# Existence, Uniqueness and Algebraic Reductibility Experiments of RANS
equations with Periodic Boundary Conditions

C. V. Valencia, S. Velasco and M. Romero de Terreros Department of Physics and
Mathematics
Universidad Iberoamericana Ciudad de México
Prolongación Paseo de la Reforma 880, Mexico City —01219, MEXICO
<EMAIL_ADDRESS>
###### Abstract

In the present work, we show the existence and uniqueness of the two-
dimensional turbulent vorticity term under conditions given on the weak
solutions to the classical incompressible Navier-Stokes problem in a compact
domain $R$ with periodic boundary conditions and initial conditions in Sobolev
space $H^{m}(R)$. To achieve this, we derive a pressureless incompressible
system and calculate kinetic energy quantities on the turbulent vorticity from
the properties of the classical curl. Finally, since the system obtained is
algebraic, we experimented with the possibilities of reducing it by applying
the Rosenfeld-Gröbner algorithm.

_K_ eywords Turbulence theory, Vortex dynamics

## Introduction

It is a well-known practice to use the rotational operator on the Navier-
Stokes equations to derive pressureless systems. However, we have not come
across similar approaches for the Navier-Stokes equations that take into
consideration velocities for the _two-scale decomposition of the flow_ , as
outlined in the _Reynolds-Averaged Navier-Stokes_ (RANS) equations statement
before the time average hypothesis is considered. The system of equations
obtained includes non-linear differential polynomials, the factorization of
which corresponds to limit cases that could also be obtained from scenarios
with small parameters. We look for terms that persist during different
algebraic simplifications, to estimate the maximum time of existence and
uniqueness of a classical solution, and to recommend using algorithms from the
theory of differential algebra to investigate the multiplicity of formal
solutions beyond that maximum time. It is our belief that this method gives a
unique perspective on exploring causal terms in various turbulent scenarios.

Olga Ladyzhenskaya, stated the: “ _Problem 1. Do the Navier-Stokes equations
together with the initial and boundary conditions actually give a
deterministic description of the dynamics of an incompressible fluid?_ ”[19,
p. 252]. However, there are inaccuracies in the description of flows given by
ideal models and real fluids [13, 14]. In particular, there is _turbulence_ ,
and study of fluid dynamics, especially _turbulent flow_ , is pivotal in
engineering applications. The Reynolds-averaged Navier-Stokes (RANS) equations
remain central in this field and offer valuable insights to increasingly
complex problems relating to the flow of fluids.

In this article, we lay the existence and uniqueness foundations for our
search for sufficient conditions to establish well-posed problems in the
context of the algebraically simplified cases of the _Reynolds-Averaged
Navier-Stokes_ equations (RANS). Our far-reaching objective is to establish
the conditions required for the existence and uniqueness of periodic
solutions, which are empirically observed, and for which, even recently, no
mathematical proofs have been reported, such as the case of a fluid flowing
past a cylinder and having Reynolds numbers between 50 and 250 [25]. Moreover,
we hope that the identification of the terms that prevail in the limit when
the viscosity tends to zero will allow us to propose explanations for why this
periodicity is lost for Reynolds numbers greater than 250 [25]. The results
obtained in this research were presented, without detailed proofs, at _the
Engineering Mechanics Institute Conference and Probabilistic Mechanics &
Reliability Conference_ (EMI/PMC 2024).

In particular, we see value in examining the theoretical implications of the
issues associated with the RANS approach to gain insight into the dynamics of
the Earth’s Atmosphere. The Intergovernmental Panel on Climate Change (IPCC)
states that “ _Atmospheric models include representations of (…) turbulence,
convection and gravity waves that are not fully represented by grid-scale
dynamics_ ”[28, p. 277]. Both the general existence and uniqueness problem and
the specific need to address this Issues point us towards the broader question
at hand: Do the limit models derived from Reynolds-Averaged Navier-Stokes
Equations’ problems could give a better understanding of the dynamics of a
turbulent fluid in the Atmosphere? We would like to highlight that there are
very few fundamental results on local and global existence and uniqueness of
solutions for a _two scale decomposition of the flow_ , as it appears on the
RANS equations [23, p. 126] and LES initial statements of the problem [26, p.
11], [21, 22].

We analyze the new system of equations with both the classical energy method,
and as a system of differential polynomials through the Rosenfeld-Gröbner
algorithm [7], [6], and [1]. In recent years, the field of differential
algebra has been extending its reach, offering a promising alternative
approach for gaining ground against complex differential systems. We propose
analyzing fluid dynamics theory from an algebraic standpoint. This is achieved
through the implementation of Rosenfeld-Gröbner algorithms, a method pioneered
by Francois Boulier. Boulier’s extensive work in differential algebra provides
the main framework for our investigation [3]. We aim to obtain decomposed
systems of from the original system of equations.

Figure 1: $R$$=[0,L_{1}]\times[0,L_{2}]\times[0,L_{3}]$

##  1 Problem Statement

Here, we present the general form obtained by applying the rotational to the
Navier-Stokes equations where the velocity is split as a linear combination of
two parts, $v+\tilde{v}$, where $v,\tilde{v}\in C^{\infty}(D)$ are infinitely
differentiable functions on $R$$=[0,L_{1}]\times[0,L_{2}]\times[0,L_{3}]$, for
fixed $L_{1},L_{2},L_{3}>0$, so that the results can be extended to
_starshaped_ real domains (where any ray with an origin in a fixed point has a
unique common point with the topological boundary of the domain) and
distributions in the Sobolev spaces $W^{k}_{p}\left(R\right)$, with
$p\in[1,\infty)$, thanks to the fact that, in particular, $C^{\infty}(R)$ is
dense in $W^{k}_{p}\left(R\right)$ if $R$ is a rectangular prism.

######  Remark 1.

While a necessary and sufficient condition for a distribution in a Sobolev
space to be the limit of a succession of analytic functions is unknown it is
possible to assume that the domain is starshaped with respect to a point to
say that the space $C^{\infty}(\bar{R})$ of infinitely differentiable
functions on a domain $D$ is dense in the Sobolev space $W^{k}_{p}(R)$ of
distributions on $R$ with derivatives of order $0\leq l\leq k$ in the Lebesgue
space $L_{p}(R)$, for $p\in[1,\infty)$. Therefore, we can just assume that the
domain is a rectangular prism $R$ and $v$ is an element of the space
$C^{\infty}(\bar{R})$ that satisfies both the Leibinz rule for the product
derivative, the Gauss-Green formula [31, p. 699], and the generalisation of
Schwarz’s Theorem [31, p. 280] to exchange the order of the generalised second
partial derivatives with respect to the spatial and temporal independent
variables [29, 30].

######  Theorem 1.

Let $p,\boldsymbol{v},\boldsymbol{\tilde{v}}\in
C^{1}\left([0,T),C^{\infty}(\bar{R})\right)$, for some $0<T<\infty$, with
periodic conditions for $\boldsymbol{v}$ and $\boldsymbol{\tilde{v}}$ at
$\partial R$, where $\partial R$ denotes the topological boundary of $R$. This
is,
$\boldsymbol{v}(x_{1},x_{2},x_{3},t)=\boldsymbol{v}(x_{1}+L_{1},x_{2}+L_{2},x_{3}+L_{3},t)$,
and
$\boldsymbol{\tilde{v}}(x_{1},x_{2},x_{3},t)=\boldsymbol{\tilde{v}}(x_{1}+L_{1},x_{2}+L_{2},x_{3}+L_{3},t)$,
$\forall(x_{1},x_{2},x_{3})\in\partial R$. If $\boldsymbol{v}$,
$\boldsymbol{\tilde{v}}$, and $p$ verify the following system of equations:

$\begin{cases}\nabla\cdot\boldsymbol{v}=0,\\\
\nabla\cdot\boldsymbol{\tilde{v}}=0,\\\
\frac{D(\boldsymbol{v}+\boldsymbol{\tilde{v}})}{Dt}=\nu\Delta(\boldsymbol{v}+\boldsymbol{\tilde{v}})-\nabla
p.\end{cases}$ (1)

Then, for at least one point $x^{0}=(x_{1}^{0},x_{2}^{0},x_{3}^{0})\in R$, $v$
and $\tilde{v}$ satisfy:

$\frac{\partial\boldsymbol{\omega}}{\partial
t}+\frac{\partial\boldsymbol{\tilde{\omega}}}{\partial
t}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{\omega}\>+\>(\boldsymbol{\tilde{v}}\cdot\nabla)\boldsymbol{\tilde{\omega}}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{\tilde{\omega}}+(\boldsymbol{\tilde{v}}\cdot\nabla)\boldsymbol{\omega}-\nu\Delta(\boldsymbol{\omega}+\boldsymbol{\tilde{\omega}})=0,$
(2)

where $\Delta$ denotes the Laplacian operator,
$\boldsymbol{\omega}=\nabla\times\boldsymbol{v}$, and
$\boldsymbol{\tilde{\omega}}=\nabla\times\boldsymbol{\tilde{v}}$.

###### Proof.

In order to obtain Eq. (10), we will integrate each term on both sides of the
Conservation of Momentum Law of Eq. (1) over the entire physical domain $R$:

$\iiint_{R}\left(\frac{D(\boldsymbol{v}+\boldsymbol{\tilde{v}})}{Dt}\right)d\boldsymbol{x}=\iiint_{R}(\nu\Delta(\boldsymbol{v}+\boldsymbol{\tilde{v}})-\nabla
p)d\boldsymbol{x},$

where $d\boldsymbol{x}=dx_{1}dx_{2}dx_{3}$ is the volume differential in
$\mathbb{R}^{3}$. By the monotone convergence theorem [31], if:

$\nabla\times\left(\iiint_{R}\left(\frac{D(\boldsymbol{v}+\boldsymbol{\tilde{v}})}{Dt}\right)d\boldsymbol{x}\right)=\nabla\times\left(\iiint_{R}\big{(}\nu\Delta(\boldsymbol{v}+\boldsymbol{\tilde{v}})-\nabla
p\big{)}d\boldsymbol{x}\right),$

then,

$\iiint_{R}\left(\nabla\times\frac{D(\boldsymbol{v}+\boldsymbol{\tilde{v}})}{Dt}\right)d\boldsymbol{x}=\iiint_{R}\left(\nabla\times\big{(}\nu\Delta(\boldsymbol{v}+\boldsymbol{\tilde{v}})-\nabla
p\big{)}\right)d\boldsymbol{x}.$ (3)

Figure 2: Periodic boundary conditions at $\partial R$

Given that $\boldsymbol{v},\boldsymbol{\tilde{v}}\in C^{\infty}(D)$ and the
Schwarz theorem, we can exchange the order of derivation. Because both
$\boldsymbol{v}$ and $\boldsymbol{\tilde{v}}$ have zero divergence and
periodic conditions, by direct calculations and integration by parts, the left
hand side of Eq. (3) becomes:

$\iiint_{R}\left(\nabla\times\frac{D(\boldsymbol{v}+\boldsymbol{\tilde{v}})}{Dt}\right)d\boldsymbol{x}=\iiint_{R}\left(\mathbf{A_{1}}+\mathbf{B_{1}}+\mathbf{M_{1}}+\mathbf{M_{2}}+\mathbf{M_{3}}+\mathbf{M_{4}}\right)d\boldsymbol{x},$

where:

$\mathbf{A_{1}}=\frac{\partial}{\partial
t}\left(\nabla\times\boldsymbol{v}\right),\quad\quad\quad\mathbf{B_{1}}=\frac{\partial}{\partial
t}\left(\nabla\times\boldsymbol{\tilde{v}}\right).$

$\mathbf{M_{2}}+\mathbf{M_{3}}=\left(\boldsymbol{v}\cdot\nabla\right)\left(\nabla\times\boldsymbol{\tilde{v}}\right)+\left(\boldsymbol{\tilde{v}}\cdot\nabla\right)\left(\nabla\times\boldsymbol{v}\right),$

and

$\mathbf{M_{4}}=\left(\boldsymbol{\tilde{v}}\cdot\nabla\right)\left(\nabla\times\boldsymbol{\tilde{v}}\right).$

On the right hand,

$\nabla\times\left(\nu\Delta(\boldsymbol{v}+\boldsymbol{\tilde{v}})-\nabla
p\right)=\nu\Delta\left(\left(\nabla\times\boldsymbol{v}\right)+\left(\nabla\times\boldsymbol{\tilde{v}}\right)\right).$

We can remove the triple integral through the Intermediate Value Theorem [32,
p. 124], knowing there is at least one $x_{0}\in R$ such that, putting
together the expressions for the left and right hand sides, we obtain Eq.
(10). ∎

##  2 Existence and Uniqueness Theorems

In light of the problem statement above, the system that arises is algebraic.
This allows for the classical energy method to be applied in order to obtain
estimates and quantities that are unaffected by viscosity, and only dependant
on the initial and boundary conditions.

######  Remark 2.

If $\boldsymbol{f}=(f_{1},f_{2},f_{3}),\ \boldsymbol{g}=(g_{1},g_{2},g_{3})\in
L^{2}(R$$;\mathbb{R}^{3})$, we denote its inner product as:
$\left(\boldsymbol{f},\boldsymbol{g}\right)_{0}=\iiint_{R}\left(\boldsymbol{f}\cdot\boldsymbol{g}\right)d\boldsymbol{x}=\iiint_{R}\left(f_{1}g_{1}+f_{2}g_{2}+f_{3}g_{3}\right)d\boldsymbol{x}$.
Similarly,
$\left(f,f\right)_{0}=\|f\|^{2}_{0}=\iiint_{R}\left(f_{1}^{2}+f_{2}^{2}+f_{3}^{2}\right)\
d\boldsymbol{x}$. Additionally, let $\partial^{k}_{\alpha}$ be a differential
operator that express the partial derivative of absolute order $k$ with order
$\alpha$ for the different spatial variables:
$\partial^{k}_{\alpha}=\frac{\partial^{k}}{\partial x^{\alpha_{1}}\partial
x^{\alpha_{2}}\partial x^{\alpha_{3}}}$, where
$\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})\in\mathbb{N}^{3}$ and
$\alpha_{1}+\alpha_{2}+\alpha_{3}=k$.

######  Lemma 1.

Let $\boldsymbol{f},\boldsymbol{g}\in
C^{1}\left([0,T),C^{\infty}(\bar{R})\right)$, for some $0<T<\infty$ satisfy
periodic conditions at $\partial R$, such that $\nabla\cdot\boldsymbol{f}=0$.
In that case, for $k\in\mathbb{N}$,
$\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})\in\mathbb{N}^{3}$ and
$\alpha_{1}+\alpha_{2}+\alpha_{3}=k$, we have that:

$\begin{cases}(\frac{\partial\boldsymbol{f}}{\partial
t},\boldsymbol{f})_{0}=\frac{1}{2}\frac{\partial}{\partial
t}(\boldsymbol{f},\boldsymbol{f})_{0}=\frac{1}{2}\frac{\partial}{\partial
t}\|f\|^{2}_{0};\\\
((\boldsymbol{f}\cdot\nabla)\boldsymbol{f},\boldsymbol{f})_{0}=0;\\\
((\boldsymbol{f}\cdot\nabla)\boldsymbol{g},\boldsymbol{f})_{0}=-((\boldsymbol{f}\cdot\nabla)\boldsymbol{f},\boldsymbol{g})_{0};\\\
((\boldsymbol{f}\cdot\nabla)\boldsymbol{f},\boldsymbol{g})_{0}=-((\boldsymbol{f}\cdot\nabla)\boldsymbol{g},\boldsymbol{f})_{0};\\\
(\partial^{k}_{\alpha}\Delta\boldsymbol{f},\partial^{k}_{\alpha}\boldsymbol{g})_{0}=-(\nabla\partial^{k}_{\alpha}\boldsymbol{f},\nabla\partial^{k}_{\alpha}\boldsymbol{g})_{0}.\\\
\end{cases}$ (4)

###### Proof.

Direct calculation, integration by parts, and applying the Schwarz theorem to
interchange the order of the partial derivatives lead to each of the above
identities. ∎

######  Lemma 2.

Assume the same conditions as the Theorem 1. Therefore, using the energy
method, we find that:

$\frac{1}{2}\frac{\partial}{\partial
t}\left\|\partial^{k}_{\alpha}\boldsymbol{\tilde{w}}\right\|^{2}_{0}+\left(\partial^{k}_{\alpha}\left(\frac{\partial\boldsymbol{\omega}}{\partial
t}\right),\partial^{k}_{\alpha}\boldsymbol{\tilde{\omega}}\right)_{0}+\left((\partial^{k}_{\alpha}\left[(\boldsymbol{v}\cdot\nabla)\boldsymbol{\omega}\right],\partial^{k}_{\alpha}\boldsymbol{\tilde{\omega}}\right)_{0}+\\\
+\left((\partial^{k}_{\alpha}\left[(\boldsymbol{\tilde{v}}\cdot\nabla)\boldsymbol{\tilde{\omega}}\right],\partial^{k}_{\alpha}\boldsymbol{\tilde{\omega}}\right)_{0}+\left(\partial^{k}_{\alpha}\left[(\boldsymbol{v}\cdot\nabla)\boldsymbol{\tilde{\omega}}\right],\partial^{k}_{\alpha}\boldsymbol{\tilde{\omega}}\right)_{0}+\left((\partial^{k}_{\alpha}\left[(\boldsymbol{\tilde{v}}\cdot\nabla)\boldsymbol{\omega}\right],\partial^{k}_{\alpha}\boldsymbol{\tilde{\omega}}\right)_{0}+\\\
+\nu\left(\partial^{k}_{\alpha}\nabla\boldsymbol{\omega},\partial^{k}_{\alpha}\nabla\boldsymbol{\tilde{\omega}}\right)_{0}+\nu\|\partial^{k}_{\alpha}\nabla\boldsymbol{\tilde{\omega}}\|^{2}_{0}=0,$
(5)

$\forall k\in\mathbb{N}$ and
$\forall\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})\in\mathbb{N}^{3}$ such that
$k=\alpha_{1}+\alpha_{2}+\alpha_{3}$.

###### Proof.

Consider the Eq. (1), and apply the differential operator
$\partial^{k}_{\alpha}$ to each of its terms:

$\partial^{k}_{\alpha}\left[\frac{\partial\boldsymbol{\omega}}{\partial
t}+\frac{\partial\boldsymbol{\tilde{\omega}}}{\partial
t}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{\omega}\>+\>(\boldsymbol{\tilde{v}}\cdot\nabla)\boldsymbol{\tilde{\omega}}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{\tilde{\omega}}+(\boldsymbol{\tilde{v}}\cdot\nabla)\boldsymbol{\omega}-\nu\Delta(\boldsymbol{\omega}+\boldsymbol{\tilde{\omega}})\right]=0.$
(6)

Now, we apply the inner product in the Lebesgue space
$L_{2}(R;$$\mathbb{R}^{3})$ of each side of Eq. (6) with
$\partial^{k}_{\alpha}\boldsymbol{\tilde{\omega}}$. According to the
properties established in Eq. (4), we reach Eq. (5). ∎

######  Corollary 1.

If $k=0$:

$\frac{1}{2}\frac{\partial}{\partial
t}\left\|\boldsymbol{\tilde{\omega}}\right\|_{0}^{2}+\left(\frac{\partial\boldsymbol{\omega}}{\partial
t}\>,\>\boldsymbol{\tilde{\omega}}\right)_{0}+\left((\boldsymbol{v}\cdot\nabla)\boldsymbol{\omega}\>,\>\boldsymbol{\tilde{\omega}}\right)_{0}+\left((\boldsymbol{\tilde{v}}\cdot\nabla)\boldsymbol{\tilde{\omega}}\>,\>\boldsymbol{\tilde{\omega}}\right)_{0}+\left((\boldsymbol{v}\cdot\nabla)\boldsymbol{\tilde{\omega}}\>,\>\boldsymbol{\tilde{\omega}}\right)_{0}+\\\
+\left((\boldsymbol{\tilde{v}}\cdot\nabla)\boldsymbol{\omega}\>,\>\boldsymbol{\tilde{\omega}}\right)_{0}+\nu\left(\nabla\boldsymbol{\omega},\nabla\boldsymbol{\tilde{\omega}}\right)_{0}+\nu\left\|\nabla\boldsymbol{\tilde{\omega}}\right\|^{2}_{0}=0.$
(7)

In particular, for planar fields $\boldsymbol{v}=(v_{1},v_{2},0)$,
$\boldsymbol{\tilde{v}}=(\tilde{v}_{1},\tilde{v}_{2},0)$:

$\frac{1}{2}\frac{\partial}{\partial
t}\left\|\boldsymbol{\tilde{\omega}}\right\|_{0}^{2}+\left(\frac{\partial\boldsymbol{\omega}}{\partial
t}\>,\>\boldsymbol{\tilde{\omega}}\right)_{0}=0.$ (8)

###### Proof.

Here, we can recall the relationships between the inner product and the norm
in the Lebesgue Space of Eq. (4) for $R$ and periodic boundary conditions for
both $\boldsymbol{v}$ and $\boldsymbol{\tilde{v}}$ over $\partial R$ to deduce
Eq. (7). Moreover, if $\boldsymbol{v}$ and $\boldsymbol{\tilde{v}}$ are two
dimensional, by definition of $\boldsymbol{\omega}$ and
$\boldsymbol{\tilde{\omega}}$, the components
$\omega_{1},\omega_{2},\tilde{\omega_{1}},\tilde{\omega_{2}}=0$, and
$\omega_{3},\tilde{\omega}_{3}$ are independent of $x_{3}$, so that the terms:
$\left\|\nabla\boldsymbol{\omega}\right\|^{2}_{0}=0$ and
$\left(\nabla\boldsymbol{\omega},\nabla\boldsymbol{\tilde{\omega}}\right)_{0}=0$.
∎

######  Theorem 2.

Let $p,\boldsymbol{v},\boldsymbol{\tilde{v}}\in
C^{1}\left([0,T),C^{\infty}(\bar{R})\right)$, for some $0<T<\infty$, with
initial conditions
$\boldsymbol{v}_{0}=\boldsymbol{v}(\cdot,0),\boldsymbol{\tilde{v}}_{0}=\boldsymbol{\tilde{v}}(\cdot,0)\in
V^{m}(R)$, for $m\geq 2$, and periodic conditions for $\boldsymbol{v}$ and
$\boldsymbol{\tilde{v}}$ at $\partial R$. Assume that
$(\boldsymbol{v},\boldsymbol{\tilde{v}},p)$ is a solution of:

$\begin{cases}\nabla\cdot\boldsymbol{v}=0,\\\
\frac{D\boldsymbol{v}}{Dt}=\nu\Delta\boldsymbol{v}-\nabla p,\\\
\nabla\cdot\boldsymbol{\tilde{v}}=0,\\\
\frac{D(\boldsymbol{v}+\boldsymbol{\tilde{v}})}{Dt}=\nu\Delta(\boldsymbol{v}+\boldsymbol{\tilde{v}})-\nabla
p.\end{cases}$ (9)

Additionally, if the dimension $n=2$, $R=(0,L_{1})\times(0,L-2)$, there exist
a unique weak solution
$\boldsymbol{\tilde{\omega}}=\nabla\times\boldsymbol{\tilde{v}}$ such that:

$\frac{\partial\boldsymbol{\omega}}{\partial
t}+\frac{\partial\boldsymbol{\tilde{\omega}}}{\partial
t}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{\omega}\>+\>(\boldsymbol{\tilde{v}}\cdot\nabla)\boldsymbol{\tilde{\omega}}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{\tilde{\omega}}+(\boldsymbol{\tilde{v}}\cdot\nabla)\boldsymbol{\omega}-\nu\Delta(\boldsymbol{\omega}+\boldsymbol{\tilde{\omega}})=0,$
(10)

for a finite time $T<0$.

###### Proof.

First, we recall the Picard Theorem for Banach Spaces, and consider the
subspace of divergence free distributions $V^{m}(R)$ of the Sobolev space
$H^{m}(R)=W^{k,2}\left(R\right)$, whose elements $\boldsymbol{f}$, and weak
derivatives $\ \partial^{k}_{\alpha}\boldsymbol{f}\ $ have finite kinetic
energy for $0\leq k\leq m$. Second, we notice that the Conservation of Volume
and Momentum laws corresponding to $\boldsymbol{v}$ have a unique weak
solution $\boldsymbol{v}\in C^{1}\left([0,T);C^{k}(\mathbb{R}^{n})\right)$,
for a finite time $T<0$, and whose vorticity
$\boldsymbol{\omega}=\nabla\times\boldsymbol{v}$ satisfies a non-linear
parabolic partial equation with a Maximum Principle, and the classical energy
constraints. Finally, we extend the application of the Gagliardo-Niremberg
inequalities to the convective terms of the form
$(\boldsymbol{f}\cdot\nabla\boldsymbol{g})$, and apply the Grönwall lemma to
verify uniqueness.

Let $\mathcal{B}$ is a Banach space with a distance $\left\|.\right\|_{B}$,
and $U\subset\mathcal{B}$ is an open subset in $\mathcal{B}$, then $\
F:U\rightarrow\mathcal{B}$ is a Lipschitz operator if there is a magnitude
$\mathtt{L}$ such that

$\left\|F(x)-F(y)\right\|_{\mathcal{B}}\leq\mathtt{L}\;\left\|x-y\right\|_{\mathcal{B}},\;\;\;\forall
x,y\in U.$

Assume a fixed initial condition $\boldsymbol{f}(0)=\boldsymbol{f}_{0}\in
C^{\infty}(\mathcal{\bar{R}})$ of $\boldsymbol{f}(t)\in\mathcal{B}$. Let
$U=B_{\boldsymbol{f}_{0}}^{R}\subset\mathcal{B}$ an open ball with center in
$\boldsymbol{f}_{0}$ and radius $R$. If $F:U\rightarrow\mathcal{B}$ is a
bounded, continuous and a Lipschitz operator in $U$, then there is a unique
generalized solution $\mathbf{f\in}C^{1}\left(\left[0,T\right);U\right)$ to:

$\frac{\partial\boldsymbol{f}}{\partial
t}=F\left(\boldsymbol{f}\left(t\right)\right),$ (11)

for that initial condition, and for $t\in\left[0,T\right)$, where
$T=\min\left(\frac{R}{M},\frac{1}{\mathtt{L}}\right)$ and $M>0$ is a bound of
$F$ is on $U$ [4].

Let $\mathcal{B}=H^{k}\left(R\right)$. This is, $\boldsymbol{f}$ and
$\partial^{m}_{\alpha}\boldsymbol{f}$ belong to the Lebesgue space
$L^{2}\left(R\right)$ for all $0\leq m\leq k$. $H^{k}\left(R\right)$ is a
Banach space [30]. In particular, $V^{k}\left(R\right)$ is the subspace of
$H^{k}\left(R\right)$ of divergence free fields. The norm is indicated as:
$\|\cdot\|_{\mathcal{B}}=\|\cdot\|_{m}$, where
$(\boldsymbol{f},\boldsymbol{g})_{m}=\sum_{0\leq k\leq
m}\iiint_{R}(D^{k}\boldsymbol{f}\cdot D^{k}\boldsymbol{g})d\boldsymbol{x}$ is
an inner product on $H^{m}(R)$, and
$\|\boldsymbol{f}\|_{m}=(\boldsymbol{f},\boldsymbol{f})_{m}$.

This result has been applied to show that the incompressible Navier-Stokes
equations of system (9) with an initial condition $\boldsymbol{v}_{0}\in
V^{m}(R)$ have a weak unique solution $\boldsymbol{v}\in V^{m}(R)$ for a
finite time $T>0$ [4].

We define the operator $F\colon H^{m}(R)\to H^{m}(R)$:

$F(\boldsymbol{\tilde{\omega}})=-\frac{\partial\boldsymbol{\omega}}{\partial
t}-(\boldsymbol{v}\cdot\nabla)\boldsymbol{\omega}\>-\>(\boldsymbol{\tilde{v}}\cdot\nabla)\boldsymbol{\tilde{\omega}}-(\boldsymbol{v}\cdot\nabla)\boldsymbol{\tilde{\omega}}-(\boldsymbol{\tilde{v}}\cdot\nabla)\boldsymbol{\omega}+\nu\Delta(\boldsymbol{\omega}+\boldsymbol{\tilde{\omega}}),$

with:

$\begin{cases}F_{1}(\boldsymbol{\tilde{\omega}})=-\frac{\partial\boldsymbol{\omega}}{\partial
t},\\\
F_{2}(\boldsymbol{v},\boldsymbol{\omega})=-(\boldsymbol{v}\cdot\nabla)\boldsymbol{\omega},\\\
F_{3}(\boldsymbol{\tilde{v}},\boldsymbol{\tilde{\omega}})=-\>(\boldsymbol{\tilde{v}}\cdot\nabla)\boldsymbol{\tilde{\omega}},\\\
F_{4}(\boldsymbol{v},\boldsymbol{\tilde{\omega}})=-(\boldsymbol{v}\cdot\nabla)\boldsymbol{\tilde{\omega}},\\\
F_{5}(\boldsymbol{\tilde{v}},\boldsymbol{\omega})=-(\boldsymbol{\tilde{v}}\cdot\nabla)\boldsymbol{\omega},\\\
F_{6}(\boldsymbol{\omega})=\nu\Delta\boldsymbol{\omega},\\\
F_{7}(\boldsymbol{\tilde{\omega}})=\nu\Delta\boldsymbol{\tilde{\omega}}.\end{cases}$
(12)

The first operator, $F_{1}$, is determined by the parabolic equation satisfied
by $\boldsymbol{\omega}$, where $\frac{\partial\boldsymbol{\omega}}{\partial
t}+(\boldsymbol{v}\cdot\nabla\boldsymbol{\omega})=\nu\Delta\boldsymbol{\omega}$,
that is naturally reduced to an equation for the third component $\omega_{3}$
of $\boldsymbol{\omega}=(\omega_{1},\omega_{2},\omega_{3})$. The last two
Laplacian operators, $F_{6}$ and $F_{7}$, are bounded and Lipschitz operators
in $V^{m}(R)$. The second operator, $F_{2}$ inherits this properties from the
classical case, where one can apply the Gagliardo-Niremberg inequalities. As
far as we know, there are no improvements of the Gagliardo-Niremberg classical
inequalities as it appear in [3, 4, 26], and [5, p. $81$].

∎

##  3 Experiments on multiplicity

Our empirical experiments suggest that the 3D Navier-Stokes equations as well
as the 3D Reynolds-averaged Navier-Stokes equations (RANS) are irreducible as
it is shown in the Figure $3$ above. After attempting to utilize Boulier’s
algorithm on either Navier-Stokes or Reynolds-averaged Navier-Stokes (RANS)
equations, the algorithm fails to converge unless the dimension is $n=2$ and
the problem is restated in terms of the _stream-function_. The complexity of
the convective derivative has proved to be a major challenge in the path
towards solving complex differential systems.

Figure 3: Experimental results

##  4 Conclusions

If we consider the existence and uniqueness of a classical velocity in the
Navier-Stokes incompressible equations over a two dimensional domain for a
finite time, then, we might think about the associated fluctuation properties
as a consequence the original classical variables. The symmetry showed in the
vorticity equations show that the fluctuation itself depends on the original
one. Moreover, the application of Differential Algebra methods through the
algorithm developed by Boulier and Lemaire allow new possible reductions and
comparisons in the turbulent regime.

From the existence and uniqueness point of view, we will review the Maximum
Principles available for quasi-linear parabolic equations (For example, those
given by Oleinik in $1963$.) to see if the result might be extended to $n=3$.
We would like to see the impact of different classical vorticity $\omega$
frequencies over $\tilde{\omega}$. In this direction, we will experiment with
the algorithm to count rational solutions over differential polynomial
systems; in particular, for periodic solutions. We greatly appreciate the
support of the Universidad Iberoamericana Ciudad de México in the framework of
the 18th call for scientific, humanistic and technological research 2023; in
particular, the Direction of Research and Graduate Studies. We deeply
appreciate the constant support of our families, friends and colleagues,
without whom any journey would only be a dream.

Declarations of interest: none

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# Statefinder Analysis of Symmetric Teleparallel Cosmology

Raja Solanki 0000-0001-8849-7688<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.

###### Abstract

Statefinder diagnostic is a convenient method that can differentiate between
the various dark energy models. In this article, we analyze the statefinder
parameters in symmetric teleparallel cosmology. The $f(Q)$ gravity theory is
an alternative theory to GR, where gravitational interactions attribute to the
non-metricity scalar $Q$. In the present work, we consider two $f(Q)$ models
which contains a linear and a non-linear form of non-metricity scalar,
specifically, $f(Q)=\alpha Q+\frac{\beta}{Q}$ and $f(Q)=\alpha Q+\beta Q^{2}$,
where $\alpha$ and $\beta$ are free parameters and then statefinder parameters
$(r,s)$ are evaluated. We plot the trajectory of our models in the $r-s$
plane. In addition, we analyze the physical behavior of different cosmological
parameters such as density, deceleration, and the EoS parameters. Further, we
use Om diagnostic to differentiate the behavior of both $f(Q)$ models. We
found that both $f(Q)$ models predicts that the present universe is
accelerating due to the dark energy component evolving due to non-metricity.
Moreover, model I represents phantom type behavior while model II follows
quintessence scenario.

$f(Q)$ gravity, statefinder parameters, non-metricity, Equation of state

##  I Introduction

In the last few decades, cosmology has faced a dramatic change as more and
more evidence [1, 2, 3, 4] support that the dark sector dominates almost all
energy content of the universe. The availability of high precision
cosmological observations such as Baryon Acoustic Oscillations [5, 6], Large
scale structure [7, 8], Cosmic Microwave Background Radiation [9, 10], and the
Wilkinson Microwave Anisotropy Probe experiment [11, 12] indicates that the
present universe is accelerating and the dark energy occupies nearly $68.3$%
of the entire universe while dark matter and baryonic matter occupies nearly
$26.8$ % and $4.9$% of the total energy content of the universe. The root
cause that triggering this accelerated expansion attributed to that dark
energy (DE). Nowadays, several cosmologists have been attracted towards
investigating the fundamental nature of dark energy.

In General Relativity (GR), the simplest candidate for DE is the Einstein’s
cosmological constant $\Lambda$ [13] which can be regarded as a fluid with
high negative pressure and constant energy density
$\varepsilon=\frac{\Lambda}{8\pi G}$ and such a fluid is characterized by the
equation of state $\omega_{\Lambda}=-1$. Also, it is found that cosmological
constant $\Lambda$ suffered from two delicate issues known as fine tuning and
the cosmic coincidence problem. In the quantum field theory, the theoretical
value of the vacuum energy [14] is $123$ order of magnitude larger than its
observed value which is $10^{-47}GeV^{4}$ [1, 2]. The absence of a response
mechanism that sets a very small value of the cosmological constant is
referred to as a fine tuning problem. Further, the coincidence problem is
referred to as the observed coincidence between the densities of dark matter
and the dark energy while it is so different during the beginning of expansion
[15].

To address the above cosmological issues, dynamical DE models were proposed.
The most popular and widely studied class of time-dependent dark energy model
is the quintessence model having EoS parameter $\omega>-1$. Ratra and Peebles
introduced the first quintessence model [16]. The quintessence description of
dark energy is much different from $\Lambda$CDM as it is time-varying while
$\Lambda$ is always constant. Another interesting dynamical DE model is
phantom models which are characterized by the equation of state $\omega<-1$
[17, 18]. The phantom model represents the growing dark energy models which
cause a big rip in the universe [19, 20]. Although plenty of dynamical DE
models such as k-essence [21, 22], Chaplygin gas [23, 24], Chameleon [25],
tachyon [26], etc have been introduced in the literature.

On the other hand, like so many dark energy models started appearing, either
qualitative or quantitative discrimination between the various dark energy
models becomes necessary. To address this issue, new geometrical diagnostic
parameters proposed by V. Sahni et al. are known as statefinder parameters
$(r,s)$ [27]. The statefinder parameters are defined as

$r=\frac{\dddot{a}}{aH^{3}}$ (1)

and

$s=\frac{(r-1)}{3(q-\frac{1}{2})}$ (2)

where $q=-a\ddot{a}/\dot{a}^{2}$ is the deceleration parameter.

The well-known geometrical parameters namely the Hubble parameter and the
deceleration parameter is generated from the cosmic scale factor and its first
and second order derivative respectively. Therefore the parameter $r$ is the
natural succeeding step beyond the deceleration parameter since it is
generated from the scale factor and its third order derivative, while $s$ is
the simple combination of $r$ and $q$. Statefinder analysis provides a useful
graphical classification for current alternative models for gravity at
cosmological scales. They are widely used to classify dark energy models in
different scenarios including theories with torsion [28], Einstein-Aether
theory as a lorentz breaking theory [29], and even from holographic point of
the view [30]. To get in touch with statefinder analysis, one can check
References [31, 32]. For the $\Lambda$CDM case, it is found that
$(r,s)=(1,0)$. Consider the sufficiently large times, when cosmological
constant dominates the universe and contribution from matter energy become
negligible, then the scale factor becomes $a(t)=exp(t)$ so that one can have
$r=1$ and $s=0$. The deviation of the evolutionary trajectories of a given
model in the $r-s$ plane from the $\Lambda$CDM point $(1,0)$ defines the
distance of that model from $\Lambda$CDM one. Thus, for a given model, the
statefinder pair $(r,s)$ can be calculated and the trajectory of that model
can be drawn in the $r-s$ plane, and the difference in the trajectory from
that of the $\Lambda$CDM gives the required discrimination. Furthermore, for
different dark energy models, the different evolutionary trajectories in the
$r-s$ plane have been found [33].

Another alternative approach to address the late-time acceleration issue and
to describe the origin of dark energy is to modify the action of GR, so called
Modified theories of gravity. Till now several modified theories have been
proposed. Some modified theories which have extensively investigated are
$f(R)$ [34, 35, 36], $f(T)$ [37, 38, 39], $f(G)$ [40, 41], $f(R,\mathcal{T})$
[42], $f(R,G)$ [43] theories, etc, with $R$, $T$, $G$ and $\mathcal{T}$ being
the Ricci, Torsion, Gauss-Bonnet and energy momentum scalars respectively. In
this article, we will work with the recently proposed $f(Q)$ theory of gravity
with $Q$ being the non-metricity scalar [44].

One can describe the gravitational interactions in the space-time manifold
using three kinds of geometrical objects namely curvature, torsion, and non-
metricity. In General Relativity (GR), gravitational interactions are
attributed to space-time curvature. Another two possibilities torsion and non-
metricity give the equivalent description of GR and the corresponding gravity
are called Teleparallel and Symmetric Teleparallel Equivalent of GR. The
$f(R)$ theory of gravity is a generalization of curvature-based gravity (GR)
with vanishing torsion and non-metricity [45]. Similarly, the $f(T)$ gravity
theory is a generalization of torsion-based gravity (the teleparallel
equivalent of GR) with vanishing non-metricity and curvature [46]. Lastly, the
$f(Q)$ gravity theory is a generalization of the symmetric teleparallel
equivalent of GR with vanishing torsion and curvature [47]. Nowadays,
investigations on the $f(Q)$ gravity theory frequently appear in the
literature. Recently, Hassan et al. have discussed the traversable wormhole in
symmetric teleparallel gravity [48]. Mandal et al. have analyzed the energy
conditions and cosmography in $f(Q)$ gravity [49, 50].

In this work, we are going to present a complete analysis of the statefinder
diagnostic for symmetric teleparallel cosmology. This manuscript is organized
as follows. In Sec II, we present the motions equations in $F(Q)$ gravity. In
Sec III, statefinder parameters are evaluated for $F(Q)$ cosmology. In Sec IV,
we consider two $F(Q)$ cosmological model and then statefinder parameters are
evaluated. Further, a detailed physical analysis of cosmological parameters is
done in this section. In Sec V, we use Om diagnostic to differentiate the
behavior of both $F(Q)$ models. Finally, we discuss our conclusions in Sec VI.

##  II Field Equation in $F(Q)$ gravity

The universe in $F(Q)$ gravity is described by following action:

$S=\int{\frac{1}{2}F(Q)\sqrt{-g}d^{4}x}+\int{L_{m}\sqrt{-g}d^{4}x}$ (3)

where $F(Q)$ is an arbitrary function of the nonmetricity scalar $Q$, $g$ is
the determinant of the metric tensor $g_{\mu\nu}$ and $L_{m}$ is the
Lagrangian density of matter.

Another key component to explain the symmetric teleparallel gravity is the
non-metricity tensor, which is given as

$Q_{\lambda\mu\nu}=\nabla_{\lambda}g_{\mu\nu}$ (4)

and its two traces are shown below

$Q_{\alpha}=Q_{\alpha}\>^{\mu}\>{}_{\mu}\>and\>\tilde{Q}_{\alpha}=Q^{\mu}\>_{\alpha\mu}$
(5)

In addition, the superpotential (non-metricity conjugate) tensor is given by

$4P^{\lambda}\>_{\mu\nu}=-Q^{\lambda}\>_{\mu\nu}+2Q_{(\mu}\>^{\lambda}\>{}_{\nu)}+(Q^{\lambda}-\tilde{Q}^{\lambda})g_{\mu\nu}-\delta^{\lambda}_{(\mu}Q_{\nu)}$
(6)

Then the trace of non-metricity tensor can be acquired as

$Q=-Q_{\lambda\mu\nu}P^{\lambda\mu\nu}$ (7)

Furthermore, the definition of the stress-energy tensor for the matter is

$T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{m})}{\delta
g^{\mu\nu}}$ (8)

For notational simpliciy, we define $F_{Q}=\frac{dF}{dQ}$

The gravitational field equation obtained by varying the action (3) with
respect to the metric is given below

$\frac{2}{\sqrt{-g}}\nabla_{\lambda}(\sqrt{-g}F_{Q}P^{\lambda}\>_{\mu\nu})+\frac{1}{2}g_{\mu\nu}F+F_{Q}(P_{\mu\lambda\beta}Q_{\nu}\>^{\lambda\beta}-2Q_{\lambda\beta\mu}P^{\lambda\beta}\>_{\nu})=-T_{\mu\nu}$
(9)

##  III State-Finder Parameters for $F(Q)$ Cosmology

Now, by assuming the cosmological principle, we describe our universe by the
spatially isotropic and homogeneous flat FLRW metric [51]:

$ds^{2}=-dt^{2}+a^{2}(t)[dx^{2}+dy^{2}+dz^{2}]$ (10)

Here, $a(t)$ is the scale factor that measures the cosmic expansion. One can
find the non-metricity scalar by taking the trace of non-metricity tensor (4)
with respect to line element given by (10) as

$Q=6H^{2}$ (11)

The stress-energy tensor for a perfect fluid distribution with respect to the
metric (10) is

$\mathcal{T}_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}$ (12)

Here, $\rho$ is the matter-energy density, $p$ is the usual pressure, and
$u^{\mu}=(1,0,0,0)$ are components of the four velocity.

Then the Friedmann equations governing the dynamics of the universe are

$3H^{2}=\frac{1}{2F_{Q}}\left(-\rho+\frac{F}{2}\right)$ (13)

and

$\dot{H}+3H^{2}+\frac{\dot{F_{Q}}}{F_{Q}}H=\frac{1}{2F_{Q}}\left(p+\frac{F}{2}\right)$
(14)

We can rewrite the Friedmann equations for a non-relativistic matter dominated
universe with the functional form $F(Q)=-Q+f(Q)$, in a more suitable form as

$H^{2}=\frac{1}{3}\left[\rho_{m}+\rho_{Q}\right]$ (15)

$\dot{H}=-\frac{1}{2}\left[\rho_{m}+\rho_{Q}+p_{Q}\right]$ (16)

where

$\rho_{Q}=-\frac{f}{2}+Qf_{Q}$ (17)

and

$p_{Q}=\frac{f}{2}-(2\dot{H}+Q)f_{Q}-2\dot{f_{Q}}H$ (18)

These equations (15) and (16) can be interpreted as the effective Friedmann
equations along with an exotic fluid part coming from the non-metricity
scalar, where $\rho_{Q}$ and $p_{Q}$ represents the energy density and
pressure of the exotic fluid due to non-metricity.

Now the dimensionless density parameters corresponding to the matter and non-
metricity density is defined by

$\Omega_{m}=\frac{\rho}{3H^{2}},\ \ \ \Omega_{Q}=\frac{\rho_{Q}}{3H^{2}},\ \ \
\Omega_{m}+\Omega_{Q}=1$ (19)

Then we can derive the expressions for the equation of state parameter and the
deceleration parameters as

$\omega=\frac{p_{Q}}{\rho_{Q}}=-1+\phi$ (20)

where

$\phi=4\dot{H}\left(\frac{f_{Q}+2Qf_{QQ}}{f-2Qf_{Q}}\right)$ (21)

and

$q=\frac{1}{2}(1+3\omega\Omega_{Q})$ (22)

The statefinder parameter $r$ in terms of deceleration parameter can be
written as

$r=2q^{2}+q-\frac{\dot{q}}{H}$ (23)

Using equations (20)-(23) , we obtain the statefinder parameters for $F(Q)$
cosmology

$r=\frac{1}{2}\left[\left\\{1+3\left(-1+\phi\right)\Omega_{Q}\right\\}^{2}+\left\\{1+3\left(-1+\phi\right)\Omega_{Q}\right\\}\right]-\frac{3}{2H}\left[\dot{\omega}\Omega_{Q}+\left(-1+\phi\right)\dot{\Omega_{Q}}\right]$
(24)

and

$s=\frac{2(r-1)}{9(-1+\phi)\Omega_{Q}}$ (25)

Furthermore, $F(Q)=-Q$ is the GR limit i.e, one can retrieve the usual
Friedmann equations of GR for this functional form and the energy density and
pressure due to non-metricity becomes zero, which is expected.

##  IV Cosmological $F(Q)$ Models

In this section, we are going to analyze our results obtained in the previous
section for some $f(Q)$ models.

Model I: We consider the following $f(Q)$ function which contains a linear and
a non-linear form of non-metricity, specifically [52]

$f(Q)=\alpha Q+\frac{\beta}{Q}$ (26)

where $\alpha$ and $\beta$ are free parameters.

Then for this specific choice of the function, one can obtain the following
differential equation

$\dot{H}\left(\alpha-1+\frac{\beta}{12H^{4}}\right)+\frac{3}{2}H^{2}\left(\alpha-1-\frac{\beta}{12H^{4}}\right)=0$
(27)

Now, one can rewrite the Friedmann equations (13) and (14) in terms of
dimensionless matter density parameter as

$H^{2}=\frac{1}{12F_{Q}}\left[-\Omega_{m}Q+F\right]$ (28)

$\dot{H}=\frac{1}{4F_{Q}}\left[\Omega_{m}Q-4F_{QQ}\dot{Q}H\right]$ (29)

The aim is to estimate the parameters of our proposed $f(Q)$ model that would
be in agreement with the observed values of the cosmographic parameters. Then
by using Friedmann equations (28) and (29) one can estimate

$\alpha=1-\frac{\Omega_{m0}}{2}\left[1+\frac{3}{2\left(1+q_{0}\right)}\right]$
(30)

and

$\beta=6\Omega_{m0}H_{0}^{4}\left[1-\frac{3}{2\left(1+q_{0}\right)}\right]$
(31)

Now, we set $E(z)=\frac{H(z)}{H_{0}}$, then by using (30) and (31) in (27) we
have the following differential equation in terms of dimensionless Hubble
parameter

$E^{\prime}\left[1-\frac{1}{E^{4}}\frac{(2q_{0}-1)}{(2q_{0}+5)}\right]-\frac{3E}{2(1+z)}\left[1+\frac{1}{E^{4}}\frac{(2q_{0}-1)}{(2q_{0}+5)}\right]=0$
(32)

Now, by using equations (19) and (20) we have

$\Omega_{Q}=1-\Omega_{m0}\frac{(-1+2q_{0}+E^{4}(5+2q_{0}))}{4E^{4}(1+q_{0})}$
(33)

$\omega=-\frac{8E^{4}(2q_{0}^{2}+q_{0}-1)}{\\{1-2q_{0}+E^{4}(5+2q_{0})\\}\\{(2q_{0}-1)\Omega_{m0}+E^{4}(5\Omega_{m0}-4+2q_{0}(\Omega_{m0}-2)\\}}$
(34)

The effective equation of state parameter for our model is given by

$w_{eff}=\frac{p_{eff}}{\rho_{eff}}=\frac{p_{Q}}{\rho_{m}+\rho_{Q}}$ (35)

Here, $p_{eff}$ and $\rho_{eff}$ correspond to the total pressure and energy
density of the universe. Then by using (17) and (18), we have

$w_{eff}=\frac{4q_{0}-2}{1-2q_{0}+E^{4}(5+2q_{0})}$ (36)

By using the definition of deceleration parameter given in equation (22), we
have

$q=\frac{E^{4}(5+2q_{0})+10q_{0}-5}{2E^{4}(5+2q_{0})-4q_{0}+2}$ (37)

Now, for our particular $f(Q)$ model we have the following expression of the
statefinder parameters

$r=\frac{10(1-2q_{0})^{2}+E^{8}(5+2q_{0})^{2}+(4q_{0}^{2}+8q_{0}-5)\\{7E^{4}-12(1+z)E^{3}E^{\prime}\\}}{\\{1-2q_{0}+E^{4}(5+2q_{0})\\}^{2}}$
(38)

and

$s=\frac{-3+3E^{4}(5+2q_{0})-4(1+z)(5+2q_{0})E^{3}E^{\prime}}{3-6q_{0}+3E^{4}(5+2q_{0})}$
(39)

Numerical Results and Physical Aspects of Cosmological Parameters:

We will solve the differential equation (32) by the numerical algorithm. In
the present work, we adopt the initial condition as $E(0)=1$, and we use
$H_{0}=67.9\>km/s/Mpc$ , $q_{0}=-0.5$, $\Omega_{0}=0.3$ [53]. The following
numerical plots are obtained using the above observed value of cosmographic
parameters.

Figure 1: Evolution profile of the cosmic density parameter vs redshift z.
Figure 2: Evolution profile of the deceleration parameter vs redshift z.
Figure 3: Evolution profile of the cosmic energy density of matter vs redshift
z. Figure 4: Evolution profile of the effective EoS parameter vs redshift z.
Figure 5: Evolution profile of the cosmic pressure due to non-metricity vs
redshift z. Figure 6: Profile of the evolution trajectories of the given
cosmological model in the $r-s$ plane .

From Fig 3 it is clear that the matter energy density of the universe
decreases and vanishes in the far future with expansion of the universe while
Fig 1 show that if the dark energy component is represented by the non-
metricity then it will overcome all the energy content of the universe.
Further, Fig 5 shows that the non-metricity component of the universe is of
high negative pressure. The deceleration parameter is a key component to
understand the expansion scenario of the universe. The plot for deceleration
parameter in Fig 2 shows a phase transition from early deceleration to present
acceleration of the universe. Further, from Fig 4 it is clear that the present
universe is in the accelerating phase. The statefinder parameter describes the
behavior of the dark energy component. A value $(r<1,s>0)$ represents
quintessence type behavior of dark energy component, $(r=1,s=0)$ represents
$\Lambda$CDM type behavior, and $(r>1,s<0)$ represents phantom type behavior.
Fig 6 represents the evolution trajectories of given model which shows that
our model lies in the chaplygin gas region $(r>1,s<0)$ and it will pass
through the $\Lambda$CDM fixed point in the far future.

Model II: We consider the following $f(Q)$ function which contains a linear
and a non-linear form of non-metricity, specifically [54]

$f(Q)=\alpha Q+\beta Q^{2}$ (40)

where $\alpha$ and $\beta$ are free parameters.

Then for this specific choice of the function, one can obtain the following
differential equation

$\dot{H}\left(\alpha-1+36\beta
H^{2}\right)+\frac{3}{2}H^{2}\left(\alpha-1+18\beta H^{2}\right)=0$ (41)

Again, by using Friedmann equations (28) and (29) one can estimate

$\alpha=1-\Omega_{m0}\left[2-\frac{3}{2\left(1+q_{0}\right)}\right]$ (42)

and

$\beta=\frac{\Omega_{m0}}{18H_{0}^{2}}\left[1-\frac{3}{2\left(1+q_{0}\right)}\right]$
(43)

Then by using (42) and (43) in (41) we have

$E^{\prime}\left[1+2E^{2}\frac{(1-2q_{0})}{(1+4q_{0})}\right]-\frac{3E}{2(1+z)}\left[1+E^{2}\frac{(1-2q_{0})}{(1+4q_{0})}\right]=0$
(44)

Now, by using equations (19) and (20) we have

$\Omega_{Q}=\frac{2q_{0}[1+\Omega_{m0}(E^{2}-2)]-\Omega_{m0}(E^{2}+1)+2}{2(1+q_{0})}$
(45)

$\omega=-\frac{2(2q_{0}^{2}+q_{0}-1)E^{2}}{\\{E^{2}(4q_{0}-2)-4q_{0}-1\\}\\{2q_{0}[1+\Omega_{m0}(E^{2}-2)]-\Omega_{m0}(E^{2}+1)+2\\}}$
(46)

and the effective equation of state parameter for this model is given by

$w_{eff}=\frac{(1-2q_{0})E^{2}}{E^{2}(4q_{0}-2)-4q_{0}-1}$ (47)

By using the definition of deceleration parameter given in equation (22), we
have

$q=\frac{E^{2}(1-2q_{0})-4q_{0}-1}{2E^{2}(2q_{0}-1)-8q_{0}-2}$ (48)

The expression of the statefinder parameters for the given $f(Q)$ model are

$r=\frac{(8q_{0}^{2}-2q_{0}-1)\\{E^{2}+6(1+z)EE^{\prime}\\}}{2\\{2E^{2}(1-2q_{0})+4q_{0}+1\\}^{2}}$
(49)

and

$s=\frac{3E^{3}(2q_{0}-1)-3E(1+4q_{0})-2(1+z)(1+4q_{0})E^{\prime}}{3E\\{2E^{2}(2q_{0}-1)-4q_{0}-1\\}}$
(50)

Numerical Results and Physical Aspects of Cosmological Parameters:

Again, we will solve the differential equation (44) by the numerical
algorithm. We adopt the initial condition as $E(0)=1$, and we use
$H_{0}=67.9\>km/s/Mpc$ , $q_{0}=-0.5$, $\Omega_{0}=0.3$ [53]. The following
numerical plots are obtained using the above observed value of cosmographic
parameters.

Figure 7: Evolution profile of the cosmic density parameter vs redshift z.
Figure 8: Evolution profile of the deceleration parameter vs redshift z.
Figure 9: Evolution profile of the cosmic energy density of matter vs redshift
z. Figure 10: Evolution profile of the effective EoS parameter vs redshift z.
Figure 11: Evolution profile of the cosmic pressure due to non-metricity vs
redshift z. Figure 12: Profile of the evolution trajectories of the given
cosmological model in the $r-s$ plane .

From Fig 9 it is clear that the matter energy density of the universe
decreases and vanishes in the far future with expansion of the universe while
Fig 7 show that if the dark energy component is represented by the non-
metricity then it will overcome all the energy content of the universe.
Further, Fig 11 shows that the non-metricity component of the universe exhibit
high negative pressure and this negative pressure decreases with the expansion
of the universe. The plot for deceleration parameter in Fig 8 shows negative
behavior that represents the accelerating expansion phase of the universe.
However this $f(Q)$ model do not show the recent transition from early
deceleration to present acceleration of the universe. Again, from Fig 10 it is
clear that the present universe is in the accelerating phase. Lastly, Fig 12
represents the evolution trajectories of the given polynomial $f(Q)$ model
that lies in the quintessence region $(r<1,s>0)$ and it will pass through the
$\Lambda$CDM fixed point in the far future.

##  V Om Diagnostics

The Om diagnostic is another effective tool to differentiate cosmological
models of dark energy [55]. It is simplest diagnostic as compared to
statefinder diagnostic since it uses only Hubble parameter which requires
first order derivative of cosmic scale factor. For spatially falt universe, it
is defined as

$Om(z)=\frac{E(z)^{2}-1}{(1+z)^{3}-1}$ (51)

Here $E(z)=\frac{H(z)}{H_{0}}$ and $H_{0}$ is the Hubble constant. The
negative slope of $Om(z)$ correspond to quintessence behavior whereas positive
slope corresponds to phantom type behavior. The constant behavior of $Om(z)$
indicates the $\Lambda$CDM model. From Fig (13) and (14) we observed that the
Om diagnostic parameter corresponding to the model I have positive slope while
corresponding to model II have negative slope on the entire domain. Thus from
Om diagnostic we can conclude that the model I represents phantom type
behavior while model II follows quintessence scenario.

Figure 13: Profile of the Om diagnostic parameter for the Cosmological Model
I.  Figure 14: Profile of the Om diagnostic parameter for the Cosmological
Model II.

##  VI Conclusion

Cosmology has been struggling mainly for two issues: dark matter and dark
energy. The dark matter does not interact electromagnetically while its
gravitational effects can be detected. Still, we have not yet found dark
matter as a particle of standard model [56, 57]. Modified gravity theory has
been used also to explain dark matter effects. Further, it is a highly counter
instinctive fact that our universe is accelerating. Although, cosmological
constant in GR can well describe this dynamical effect the aforementioned
issues related to $\Lambda$ motivate the search for an alternative explanation
of the dark energy. As a consequence, so many dark energy models started
appearing. The discrimination between the various dark energy models becomes
necessary. Statefinder diagnostic is a convenient method that can
differentiate between the various dark energy models.

In this article, we have performed a complete analysis of statefinder
parameters for $f(Q)$ cosmology. We considered two $f(Q)$ models which
contains a linear and a non-linear term of non-metricity scalar, specifically,
$f(Q)=\alpha Q+\frac{\beta}{Q}$ and $f(Q)=\alpha Q+\beta Q^{2}$, where
$\alpha$ and $\beta$ are free parameters and then we performed a complete
diagnosis of the statefinder parameters. We have plotted the evolution
trajectory of our model in the $r-s$ plane and analyzed the behavior of
different cosmological parameters. For both the models (see fig 1,3, 7, and 9
), we found that the matter energy density falls off as the universe expands
while the dark energy due to non-metricity will overcome all the energy
content of the universe. From Fig 4 and 10, we found that the present universe
is in the accelerating phase. In addition, Fig 2 show that our universe has
experienced a transition from decelerated to accelerated phase in the recent
past while Fig 8 do not show the transition phase. Also, From Fig 5 and 11 we
observed that the non-metricity component of the universe exhibit high
negative pressure. Further, Fig 6 represents the evolution trajectories of the
model I which lies in the chaplygin gas region $(r>1,s<0)$ and it will pass
through the $\Lambda$CDM fixed point in the far future whereas the evolution
trajectories of the model II in Fig 12 lies in the quintessence region
$(r<1,s>0)$. Finally, from Om diagnostic (see fig 13 and 14) we can conclude
that the model I represents phantom type behavior while model II follows
quintessence scenario. The cosmological $f(Q)$ models presented in this work
have great significance. A power law correction to the STEGR will give rise to
branches of solution applicable either to the early universe or to late-time
cosmic acceleration. Our cosmological model I provides a correction to the
late-time cosmology, where they can give rise to the dark energy, whereas
model II is relevant to the early universe with potential applications to
inflationary solutions [58]. Furthermore, another important aspect of our
considered $f(Q)$ models is that throughout the evolution of the universe, the
varying growth index of matter perturbations is smaller than that of the
$\Lambda$CDM for our cosmological model I, whereas larger for model II, which
can be helpful to estimate the distribution of matter in the universe [59].

##  VII Data Availability Statement

There are no new data associated with this article.

## Acknowledgments

RS acknowledges University Grants Commission(UGC), New Delhi, India for
awarding Junior Research Fellowship (UGC-Ref. No.: 191620096030). We are very
much grateful to the honorable referees and to the editor for the illuminating
suggestions that have significantly improved our work in terms of research
quality, and presentation.

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