilon}dx,$

which leads to

$\displaystyle\int_{\Omega}\theta^{\varepsilon}(x,t)\partial_{i}\theta^{\varepsilon}(x,t)dx-\int_{\Omega}\theta^{\varepsilon}(x,0)\partial_{i}\theta^{\varepsilon}(x,0)dx$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{\Omega}\partial_{j}\theta^{\varepsilon}\big{[}(\partial_{i}v_{j}\theta)^{\varepsilon}-(\partial_{i}v_{j}^{\varepsilon}\theta^{\varepsilon})\big{]}dxds+\int_{0}^{t}\int_{\Omega}\big{[}(v_{j}\partial_{i}\theta)^{\varepsilon}-(v_{j}^{\varepsilon}\partial_{i}\theta^{\varepsilon})\big{]}\partial_{j}\theta^{\varepsilon}dxds$
$\displaystyle+\int_{0}^{t}\int_{\Omega}\big{[}(v_{j}\theta)^{\varepsilon}-\theta^{\varepsilon}v_{j}^{\varepsilon}\big{]}\partial_{i}\partial_{j}\theta^{\varepsilon}dxds$
$\displaystyle=:$ $\displaystyle I+II+III.$

For the first term $I$, we conclude by the Hölder inequality that

$|I|\leq\|(\partial_{i}v_{j}\theta)^{\varepsilon}-(\partial_{i}v_{j}^{\varepsilon}\theta^{\varepsilon})\|_{L^{\frac{3}{2}}(0,T;L^{\frac{6}{5}}(\Omega))}\|\partial_{j}\theta^{\varepsilon}\|_{L^{3}(0,T;L^{6}(\Omega))}.$

Take $q=\frac{6}{5},d=2,q_{1}=\frac{3}{2},q_{2}=\frac{3}{2}<2.$ Note that
$\frac{1}{2}+\frac{5}{6}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$ and $\nabla\theta\in
L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{\frac{3}{2},c(\mathbb{N})})$, we can invoke
(3) in Lemma 2.6 to get

$\|(\partial_{i}v_{j}\theta)^{\varepsilon}-(\partial_{i}v_{j}^{\varepsilon}\theta^{\varepsilon})\|_{L^{\frac{3}{2}}(0,T;L^{\frac{6}{5}}(\Omega))}\leq
o(\varepsilon^{\frac{2}{3}}).$

Combining Sobolev’s inequality and Lemma 2.4, we deduce from $\nabla\theta\in
L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{\frac{3}{2},c(\mathbb{N})})$ that

$\|\partial_{j}\theta^{\varepsilon}\|_{L^{3}(0,T;L^{6}(\Omega))}\leq
C\|\nabla\partial_{j}\theta^{\varepsilon}\|_{L^{3}(0,T;L^{\frac{3}{2}}(\Omega))}\leq
o(\varepsilon^{-\frac{2}{3}}).$

Collecting the above estimates, we get

$|I|\leq o(1).$

Likewise, we have

$|II|\leq o(1).$

It remains to estimate the third term $III$. Using the Hölder inequality, we
find

$|III|\leq\|(v_{j}\theta)^{\varepsilon}-\theta^{\varepsilon}v_{j}^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{3}(\Omega))}\|\partial_{i}\partial_{j}\theta^{\varepsilon}\|_{L^{3}(0,T;L^{\frac{3}{2}}(\Omega))}.$

Choosing $q=3,d=2,q_{1}=\frac{3}{2}=q_{2}=\frac{3}{2}<2,$ (2) in Lemma 2.6
enables us to get

$\|(v_{j}\theta)^{\varepsilon}-\theta^{\varepsilon}v_{j}^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{3}(\Omega))}\leq
o(\varepsilon^{\frac{2}{3}}),$

where we have used the condition that $\nabla\theta\in
L^{3}(0,T;\dot{B}^{\frac{1}{3}}_{\frac{3}{2},c(\mathbb{N})})$. Besides, Lemma
2.4 allows us to conclude that

$\|\partial_{i}\partial_{j}\theta^{\varepsilon}\|_{L^{3}(0,T;L^{\frac{3}{2}}(\Omega))}\leq
o(\varepsilon^{-\frac{2}{3}}).$

In summary, we know that

$|III|\leq o(1).$

We achieve the proof of this theorem. ∎

## Acknowledgements

Wang was partially supported by the National Natural Science Foundation of
China under grant (No. 11971446, No. 12071113 and No. 11601492). Wei was
partially supported by the National Natural Science Foundation of China under
grant (No. 11601423, No. 11871057). Ye was partially supported by the National
Natural Science Foundation of China under grant (No.11701145) and China
Postdoctoral Science Foundation (No. 2020M672196).

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# The relation of metal-poor stars to nearby solar analogues

Charles R. Cowley  Department of Astronomy, University of Michigan, 1085 S.
University, Ann Arbor, MI 481090-1107
orcid I.D.:0000-0001-9837-3662  Robert E. Stencel  Chamberlin Observatory,
University of Denver, 2930 E Warren Ave., Denver, CO 80210, USA,
orcid I.D.:0000-0001-8217-9435
e-mail<EMAIL_ADDRESS>
###### Abstract

Sunlike dwarf stars in the solar neighborhood reflect ages, an “average”
chemical evolution, and departures from that average. We show the chemical,
and kinematic properties of four groups of Sunlike dwarfs form a continuum
related to age. We plot $[Fe/H]$ vs. age, as well as kinematical values for
the four groups. The vertical (negative) scatter in $[Fe/H]$ increases with
age in a systematic way: as the age increases, $[Fe/H]$ decreases. The sets of
Solar and metal-poor stars in the solar neighborhood are related by
distributions in $[Fe/H]$ vs. age, as well as in Galactic position (XYZ) and
velocity space (UVW). Among the samples there are no clusters of points that
set one sample apart from the others. The distributions vary slowly from one
set to the next, suggesting a mixture of stellar populations. A plot in Energy
vs angular momentum phase space, with coordinate origin moved to the Galactic
center, highlights different aspects of the kinematics of the four groups of
stars. We finally compare the kinematic properties of these four groups with
those of two sets of ultra metal-poor stars.

stars: abundances – stars: solar-type –Galaxy: abundances – Galaxy: solar
neighborhood

##  1 Introduction

Figure 1: [Fe/H] vs. age (Gyr) from DM19 (black points), BD18 (red plus
signs), RO14 (blue triangles; $[Y/Mg]$ ages, see text), and GIR21 (orange
squares). The large dark diamond is for 2MASS J18082002-5104378, the oldest
known thin-disk star. We use its age, 13.5 Gyr (Schlaufman, et al., 2018) as a
proxy for the age of the Galaxy. The horizontal bar indicates typical age
uncertainties. The vertical bar shows is a liberal estimate ($\pm 0.2$ dex) of
the largest errors of the samples (See text for further details.) The blue and
orange lines are respectively an unconstrained least squares fit to the BD18
stars, while the wider orange line is a quadratic fit, constrained to have
$[Fe/H]=0$ when age is 4.6 Gyr. See CY22 for further details.

This is an archival study based on four surveys of FGK dwarf and subgiant
stars by Bedell, et al. (2018, henceforth, BD18), Delgado Mena et al. (2019,
henceforth, DM19), Giribaldi et al. (2021, henceforth, GIR21), and Roederer et
al. (2014, henceforth, RO14). We examine their metallicity with age, their
galactic positions and motions, and compare their relative kinematics within
the Galaxy, in the following sections. Finally, we contrast our four sets of
stars with results from ultra metal-poor stars.

##  2 The data sets

We focus on $[Fe/H]$111We use the notation $[Y/X]$ = $\log(Y/X)_{\rm
star}-\log(Y/X)_{\rm Sun}$ in our consideration of metallicity vs. age.
Typical values of $[Fe/H]$ decrease, and stellar age increase from the first
to the last sample studied.

###  2.1 BD18

BD18, and Spina et al. (2018) obtained precision differential abundances for
79 solar (or near solar) twins, all located within 100 pc of the Sun. Ages,
from Spina, et al. were considered uncertain by Cowley & Yüce (2022,
henceforth, CY22) roughly 1.4 Gyr, from a comparison of overlapping stars in
BD18 and Nissen et al. (2020). The oldest stars in this set are $\leq 9$ Gyr,
and the lowest $[Fe/H]$ is -0.15 dex222The notation “dex” means $10^{\rm
dex}$. Within this age spread, there is scatter in $[Fe/H]$, but little
indication of of an overall trend. Abundance error estimates are very small
($\pm 0.02$ dex or less). It was also shown by CY22 that mean elemental
abundances in this sample are well defined as a function of age, and that
individual deviations from the mean abundances for 30 elements may be shown
graphically (see Fig.2 of CY22). We also note the useful correlation of
[Eu/Ba] with age among these stars (Cowley & Stencel (2022)). These abundance
deviations may be attributed to pollution by Galactic streams and/or diffusion
into and out of the solar neighborhood (see Helmi, 2020; Bonaca et al., 2021).

###  2.2 DM19

The major work by DM19 involved more than 1,000 FGK stars. We examine a subset
of the oldest (277) sunlike dwarfs with age errors $\leq 1.5$ Gyr. Most (79%)
of the stars are thin disk, but a few thick disk and halo objects occur among
these stars. Uncertainties in $[Fe/H]$ were estimated by CY22 to be $\approx
0.11$ dex from a comparison of stars in common with BD18. Importantly, the age
range of the DM19 stars extends to slightly more than 13 Gyr, and the expected
average decrease of $[Fe/H]$ with age becomes evident. Fitting parameters to
the DM19 data also reflect this age-dependence (see Fig.1 and remarks below).

###  2.3 Titans

GIR21 provide high quality stellar parameters and abundances and ages for 48
dwarfs and subgiants chosen from Casagrande et al. (2010) and the GAIA
Benchmark stars (Heiter et al., 2015). Age uncertainties average 1.2 Gyr. We
estimate errors in $[Fe/H]$ to be 0.02 dex from abundance differences in Fe i
and Fe ii lines. The ’Titans’ set is the most accurate in $[Fe/H]$ of our four
samples. GIR21 note that Titans were the offspring of Gaia in Greek mythology.

###  2.4 RO14

We examine a subset of 19 dwarfs and subgiants from RO14’s analysis of 313
metal-poor stars from the survey of Beers et al. (1992). We assess typical
$[Fe/H]$ uncertainties in this pre-GAIA work to be of the order of $\pm 0.2$
dex for $[Fe/H]$, based on several estimates in RO14, as well as a comparison
with the one star, HD 193901, in common with the Titans. Individual stellar
ages were not determined in RO 14. We have made crude estimates by requiring
the mean ages and their variances to be equal to those of the Titans. We used
the time-sensitive (Berger et al., 2022) $[Y/Mg]$ rato, with constants
appropriate to a match with the Titans: ${\rm age}=-4.0*[Y/Mg]+10.0$. The
standard deviation of the plotted RO14 points is 1.03 Gyr, and is surely
uncertain by -2 or \+ 1.5 Gyr.

Figure 2: Positions in the Galactic plane for four stellar data sets. The
positive X-axis is in the direction of the Galactic center while the positive
Y-axis is the direction of Galactic rotation. The circle marks 100 parsecs
distance. Color coding and symbols are as in Fig. 1 – DM19 (black points),
BD18 (red plus signs), RO14 (blue triangles), and GIR21 (orange squares).

##  3 More distant metal-poor stars

Plotnikova et al. (2023, henceforth, PL23) is a study of a selection of stars
from the metal-poor survey of Barklem et al. (2005, henceforth, BAR05). The
latter study included some 250 stars with $[Fe/H]<-1.5$ as estimated from
medium resolution spectra. In an earlier version, PL22, selected 28 stars for
which age estimates with uncertainties under 1 Gyr could be obtained (see
their Fig.1 and Table 10). Uncertainties in $[Fe/H]$ were estimated as under
0.2 dex by Barklem et al. (2005). This uncertainty is small relative to the
spread in the values of the PL23 stars, but is consonant with the vertical
error bar in our Fig.1.

Although the distribution of their 28 stars would fit in Fig.1, the kinematics
of the larger set do not. The spread in both position and velocity space of
the 252 stars of Barklem et al. (2005) dwarf the spread of $Y$ vs $X$ and $U$
vs. $V$ among our four sets (see Figs. 2 and 3). The distribution in space of
the 252 BAR05 stars is shown in Fig. 1 of PL22. The distances scatter to some
10 kpc. In velocity space, the spread in V is beyond -1000 km s-1, well beyond
the scatter of the DM19 or RO14 points. Further consideration is left for
future work.

Figure 3: Four sets of stars in velocity space. The inner (distorted) circle
has a radius of 63 km s-1, the outer circle’s radius is 240 km s-1. Positive
$U$ is in the direction of the Galactic center, while positive $V$ is in the
direction of Galactic rotation. Color coding and symbols are as in Fig.1 –
DM19 (black points), BD18 (red plus signs), RO14 (blue triangles), and GIR21
(orange squares).

##  4 Ages

The overall structure of the points in Fig.1 is a downward sloping scatter of
$[Fe/H]$ with age. The scatter increases with age. It is uncertain whether the
abrupt increase near ages of 7-8 Gyr is real or due to systematic effects,
some of which are:

  * •

The BD18 and DM19 stars were chosen to be similar to the sun. Their vertical
scatter is small but indicates real abundance variations due to pollution from
Galactic streams or diffusion of stars into and out of the solar neighborhood.

  * •

The nineteen star RO14 sample was chosen from surveys for metal-poor stars. A
vertical separation of these stars from the BD18 and DM19 objects is therefore
expected–mildly metal-weak objects were not selected for study.

  * •

The Titans (GIR21) were selected from the Gaia benchmark stars Heiter et al.
(2015), but they do include benchmark stars with essentially solar $[Fe/H]$,
much like Procyon.

##  5 KINEMATICS & DYNAMICS

The relationships among the four sets of stars may be further illustrated by
their kinematics. Fig.2 shows the distribution of the stars in the Galactic
plane.

The BD18 (crosses) and nearly all of the DM19 points are within 100 pc. None
of the stars are as much as 1 kpc from the Sun. The spread of points is
smallest for the youngest (BD18) population, and widens for the oldest two.
The older DM19 stars are intermediate, as expected.

The population associations are even clearer in Fig.3. Corrections to the
local standard of rest (8.5, 13.58, and 6.49 km s-1) were taken from
PyAstronomy333Code site: https://github.com/sczesla/PyAstronomy , whose code
was also used to check our independent routine for transformation from
equatorial to Galactic coordinates.

The three DM19 stars outside the outer (240 km s-1 ring are: HD 113679, HD
121004, and HD 148816. They were classified halo, halo, and thick disk
respectively by DM19. However all three are within 140 pc of the Sun.

Mean abundances of the majority of stars, which are thin-disk objects, are
well defined as a function of age from 0 to $\approx 10$ Gyr. Departures from
those means abundances have been illustrated by CY22 for the 30 elements in
the BD18 sample (see CY22, their Figure 2).444cf.
https://zenodo.org/record/6077735.ZBtNBNLMLlg
– see ObsandPredPlots .pptx (or .pdf)  .

These departures from the mean, for a given age, may be attributed to the
influence of streams and perhaps diffusion into and out of the solar
neighborhood. The transition from the solar neighborhood to the thin and thick
disk, including a few halo stars, is illustrated using the samples of BD18,
DM19, RO14, andGIR21.

###  5.1 Phase space

Figure 4: Four sets of stars in parameter space. The parabola is the locus of
points having circular velocity at the LSR. See text for discussion. Color
coding and symbols are as in Fig. 1 – DM19 (black points), BD18 (red plus
signs), RO14 (blue triangles), and GIR21 (orange squares).

Plots of kinetic energy, En, versus angular momentum, Lz, highlight different
aspects of Galactic kinematics (Bonaca et al., 2021). For a central potential
Lz is one of the constant integrals of motion. The kinetic energy, of course,
differs from the constant total energy by the potential at the position of the
individual stars. Its use here avoids the problem of estimating that potential
energy, and enables a simple, parabolic relation between energy and Lz. While
Figs. 2 and 3 have origins at the sun and the local standard of rest (LSR),
the coordinate origin for Fig. 4 is the Galactic center. For stars in the
solar neighborhood,

$En=\frac{1}{2R_{c}}\times Lz^{2},$ (1)

where $R_{c}$ is the distance to the Galactic center.

In order to transform the coordinates of Fig. 1 and Fig. 2 to the Galactic
center, it is only necessary to substitute $X=X-R_{c}$, for the X-coordinate,
and $V=V+V_{c}$, where $V_{c}$ is the circular velocity (220 km s-1 at the
Local Center of Rest (LSR). We use 8 kpc for $R_{c}$ (GRAVITY Collaboration et
al., 2018), and 229 km/sec for the circular velocity at that distance (Eilers
et al., 2019).

Figure 5: Two sets of ultra metal-poor stars in the Galactic plane (Y vs X).
Blue triangles are from RO14, including giants as well as dwarfs. Black stars
are from the Hamburg survey (Barklem et al., 2005). The stars of these surveys
reach distances significantly greater than the stars discussed earlier.
Figure 6: En vs Lz distributions of of the RO14 and Barklem et al. (2005)
stars fill the parabola of Eq. 1.

The BD18 points in Figure 4 (bright red plus signs) cling closely to the left
branch of that parabola, as expected for objects with velocities close to the
solar velocity. The right-hand branch of the parabola is mostly unoccupied, as
would be expected since it indicates retrograde velocities.

Many of the DM points (black circles) also fall near the the BD18 clump, but
some diverge at both higher and lower energies. Points within the parabola
have, from left to right, increasingly retrograde orbits. This is the case for
the RO14 and GIR21 stars.

Many of the GIR21 stars are members of well-known streams. Possible groups
including RO14 stars are not noticable in this plot.

##  6 Ultra metal-poor stars

In order to put the preceeding sets of stars in perspective, we show two plots
with the distributions of ultra metal-poor stars in the Galactic plane and in
phase space. In these plots, we use stars of all types, giants as well as
dwarfs, from RO14 and the Hamburg survey (Barklem et al., 2005). The RO14
stars plotted here include the full complement of stars, not just dwarfs and
subgiants as in our earlier plot, Fig.2. Fig. 5 shows that the positions of
these stars range from the solar neighborhood to distances of several
kiloparsecs. Similarly, the distributions in phase space, Fig. 6, virtually
fill the phase-space parabola, having nearly as many retrograde as prograde
orbits. Note the accumulation of points toward the base of the parabola,
indicating stars with both low kinetic energy and z-component angular
momentum.

##  7 Summary

We conclude that the sample stars discussed vary systematically in $[Fe/H]$
versus age, as well as in position and velocity space. The plot in phase space
shows a systematic shift from nearly circular orbits in the solar
neighborhood, to sub-solar and retrograde orbits.

##  8 Data availability

The data underlying this article are available in the articles referenced and
new data may be found at in Zenodo at
https://zenodo.org/record/7215968.ZBtNrNLMLlg .

CRC thanks E. Delgado Mena for pointing out that her (DM19) $[Fe/H]$ vs. age
decreases for the highest ages, as is expected from Galactic chemical
evolution. We thank L. Spina for advice on the use of $[Y/Mg]$ as an age
indicator. We also thank Ian Roederer for comments and advice. We are grateful
for the advice of M. Valluti and M Chiba for help with the phase space
section. This work made use of PyAstronomy
(https://github.com/sczesla/PyAstronomy ). RES thanks the Simons Foundation
for support of ArXiv.org. This work presents results from the European Space
Agency (ESA) space mission Gaia. The Gaia archive website is
https://archives.esac.esa.int/gaia. We also acknowledge use of the SIMBAD
database (Wenger et al., 2000) operated at CDS, Strasbourg, France.

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