# Energy equality for the compressible Primitive Equations with vacuum

Šárka Nečasová1 111Email<EMAIL_ADDRESS>María Ángeles Rodríguez-
Bellido2222Email<EMAIL_ADDRESS>
Tong Tang3 333Email<EMAIL_ADDRESS>
1\. Institute of Mathematics of the Academy of Sciences of the Czech Republic,
Žitná 25, 11567, Praha 1, Czech Republic
2\. Dpto. Ecuaciones Diferenciales y Análisis Numérico and IMUS, Universidad
de Sevilla
Campus de Reina Mercedes, Facultad de Matemáticas, C/ Tarfia, s/n- 41012
Sevilla, Spain
3\. School of Mathematical Science
Yangzhou University, Yangzhou 225002, P.R. China

###### Abstract

We study the energy conservation for the weak solutions to the compressible
Primitive Equations (CPE) system with degenerate viscosity. We give sufficient
conditions on the regularity of weak solutions for the energy equality to
hold, even for solutions that may include vacuum. In this paper, we give two
theorems, the first one gives regularity in the classical isotropic Sobolev
and Besov spaces. The second one state result in the anisotropic spaces. We
get new regularity results in the second theorem because of the special
structure of CPE system, which are in contrast to compressible Navier-Stokes
equations.

Key words: energy conservation, compressible, Primitive Equations, Onsager’s
conjecture

2010 Mathematics Subject Classifications: 35Q30, 35Q86.

##  1 Introduction

In this paper, we investigate the relationship between regularity and
conservation of energy for the compressible Primitive Equations (CPE) system
in the periodic domain $\mathbb{T}^{3}$. The system has the following form

$\left\\{\begin{array}[]{llll}\partial_{t}\rho+\text{div}_{\bf
x}(\rho\mathbf{v})+\partial_{z}(\rho w)=0,\\\
\partial_{t}(\rho\mathbf{v})+\textrm{div}_{\bf
x}(\rho\mathbf{v}\otimes\mathbf{v})+\partial_{z}(\rho\mathbf{v}w)+\nabla_{\bf
x}p(\rho)={\rm div}_{\bf x}(\rho\nabla_{\bf
x}\mathbf{v})+\partial_{z}(\rho\partial_{z}\mathbf{v}),\\\
\partial_{z}p(\rho)=0,\end{array}\right.$ (1.1)

where $\rho,\mathbf{u},p$ represent the density, velocity and pressure,
respectively. The velocity can be defined as $\mathbf{u}=(\mathbf{v},w)$,
$\mathbf{v}(t,{\bf x},z)\in\mathbb{R}^{2}$ and $w(t,{\bf x},z)\in\mathbb{R}$
represent the horizonal velocity and vertical velocity respectively and where
${{\bf x}\in\mathbb{R}^{2}}$ denotes the horizontal direction and $z$ denotes
the vertical direction. From $(\ref{1.1})_{3}$, we can assume that the density
is independent of $z$, which means $\rho=\rho(t,{\bf x})$. Such assumption can
be seen also in [23, 31], where the authors use the change of variables to
derive the density $\rho$ is independent of $z$. Moreover we consider pressure
as $p(\rho)=\rho^{\gamma}\hskip 5.0pt(\gamma>1)$.

Primitive Equations (PE)444PE we mean incompressible primitive equations.
system is an important model which is widely used in the geophysical research
to describe and analyze the phenomena of atmosphere and ocean. It is derived
from the Navier-Stokes or Euler system by asymptotic analysis or numerical
approximation. Let us briefly recall some important results for such a system.

During the last decades, there is a vast body of literature about rigourous
mathematical justification of deriving PE model. More precisely, for
incompressible (PE) system, Azérad and Guillén [2] proved that the
incompressible Navier-Stokes equations converge to PE in the sense of weak
solutions. Further, Li and Titi [40] proved convergence of the weak solutions
of incompressible Navier-Stokes equations to the strong solutions of PE. Based
on [2, 40], Donatelli and Juhasz [20] give a justification that PE model with
the pollution effect is the hydrostatic limit of the Navier-Stokes equations
with an advection-diffusion equation. Grenier [32] used the energy estimates
and Brenier [7] used the relative entropy inequality to prove that the smooth
solutions of incompressible Euler system converge to smooth solutions of
inviscid PE. Precisely, Brenier [6] proved the existence of smooth solutions
in two-dimensions under the convex horizontal velocity assumptions. Later,
Masmoudi and Wong [47] extended Brenier’s result, removing the convex
horizontal velocity assumptions.

On the other hand, Ersoy et al. [23] used the asymptotic analysis to deduce
the compressible primitive system (CPE) with degenerate viscosity
coefficients. Gao, Nečasová and Tang [30] deduced the CPE from anisotropic
Navier-Stokes equations with constant viscosity coefficient.

One of the typical features of the PE model is that there is no information
for the vertical velocity in the momentum equation and the vertical velocity
is determined by the horizontal velocity through the continuity equation that
here reduces to incompressible constraint. Therefore, the mathematical and
numerical study of the PE model was unsolved until 1990s when Lions, Teman and
Wang [41, 42] were first to study the PE and received fundamental results in
this field. Then Guillén-González, Masmoudi and Rodríguez-Bellido [33] proved
the local existence of strong solutions and uniqueness with some interesting
anisotropic estimates. The celebrated result was made by Cao and Titi [12],
where they first proved the global well-posedness of PE in the three
dimensional case. Then, by virtue of semigroup method, Hieber and Kashiwabara
[35] extended this result relaxing the smoothness on the initial data.

Compressible primitive equations system (CPE) was studied recently. Gatapov
and Kazhikhov [31] proved the global existence of weak solutions with the
constant viscosity coefficients in 2D case. Liu and Titi [44, 46] proved the
local existence of strong solutions in 3D case and considered the zero Mach
number limit of CPE. Regarding with degenerate viscosity case, Ersoy et al.
[23], and Tang and Gao [52] showed the stability of weak solutions. The
stability means that a subsequence of weak solutions will converge to another
weak solutions if it satisfies some uniform bounds. Liu and Titi [45] and Wang
et al. [54], independently, used the B-D entropy to prove the global existence
of weak solutions. Readers can refer to Bresch et al. [8], Cao et al. [13], Li
and Titi [40], the book of Temam and Ziane [53] and references therein for
more physical background and other interesting mathematical results.

The existence of weak solutions is a fundamental question in PDE, especially
for fluids models. The case of the incompressible fluids was investigated
already in thirties by famous works of Leray [38]. He proved the global
existence of weak solutions to incompressible Navier-Stokes equations. On the
other hand, the proof of existence for the compressible case is going back to
nineties years by Lions [43] and Feireisl [25, 26, 27] which proved the
existence of global weak solutions of compressible isentropic case and later
to full system.

Generally, it is not clear known whether weak solutions satisfy the principle
of conservation of energy for both incompressible and compressible fluids. It
is a nature question how high regularity for weak solutions is needed to
obtain the energy equality.

Onsager [49] gave a famous conjecture that the three dimensional
incompressible Euler equations conserve energy if the velocity $\mathbf{u}\in
L^{3}((0,T);C^{0,\alpha}(\mathbb{T}^{3}))$ with $\alpha>\frac{1}{3}$. The
second part of conjecture said that there exist weak solutions of the Euler
equation for $\alpha\leq\frac{1}{3}$ which do not conserve energy.

The second part of Onsager’s conjecture has been underlined by the celebrated
work of Scheffer [50] and Shnirelman [51]. And a series of breakthrough papers
were done by De Lellis and Székelyhidi [17, 18, 19] by virtue of convex
integration. Recently, this part is fully solved by Isett [36], Buckmaster et.
al. [11].

The first part of Onsager’s conjecture was expressed as energy conservation,
which is a lively direction of research at the contemporary mathematical
society. It was proved by Constantin, E, Titi et [15] (also by Eyink [24], and
the work of Duchon and Robert [22]), stating that if ${\bf u}$ belongs to
$L^{3}([0,T];B^{\alpha,\infty}_{3}(\mathbb{T}^{3}))\cap
C([0,T];L^{2}(\mathbb{T}^{3}))$ with $\alpha>\frac{1}{3}$, then the energy is
conserved. Cheskidov et al. [14], and Fjordhlom and Wiedemann [29] made the
sharpest result in optimal Besov spaces. Their main idea is using the suitable
commutator estimates for incompressible Euler system. These results were
extended to the bounded domain by Bardos, Titi and Wiedemann [4], Drivas and
Nguyen [21]. Later this result was extended to incompressible inhomogeneous
Euler equations by Feireisl et al. [28]. Precisely speaking, the authors in
[28] mollified the weak solutions of the density $\rho$ and velocity
$\mathbf{u}$ and stated that if ${\bf u}$ belongs to
$B^{\alpha,\infty}_{p}([0,T];\times\mathbb{T}^{3})$,$\varrho,\varrho{\bf u}\in
B^{\beta,\infty}_{q}([0,T];\times\mathbb{T}^{3})$, $p\in
L^{p^{*}}_{loc}((0,T)\times\mathbb{T}^{3})$ for some $1\leq p,q\leq\infty$,
$p^{*}$ is the conjugate of $p$, and $0\leq\alpha,\beta\leq 1$, then the
energy is locally conserved. On the other hand, Leslie and Shvydkoy [39]
extended these results into the inhomogeneous Navier-Stokes case. Moreover,
Gwiazda et al. [34] proved the corresponding result to general system of first
order conservation laws.

Concerning compressible Euler system such result goes to Feireisl et al [28].
Inspired by Constantin et al. [15] work, Feireisl et al. [28] proved the
energy conservation for compressible Euler system with initial data containing
vacuum. Yu [55] used the Lions’s commutator estimates to show energy
conservation for compressible Navier-Stokes equations with degenerate
viscosity but without vacuum. Nguyen et al. [48] extended Yu’s result with
weaker regularity condition in bounded domain. Recently, Wiedmann et al. [1]
proved the corresponding results for compressible Euler and Navier-Stokes
system by removing the key assumptions that the pressure is a twice
continuously differentiable function. One can refer to [5, 10, 16] for other
related works.

Comparing with fruitful results for either incompressible Euler, Navier-Stokes
system or compressible case, there is a few results about PE model due to its
special structure. There are many differences at the mathematical structure
between Navier-Stokes equations and PE model, due to the hydrostatic
approximation. There is no information for the vertical velocity in the
momentum equation of PE model, so it is very difficult to analyze the PE
model. As far as we know, there is only one result due to Boutros, Markfelder
and Titi [9], in which they showed the energy conservation for the
incompressible inviscid PE. They gave three types of weak solutions, where
their vertical velocity is determined by horizontal velocity by virtue of
incompressible conditions. One of the sufficient condition for energy
conservation is horizonal velocity $\mathbf{v}\in
L^{4}((0,T);B^{\alpha}_{4,\infty}(\mathbb{T}^{3}))$ ($\alpha>\frac{1}{2}$) and
vertical velocity $w\in L^{2}((0,T);L^{2}(\mathbb{T}^{3}))$. It is interesting
to find that the index for velocity regularity is different from conventional
Onsager’s exponent, which coincides what stated in [10], ”Several of these
exponents were different from $\frac{1}{3}$, which is the Onsager exponent for
the Euler equations. This means that the Onsager exponent is not universal and
is plausibly determined by the regularity of the coefficient functions of the
nonlinearity…” It also shows the PE has different mathematical structure and
corresponding different results compared with Navier-Stokes equations.

There is no result concerning the energy equality for CPE system. The goal of
this work is to investigate the energy equality for CPE system under some
regularity conditions.

The paper is organized as follows: In Section 2, we introduce the definition
of weak solutions, give some useful lemmas and state the main theorems.
Section 3 is devoted to the proof of the energy conservation.

##  2 Definition and Preliminaries

###  2.1 Definition of weak solution

######  Definition 2.1.

The $(\rho,\mathbf{u})$ is called a weak solution to CPE system (1.1), if
$(\rho,\mathbf{u})$ satisfies the following assumptions for any time
$t\in[0,T]$

$\bullet$ $\rho$ and $\mathbf{v}$ have the following regularity:

$\displaystyle\rho\geq 0,\hskip 2.0pt\rho\in
L^{\infty}(0,T;L^{1}(\mathbb{T}^{3})\cap L^{\gamma}(\mathbb{T}^{3})),\hskip
2.0pt\nabla_{\bf x}\sqrt{\rho}\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{3})),\hskip
2.0pt\nabla_{\bf x}\rho^{\frac{\gamma}{2}}\in
L^{2}((0,T)\times\mathbb{T}^{3}),$ $\displaystyle\sqrt{\rho}\nabla_{\bf
x}\mathbf{v}\in L^{2}(0,T;L^{2}(\mathbb{T}^{3})),\hskip
2.0pt\sqrt{\rho}\partial_{z}\mathbf{v}\in
L^{2}(0,T;L^{2}(\mathbb{T}^{3})),\hskip 3.0pt\sqrt{\rho}\partial_{z}w\in
L^{2}(0,T;L^{2}(\mathbb{T}^{3}));$

$\bullet$ the continuity equation is satisfied in the sense of distributions
as follows

$\displaystyle\left[\int_{\mathbb{T}^{3}}\rho\varphi\,d{\bf
x}dz\right]\Big{|}^{t=\tau}_{t=0}=\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\left(\rho\partial_{t}\varphi+\rho\mathbf{v}\nabla_{\bf
x}\varphi+\rho w\partial_{z}\varphi\right)\\-d{\bf x}dzdt,$

for all $\varphi\in C^{\infty}_{c}((0,T)\times\mathbb{T}^{3})$;

$\bullet$ the momentum equation is satisfied as

$\displaystyle\left[\int_{\mathbb{T}^{3}}\rho\mathbf{v}\varphi\,d{\bf
x}dz\right]\Big{|}^{t=\tau}_{t=0}$
$\displaystyle=\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\left(\rho\mathbf{v}\partial_{t}\varphi+\rho\mathbf{v}\otimes\mathbf{v}:\nabla\varphi+\rho\mathbf{v}w\partial_{z}\varphi+p(\rho)\text{div}_{\bf
x}\varphi\right)\\-d{\bf x}dzdt$
$\displaystyle-\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\left(\rho\nabla_{\bf
x}\mathbf{v}:\nabla_{\bf
x}\varphi+\rho\partial_{z}\mathbf{v}\partial_{z}\varphi\right)\\-d{\bf
x}dzdt=0,$

for all $\varphi\in C^{\infty}_{c}([0,T)\times\Omega)$;

$\bullet$ the energy inequality

$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{3}}\left(\frac{1}{2}\rho|\mathbf{v}|^{2}+P(\rho)\right)\,d{\bf
x}dz\Big{|}^{t=\tau}_{t=0}+\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\left(\rho|\nabla_{\bf
x}\mathbf{v}|^{2}+\rho|\partial_{z}\mathbf{v}|^{2}\right)\,d{\bf x}dz\leq 0$

for a.a $\tau\in(0,T)$, where
$P(\rho)=\rho\displaystyle\int^{\rho}_{1}\frac{p(s)}{s^{2}}ds$.

######  Remark 2.1.

By virtue of the B-D entropy, Liu and Titi [45] obtained the extra estimates
and the existence of global weak solutions to CPE with degenerate viscosity.
However, it is still an open problem the existence of weak solutions to CPE
with constant viscosity.

###  2.2 Function spaces

Let $\Omega:=(0,T)\times\mathbb{T}^{3}$, and we define the Besov space:
$B^{\alpha,\infty}_{p}(\Omega)$ with $1\leq p<\infty$, $0<\alpha<1$, normed by

$\displaystyle\|w\|_{B^{\alpha,\infty}_{p}(\Omega)}=\|w\|_{L^{p}(\Omega)}+\sup_{{\boldsymbol{\xi}}\in\Omega}\\{|{\boldsymbol{\xi}}|^{-\alpha}\|w(\cdot+{\boldsymbol{\xi}})-w\|_{L^{p}(\Omega\cap(\Omega-{\boldsymbol{\xi}}))}\\}$

is finite. In order to understand the anisotropic structure in CPE, inspired
by [9, 33], we introduce the following anisotropic spaces:

$\displaystyle\|w\|_{H^{1}_{\bf
x}(\mathbb{T}^{2})L^{2}_{z}(\mathbb{T}^{1})}=\|\|w(x,\cdot)\|_{L^{2}(\mathbb{T}^{1})}\|_{H^{1}_{\bf
x}(\mathbb{T}^{2})}$

and considering ${\boldsymbol{\xi}}=(\tau,{\boldsymbol{\xi}}_{h},\xi_{z})$,

$\displaystyle\|w\|_{B^{\alpha,\infty}_{p}((0,T);B^{\alpha,\infty}_{p}(\mathbb{T};B^{\beta,\infty}_{p}(\mathbb{T}^{2}))}=$
$\displaystyle\|w\|_{L^{p}(\Omega)}+\sup_{{\boldsymbol{\xi}}\in\Omega}\\{|{\boldsymbol{\xi}}|^{-\alpha}\|w(\cdot+{\boldsymbol{\xi}})-w\|_{L^{p}(\Omega)\cap(\Omega-{\boldsymbol{\xi}}))}\\}$
$\displaystyle+\sup_{{\boldsymbol{\xi}}_{h}\in\mathbb{T}^{2}}\\{|{\boldsymbol{\xi}}_{h}|^{-\beta}\|w(\cdot+(0,{\boldsymbol{\xi}}_{h},0))-w\|_{L^{p}(\Omega)\cap(\Omega-(0,{\boldsymbol{\xi}}_{h},0))}\\},$

where $0<\alpha<\beta<1$. The space $H^{1}_{\bf
x}(\mathbb{T}^{2})L^{2}_{z}(\mathbb{T}^{1})$ means its horizontal regularity
is $H^{1}$ and vertical regularity is $L^{2}$. On the other hand, the space
$B^{\alpha,\infty}_{p}(\mathbb{T};B^{\beta,\infty}_{p}(\mathbb{T}^{2}))$ means
that it is Besov regular in the $z-$ direction with exponent $\alpha$ while
Besov regular in the horizontal direction with exponent $\beta$.

Let $\eta\in C^{\infty}_{c}(\mathbb{R}^{d+1})$ ($d$ is the number of the space
dimension) be a standard mollification kernel and set

$\displaystyle\eta^{\varepsilon}(x)=\frac{1}{\varepsilon^{d+1}}\eta\left(\frac{x}{\varepsilon}\right),\hskip
3.0ptw^{\epsilon}=\eta^{\varepsilon}\ast w,\hskip
3.0ptf^{\varepsilon}(w)=f(w)\ast\eta^{\varepsilon}.$

We should notice that $w^{\varepsilon}$ is well-defined on
$\Omega^{\varepsilon}=\\{x\in\Omega:d(x,\partial\Omega)>\varepsilon\\}$. We
first recall some useful lemmas which will be frequently used throughout the
paper.

######  Lemma 2.1.

([28, 16]) For any function $u\in B^{\alpha,\infty}_{p}(\Omega)$, we have

$\displaystyle\|u(\cdot+{\boldsymbol{\xi}})-u\|_{L^{p}(\Omega\cap(\Omega-{\boldsymbol{\xi}}))}\leq|{\boldsymbol{\xi}}|^{\alpha}\|u\|_{B^{\alpha,\infty}_{p}(\Omega)},$
$\displaystyle\|u^{\varepsilon}-u\|_{L^{p}(\Omega)}\leq\varepsilon^{\alpha}\|u\|_{B^{\alpha,\infty}_{p}(\Omega)},$
$\displaystyle\|\nabla
u^{\varepsilon}\|_{L^{p}(\Omega)}\leq\varepsilon^{\alpha-1}\|u\|_{B^{\alpha,\infty}_{p}(\Omega)},$

where $\nabla$ stands for space-time gradient.

######  Remark 2.2.

It is easy to prove the following equality [28]

$\displaystyle
f^{\varepsilon}g^{\varepsilon}-(fg)^{\varepsilon}=(f^{\varepsilon}-f)(g^{\varepsilon}-g)$
$\displaystyle-\displaystyle\int^{\varepsilon}_{-\varepsilon}\int_{\mathbb{T}^{3}}\\-\\-\eta^{\varepsilon}(\tau,{\boldsymbol{\xi}}_{h},\xi_{z})\left(f(t-\tau,{\bf
x}-{\boldsymbol{\xi}}_{h},z-\xi_{z})-f(t,x.,z)\right)\left(g(t-\tau,{\bf
x}-{\boldsymbol{\xi}}_{h},z-\xi_{z})-g(t,{\bf
x,z})\right)d{\boldsymbol{\xi}}_{h}d\xi_{z}d\tau.$

######  Lemma 2.2.

([3] Besov space embedding) Let $s\in\mathbb{R}$, $1\leq p,r\leq\infty$, then

$\displaystyle B^{s,r}_{p}\hookrightarrow
B^{s-d(\frac{1}{p}-\frac{1}{p_{1}}),r}_{p_{1}},\hskip 3.0ptp_{1}\geq p.$

It is easy to obtain the following remark

######  Remark 2.3.

$H^{1}(\mathbb{T}^{3})\hookrightarrow
B^{1,\infty}_{2}(\mathbb{T}^{3})\hookrightarrow
B^{\alpha,\infty}_{3}(\mathbb{T}^{3})$ ($\alpha>\frac{1}{2}$).

The following two lemmas can be proved as [16]

######  Lemma 2.3.

Let $u\in B^{\alpha,\infty}_{p}(\Omega,\mathbb{R}^{m})$. Suppose
$f:\mathbb{R}^{m}\rightarrow\mathbb{R}$ is a $C^{1}$ function with
$\frac{\partial f}{\partial u_{i}}\in L^{\infty}$ $i=1...m$. Then

$\displaystyle\|\nabla f(u^{\varepsilon})\|_{L^{p}(\Omega)}\leq
C\varepsilon^{\alpha-1}\|u\|_{B^{\alpha,\infty}_{p}(\Omega)},$

where $\nabla$ stands for space-time gradient

######  Lemma 2.4.

Let $1\leq q<\infty$ and suppose $v\in
L^{2q}((0,T)\times\mathbb{T}^{d};\mathbb{R}^{k})$ and $f\in
C^{2}(\mathbb{R}^{k};\mathbb{R}^{N})$. If

$\displaystyle\sup_{i,j}\|\frac{\partial^{2}f}{\partial v_{i}\partial
v_{j}}\|_{L^{\infty}}<\infty,$

then there exists a constant $C>0$ such that

$\displaystyle\|f(v^{\varepsilon})-f^{\varepsilon}(v)\|_{L^{q}}<C\big{(}\|v^{\varepsilon}-v\|_{L^{2q}}+\sup_{(s,{\bf
y})\in{\rm supp}\eta_{\varepsilon}}\|v(\cdot,\cdot)-v(\cdot-s,\cdot-{\bf
y})\|^{2}_{L^{2q}}\big{)}.$

The following lemma, known as Lions’s commutator lemma ([55] Lemma 2.3 and
[37] Lemma 3.2) will be very useful for the estimates below

######  Lemma 2.5.

Let $\partial$ be a partial derivative in space or time. Let $f$, $\partial
f\in L^{p}(\mathbb{R}^{+}\times\Omega)$, $g\in
L^{q}(\mathbb{R}^{+}\times\Omega)$ with $1\leq p,q\leq\infty$, and
$\frac{1}{p}+\frac{1}{q}\leq 1$. Then, we have

$\displaystyle\|\partial(fg)^{\varepsilon}-\partial(fg^{\varepsilon})\|_{L^{r}}\leq
C\|\partial f\|_{L^{p}}\|g\|_{L^{q}}$

for some constant $C>0$ independent of $\varepsilon$, $f$ and $g$, with
$\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. In addition,

$\displaystyle\partial(fg)^{\varepsilon}-\partial(fg^{\varepsilon})\rightarrow
0,\hskip 3.0pt\text{in}\hskip 2.0ptL^{r},$

as $\varepsilon\rightarrow 0$, if $r<\infty$.

######  Remark 2.4.

The original version in [55] is
$\|(\partial(fg))^{\varepsilon}-\partial(fg^{\varepsilon})\|_{L^{r}}\leq
C\|\partial f\|_{L^{p}}\|g\|_{L^{q}}$. Using Lemma 3.1 in [37],
$\partial(fg)^{\varepsilon}=(\partial(fg))^{\varepsilon}$, we get the
statement of Lemma 2.5 as

$\|\partial(fg)^{\varepsilon}-\partial(fg^{\varepsilon})\|_{L^{r}}=\|(\partial(fg))^{\varepsilon}-\partial(fg^{\varepsilon})\|_{L^{r}}.$

###  2.3 Main result

Our main results can be stated as follows:

######  Theorem 2.1.

Let $(\rho,\mathbf{u})$ be a weak solution of CPE (1.1) with $\rho>0$. Let us
assume $\mathbf{v}\in
B^{\alpha,\infty}_{3}((0,T);B^{\alpha,\infty}_{3}(\mathbb{T}^{3}))$,
$\rho,\rho\mathbf{v}\in
B^{\beta,\infty}_{3}((0,T);B^{\beta,\infty}_{3}(\mathbb{T}^{3})$, and

$\displaystyle 0\leq\underline{\rho}\leq\rho\leq\overline{\rho},\hskip
3.0pt{\text{a}.e.in}\hskip 2.0pt(0,T)\times\Omega$

for some constants $\underline{\rho},\overline{\rho}$ and
$0\leq\alpha,\beta\leq 1$, $\alpha>\frac{3}{4}$ and
$\alpha+\beta>\frac{3}{2}$. Moreover, we assume the additional regularity
$w\in L^{2}(0,T;L^{2}(\mathbb{T}^{3}))$, $\mathbf{v}\in
L^{2}((0,T);H^{1}(\mathbb{T}^{3}))$, $\rho\mathbf{v}\in
L^{3}((0,T);H^{1}(\mathbb{T}^{3}))$ and $p\in
C[\underline{\rho},\overline{\rho}]$. Then the energy is locally conserved
i.e.

$\begin{array}[]{r}\displaystyle\partial_{t}\left(\frac{1}{2}\rho|\mathbf{v}|^{2}+P(\rho)\right)+\rho|\nabla_{\bf
x}\mathbf{v}|^{2}+\rho|\partial_{z}\mathbf{v}|^{2}+{\rm div}_{\bf
x}\left[(\frac{1}{2}\rho|\mathbf{v}|^{2}+p(\rho)+P(\rho)-\rho\nabla_{\bf
x}\mathbf{v})\mathbf{v}\right]\\\ \vspace{-1ex}\cr\\\
\displaystyle+\partial_{z}\left[\left(\frac{1}{2}\rho|\mathbf{v}|^{2}+p(\rho)+P(\rho)\right)w-\rho\partial_{z}\mathbf{v}\cdot\mathbf{v}\right]=0.\end{array}$
(2.1)

######  Remark 2.5.

Our result is still valid for the constant viscosity coefficient whose
corresponding energy is

$\displaystyle\partial_{t}\left(\frac{1}{2}\rho|\mathbf{v}|^{2}+P(\rho)\right)+|\nabla_{\bf
x}\mathbf{v}|^{2}+|\partial_{z}\mathbf{v}|^{2}+{\rm div}_{\bf
x}\left[\left(\frac{1}{2}\rho|\mathbf{v}|^{2}+p(\rho)+P(\rho)-\nabla_{\bf
x}\mathbf{v}\right)\mathbf{v}\right]$
$\displaystyle+\partial_{z}\left[\left(\frac{1}{2}\rho|\mathbf{v}|^{2}+p(\rho)+P(\rho)\right)w-\partial_{z}\mathbf{v}\cdot\mathbf{v}\right]=0.$

######  Remark 2.6.

There are some major differences between our work and Feireisl et al. [28],
Wiedemann et al. [1], Titi et al.[9]. Theorem 2.1 seems similar and parallel
to the results in [28, 1]. However, in contrast with compressible Euler and
Navier-Stokes system, we have only information on the horizontal velocity in
CPE. It means it is a hurdle to control the estimate of vertical velocity in
the isotropic space. In order to overcome this difficulty, we need the
additional assumption $\rho\mathbf{v}\in L^{3}((0,T);H^{1})$ in Theorem 2.1.
Fortunately, by deep investigation of the structure of CPE, we find that we
can work with the anisotropic space to solve the corresponding problems.
Therefore, we are able to remove this restriction by virtue of the anisotropic
regularity in Theorem 2.2.

Compared with [9], we deal with the compressible situation which completely
changes the behaviour of fluid: Firstly, the only way to obtain the
information of vertical velocity is through continuity equation instead of
incompressible constraint. Moreover, the pressure depends on density and needs
to be controlled. Last but not least, we have dissipative term and the
viscosity depends on density, which brings the new commutator estimates.
Therefore, the conventional method used in incompressible PE [9] can not be
applied straightforwardly to the CPE model. Here, we find a new observation
and use the Lions’s commutator lemma (Lemma 2.5) to control the convergence of
vertical viscosity commutator which is different to [1].

Inspired by [9, 33], we introduce the horizontal and vertical regularity and
obtain the following theorem:

######  Theorem 2.2.

Let $(\rho,\mathbf{u})$ be a weak solution of CPE (1.1) with $\rho>0$. Let us
assume $\mathbf{v}\in
B^{\alpha,\infty}_{3}((0,T);B^{\alpha,\infty}_{3}(\mathbb{T};B^{\beta,\infty}_{3}(\mathbb{T}^{2})))$,
$\rho,\rho\mathbf{v}\in
B^{\alpha,\infty}_{3}((0,T);B^{\alpha,\infty}_{3}(\mathbb{T};B^{\beta,\infty}_{3}(\mathbb{T}^{2})))$,
where $0<\alpha<\beta\leq 1$, $\alpha>\frac{1}{3}$ and $2\alpha+\beta>2$.
Moreover, we assume future

$\displaystyle 0\leq\underline{\rho}\leq\rho\leq\overline{\rho},\hskip
3.0pt{\text{a}.e.in}\hskip 2.0pt(0,T)\times\Omega$

for some constants $\underline{\rho},\overline{\rho}$. Moreover, we assume
$w\in L^{2}(0,T;L^{2}(\mathbb{T}^{3}))$ $\mathbf{v}\in L^{2}((0,T);H^{1}_{\bf
x}(\mathbb{T}^{2})L^{2}_{z}(\mathbb{T}^{1}))$ and $p\in
C[\underline{\rho},\overline{\rho}]$. Then the energy is locally conserved
i.e.

$\displaystyle\partial_{t}\left(\frac{1}{2}\rho|\mathbf{v}|^{2}+P(\rho)\right)+\rho|\nabla_{\bf
x}\mathbf{v}|^{2}+\rho|\partial_{z}\mathbf{v}|^{2}+{\rm div}_{\bf
x}\left[\left(\frac{1}{2}\rho|\mathbf{v}|^{2}+p(\rho)+P(\rho)-\rho\nabla_{\bf
x}\mathbf{v}\right)\mathbf{v}\right]$
$\displaystyle+\partial_{z}\left[\left(\frac{1}{2}\rho|\mathbf{v}|^{2}+p(\rho)+P(\rho)\right)w-\rho\partial_{z}\mathbf{v}\cdot\mathbf{v}\right]=0.$

######  Remark 2.7.

The corresponding constant viscosity coefficient result is also valid as in
Remark 2.5.

######  Remark 2.8.

Notice that our assumption $\beta>\alpha$, in the definition of the
anisotropic Besov space and Theorem 2.2, satisfies the special structure of
CPE, which means that the horizontal regularity is higher than vertical one.
Moreover, from the assumptions $\beta>\alpha$ and $2\alpha+\beta>2$, we can
deduce $\beta>\frac{2}{3}$. It is important to point out that our result could
be extended to different regularity with $\mathbf{v}$ and $\rho$, such as
$\mathbf{v}\in
B^{\alpha,\infty}_{3}((0,T);B^{\alpha_{1},\infty}_{3}(\mathbb{T};B^{\beta_{1},\infty}_{3}(\mathbb{T}^{2})))$
and $\rho\in
B^{\beta,\infty}_{3}((0,T);B^{\alpha_{2},\infty}_{3}(\mathbb{T};B^{\beta_{2},\infty}_{3}(\mathbb{T}^{2})))$.
But it will produce many complex index relationships. In order not to bring
confuse to readers, we just uniform the regularity and assume they are in the
same regularity space.

######  Remark 2.9.

It should be noticed that vertical regularity of $\mathbf{v}$ in the Theorem
2.2 is lower than in [1], in which they assume $\mathbf{v}\in
L^{2}((0,T);H^{1})$, by virtue of the anisotropic regularity and the special
structure of CPE.

##  3 Proof of Theorem 2.1 and Theorem 2.2

###  3.1 Energy equality

We follow the strategy from [1, 28]. Precisely speaking, we mollify the CPE in
both space and time, then derive the local energy equality for regularized
quantities, estimate commutator errors generated by nonlinear terms and,
passing $(\varepsilon,\delta)\rightarrow 0$ (see below), commutators tend to
zero and the result is obtained in the original quantities.

Firstly, we mollify the momentum equation $\eqref{1.1}_{2}$ and obtain

$\displaystyle\partial_{t}(\rho\mathbf{v})^{\varepsilon}+\textrm{div}_{\bf
x}(\rho\mathbf{v}\otimes\mathbf{v})^{\varepsilon}+\partial_{z}(\rho\mathbf{v}w)^{\varepsilon}+\nabla_{\bf
x}p^{\varepsilon}(\rho)={\rm div}_{\bf x}(\rho\nabla_{\bf
x}\mathbf{v})^{\varepsilon}+\partial_{z}(\rho\partial_{z}\mathbf{v})^{\varepsilon}.$
(3.1)

Notice that the pressure term in (3.1) does not belong to $C^{2}$, thus it
does not satisfy the conditions in [28]. Therefore, we need mollify the
pressure as the following (based on [1]): Take a sequence $p^{\delta}\in
C^{2}[\underline{\rho},\overline{\rho}]$ that converges uniformly to $p\in
C[\underline{\rho},\overline{\rho}]$, that is for each $\delta>0$

$\displaystyle\|p^{\delta}-p\|_{L^{\infty}}\leq\delta.$

Replacing $p^{\delta}$ in (3.1), we obtain

$\displaystyle\partial_{t}(\rho\mathbf{v})^{\varepsilon}$
$\displaystyle+\textrm{div}_{\bf
x}(\rho\mathbf{v}\otimes\mathbf{v})^{\varepsilon}+\partial_{z}(\rho\mathbf{v}w)^{\varepsilon}+\nabla_{\bf
x}\left(p^{\delta}(\rho)\right)^{\varepsilon}$ $\displaystyle={\rm div}_{\bf
x}(\rho\nabla_{\bf
x}\mathbf{v})^{\varepsilon}+\partial_{z}(\rho\partial_{z}\mathbf{v})^{\varepsilon}+\nabla_{\bf
x}\Big{(}(p^{\delta}(\rho))^{\varepsilon}-p^{\varepsilon}(\rho)\Big{)}.$

It is easy to obtain

$\displaystyle\partial_{t}(\rho^{\varepsilon}\mathbf{v}^{\varepsilon})+\textrm{div}_{\bf
x}\big{(}(\rho\mathbf{v})^{\varepsilon}\otimes\mathbf{v}^{\varepsilon}\big{)}$
$\displaystyle+\partial_{z}\big{(}(\rho
w)^{\varepsilon}\mathbf{v}^{\varepsilon}\big{)}+\nabla_{\bf
x}p^{\delta}(\rho^{\varepsilon})-{\rm div}_{\bf
x}\left(\rho^{\varepsilon}\nabla_{\bf
x}\mathbf{v}^{\varepsilon}\right)-\partial_{z}\left(\rho^{\varepsilon}\partial_{z}\mathbf{v}^{\varepsilon}\right)$
$\displaystyle=\partial_{t}\Big{(}\rho^{\varepsilon}\mathbf{v}^{\varepsilon}-(\rho\mathbf{v})^{\varepsilon}\big{)}+\textrm{div}_{\bf
x}\big{(}(\rho\mathbf{v})^{\varepsilon}\otimes\mathbf{v}^{\varepsilon}-(\rho\mathbf{v}\otimes\mathbf{v})^{\varepsilon}\Big{)}$
$\displaystyle\hskip 10.0pt+\partial_{z}\Big{(}(\rho
w)^{\varepsilon}\mathbf{v}^{\varepsilon}-(\rho\mathbf{v}w)^{\varepsilon}\Big{)}$
$\displaystyle\hskip 10.0pt-{\rm div}_{\bf
x}\Big{(}\rho^{\varepsilon}\nabla_{\bf
x}\mathbf{v}^{\varepsilon}-(\rho\nabla_{\bf
x}\mathbf{v})^{\varepsilon}\big{)}-\partial_{z}\big{(}\rho^{\varepsilon}\partial_{z}\mathbf{v}^{\varepsilon}-(\rho\partial_{z}\mathbf{v})^{\varepsilon}\Big{)}$
$\displaystyle\hskip 10.0pt+\nabla_{\bf
x}\Big{(}p^{\delta}(\rho^{\varepsilon})-(p^{\delta}(\rho))^{\varepsilon}\big{)}+\nabla_{\bf
x}\big{(}(p^{\delta}(\rho))^{\varepsilon}-p^{\varepsilon}(\rho)\Big{)}$
$\displaystyle=R^{\varepsilon}=\displaystyle\sum_{i=1}^{7}R^{\varepsilon}_{i}.$
(3.2)

Multiplying with $\mathbf{v}^{\varepsilon}$, we get

$\displaystyle\partial_{t}\rho^{\epsilon}|\mathbf{v}^{\epsilon}|^{2}$
$\displaystyle+\rho^{\epsilon}\partial_{t}\frac{|\mathbf{v}^{\epsilon}|^{2}}{2}+{\rm
div}_{\bf
x}{(\rho\mathbf{v})^{\varepsilon}}|\mathbf{v}^{\epsilon}|^{2}+(\rho\mathbf{v})^{\varepsilon}\nabla_{\bf
x}\frac{|\mathbf{v}^{\epsilon}|^{2}}{2}+\partial_{z}(\rho
w)^{\epsilon}|\mathbf{v}^{\epsilon}|^{2}+(\rho
w)^{\varepsilon}\partial_{z}\frac{|\mathbf{v}^{\epsilon}|^{2}}{2}$
$\displaystyle+\rho^{\epsilon}\mathbf{v}^{\epsilon}\cdot\nabla_{\bf
x}(P^{\delta}(\rho^{\varepsilon}))^{\prime}+\rho^{\varepsilon}|\nabla_{\bf
x}\mathbf{v}^{\varepsilon}|^{2}+\rho^{\varepsilon}|\partial_{z}\mathbf{v}^{\varepsilon}|^{2}-{\rm
div}_{\bf x}(\rho^{\varepsilon}\nabla_{\bf
x}\mathbf{v}^{\epsilon}\cdot\mathbf{v}^{\epsilon})-\partial_{z}(\rho^{\epsilon}\partial_{z}\mathbf{v}^{\epsilon}\cdot\mathbf{v}^{\varepsilon})$
$\displaystyle=R^{\varepsilon}\mathbf{v}^{\varepsilon}$ (3.3)

where

$\displaystyle\nabla_{\bf
x}p^{\delta}(\rho^{\varepsilon})=(p^{\delta}(\rho^{\varepsilon}))^{\prime}\nabla_{\bf
x}\rho^{\epsilon}=\rho^{\varepsilon}(P^{\delta}(\rho^{\varepsilon}))^{\prime\prime}\nabla_{\bf
x}\rho^{\epsilon},=\rho^{\varepsilon}\nabla_{\bf
x}(P^{\delta}(\rho^{\varepsilon}))^{\prime}.$

Mollifying the continuity equation $\eqref{1.1}_{1}$ with space and time, we
have

$\displaystyle\partial_{t}\rho^{\varepsilon}+{\rm div}_{\bf
x}(\rho\mathbf{v})^{\varepsilon}+\partial_{z}(\rho w)^{\varepsilon}=0.$ (3.4)

It is easy to obtain the following equality

$\displaystyle(\partial_{t}\rho^{\varepsilon}+{\rm div}_{\bf
x}(\rho\mathbf{v})^{\varepsilon}+\partial_{z}(\rho
w)^{\varepsilon})\frac{|\mathbf{v}^{\varepsilon}|^{2}}{2}=0.$ (3.5)

We rewrite (3.5) and get

$\displaystyle\partial_{t}\rho^{\varepsilon}+{\rm div}_{\bf
x}(\rho^{\varepsilon}\mathbf{v}^{\varepsilon})+\partial_{z}(\rho
w)^{\varepsilon}={\rm div}_{\bf
x}(\rho^{\varepsilon}\mathbf{v}^{\varepsilon}-(\rho\mathbf{v})^{\varepsilon}),$

then

$\displaystyle\partial_{t}P^{\delta}(\rho^{\varepsilon})+{\rm div}_{\bf
x}(\rho^{\varepsilon}\mathbf{v}^{\varepsilon})(P^{\delta}(\rho^{\varepsilon}))^{\prime}+\partial_{z}(\rho
w)^{\varepsilon}(P^{\delta}(\rho^{\varepsilon}))^{\prime}={\rm div}_{\bf
x}\Big{(}\rho^{\varepsilon}\mathbf{v}^{\varepsilon}-(\rho\mathbf{v})^{\varepsilon}\Big{)}(P^{\delta}(\rho^{\varepsilon}))^{\prime}.$
(3.6)

We define $s^{\varepsilon}={\rm div}_{\bf
x}(\rho^{\varepsilon}\mathbf{v}^{\varepsilon}-(\rho\mathbf{v})^{\varepsilon})(P^{\delta}(\rho^{\varepsilon}))^{\prime}$,
and put (3.3)-(3.6) together to obtain

$\displaystyle\partial_{t}\left(\frac{1}{2}\rho^{\varepsilon}|\mathbf{v}^{\varepsilon}|^{2}+P^{\delta}(\rho^{\varepsilon})\right)+$
$\displaystyle\rho^{\varepsilon}|\nabla_{\bf
x}\mathbf{v}^{\varepsilon}|^{2}+\rho^{\varepsilon}|\partial_{z}\mathbf{v}^{\varepsilon}|^{2}$
$\displaystyle+$ $\displaystyle{\rm div}_{\bf
x}\left((\rho\mathbf{v})^{\varepsilon}\frac{1}{2}|\mathbf{v}^{\varepsilon}|^{2}+(\rho^{\varepsilon}\mathbf{v}^{\varepsilon})(P^{\delta}(\rho^{\varepsilon}))^{\prime}\mathbf{-}\rho^{\varepsilon}\nabla_{\bf
x}\mathbf{v}^{\varepsilon}\mathbf{v}^{\varepsilon}\right)$ $\displaystyle+$
$\displaystyle\partial_{z}\left((\rho
w)^{\varepsilon}\frac{1}{2}|\mathbf{v}^{\varepsilon}|^{2}+(\rho
w)^{\varepsilon}(P^{\delta}(\rho^{\varepsilon}))^{\prime}-\rho^{\epsilon}\partial_{z}\mathbf{v}^{\epsilon}\cdot\mathbf{v}^{\varepsilon}\right)=R^{\varepsilon}\mathbf{v}^{\varepsilon}+s^{\varepsilon}.$
(3.7)

Therefore, we just need to prove the limit $\varepsilon\rightarrow 0$ for each
fixed $\delta>0$, then consider $\delta\rightarrow 0$ in the following
section.

###  3.2 Commutator estimates

Section 3.1 can be seen as the prior estimates of energy equality. We will
deduce the local energy equality for the regularized quantities as follows:

Let $\varphi\in C^{\infty}_{c}(\Omega)$ be a test function and take
$\varepsilon>0$ small enough so that
$supp(\varphi)\subset(\varepsilon,T-\varepsilon)\times\mathbb{T}^{3}$.
Multiplying (3.1) with $\varphi\mathbf{v}^{\varepsilon}$ and integrating in
time and space, we have

$\displaystyle\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\partial_{t}(\rho^{\varepsilon}\mathbf{v}^{\varepsilon})\cdot\varphi\mathbf{v}^{\varepsilon}\,d{\bf
x}dzdt$ $\displaystyle+\int^{\tau}_{0}\int_{\mathbb{T}^{3}}{\rm div}_{\bf
x}\big{(}(\rho\mathbf{v})^{\varepsilon}\otimes\mathbf{v}^{\varepsilon})\cdot\varphi\mathbf{v}^{\varepsilon}+\partial_{z}\big{(}(\rho
w)^{\varepsilon}\mathbf{v}^{\varepsilon})\cdot\varphi\mathbf{v}^{\epsilon}\,d{\bf
x}dzdt$ $\displaystyle+\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\nabla_{\bf
x}p^{\delta}(\rho^{\varepsilon})\cdot\varphi\mathbf{v}^{\varepsilon}\,d{\bf
x}dzdt$ $\displaystyle=\int^{\tau}_{0}\int_{\mathbb{T}^{3}}{\rm div}_{\bf
x}(\rho^{\varepsilon}\nabla_{\bf
x}\mathbf{v}^{\varepsilon})\cdot\varphi\mathbf{v}^{\varepsilon}+\partial_{z}(\rho^{\varepsilon}\partial_{z}\mathbf{v}^{\varepsilon})\varphi\mathbf{v}^{\varepsilon}\,d{\bf
x}dzdt$
$\displaystyle+\displaystyle\sum_{i=1}^{7}\int^{\tau}_{0}\int_{\mathbb{T}^{3}}R^{\varepsilon}_{i}\cdot\varphi\mathbf{v}^{\varepsilon}\,d{\bf
x}dzdt.$

Then, multiplying (3.4) by
$\displaystyle\frac{1}{2}\,\varphi|\bf{v}^{\varepsilon}|^{2}$, multiplying
(3.6) by $\varphi$ and combining together, so that we obtain the form of (3.1)
in the distribution sense. By the same taken as [1, 28], we just need to show
that each commutator error term converges to zero as

$\displaystyle\int^{\tau}_{0}\int_{\mathbb{T}^{3}}R^{\varepsilon}_{i}\varphi\,d{\bf
x}dzdt\rightarrow 0,\hskip
3.0pt\int^{\tau}_{0}\int_{\mathbb{T}^{3}}s^{\varepsilon}\varphi\,d{\bf
x}dzdt\rightarrow 0.$

In the following sections, we will divide into two parts to prove the above
convergence. Therefore, we complete the proof of both theorems.

####  3.2.1 Proof of Theorem 2.1

The terms $R^{\varepsilon}_{i}$ $(i=1,2)$ and $s^{\varepsilon}$ is dealt by
the same way as [28]. For readers’ convenience, we recall the main steps and
take $R^{\varepsilon}_{2}$ as an example to show the proof. Using Remark 2.2,
we have

$\displaystyle(\rho\mathbf{v})^{\varepsilon}\otimes\mathbf{v}^{\varepsilon}-(\rho\mathbf{v}\otimes\mathbf{v})^{\varepsilon}$
$\displaystyle=\big{(}(\rho\mathbf{v})^{\varepsilon}-\rho\mathbf{v}\big{)}\otimes(\mathbf{v}^{\varepsilon}-\mathbf{v})\hfill$
$\displaystyle-\int^{\varepsilon}_{-\varepsilon}\int_{\mathbb{T}^{3}}\\-\\-\eta^{\varepsilon}(\tau,{\boldsymbol{\xi}_{h}},\xi_{z})\left(\rho\mathbf{v}(t-\tau,{\bf
x}-{\boldsymbol{\xi}}_{h},z-\xi_{z})-\rho\mathbf{v}(t,{\bf x},z)\right)$
$\displaystyle\hskip 56.9055pt\otimes\left(\mathbf{v}(t-\tau,{\bf
x}-{\boldsymbol{\xi}}_{h},z-\xi_{z})-\mathbf{v}(t,{\bf
x},z)\right)\,d{\boldsymbol{\xi}}_{h}d\xi_{z}d\tau.$ (3.8)

Therefore, for the first term we split it into two parts according to (3.2.1),

$\displaystyle\int^{\tau}_{0}\int_{\mathbb{T}^{3}}$ $\displaystyle{\rm
div}_{\bf
x}\big{(}(\rho\mathbf{v})^{\varepsilon}-\rho\mathbf{v})\otimes(\mathbf{v}^{\varepsilon}-\mathbf{v})\big{)}\varphi\mathbf{v}^{\varepsilon}d{\bf
x}dzdt$
$\displaystyle\leq\|\varphi\|_{C^{1}}\|(\rho\mathbf{v})^{\varepsilon}-\rho\mathbf{v}\|_{L^{3}}\|\mathbf{v}^{\varepsilon}-\mathbf{v}\|_{L^{3}}\|\mathbf{v}^{\varepsilon}\|_{L^{3}}+\|\varphi\|_{C^{0}}\|(\rho\mathbf{v})^{\varepsilon}-\rho\mathbf{v}\|_{L^{3}}\|\mathbf{v}^{\varepsilon}-\mathbf{v}\|_{L^{3}}\|\nabla_{\bf
x}\mathbf{v}^{\varepsilon}\|_{L^{3}}$ $\displaystyle\leq
C\left(\varepsilon^{\alpha+\beta}+\varepsilon^{2\alpha+\beta-1}\right)\|\rho\mathbf{v}\|_{B_{3}^{\beta,\infty}}\|\mathbf{v}\|^{2}_{B_{3}^{\alpha,\infty}}.$

From the assumption in Theorem 2.1 $\alpha+\beta>\frac{3}{2}$, it is easy to
deduce $2\alpha+\beta>1$, so that we can obtain the term vanishes when
$\varepsilon\rightarrow 0$. For another part of $R^{\varepsilon}_{2}$, we use
the Fubini’s theorem and Holder’s inequality, and get

$\displaystyle\int^{\tau}_{0}\int_{\mathbb{T}^{3}}{\rm
div}\left(\int^{\varepsilon}_{\varepsilon}\int_{\mathbb{T}^{3}}\eta^{\varepsilon}(\tau,\xi)\right.$
$\displaystyle\Big{(}\rho\mathbf{v}(t-\tau,{\bf
x}-{\boldsymbol{\xi}}_{h},z-\xi_{z})-\rho\mathbf{v}(t,{\bf x},z)\Big{)}$
$\displaystyle\left.\cdot\Big{(}\mathbf{v}(t-\tau,{\bf
x}-{\boldsymbol{\xi}}_{h},z-\xi)-\mathbf{v}(t,{\bf
x},z)\Big{)}d{\boldsymbol{\xi}}_{h}\,d\xi_{z}\,d\tau\right)\cdot\varphi\mathbf{v}^{\varepsilon}d{\bf
x}dzdt$ $\displaystyle\leq
C\left(\varepsilon^{\alpha+\beta}+\varepsilon^{2\alpha+\beta-1}\right)\|\rho\mathbf{v}\|_{B_{3}^{\beta,\infty}}\|\mathbf{v}\|^{2}_{B_{3}^{\alpha,\infty}}.$
(3.9)

Next, we turn to estimate the commutators $R^{\varepsilon}_{6}$ and
$R^{\varepsilon}_{7}$, the proof is inspired by [1] and we include the proof
here for completeness

$\displaystyle\int^{\tau}_{0}\int_{\mathbb{T}^{3}}$ $\displaystyle\nabla_{\bf
x}\big{(}p^{\delta}(\rho^{\varepsilon})-(p^{\delta}(\rho))^{\varepsilon}\big{)}\varphi\mathbf{v}^{\varepsilon}d{\bf
x}dzdt$
$\displaystyle=-\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\big{(}p^{\delta}(\rho^{\varepsilon})-(p^{\delta}(\rho))^{\varepsilon}\big{)}(\varphi{\rm
div}_{\bf x}\mathbf{v}^{\varepsilon}+\nabla_{\bf
x}\varphi\mathbf{v}^{\varepsilon})d{\bf x}dzdt$
$\displaystyle\leq\|\varphi\|_{C^{0}}\|p^{\delta}(\rho^{\varepsilon})-(p^{\delta}(\rho))^{\varepsilon}\|_{L^{\frac{3}{2}}}\|{\rm
div}_{\bf
x}\mathbf{v}^{\varepsilon}\|_{L^{3}}+C\|\varphi\|_{C^{1}}\|p^{\delta}(\rho^{\varepsilon})-(p^{\delta}(\rho))^{\varepsilon}\|_{L^{\frac{3}{2}}}\|\mathbf{v}^{\varepsilon}\|_{L^{3}}$
$\displaystyle\leq
C\|p^{\delta}(\rho^{\varepsilon})-(p^{\delta}(\rho))^{\varepsilon}\|_{L^{\frac{3}{2}}}\varepsilon^{\alpha-1}\|\mathbf{v}\|_{B^{\alpha,\infty}_{3}}+C\|p^{\delta}(\rho^{\varepsilon})-(p^{\delta}(\rho))^{\varepsilon}\|_{L^{\frac{3}{2}}}\|\mathbf{v}^{\varepsilon}\|_{L^{3}}$
$\displaystyle\leq C\big{(}\|\rho^{\varepsilon}-\rho\|^{2}_{L^{3}}+\sup_{{\bf
y}\in{\rm supp}\eta^{\varepsilon}}\|\rho(\cdot)-\rho(\cdot-{\bf
y})\|^{2}_{L^{3}}\big{)}(\varepsilon^{\alpha-1}+1)\|\mathbf{v}\|_{B^{\alpha,\infty}_{3}}$
$\displaystyle\leq
C\,\varepsilon^{2\beta}(\varepsilon^{\alpha-1}+1)\|\rho\|^{2}_{B_{3}^{\beta,\infty}}\|\mathbf{v}\|_{B^{\alpha,\infty}_{3}},$

where we have used Lemma 2.4 in the second inequality and

$\displaystyle\int^{\tau}_{0}\int_{\mathbb{T}^{3}}$ $\displaystyle\nabla_{\bf
x}\big{(}(p^{\delta}(\rho))^{\varepsilon}-p^{\varepsilon}(\rho)\big{)}\varphi\mathbf{v}^{\varepsilon}d{\bf
x}dzdt$ $\displaystyle=$
$\displaystyle-\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\big{(}p^{\delta}(\rho)-p(\rho)\big{)}^{\varepsilon}\varphi{\rm
div}_{\bf x}\mathbf{v}^{\varepsilon}d{\bf
x}dzdt-\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\big{(}p^{\delta}(\rho)-p(\rho)\big{)}^{\varepsilon}\nabla_{\bf
x}\varphi\mathbf{v}^{\varepsilon}d{\bf x}dzdt$ $\displaystyle\leq
C\,\|\varphi\|_{C^{0}}\|\big{(}p^{\delta}(\rho)-p(\rho)\big{)}^{\varepsilon}\|_{L^{\infty}}\|{\rm
div}_{\bf
x}\mathbf{v}^{\varepsilon}\|_{L^{2}}+C\|\varphi\|_{C^{1}}\|\big{(}p^{\delta}(\rho)-p(\rho)\big{)}^{\varepsilon}\|_{L^{\infty}}\|\mathbf{v}^{\varepsilon}\|_{L^{3}}$
$\displaystyle\leq C\delta.$

The remaining is to estimate $R^{\varepsilon}_{4}$ and $R^{\varepsilon}_{5}$.
We give a different proof compared with [1] as the following:

$\displaystyle\int^{\tau}_{0}$ $\displaystyle\int_{\mathbb{T}^{3}}{\rm
div}_{\bf x}(\rho^{\varepsilon}\nabla_{\bf
x}\mathbf{v}^{\varepsilon}-(\rho\nabla_{\bf
x}\mathbf{v})^{\varepsilon})\varphi\mathbf{v}^{\varepsilon}d{\bf x}dzdt$
$\displaystyle=-\int^{\tau}_{0}\int_{\mathbb{T}^{3}}(\rho^{\varepsilon}\nabla_{\bf
x}\mathbf{v}^{\varepsilon}-(\rho\nabla_{\bf
x}\mathbf{v})^{\varepsilon})\nabla_{\bf x}\varphi\mathbf{v}^{\varepsilon}d{\bf
x}dzdt-\int^{\tau}_{0}\int_{\mathbb{T}^{3}}(\rho^{\varepsilon}\nabla_{\bf
x}\mathbf{v}^{\varepsilon}-(\rho\nabla_{\bf
x}\mathbf{v})^{\varepsilon})\varphi\nabla_{\bf x}\mathbf{v}^{\varepsilon}d{\bf
x}dzdt.$ (3.10)

Using Remark 2.2, we split the two terms on the right hand side of (3.10) into
four terms:

$\displaystyle\int^{\tau}_{0}\int_{\mathbb{T}^{3}}(\rho^{\varepsilon}\nabla_{\bf
x}\mathbf{v}^{\varepsilon}-(\rho\nabla\mathbf{v})^{\varepsilon})\nabla_{\bf
x}\varphi\mathbf{v}^{\varepsilon}d{\bf x}dzdt$
$\displaystyle=\int^{\tau}_{0}\int_{\mathbb{T}^{3}}(\rho^{\varepsilon}-\rho)(\nabla_{\bf
x}\mathbf{v}^{\varepsilon}-\nabla_{\bf
x}\mathbf{v})\mathbf{v}^{\epsilon}\cdot\nabla_{\bf x}\varphi\,d{\bf x}dzdt$
$\displaystyle\hskip
3.0pt+\int^{\varepsilon}_{-\varepsilon}\int_{\mathbb{T}^{3}}\\-\\-\eta^{\varepsilon}(\tau,\xi)\Big{(}\rho(t-\tau,{\bf
x}-{\boldsymbol{\xi}}_{h})-\rho(t,{\bf x})\Big{)}\\-\Big{(}\nabla_{\bf
x}\mathbf{v}(t-\tau,{\bf x}-{\boldsymbol{\xi}}_{h},z-\xi_{z})-\nabla_{\bf
x}\mathbf{v}(t,x,z)\Big{)}\mathbf{v}^{\epsilon}\cdot\nabla_{\bf
x}\varphi\,d{\boldsymbol{\xi}}_{h}d\xi_{z}d\tau$ $\displaystyle=J_{1}+J_{2},$

and

$\displaystyle\int^{\tau}_{0}\int_{\mathbb{T}^{3}}(\rho^{\varepsilon}\nabla_{\bf
x}\mathbf{v}^{\varepsilon}-(\rho\nabla_{\bf
x}\mathbf{v})^{\varepsilon})\varphi\nabla_{\bf x}\mathbf{v}^{\varepsilon}d{\bf
x}dzdt$
$\displaystyle=\int^{\tau}_{0}\int_{\mathbb{T}^{3}}(\rho^{\varepsilon}-\rho)(\nabla_{\bf
x}\mathbf{v}^{\varepsilon}-\nabla_{\bf x}\mathbf{v}):\nabla_{\bf
x}\mathbf{v}^{\epsilon}\varphi\,d{\bf x}dzdt$ $\displaystyle\hskip
3.0pt+\int^{\varepsilon}_{-\varepsilon}\int_{\mathbb{T}^{3}}\eta^{\varepsilon}(\tau,\xi)\Big{(}\rho(t-\tau,{\bf
x}-{\boldsymbol{\xi}}_{h})-\rho(t,{\bf x})\Big{)}\Big{(}\nabla_{\bf
x}\mathbf{v}(t-\tau,{\bf x}-{\boldsymbol{\xi}}_{h},z-\xi_{z})-\nabla_{\bf
x}\mathbf{v}(t,{\bf x},z)\Big{)}\\-:\\-\nabla_{\bf
x}\mathbf{v}^{\epsilon}\varphi\,d{\boldsymbol{\xi}}_{h}d{\xi_{z}}d\tau$
$\displaystyle=J_{3}+J_{4}.$

Observe that $J_{2}$ and $J_{4}$ has been written taking into account that
$\rho$ is independent of $z$.

We just concern on the terms $J_{1}$ and $J_{3}$, the remaining ones can be
estimated similarly by applying the Fubini’s theorem (see in [1, 28]).

$\displaystyle|J_{1}|+|J_{3}|$
$\displaystyle=\left|\int^{\tau}_{0}\int_{\mathbb{T}^{3}}(\rho^{\varepsilon}-\rho)(\nabla_{\bf
x}\mathbf{v}^{\varepsilon}-\nabla_{\bf
x}\mathbf{v})\mathbf{v}^{\epsilon}\cdot\nabla_{\bf x}\varphi\,d{\bf
x}dzdt\right|+\left|\int^{\tau}_{0}\int_{\mathbb{T}^{3}}(\rho^{\varepsilon}-\rho)(\nabla_{\bf
x}\mathbf{v}^{\varepsilon}-\nabla_{\bf x}\mathbf{v}):\nabla_{\bf
x}\mathbf{v}^{\epsilon}\varphi\,d{\bf x}dzdt\right|$
$\displaystyle\leq\|\varphi\|_{C^{1}}\|\rho^{\varepsilon}-\rho\|_{L^{3}}\|\nabla_{\bf
x}(\mathbf{v}^{\varepsilon}-\mathbf{v})\|_{L^{2}}\|\mathbf{v}^{\varepsilon}\|_{L^{6}}+\|\varphi\|_{C^{0}}\|\rho^{\varepsilon}-\rho\|_{L^{3}}\|\nabla_{\bf
x}(\mathbf{v}^{\varepsilon}-\mathbf{v})\|_{L^{2}}\|\nabla_{\bf
x}\mathbf{v}^{\varepsilon}\|_{L^{6}}$
$\displaystyle\leq\|\varphi\|_{C^{1}}\varepsilon^{\beta}\|\rho\|_{B_{3}^{\beta,\infty}}\|\nabla_{\bf
x}(\mathbf{v}^{\varepsilon}-\mathbf{v})\|_{L^{2}}\|\mathbf{v}\|_{H^{1}}+\|\varphi\|_{C^{0}}\varepsilon^{\beta+\alpha-3/2}\|\rho\|_{B_{3}^{\beta,\infty}}\|\nabla_{\bf
x}(\mathbf{v}^{\varepsilon}-\mathbf{v})\|_{L^{2}}\|\mathbf{v}\|_{B_{6}^{\alpha-1/2,\infty}}$
$\displaystyle\leq\|\varphi\|_{C^{1}}\varepsilon^{\beta}\|\rho\|_{B_{3}^{\beta,\infty}}\|\nabla_{\bf
x}(\mathbf{v}^{\varepsilon}-\mathbf{v})\|_{L^{2}}\|\mathbf{v}\|_{H^{1}}+\|\varphi\|_{C^{0}}\varepsilon^{\beta+\alpha-3/2}\|\rho\|_{B_{3}^{\beta,\infty}}\|\nabla_{\bf
x}(\mathbf{v}^{\varepsilon}-\mathbf{v})\|_{L^{2}}\|\mathbf{v}\|_{B_{3}^{\alpha,\infty}}.$

where we have used Lemma 2.1 and Lemma 2.2 to get

$\displaystyle\|\nabla_{\bf
x}\mathbf{v}^{\varepsilon}\|_{L^{6}}\leq\varepsilon^{\alpha-\frac{3}{2}}\|\mathbf{v}\|_{B^{\alpha-\frac{1}{2},\infty}_{6}}\leq\varepsilon^{\alpha-\frac{3}{2}}\|\mathbf{v}\|_{B^{\alpha,\infty}_{3}}.$
(3.11)

Because we deduce $\mathbf{v}\in B^{\alpha-\frac{1}{2},\infty}_{6}$ by virtue
of the assumption $\mathbf{v}\in B^{\alpha,\infty}_{3}$ and the embedding
$B^{\alpha,\infty}_{3}\hookrightarrow B^{\alpha-\frac{1}{2},\infty}_{6}$ from
Lemma 2.2.

######  Remark 3.1.

In (3.11) we applied Lemma $(2.1)_{3}$ with $\omega$

$\|\nabla_{\bf
x}\mathbf{v}^{\varepsilon}\|_{L^{6}}\leq\varepsilon^{\omega-1}\|\mathbf{v}\|_{B^{\omega,\infty}_{6}}.$

Setting $\omega=\alpha-1/2$ we get (3.11). Since $\alpha>\frac{3}{4}$, it
converges to zero.

Recalling $\mathbf{v}\in L^{2}((0,T);H^{1})$, we have
$R^{\varepsilon}_{4}\rightarrow 0$ as $\varepsilon\rightarrow 0$. For the term
$R^{\varepsilon}_{5}$, it is similar as $R^{\varepsilon}_{4}\rightarrow 0$,
hence we omit the detail.

Finally, we turn to analysis the difficult commutator error
$R^{\varepsilon}_{3}$. As usual, the decomposition to $R^{\varepsilon}_{3}$ is
similar to $R^{\varepsilon}_{2}$, we divide it into two parts as the
following:

$\displaystyle(\rho w)^{\varepsilon}\mathbf{v}^{\varepsilon}-(\rho
w\mathbf{v})^{\varepsilon}=((\rho w)^{\varepsilon}-\rho
w)(\mathbf{v}^{\varepsilon}-\mathbf{v})$
$\displaystyle-\int^{\varepsilon}_{\varepsilon}\int_{\mathbb{T}^{3}}\eta^{\varepsilon}(\tau,\xi)\big{(}\rho
w(t-\tau,{\bf x}-{\boldsymbol{\xi}}_{h},z-\xi_{z})-\rho w(t,{\bf
x},z)\big{)}\big{(}\mathbf{v}(t-\tau,{\bf
x}-{\boldsymbol{\xi}}_{h},z-\xi_{z})-\mathbf{v}(t,{\bf
x},z)\big{)}\,d{\boldsymbol{\xi}}_{h}d\xi_{z}d\tau.$

We only focus on the first part only, as the second part produce the same
estimates after applying Fubin’s theorem. It is difficult to estimate this
term directly, because we do not have enough regularity for the regularity of
vertical velocity. This is the essential difficulty and distinction between
Navier-Stokes system and CPE system.

######  Remark 3.2.

The following equality plays a key role in the existence of CPE [45], which is
also helpful and important in our analysis,

$\displaystyle(\rho w)$ $\displaystyle=-\int^{z}_{0}{\rm div}_{\bf
x}[\rho(v(x,z^{\prime},t)-\int^{1}_{0}v(x,s,t)ds)]dz^{\prime}.$
$\displaystyle=-\int^{z}_{0}{\rm div}_{\bf
x}(\rho\widetilde{\mathbf{v}})dz^{\prime},$ (3.12)

where
$\widetilde{\mathbf{v}}=\mathbf{v}-\overline{\mathbf{v}}=v-\int^{1}_{0}\mathbf{v}(x,s,t)ds$.
Observe that here it is crucial the fact that $\rho$ is independent of the
$z$-variable (see [45]).

Since $\rho\mathbf{v}\in L^{3}((0,T);H^{1}(\mathbb{T}^{3}))$, which means
$\rho w\in L^{3}((0,T);L^{2}(\mathbb{T}^{3}))$ from (3.12), thus we get

$\displaystyle\int^{\tau}_{0}\int_{\mathbb{T}^{3}}$ $\displaystyle((\rho
w)^{\varepsilon}-\rho
w)(\mathbf{v}^{\varepsilon}-\mathbf{v})(\partial_{z}\varphi\mathbf{v}^{\varepsilon}+\varphi\partial_{z}\mathbf{v}^{\varepsilon})\,d{\bf
x}dzdt$ $\displaystyle\leq\|(\rho w)^{\varepsilon}-\rho
w\|_{L^{2}}\|\mathbf{v}^{\varepsilon}-\mathbf{v}\|_{L^{3}}\Big{(}\|\varphi\|_{C^{1}}\|\mathbf{v}^{\varepsilon}\|_{L^{6}}+\|\varphi\|_{C^{0}}\|\partial_{z}\mathbf{v}^{\varepsilon}\|_{L^{6}}\Big{)}$
$\displaystyle\leq C\|\rho
w\|_{L^{2}}\varepsilon^{\alpha}\|\mathbf{v}\|_{B^{\alpha,\infty}_{3}}\Big{(}\|\mathbf{v}\|_{B^{\alpha-\frac{1}{2},\infty}_{6}}+\varepsilon^{\alpha-\frac{3}{2}}\|\mathbf{v}\|_{B^{\alpha-\frac{1}{2},\infty}_{6}}\Big{)}$
$\displaystyle\leq
C\left(\varepsilon^{\alpha}+\varepsilon^{2\alpha-\frac{3}{2}}\right)\|\mathbf{v}\|^{2}_{B^{\alpha,\infty}_{3}},$

where we use similar method from (3.11) to get

$\displaystyle\|\partial_{z}\mathbf{v}^{\varepsilon}\|_{L^{6}}\leq\varepsilon^{\alpha-\frac{3}{2}}\|\mathbf{v}\|_{B^{\alpha-\frac{1}{2},\infty}_{6}}\leq\varepsilon^{\alpha-\frac{3}{2}}\|\mathbf{v}\|_{B^{\alpha,\infty}_{3}}.$
(3.13)

Recalling the assumption $\alpha>\frac{3}{4}$, we could deduce this term
converges to zero.

######  Remark 3.3.

We used similar argument as in (3.11). From Lemma 2.1, it is easy to obtain
that
$\|\partial_{z}\mathbf{v}^{\varepsilon}\|_{L^{6}}\leq\varepsilon^{\omega-1}\|\mathbf{v}\|_{B^{\omega,\infty}_{6}}$.
Then, setting again $\omega=\alpha-1/2$ (since $\alpha>3/4$), we can deduce
(3.13).

In order to complete the proof of Theorem 2.1, we need to show the final level
of convergence for $\delta$. For each fixed $\delta>0$, we have the in the
limit (3.1) as $\varepsilon\rightarrow 0$

$\displaystyle\partial_{t}\left(\frac{1}{2}\rho|\mathbf{v}|^{2}+P^{\delta}(\rho)\right)+\rho|\nabla_{\bf
x}\mathbf{v}|^{2}+\rho|\partial_{z}\mathbf{v}|^{2}+{\rm div}_{\bf
x}\left((\rho\mathbf{v})\frac{1}{2}|\mathbf{v}|^{2}+(p^{\delta}(\rho)+P^{\delta}(\rho))\mathbf{v}-\rho\nabla_{\bf
x}\mathbf{v}\mathbf{v}\right)$ $\displaystyle+\partial_{z}\left((\rho
w)\frac{1}{2}|\mathbf{v}|^{2}+(p^{\delta}(\rho)+P^{\delta}(\rho))w-\rho\partial_{z}\mathbf{v}\cdot\mathbf{v}\right),$
(3.14)

where
$P^{\delta}(\rho)=\rho^{\delta}\displaystyle\int^{\rho}_{1}\frac{p^{\delta}(r)}{r^{2}}dr$.

We will prove (3.2.1) converges in the sense of distributions on $\Omega$ as
$\delta\rightarrow 0$ to (2.1). This limit process of $\delta$ is similar to
Wiedemann et al. [1]. We just show the slight difference in contrast to [1],
since our system has different behavior in vertical direction, that is $w\in
L^{2}(0,T;L^{2}(\mathbb{T}^{3}))$.

Taking $\varphi\in C^{\infty}_{c}(\Omega)$, we have

$\displaystyle\Big{|}\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\partial_{z}\varphi(p^{\delta}(\rho)-p(\rho))w\,d{\bf
x}dzdt\Big{|}\leq
C\,\|\varphi\|_{C^{1}}\|p^{\delta}(\rho)-p(\rho)\|_{L^{\infty}}\|w\|_{L^{2}}\leq
C\delta.$

Recalling the estimates from [1]

$\displaystyle|P^{\delta}(\rho)-P(\rho)|\leq\rho\int^{\rho}_{1}\frac{|p^{\delta}(r)-p(r)|}{r^{2}}dr\leq\|p^{\delta}-p\|_{L^{\infty}}\rho\left|\int^{\rho}_{1}\frac{1}{r^{2}}dr\right|\leq(1+\rho)\|p^{\delta}-p\|_{L^{\infty}},$

we can obtain that

$\displaystyle\Big{|}\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\partial_{z}\varphi(P^{\delta}(\rho)-P(\rho))w\,d{\bf
x}dzdt\Big{|}\leq\|\varphi\|_{C^{1}}\Big{(}1+\|\rho\|_{L^{2}}\Big{)}\|p^{\delta}-p\|_{L^{\infty}}\|w\|_{L^{2}}\leq
C\delta.$

####  3.2.2 Proof for Theorem 2.2

Comparing with the Theorem 2.1, we delicately combine the anisotropic
regularity with the special structure of CPE system, and observe some new
results. The process of proving Theorem 2.2 is similar to Theorem 2.1. We just
focus on some different estimates for commutators $R^{\varepsilon}_{2}$ and
$R^{\varepsilon}_{6}$. First we observe for $R^{\varepsilon}_{2}$

$\displaystyle\int^{\tau}_{0}\int_{\mathbb{T}^{3}}$ $\displaystyle{\rm
div}_{\bf
x}\big{(}(\rho\mathbf{v})^{\varepsilon}-\rho\mathbf{v})\otimes(\mathbf{v}^{\varepsilon}-\mathbf{v})\big{)}\varphi\mathbf{v}^{\varepsilon}\,d{\bf
x}dzdt$
$\displaystyle\leq\|\varphi\|_{C^{1}}\|(\rho\mathbf{v})^{\varepsilon}-\rho\mathbf{v}\|_{L^{3}}\|\mathbf{v}^{\varepsilon}-\mathbf{v}\|_{L^{3}}\|\mathbf{v}^{\varepsilon}\|_{L^{3}}+\|\varphi\|_{C^{0}}\|(\rho\mathbf{v})^{\varepsilon}-\rho\mathbf{v}\|_{L^{3}}\|\mathbf{v}^{\varepsilon}-\mathbf{v}\|_{L^{3}}\|\nabla_{\bf
x}\mathbf{v}^{\varepsilon}\|_{L^{3}}$ $\displaystyle\leq
C\varepsilon^{2\alpha}\|\rho\mathbf{v}\|_{B_{3,t}^{\alpha}(B_{3,z}^{\alpha}(B_{3,h}^{\beta}))}\|\mathbf{v}\|^{2}_{B_{3,t}^{\alpha}(B_{3,z}^{\alpha}(B_{3,h}^{\beta}))}$
$\displaystyle\hskip
10.0pt+C\varepsilon^{2\alpha}\|\rho\mathbf{v}\|_{B_{3,t}^{\alpha}(B_{3,z}^{\alpha}(B_{3,h}^{\beta}))}\|\mathbf{v}\|_{B_{3,t}^{\alpha},B_{3,z}^{\alpha}(B_{3,h}^{\beta})}\varepsilon^{\alpha-1}\|\mathbf{v}\|_{B_{3,t}^{\alpha}(B_{3,z}^{\alpha}(B_{3,h}^{\beta}))}$
$\displaystyle\leq
C\left(\varepsilon^{2\alpha}+\varepsilon^{3\alpha-1}\right)\|\rho\mathbf{v}\|_{B_{3,t}^{\alpha}(B_{3,z}^{\alpha}(B_{3,h}^{\beta}))}\|\mathbf{v}\|^{2}_{B_{3,t}^{\alpha}(B_{3,z}^{\alpha}(B_{3,h}^{\beta}))}.$

We can use the same the idea on the commutator of horizontal dissipative term
$R^{\varepsilon}_{4}$. For the vertical dissipative commutator term
$R^{\varepsilon}_{5}$, we utilize the anisotropic regularity to give an
different and interesting proof. Recalling the density is independent of $z$,
we integrate $R^{\varepsilon}_{5}$ by parts and rewrite it as

$\displaystyle\int^{\tau}_{0}$
$\displaystyle\int_{\mathbb{T}^{3}}\partial_{z}\Big{(}\rho^{\varepsilon}\partial_{z}\mathbf{v}^{\varepsilon}-(\rho\partial_{z}\mathbf{v})^{\varepsilon}\Big{)}\cdot\varphi\mathbf{v}^{\varepsilon}\\-d{\bf
x}dzdt$
$\displaystyle=\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\partial_{z}\Big{(}\partial_{z}(\rho^{\varepsilon}\mathbf{v}^{\varepsilon})-\partial_{z}(\rho\mathbf{v})^{\varepsilon}\Big{)}\cdot\varphi\mathbf{v}^{\varepsilon}\\-d{\bf
x}dzdt$
$\displaystyle=-\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\Big{(}\partial_{z}(\rho^{\varepsilon}\mathbf{v}^{\varepsilon})-\partial_{z}(\rho\mathbf{v})^{\varepsilon}\Big{)}\cdot\Big{(}\mathbf{v}^{\epsilon}\partial_{z}\varphi+\partial_{z}\mathbf{v}^{\epsilon}\varphi\Big{)}\,d{\bf
x}dzdt$
$\displaystyle=-\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\Big{(}\partial_{z}(\rho\mathbf{v}^{\varepsilon})-\partial_{z}(\rho\mathbf{v})^{\varepsilon}\Big{)}\cdot\Big{(}\mathbf{v}^{\epsilon}\partial_{z}\varphi+\partial_{z}\mathbf{v}^{\epsilon}\varphi\Big{)}\,d{\bf
x}dzdt$ $\displaystyle\hskip
5.0pt-\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\Big{(}\partial_{z}(\rho^{\varepsilon}-\rho)\mathbf{v}^{\varepsilon}\Big{)}\cdot\Big{(}\mathbf{v}^{\epsilon}\partial_{z}\varphi+\partial_{z}\mathbf{v}^{\epsilon}\varphi\Big{)}\,d{\bf
x}dzdt.$

For the first term, we use Holder’s inequality to obtain

$\displaystyle\Big{|}\int^{\tau}_{0}$
$\displaystyle\int_{\mathbb{T}^{3}}\Big{(}\partial_{z}(\rho\mathbf{v}^{\varepsilon})-\partial_{z}(\rho\mathbf{v})^{\varepsilon}\Big{)}\cdot\Big{(}\mathbf{v}^{\epsilon}\partial_{z}\varphi+\partial_{z}\mathbf{v}^{\epsilon}\varphi\Big{)}\,d{\bf
x}dzdt\Big{|}$
$\displaystyle\leq\|\partial_{z}(\rho\mathbf{v}^{\varepsilon})-\partial_{z}(\rho\mathbf{v})^{\varepsilon}\|_{L^{3}}\Big{(}\|\mathbf{v}^{\epsilon}\|_{L^{3}}\|\partial_{z}\varphi^{\epsilon}\|_{L^{3}}+\|\partial_{z}\mathbf{v}^{\epsilon}\|_{L^{3}}\|\varphi^{\epsilon}\|_{L^{3}}\Big{)}.$

Recalling Lemma 2.5, we have
$\|\partial_{z}(\rho\mathbf{v}^{\varepsilon})-\partial_{z}(\rho\mathbf{v})^{\varepsilon}\|_{L^{3}}\rightarrow
0$ as $\varepsilon\rightarrow 0$, where we take $\rho=f$ and
$\mathbf{v}=\mathbf{g}$.

Due to the independence of $\rho$ with respect to $z$, Besov estimates for
$\rho$ only depends on variations in the $\beta$-index, which is the index
related to the horizontal components. Based on this fact, we observe the
second term as the following

$\displaystyle\Big{|}\int^{\tau}_{0}$
$\displaystyle\int_{\mathbb{T}^{3}}\Big{(}\partial_{z}(\rho^{\varepsilon}-\rho)\mathbf{v}^{\varepsilon}\Big{)}\cdot\Big{(}\mathbf{v}^{\epsilon}\partial_{z}\varphi+\partial_{z}\mathbf{v}^{\epsilon}\varphi\Big{)}\,d{\bf
x}dzdt\Big{|}$
$\displaystyle=\Big{|}\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\Big{(}\rho^{\varepsilon}-\rho)(\partial_{z}\mathbf{v}^{\varepsilon}\Big{)}\cdot\Big{(}\mathbf{v}^{\epsilon}\partial_{z}\varphi+\partial_{z}\mathbf{v}^{\epsilon}\varphi\Big{)}\,d{\bf
x}dzdt\Big{|}$ $\displaystyle\leq
C\|\rho^{\epsilon}-\rho\|_{L^{3}}\|\partial_{z}\mathbf{v}^{\epsilon}\|_{L^{3}}\Big{(}\|\mathbf{v}^{\epsilon}\|_{L^{3}}+\|\partial_{z}\mathbf{v}^{\epsilon}\|_{L^{3}}\Big{)}$
$\displaystyle\leq
C\varepsilon^{\beta}\|\rho\|_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}\varepsilon^{\alpha-1}\|\mathbf{v}\|_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}\left(\|\mathbf{v}\|_{L^{3}}+\varepsilon^{\alpha-1}\|\mathbf{v}\|_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}\right)$
$\displaystyle\leq
C\Big{(}\varepsilon^{\alpha+\beta-1}+\varepsilon^{2\alpha+\beta-2}\Big{)}\|\rho\|_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}\|\mathbf{v}\|^{2}_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}.$

Then for $R^{\varepsilon}_{6}$, it is similar to obtain

$\displaystyle\Big{|}\int^{\tau}_{0}\int_{\mathbb{T}^{3}}$
$\displaystyle\nabla_{\bf
x}\big{(}p^{\delta}(\rho^{\varepsilon})-(p^{\delta}(\rho))^{\varepsilon}\big{)}\varphi\mathbf{v}^{\varepsilon}\,d{\bf
x}dzdt\Big{|}$
$\displaystyle=\Big{|}-\int^{\tau}_{0}\int_{\mathbb{T}^{3}}\big{(}p^{\delta}(\rho^{\varepsilon})-(p^{\delta}(\rho))^{\varepsilon}\big{)}(\varphi{\rm
div}_{\bf x}\mathbf{v}^{\varepsilon}+\nabla_{\bf
x}\varphi\mathbf{v}^{\varepsilon})\,d{\bf x}dzdt\Big{|}$
$\displaystyle\leq\|\varphi\|_{C^{0}}\|p^{\delta}(\rho^{\varepsilon})-(p^{\delta}(\rho))^{\varepsilon}\|_{L^{\frac{3}{2}}}\|{\rm
div}_{\bf
x}\mathbf{v}^{\varepsilon}\|_{L^{3}}+C\|\varphi\|_{C^{1}}\|p^{\delta}(\rho^{\varepsilon})-(p^{\delta}(\rho))^{\varepsilon}\|_{L^{\frac{3}{2}}}\|\mathbf{v}^{\varepsilon}\|_{L^{3}}$
$\displaystyle\leq
C\|p^{\delta}(\rho^{\varepsilon})-(p^{\delta}(\rho))^{\varepsilon}\|_{L^{3/2}}\varepsilon^{\alpha-1}\|\mathbf{v}^{\varepsilon}\|_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}+C\|p^{\delta}(\rho^{\varepsilon})-(p^{\delta}(\rho))^{\varepsilon}\|_{L^{3/2}}\|\mathbf{v}\|_{L^{3}}$
$\displaystyle\leq
C\big{(}\|\rho^{\varepsilon}-\rho\|^{2}_{L^{3}}+\sup_{y\in{\rm
supp}\eta^{\varepsilon}}\|\rho(\cdot)-\rho(\cdot-y)\|^{2}_{L^{3}}\big{)}\Big{(}\varepsilon^{\alpha-1}+1\Big{)}\|\mathbf{v}\|_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}$
$\displaystyle\leq
C\varepsilon^{2\beta}\Big{(}\varepsilon^{\alpha-1}+1\Big{)}\|\rho\|^{2}_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}\|\mathbf{v}\|_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}$
$\displaystyle\leq
C\Big{(}\varepsilon^{2\beta+\alpha-1}+\varepsilon^{2\beta}\Big{)}\|\rho\|^{2}_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}\|\mathbf{v}\|_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))},$

where we use the assumption $2\alpha+\beta-2>0$ and $\beta>\alpha$ to deduce
$\alpha+2\beta-1>0$.

Let us stress that for the term $R^{\varepsilon}_{3}$, we need to apply the
important property of (CPE), see Remark 3.2 and the equality (3.12). Due to
the assumption $\rho\mathbf{v}\in
B^{\alpha,\infty}_{3}((0,T);B^{\alpha,\infty}_{3}(\mathbb{T};B^{\beta,\infty}_{3}(\mathbb{T}^{2})))$,
then we can deduce $\rho w\in
B^{\alpha,\infty}_{3}$$((0,T);B^{\alpha+1,\infty}_{3}(\mathbb{T};B^{\beta-1,\infty}_{3}(\mathbb{T}^{2}))$.
It means $\rho w$ has $B^{\alpha+1,\infty}_{3}$ regularity in the $z-$
direction, $B^{\beta-1,\infty}_{3}$ regularity in the horizontal directions,
and $B^{\alpha,\infty}_{3}$ regularity in time. Therefore, we can estimate
$R^{\varepsilon}_{3}$ as follows

$\displaystyle\int^{\tau}_{0}\int_{\mathbb{T}^{3}}$ $\displaystyle\left((\rho
w)^{\varepsilon}-\rho
w)(\mathbf{v}^{\varepsilon}-\mathbf{v})(\partial_{z}\varphi\mathbf{v}^{\varepsilon}+\varphi\partial_{z}\mathbf{v}^{\varepsilon}\right)\,d{\bf
x}dzdt$ $\displaystyle\leq\left(\int^{t}_{0}\|(\rho w)^{\varepsilon}-\rho
w\|^{3}_{L^{3}(\mathbb{T}^{3})}dt\right)^{\frac{1}{3}}\cdot\|\mathbf{v}^{\varepsilon}-\mathbf{v}\|_{L^{3}}\Big{(}\|\varphi\|_{C^{1}}\|\|\mathbf{v}^{\varepsilon}\|_{L^{3}}+\|\varphi\|_{C^{0}}\|\|\partial_{z}\mathbf{v}^{\varepsilon}\|_{L^{3}}\Big{)}$
$\displaystyle\leq C\varepsilon^{\beta-1}\left(\int^{t}_{0}\|\rho
w\|^{3}_{B^{\alpha+1}_{3,z}(B^{\beta-1}_{3,h})}dt\right)^{\frac{1}{3}}\varepsilon^{\alpha}\|\mathbf{v}\|^{2}_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}$
$\displaystyle\hskip 10.0pt+C\varepsilon^{\beta-1}\left(\int^{t}_{0}\|\rho
w\|^{3}_{B^{\alpha+1}_{3,z}(B^{\beta-1}_{3,h})}dt\right)^{\frac{1}{3}}\varepsilon^{\alpha}\|\mathbf{v}\|_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}\varepsilon^{\alpha-1}\|\mathbf{v}\|_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}$
$\displaystyle=C\left(\varepsilon^{\alpha+\beta-1}+\varepsilon^{2\alpha+\beta-2}\right)\Big{(}\int^{t}_{0}\|\rho
w\|^{3}_{B^{\alpha+1}_{3,z}(B^{\beta-1}_{3,h})}dt\Big{)})^{\frac{1}{3}}\|\mathbf{v}\|^{2}_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}$
$\displaystyle=\Big{(}\varepsilon^{\alpha+\beta-1}+\varepsilon^{2\alpha+\beta-2}\Big{)}\|\rho
w\|_{B^{\alpha}_{3,t}(B^{\alpha+1}_{3,z}(B^{\beta-1}_{3,h}))}\|\mathbf{v}\|^{2}_{B^{\alpha}_{3,t}(B^{\alpha}_{3,z}(B^{\beta}_{3,h}))}$

##  4 Conclusion

Theorem 2.1 can be seen as a kind of generalization of Feireisl et al. [28]
and Wiedeman et al. [1], by different proof of some part of estimates (like
$R^{\varepsilon}_{4}$) together by observation of the special structure of
CPE. But the conventional isotropic spaces in Theorem 2.1 do not reflect the
different regularity in the horizontal and the vertical directions of the
velocity field. Since the regularity of the $w$ follows from the continuity
equation under assumptions that the density is independent on the vertical
direction, we need additional regularity condition about $\rho\mathbf{v}$ to
control the convergence of $R^{\varepsilon}_{3}$. In Theorem 2.2, we show that
we can solve the corresponding problem and remove this additional regularity
condition by virtue of anisotropic space. Theorem 2.2 not only considers the
continuity of the pressure but also reflects the anisotropy of the velocity
field in the vertical and horizontal directions of the velocity field, the
density and also the momentum. Moreover, our CPE model also contains
dissipation term, vacuum and it is compressible which means different from the
structure of the problem investigated by Titi et al. [9].

Acknowledgements

The research of Š. Nečasová has been supported by the Czech Science Foundation
(GAČR) project 22-01591S. Moreover, Š. N. has been supported by Praemium
Academiae of Š. Nečasová. The Institute of Mathematics, CAS is supported by
RVO:67985840. M.A. Rodríguez-Bellido has been supported by Grant
PGC2018-098308-B-I00 (MCI/AEI/FEDER, UE), Grant US-1381261 (US/JUNTA/FEDER,
UE) and Grant P20-01120 (PAIDI/JUNTA/FEDER, UE). T. Tang is partially
supported by NSF of Jiangsu Province Grant No. BK20221369.

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# Indel Error Correction Codes for DNA Digital Data Storage and Retrieval

NallappaBhavithran G
Department of Mathematics
Vellore Institute of Technology
Vellore.
<EMAIL_ADDRESS>
& Selvakumar R
Department of Mathematics
Vellore Institute of Technology
Vellore.
<EMAIL_ADDRESS>
Corresponding author

###### Abstract

A procedure for storage and retrieval of Digital information in DNA strings is
discussed by constructing an error correction code which is capable of
correcting indel errors and maintaining the stability of DNA strings. For
correction of indel errors, Varshamov–Tenengolts algorithm is used. To
maintain the stability of DNA code, Reverse-complementary property, and GC-
content constraints are implemented by appropriate choices of group
homomorphisms. Code for any desired length(n) can be created using the
construction methods presented in this paper. The code’s reverse complement
distance is calculated using the value of n.

_Keywords_ Coding theory $\cdot$ Kernel code $\cdot$ DNA code $\cdot$ VT
codes.

##  1 Introduction

Over the last few years, there has been significant growth in the creation and
replication of data, and this tendency will continue. Currently, these large-
scale data are being stored on silica-based storage systems, which have high
maintenance costs. One of the factors resulting in the high maintenance costs
of storage is that the maximum lifespan of silica-based storage devices is
almost thirty years. As a result, the storage devices must be updated every
ten to twenty years. And the disposal of the outdated system is a complex
process due to silicon’s non-biodegradability. To store exponentially growing
data, it is essential to find a storage technology that is convenient and
affordable to keep data. The finding that information may be recovered from
sources that date back thousands of years led to the development of a DNA-
based storage system. Compared to silica-based storage systems, DNA-based
storage systems are more durable and capable of handling exponentially growing
data. It is possible to store exabytes of data in a single gram of DNA. The
cost of synthesising and sequencing a DNA storage system is substantial;
however, this cost is expected to go down in the next few years.Dong et al.
(2020)
The structure of DNA is a double helix, which is made up of two connected
strands that loop around one another to resemble a twisted ladder. The
backbone of each strand is composed of alternating deoxyribose and phosphate
groups. Adenine (A), cytosine (C), guanine (G), or thymine are the four bases
that are joined to each sugar (T). Those two strands of nucleotides are held
together by hydrogen bonds between the complementary bases A = T (double bond)
and C $\equiv$ G(triple Bonds) and denoted as Ac = T, Gc = C and vice versa.
Information is encoded into A, G, T, and C sequences, and the corresponding
nucleotides are synthesised(oligos). These oligos are stored in e.coli and can
be retrieved later. Figure 1 illustrates the entire process of storing and
retrieval of data. The oligos typically include 100 to 1000 nucleotide bases
since synthesising them is expensive and prone to mistakes. Polymerase Chain
Reaction(PCR) is used to amplify the stored information and create numerous
copies of the same information. A DNA sequencer aligns these sequences with
retrieving the data.Kannan and McGregor (2005) Singh and Selvakumar (2023)

Figure 1: Systematic procedure of DNA-based storage systems

Starting from kilobytes in 2012 Church et al. (2012) Blawat et al. (2016)
researchers created coding schemes that managed to store and retrieve
megabytes Carmean et al. (2018) Organick et al. (2018) of information from
DNA. In 2016, Yazdi Tabatabaei Yazdi et al. (2015) devised a system to store
six universities’ budgets that could be randomly accessed and modified. To get
random access, they placed addresses on both ends of 27 DNA strands, which
were used to store the university information. These addresses contain 20
nucleotide bases and are self-uncorrelated to each other. Additionally, each
piece of information is encoded into three nucleotide bases(A, T and C)
dependent on their address, and the modification is done using OE-PCR and
g-block technologies.
To rectify errors made during DNA synthesis and sequencing, a suitable error-
correcting code Berlekamp (2015) must be devised. The error-correcting codes
are created by adding redundancy bits to the information while encoding the
information. These redundancy bits are helpful in retrieving information
without errors. The set of all encoded information is called the code, and
each element of the code is called a codeword. The encoding is done in such a
way as to preserve some of the algebraic characteristics of the information.
One of the primary focuses of error in DNA systems is indel errors. Many
bounds have been achieved for the indel errors with the usage of error
correction codes. LevenshteinLevenshtein (1965) suggested a method for
recovering the original data even after the codeword had a single bit removed.
After half a century Brakensiek Brakensiek et al. (2016) generalised the
result for multiple deletions. However, for more than 2 deletions the code is
hard to design. Sima Sima and Bruck (2021) and other researchers Song et al.
(2021)also suggested different coding schemes and attain an optimal bound for
retrieving k-deleted bits from the codeword.
This article presents a new technique for eliminating Indel errors in DNA
strings. Additionally, this technique imposes GC-content and reverse
complement constraints on the code to improve DNA string stability and prevent
secondary structure formation. In section 2, we discuss the problems which
arise while using DNA as a storage system. In section 3, we analyse those
problems discussed in section 2 by imposing constraints. Finally, in section
4, we gave a procedure by using VT and kernel code to construct an indel
error-free DNA code.

##  2 DNA storage model

A DNA code $\mathcal{C}_{DNA}(n,M)\subset\\{A,G,T,C\\}^{n}$ (nucleotide bases)
with each DNA codeword of length n and size M. Let
$x,y\in\mathcal{C}_{DNA}(n,M)$ The Hamming distance H($x,y$) between two
codewords is the number of distinct elements in those two codewords. The
reverse of a codeword $x=(x_{1},x_{2},\dots,x_{n})$ is
$(x_{n},x_{n-1},\dots,x_{2},x_{1})$ and denoted by $x^{R}$. Similarly the
reverse compliment of a codeword $x=(x_{1},x_{2},\dots,x_{n})$ is
$(x_{n}^{c},x_{n-1}^{c},\dots,x_{2}^{c},x_{1}^{c})$ and denoted by $x^{RC}$.
For example: Let us consider n = 3, x = (A, G, C), y = (A, T, G) then H(x, y)
= 2 since second and third position of x and y are different, $x^{R}$ = (C, G,
A) and $x^{RC}$ = (Cc, Gc, Ac) = (G, C, T) Whenever a storage system is
considered, the occurrence of errors is unavoidable. In DNA-based storage
systems, errors occurring during synthesis, sequencing, and PCR amplification
are referred to as indel, sequencing, and storing errors, respectively.

###  2.1 Indel Errors

The deletion, insertion, and substitution errors that take place during DNA
synthesis are referred to as indel errors. In the deletion, a nucleotide base
is deleted from the original DNA string, and the position of deletion is
random. In the insertion, a nucleotide base is inserted into a random place in
the original DNA string. The insertion and deletion change the length of the
original sequence. Substitution errors randomly replace some of the bases in
the original string. For example: Let x = ATACGTTCA, and a base G is inserted
into the 2nd position. The resulting word would be AGTACGTTCA. If the 4th base
is deleted, the resulting string will be ATAGTTCA. If a subsection of A to G
is in third place will be ATGCGTTCA. Levenshtein distance$(d_{l}(x,y))$ is
introduced to find the relation between two strings with respect to indel
errors. For example, let x = ATACGTTCA, y = AGTAGTTCA, then $d_{l}(x,y)$ = 2,
since deleting the second position ‘G’ and inserting a ‘C’ in the fifth
position converts y to x. However, Hamming distance is used to determine the
relationship between two DNA sequences since calculating the Levenshtein
distance is NP-hard.

###  2.2 Sequencing error

In sequencing, small fragments are pieced together to create the whole
sequence. Tatum error is a sequencing error that occurs when a single base
pair or section of the DNA string is replicated and placed adjacent to the
original. As an example, suppose ATACGTCA is a DNA string. Upon sequencing,
one would receive ATATACGTCA. The length two tantum error occurred as the
first two base pairs “AT”. Most of the time, this error correction is done for
therapy and other medical purposes. But we also need to consider this when
talking about the longevity of DNA and other materials that need to be stored
for an extended period.Jain et al. (2017)

###  2.3 Storing Errors

For replicating the DNA, a process called hybridisation is used. However, the
unwanted hybridisation in long strands results in the formation of a secondary
structure. Since DNA strings are sequenced to form the long strand of a DNA
molecule, complementary substrings of the concatenation of two DNA strings
should not be in the long strand. (i.e.) Let x = AGT, y = CAG, then any
substring of (xy)${}^{R}C$ = CTGACT should not be a codeword of
$\mathcal{C}_{DNA}(n,M)$. For the stability of DNA proportionate amount of G
and C has to be present in the codeword. The number of occurrences of G or C
in a codeword(say x) is called the GC weight of the codeword and is denoted by
$w_{GC}(x)$. The GC weight ratio to the codeword’s length is called the
codeword’s GC content.

##  3 Constraints on DNA codes

###  3.1 Hamming Distance constraint

Let $\mathcal{C}$ be a DNA code, $H(x,y)\geq d,$ $\forall x,y\in\mathcal{C}$
with $x\neq y$. We refer to this distance as the minimum distance for Code
$\mathcal{C}$. This acts as an alternative to the Levenshtein distance.

###  3.2 Reverse constraint

Let $\mathcal{C}$ be a DNA code, $H(x^{R},y)\geq d,$ $\forall
x,y\in\mathcal{C}$.Also considering the case $x=y$. This step acts as a bridge
for constructing the reverse complement constraint. One can use the bound of
this constraint and implement on the other.

###  3.3 Reverse Complement constraint

Let $\mathcal{C}$ be a DNA code, $H(x^{RC},y)\geq d,$ $\forall
x,y\in\mathcal{C}$.Also considering the case $x=y$. This constraint helps in
the reduction of unwanted hybridization errors.

###  3.4 GC-content constraint

PCR amplification is difficult if the GC content is high, whereas a low GC
content results in a non-stable gene. So, the GC content of all the codewords
in a DNA code C should be a fixed number, say w.
These constraints are imposed in the DNA code to eradicate the errors
discussed in section 2. Here, to impose these constraints, we are using VT and
concatenated kernel code Ramachandran and P Balasubramanie (2003) Selvakumar
and Pavan Kumar (2020) to satisfy these constraints.

##  4 Construction of DNA codes

Here, a code of length n is constructed with a message block of length:

$l=n-log_{2}(2*n-1)-1$ (1)

Each element of the block is from the same finite abelian group $\mathcal{G}$.
To construct an indel error-free DNA code with GC-constraint(section 3.4) and
Reverse Complement constraint(section 3.3) VT code and kernel code have been
used to encode the information (figure2). The algorithm goes as follows:

  1. 1.

The set of all information of length $l$ is generated.

  2. 2.

Each of that information is encoded using VT code to restrict indel errors.

  3. 3.

Each VT-encoded information is then mapped to a unique element in the Kernel
code of length n+1.

  4. 4.

Now, the mapped elements of the sets are again outer encoded using the
proposed homomorphism to address the constraints (section 3).

  5. 5.

Then, the obtained word is mapped onto the corresponding base pairs.

Figure 2: Our encoding and decoding procedure

The construction begins by encoding the information using VT codes. VT codes
are used for correcting single deletion, insertion or substitution(indel)
errors. Goldman Goldman et al. (2013) has suggested a way for correcting
multiple indel errors by segmenting codewords into smaller lengths.
The first step of the algorithm is calculating the maximum length for the
message bits($l$) equation (1). All possible information on length $l$ is
encoded using VT code. Then, the set of all VT encoded information(say Y) is
obtained.

###### Example 4.1

Example: For $n=8$, the value $l$ is 3. Since, $n-1=7$ is not a power of 2,
the par_pos value is [$2^{0},2^{1},2^{2},7$].

Let our information be 010, and $y$ is initialised as 0000100. The syndrome of
$y$ (i.e. 10) is partitioned as 7+2+1. So, $y$ is finalised to 1100101.
Then in the second step, Kernel code(K) of length n+1 is generated. All the
elements from K that starts with 1 are collected into a subset(S). For
example: For n = 3 , $\mathcal{G}=\mathcal{Z}_{2}$. K = { 0000, 0001, 0010,
0011, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1101, 1110, 1111} and S =
{1001, 1010, 1111}.
In the third step, Each element of set S is mapped to an element of Y. Let $s$
be an element in S, $s[i:j]$ represents the sub-sequence starting from
$i^{th}$ position to the $j^{th}$ position of the sequence.
(i.e) if $s=01101,$ $s[2:4]=110$. The element $s_{y}\in S$ is mapped to $y\in
Y$ if $s_{y}[2:n-1]=y$.

###### Example 4.2

Considering the example (4.1), $n=8$ and $y=1100101$ then kernel map of $y$ is
$111001011$. Clearly, $s_{y}=111001011$ is in K.

. In the fourth step, n - 1 redundancy bits are added to each element $s_{y}$
of the kernel subset S and denoted as $E_{y}$. Here, $i^{th}$ redundancy bit
is obtained using the homomorphism $h_{i}$. The homomorphism goes as follows.

$h_{i}(g_{1},g_{2},\dots,g_{n})=\begin{cases}g_{i+1}&1\leq
i\leq\lfloor{\frac{n-1}{2}}\rfloor\\\
g_{1}+g_{i+1}&\lceil{\frac{n+1}{2}}\rceil\leq i\leq n-1\\\
{g_{1}+g_{i+1}+g_{n+1}}&i=\frac{n}{2},\\\ &\frac{n}{2}\text{ is an
integer}\end{cases}$ (2)

###### Example 4.3

From the example (4.2) $s_{y}=111001011$ the concatenated Kernel encoding is
$E_{y}=1110010111101010$

Considering the previous example, In the final step, the binary codeword is
separated into small strings of length two with the idea portrayed. This
binary conversion is dependent on an element from each half. In this way, $l$
length information is mapped onto the n-length bps.

###### Example 4.4

For $E_{y}=1110010111101010$(example 4.3), the DNA encoded codeword is
$GGGCATAT$.

Figure 3: Encoding for three-length information set

Source: It is the python code output which was executed on the Pycharm
version: 2021.2.2 (open source software) on the computer with the processor
Intel(R)Xeon(R) E3-1225 v5@ 3.30GHz.

For decoding, the second to last base pairs of the received word are mapped to
$\mathcal{Z}_{2}$. Here, ’A’ and ’T’ are mapped to ’0’ and ’G’ and ’C’ are
mapped to ’1’. Suppose the DNA codeword from the example 4.4($GGGCATAT$) is
received as $GGGATAT$. Then, $\hat{a}=110101$ and the value of $y$ is
initialized to $110101$, syndrome = $2$, weight $w=4>2=$ syndrome. Since
weight is less than syndrome 0 has to be added after the two 1’s $0$.
Therefore $y$ becomes $110101$. Then, the information is taken without parity
positions from y(i.e.) $010$ which is the stored information. Figure 3 depicts
how the encoding has been done for an information set of length(l) 3 and
codeword(n) of length 8 using python3.
This way of encoding preserves the GC-content of $50\%$ for even length n and
odd length, it ranges from $40\%$ to $60\%$. This determines the stability of
the DNA code. This also gives a greater distance for reverse complement
constraint. For an n-length DNA code, the RC-distance is

$d_{RC}=2\times\Bigg{\lfloor}{\frac{n-3}{2}}\Bigg{\rfloor}$

This acts as an improvised coding scheme for Goldman et al. (2013) where the
$3^{l}$ and other $S_{l}$ methods have been used. This meets the constraints
described in 3 and serves as a solution to the DNA storage problem.

##  5 Conclusion

As DNA has a high chance of occurring for mutation errors and instability,
algorithms for generating balanced GC content code have been proposed. The
balanced GC- content ensures the stability of DNA strings. Further, the VT
algorithm used here corrects a single indel error that occurs in DNA strings
during retrieval.

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