# A new approach toward locally bounded global solutions to a $3D$ chemotaxis-
stokes system with nonlinear diffusion and rotation

Jiashan Zheng
School of Mathematics and Statistics Science,
Ludong University, Yantai 264025, P.R.China
Corresponding author. E-mail address<EMAIL_ADDRESS>(J.Zheng)

###### Abstract

We consider a degenerate quasilinear chemotaxis–Stokes type involving rotation
in the aggregative term,

$\left\\{\begin{array}[]{l}n_{t}+u\cdot\nabla n=\Delta
n^{m}-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in\Omega,t>0,\\\
c_{t}+u\cdot\nabla c=\Delta c-nc,\quad x\in\Omega,t>0,\\\ u_{t}+\nabla
P=\Delta u+n\nabla\phi,\quad x\in\Omega,t>0,\\\ \nabla\cdot u=0,\quad
x\in\Omega,t>0,\\\ \end{array}\right.$ $None$

where $\Omega\subseteq\mathbb{R}^{3}$ is a bounded convex domain with smooth
boundary. Here $S\in
C^{2}(\bar{\Omega}\times[0,\infty)^{2};\mathbb{R}^{3\times 3})$ is a matrix
with $s_{i,j}\in C^{1}(\bar{\Omega}\times[0,\infty)\times[0,\infty)).$
Moreover, $|S(x,n,c)|\leq S_{0}(c)$ for all
$(x,n,c)\in\bar{\Omega}\times[0,\infty)\times[0,\infty)$ with $S_{0}(c)$
nondecreasing on $[0,\infty)$. If

$m>\frac{9}{8},$

then for all reasonably regular initial data, a corresponding initial-boundary
value problem for $(CF)$ possesses a globally defined weak solution $(n,c,u)$.
Moreover, for any fixed $T>0$ this solution is bounded in $\Omega\times(0,T)$
in the sense that

$\|u(\cdot,t)\|_{L^{\infty}(\Omega)}+\|c(\cdot,t)\|_{W^{1,\infty}(\Omega)}+\|n(\cdot,t)\|_{L^{\infty}(\Omega)}\leq
C~{}~{}\mbox{for all}~{}~{}t\in(0,T)$

is valid with some $C(T)>0$. In particularly, if $S(x,n,c):=C_{S}$, this
result extends of Tao and Winkler ([15]), while, if fluid-free subcase of the
flow of fluid is ignored or the fluid is stationary in $(CF)$,
$S(x,n,c):=C_{S}$ and $N=3$, this results is consistent with the result of
Theorem 2.1 of Zheng and Wang ([28]). In view of some carefully analysis, we
can establish some natural gradient-like structure of the functional
$\int_{\Omega}n(\cdot,t)\ln
n(\cdot,t)+\int_{\Omega}|\nabla\sqrt{c}(\cdot,t)|^{2}+\int_{\Omega}|u(\cdot,t)|^{2}$
of $(CF),$ which is a new estimate of chemotaxis–Stokes system with rotation
(see [1, 2, 22, 19]).

Key words: Chemotaxis–fluid system; Global existence; Tensor-valued
sensitivity

2010 Mathematics Subject Classification: 35K55, 35Q92, 35Q35, 92C17

##  1 Introduction

We consider the following chemotaxis-Stokes system with porous medium
diffusion and rotation in the aggregation term:

$\left\\{\begin{array}[]{ll}n_{t}+u\cdot\nabla n=\nabla\cdot(D(n)\nabla
n)-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in\Omega,t>0,\\\
c_{t}+u\cdot\nabla c=\Delta c-nc,\quad x\in\Omega,t>0,\\\ u_{t}+\nabla
P=\Delta u+n\nabla\phi,\quad x\in\Omega,t>0,\\\ \nabla\cdot u=0,\quad
x\in\Omega,t>0,\\\ \displaystyle{(\nabla n-nS(x,n,c)\cdot\nabla
c)\cdot\nu=\nabla c\cdot\nu=0,u=0,}\quad x\in\partial\Omega,t>0,\\\
\displaystyle{n(x,0)=n_{0}(x),c(x,0)=c_{0}(x),u(x,0)=u_{0}(x),}\quad
x\in\Omega,\\\ \end{array}\right.$ (1.1)

where $\Omega$ is a bounded convex domain in $\mathbb{R}^{3}$ with smooth
boundary $\partial\Omega$, $S(x,n,c)$ is a chemotactic sensitivity tensor
satisfying

$S\in C^{2}(\bar{\Omega}\times[0,\infty)^{2};\mathbb{R}^{3\times 3})$ (1.2)

and

$|S(x,n,c)|\leq S_{0}(c)~{}~{}~{}~{}\mbox{for
all}~{}~{}(x,n,c)\in\Omega\times[0,\infty)^{2}$ (1.3)

with some nondecreasing $S_{0}:[0,\infty)\rightarrow\mathbb{R},$ porous medium
diffusion function $D$ satisfies

$D\in C^{\iota}_{loc}([0,\infty))~{}~{}\mbox{for
some}~{}~{}\iota>0,~{}~{}~{}D(n)\geq C_{D}n^{m-1}~{}~{}\mbox{for
all}~{}~{}n>0$ (1.4)

with some $m\geq 1$. Here $n$ and $c$ denote the bacterium density and the
oxygen concentration, respectively, and $u$ represents the velocity field of
the fluid subject to an incompressible Navier-Stokes equation with pressure
$P$ and viscosity $\eta$ and a gravitational force $\nabla\phi$. This type of
the system arises in mathematical biology to model the evolution of oxygen-
driven swimming bacteria in an impressible fluid.

To motivate our study, let us first recall the following fluid-free subcase of
system (1.1):

$\left\\{\begin{array}[]{ll}n_{t}=\nabla\cdot(D(n)\nabla
n)-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in\Omega,t>0,\\\ c_{t}=\Delta
c-nc,\quad x\in\Omega,t>0,\\\ \displaystyle{\nabla n\cdot\nu=\nabla
c\cdot\nu=0,}\quad x\in\partial\Omega,t>0,\\\
\displaystyle{n(x,0)=n_{0}(x),c(x,0)=c_{0}(x),}\quad x\in\Omega.\\\
\end{array}\right.$ (1.5)

There are only few rigorous results on global existence and qualitative
behavior of solutions to (1.5) with either a matrix-valued function
($S(x,n,c)$) or a scalar one $(S(x,n,c):=S(c))$ (see e.g. [13, 28, 23]).

The following chemotaxis-(Navier)-Stokes model which is a generalized version
of the model proposed in [16], describes the motion of oxygen-driven swimming
cells in an incompressible fluid:

$\left\\{\begin{array}[]{ll}n_{t}+u\cdot\nabla n=\nabla\cdot(D(n)\nabla
n)-C_{S}\nabla\cdot(nS(c)\cdot\nabla c),\quad x\in\Omega,t>0,\\\
c_{t}+u\cdot\nabla c=\Delta c-nc,\quad x\in\Omega,t>0,\\\ u_{t}+(u\cdot\nabla
u)=\nabla P+\Delta u+n\nabla\phi,\quad x\in\Omega,t>0,\\\ \nabla\cdot
u=0,\quad x\in\Omega,t>0,\\\ \end{array}\right.$ (1.6)

where compared with (1.1), the nonlinear convective term $u\nabla u$ exists
$u$-equation of (1.6), moreover $S(x,n,c):=S(c)$ is scalar function in (1.6)
and hence there is a certain natural quasi-Lyapunov functionals of (1.6).
Hence, by making use of energy-type functionals, some local and global
solvability of corresponding initial value problem for (1.6) in either bounded
or unbounded domains have been obtained in the past years (see e.g. Lorz et
al. [6, 10], Winkler et al. [1, 14, 21], Chae et al. [4, 5], Di Francesco et
al. ([7], Zhang, Zheng [25] and references therein).

If the chemotactic sensitivity $S(x,n,c)$ is regarded as a tensor rather than
a scalar one ([24]), (1.1) turns into a chemotaxis–Stokes system with
rotational flux which implies that chemotactic migration need not be directed
along the gradient of signal concentration. In contrast to the chemotaxis-
fluid system (1.6), chemotaxis-fluid systems with tensor-valued sensitivity
lose some natural gradient-like structure (see Cao [2], Wang et. al [18, 19],
Winkler [22]). This gives rise to considerable mathematical difficulties.
Therefore, only very few results appear to be available on chemotaxis–Stokes
system with such tensor-valued sensitivities (Cao et al. [2], Ishida [8], Wang
et al. [18, 20, 19], Winkler [22]). In fact, assuming (1.2)–(1.3) holds,
Ishida ([8]) showed that the corresponding full chemotaxis Navier-Stokes
system with porous-medium-type diffusion model possesses a bounded global weak
solution in two space dimensions. While, in three space dimensions, if the
initial data satisfy certain smallness conditions and $D(n)=1$, Cao and
Lankeit [3] showed that (1.1) has global classical solutions and give decay
properties of these solutions. In this paper, the core step is to establish
the estimates of the functional

$\int_{\Omega}n^{p}(\cdot,t)+\int_{\Omega}|\nabla{c}(\cdot,t)|^{2{q}}$ (1.7)

for suitably chosen but arbitrarily large numbers $p>1$ and $q>1$. In fact,
one of our main tool is consideration of the natural gradient-like energy
functional

$\int_{\Omega}n_{\varepsilon}(\cdot,t)\ln
n_{\varepsilon}(\cdot,t)+\int_{\Omega}|\nabla\sqrt{c_{\varepsilon}}(\cdot,t)|^{2}+\int_{\Omega}|u_{\varepsilon}(\cdot,t)|^{2},$
(1.8)

which is new estimate of chemotaxis–Stokes system with rotation (see Lemmata
2.7–2.10), although, (1.8) has been used to solve the
chemotaxis-(Navier)-Stokes system without rotation (see [1, 9, 21]). Here
$(n_{\varepsilon},c_{\varepsilon},u_{\varepsilon})$ is solution of the
approximate problem of (1.1). We guess that (1.8) can also be dealt with other
types of systems, e.g., quasilinear chemotaxis system with rotation,
chemotaxis-(Navier)-Stokes system with rotation. Then, in view of the
estimates (1.8), the suitable interpolation arguments (see Lemma 2.4) and the
basic a priori information (see Lemma 2.5), we can get the the estimates of
the functional

$\int_{\Omega}n^{p}_{\varepsilon}+\int_{\Omega}|\nabla{c}_{\varepsilon}|^{2{q_{0}}}+\int_{\Omega}|A^{\frac{1}{2}}u_{\varepsilon}|^{2},$
(1.9)

where $p:=p(q_{0},m)\geq\frac{7}{4}$ and $q_{0}<2$. Next, (1.9) and some other
carefully analysis (Lemma 2.18–2.19) yield the estimates of the functional
(1.7). Indeed, the paper makes sure that under the assumption $m$ satisfies
(1.4), $S$ satisfies (1.2) and (1.3) with some

$m>\frac{9}{8},$ (1.10)

then problem (1.1) possesses a global weak solution, which extends the results
of Tao and Winkler ([15]), who showed the global existence of solutions in the
cases $S(x,n,c):=C_{S}$, $m$ satisfies (1.4) with $m>\frac{8}{7}$.

##  2 Preliminaries and main results

Due to the hypothesis (1.4), the problem (1.1) has no classical solutions in
general, and thus we consider its weak solutions in the following sense.

######  Definition 2.1.

(weak solutions) Let $T\in(0,\infty)$ and

$H(s)=\int_{0}^{s}D(\sigma)d\sigma~{}~{}\mbox{for}~{}~{}s\geq 0.$

Suppose that $(n_{0},c_{0},u_{0})$ satisfies (2.7). Then a triple of functions
$(n,c,u)$ defined in $\Omega\times(0,T)$ is called a weak solution of model
(1.1), if

$\left\\{\begin{array}[]{ll}n\in L_{loc}^{1}(\bar{\Omega}\times[0,T)),\\\ c\in
L_{loc}^{\infty}(\bar{\Omega}\times[0,T))\cap
L_{loc}^{1}([0,T);W^{1,1}(\Omega)),\\\ u\in
L_{loc}^{1}([0,T);W^{1,1}(\Omega)),\\\ \end{array}\right.$ (2.1)

where $n\geq 0$ and $c\geq 0$ in $\Omega\times(0,T)$ as well as $\nabla\cdot
u=0$ in the distributional sense in $\Omega\times(0,T)$, in addition,

$H(n),~{}~{}n|\nabla c|~{}~{}~{}\mbox{and}~{}~{}~{}n|u|~{}~{}\mbox{belong
to}~{}~{}L^{1}_{loc}(\bar{\Omega}\times[0,T)),$ (2.2)

and

$\begin{array}[]{rl}\displaystyle{-\int_{0}^{T}\int_{\Omega}n\varphi_{t}-\int_{\Omega}n_{0}\varphi(\cdot,0)}=&\displaystyle{\int_{0}^{T}\int_{\Omega}H(n)\Delta\varphi+\int_{0}^{T}\int_{\Omega}n(S(x,n,c)\cdot\nabla
c)\cdot\nabla\varphi}\\\
&+\displaystyle{\int_{0}^{T}\int_{\Omega}nu\cdot\nabla\varphi}\\\ \end{array}$
(2.3)

for any $\varphi\in C_{0}^{\infty}(\bar{\Omega}\times[0,T))$ satisfying
$\frac{\partial\varphi}{\partial\nu}=0$ on $\partial\Omega\times(0,T)$ as well
as

$\begin{array}[]{rl}\displaystyle{-\int_{0}^{T}\int_{\Omega}c\varphi_{t}-\int_{\Omega}c_{0}\varphi(\cdot,0)}=&\displaystyle{-\int_{0}^{T}\int_{\Omega}\nabla
c\cdot\nabla\varphi-\int_{0}^{T}\int_{\Omega}nc\cdot\varphi+\int_{0}^{T}\int_{\Omega}cu\cdot\nabla\varphi}\\\
\end{array}$ (2.4)

for any $\varphi\in C_{0}^{\infty}(\bar{\Omega}\times[0,T))$ and

$\begin{array}[]{rl}\displaystyle{-\int_{0}^{T}\int_{\Omega}u\varphi_{t}-\int_{\Omega}u_{0}\varphi(\cdot,0)}=&\displaystyle{-\int_{0}^{T}\int_{\Omega}\nabla
u\cdot\nabla\varphi-\int_{0}^{T}\int_{\Omega}n\nabla\phi\cdot\varphi}\\\
\end{array}$ (2.5)

for any $\varphi\in C_{0}^{\infty}(\bar{\Omega}\times[0,T);\mathbb{R}^{3})$
fulfilling $\nabla\varphi\equiv 0$ in $\Omega\times(0,T)$. If $(n,c,u)$ is a
weak solution of (1.1) in $\Omega\times(0,T)$ for any $T\in(0,\infty),$ then
we call $(n,c,u)$ a global weak solution.

In this paper, we assume that

$\phi\in W^{1,\infty}(\Omega).$ (2.6)

Moreover, let the initial data $(n_{0},c_{0},u_{0})$ fulfill

$\left\\{\begin{array}[]{ll}\displaystyle{n_{0}\in
C^{\kappa}(\bar{\Omega})~{}~{}\mbox{for
certain}~{}~{}\kappa>0~{}~{}\mbox{with}~{}~{}n_{0}\geq
0~{}~{}\mbox{in}~{}~{}\Omega},\\\ \displaystyle{c_{0}\in
W^{1,\infty}(\Omega)~{}~{}\mbox{with}~{}~{}c_{0}\geq
0~{}~{}\mbox{in}~{}~{}\bar{\Omega},}\\\ \displaystyle{u_{0}\in
D(A^{\gamma}_{r})~{}~{}\mbox{for~{}~{}
some}~{}~{}\gamma\in(\frac{3}{4},1)~{}~{}\mbox{and
any}~{}~{}{r}\in(1,\infty),}\\\ \end{array}\right.$ (2.7)

where $A_{r}$ denotes the Stokes operator with domain
$D(A_{r}):=W^{2,{r}}(\Omega)\cap W^{1,{r}}_{0}(\Omega)\cap
L^{r}_{\sigma}(\Omega)$, and $L^{r}_{\sigma}(\Omega):=\\{\varphi\in
L^{r}(\Omega)|\nabla\cdot\varphi=0\\}$ for ${r}\in(1,\infty)$ ([12]).

######  Theorem 2.1.

Let (2.6) hold, and suppose that $m$ and $S$ satisfies (1.4) and (1.2)–(1.3),
respectively. Suppose that the assumptions (2.7) hold. If

$m>\frac{9}{8},$ (2.8)

then there exists at least one global weak solution (in the sense of
Definition 2.1 above) of problem (1.1). Moreover, for any fixed $T>0$, there
exists a positive constant $C:=C(T)$ such that

$\|u(\cdot,t)\|_{L^{\infty}(\Omega)}+\|c(\cdot,t)\|_{W^{1,\infty}(\Omega)}+\|n(\cdot,t)\|_{L^{\infty}(\Omega)}\leq
C~{}~{}\mbox{for all}~{}~{}t\in(0,T).$ (2.9)

######  Remark 2.1.

(i) If $S(x,n,c):=C_{S},$ Theorem 2.1 extends the results of Theorem 1.1 of
Tao and Winkler [15], who proved the possibility of
$\mathbf{global~{}~{}existence}$, in the case that $m>\frac{8}{7}$.

(ii) In view of Theorem 2.1, if the flow of fluid is ignored or the fluid is
stationary in (1.1), $S(x,n,c):=C_{S},$ and $N=3$, Theorem 2.1 is consistent
with the result of Theorem 2.1 of Zheng and Wang([28]), who proved the
possibility of $\mathbf{global~{}~{}existence}$, in the case that
$m>\frac{9}{8}$.

(iii) From Theorem 2.1, if the flow of fluid is ignored or the fluid is
stationary in (1.1) and $S(x,n,c):=C_{S},$ and $N=3$, our results improve of
Wang et al. [17], who proved the possibility of
$\mathbf{global~{}~{}existence}$, in the case that $m>\frac{4}{3}$.

(iv) By Theorem 2.1, we also derive that the large diffusion exponent $m$
($>\frac{10}{9}$) yields the existence of solutions to (1.1). Moreover, no
smallness condition on either $\phi$ or on the initial data needs to be
fulfilled here, which is different from [3].

######  Lemma 2.1.

([22]) Let $l\in[1,+\infty)$ and $r\in[1,+\infty]$ be such that

$\left\\{\begin{array}[]{ll}l<\frac{3r}{3-r}~{}~{}\mbox{if}~{}~{}r\leq 3,\\\
l\leq\infty~{}~{}\mbox{if}~{}~{}r>3.\end{array}\right.$ (2.10)

Then for all $K>0$ there exists $C:=C(l,r,K)$ such that if

$\|n{}(\cdot,t)\|_{L^{r}(\Omega)}\leq K~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),$ (2.11)

then

$\|Du{}(\cdot,t)\|_{L^{l}(\Omega)}\leq C~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).$ (2.12)

In general, the degenerate diffusion case of (1.1) might not have classical
solutions, thus in order to justify all the formal arguments, we need to
introduce the following approximating system of (1.1):

$\left\\{\begin{array}[]{ll}n_{\varepsilon t}+u_{\varepsilon}\cdot\nabla
n_{\varepsilon}=\nabla\cdot(D_{\varepsilon}(n_{\varepsilon})\nabla
n_{\varepsilon})-\nabla\cdot(n_{\varepsilon}S_{\varepsilon}(x,n_{\varepsilon},c_{\varepsilon})\nabla
c_{\varepsilon}),\quad x\in\Omega,t>0,\\\ c_{\varepsilon
t}+u_{\varepsilon}\cdot\nabla c_{\varepsilon}=\Delta
c_{\varepsilon}-n_{\varepsilon}c_{\varepsilon},\quad x\in\Omega,t>0,\\\
u_{\varepsilon t}+\nabla P_{\varepsilon}=\Delta
u_{\varepsilon}+n_{\varepsilon}\nabla\phi,\quad x\in\Omega,t>0,\\\ \nabla\cdot
u_{\varepsilon}=0,\quad x\in\Omega,t>0,\\\ \displaystyle{\nabla
n_{\varepsilon}\cdot\nu=\nabla
c_{\varepsilon}\cdot\nu=0,u_{\varepsilon}=0,\quad x\in\partial\Omega,t>0,}\\\
\displaystyle{n_{\varepsilon}(x,0)=n_{0}(x),c_{\varepsilon}(x,0)=c_{0}(x),u_{\varepsilon}(x,0)=u_{0}(x)},\quad
x\in\Omega,\\\ \end{array}\right.$ (2.13)

where a family $(D_{\varepsilon})_{\varepsilon\in(0,1)}$ of functions

$D_{\varepsilon}\in C^{2}((0,\infty))~{}\mbox{such
that}~{}~{}D_{\varepsilon}(n)\geq\varepsilon~{}\mbox{for all}~{}n>0$
$~{}~{}\mbox{and}~{}D(n)\leq D_{\varepsilon}(n)\leq
D(n)+2\varepsilon~{}\mbox{for all}~{}n>0~{}\mbox{and}~{}\varepsilon\in(0,1),$
$\begin{array}[]{ll}S_{\varepsilon}(x,n,c):=\rho_{\varepsilon}(x)S(x,n,c),~{}~{}x\in\bar{\Omega},~{}~{}n\geq
0,~{}~{}c\geq 0~{}~{}\mbox{and}~{}~{}\varepsilon\in(0,1).\end{array}$ (2.14)

Here $(\rho_{\varepsilon})_{\varepsilon\in(0,1)}\in C^{\infty}_{0}(\Omega)$ be
a family of standard cut-off functions satisfying $0\leq\rho_{\varepsilon}\leq
1$ in $\Omega$ and $\rho_{\varepsilon}\rightarrow 1$ in $\Omega$ as
$\varepsilon\rightarrow 0$.

Let us begin with the following statement on local well-posedness of (2.13),
along with a convenient extensibility criterion. For a proof we refer to (see
[22], Lemma 2.1 of [21]):

######  Lemma 2.2.

Let $\Omega\subseteq\mathbb{R}^{3}$ be a bounded convex domain with smooth
boundary. Assume that the initial data $(n_{0},c_{0},u_{0})$ fulfills (2.7).
Then there exist $T_{max}\in(0,\infty]$ and a classical solution
$(n_{\varepsilon},c_{\varepsilon},u_{\varepsilon},P_{\varepsilon})$ of (2.13)
in $\Omega\times(0,T_{max})$ such that

$\left\\{\begin{array}[]{ll}n_{\varepsilon}\in
C^{0}(\bar{\Omega}\times[0,T_{max}))\cap
C^{2,1}(\bar{\Omega}\times(0,T_{max})),\\\ c_{\varepsilon}{}\in
C^{0}(\bar{\Omega}\times[0,T_{max}))\cap
C^{2,1}(\bar{\Omega}\times(0,T_{max})),\\\ u_{\varepsilon}{}\in
C^{0}(\bar{\Omega}\times[0,T_{max}))\cap
C^{2,1}(\bar{\Omega}\times(0,T_{max})),\\\ P_{\varepsilon}{}\in
C^{1,0}(\bar{\Omega}\times(0,T_{max})),\\\ \end{array}\right.$ (2.15)

classically solving (2.13) in $\Omega\times[0,T_{max})$. Moreover,
$n_{\varepsilon}$ and $c_{\varepsilon}$ are nonnegative in
$\Omega\times(0,T_{max})$, and

$\limsup_{t\nearrow
T_{max}}(\|n_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}+\|c_{\varepsilon}(\cdot,t)\|_{W^{1,\infty}(\Omega)}+\|A^{\gamma}u_{\varepsilon}(\cdot,t)\|_{L^{2}(\Omega)})=\infty,$
(2.16)

where $\gamma$ is given by (2.7).

######  Lemma 2.3.

([9]) Let $w\in C^{2}(\bar{\Omega})$ satisfy $\nabla w\cdot\nu=0$ on
$\partial\Omega$.

(i) Then

$\frac{\partial|\nabla w|^{2}}{\partial\nu}\leq C_{\partial\Omega}|\nabla
w|^{2},$

where $C_{\partial\Omega}$ is an upper bound on the curvature of
$\partial\Omega$.

(ii) Furthermore, for any $\delta>0$ there is $C(\delta)>0$ such that every
$w\in C^{2}(\bar{\Omega})$ with $\nabla w\cdot\nu=0$ on $\partial\Omega$
fulfils

$\|w\|_{L^{2}(\partial\Omega)}\leq\delta\|\Delta
w\|_{L^{2}(\Omega)}+C(\delta)\|w\|_{L^{2}(\Omega)}.$

(iii) For any positive $w\in C^{2}(\bar{\Omega})$

$\|\Delta
w^{\frac{1}{2}}\|_{L^{2}(\Omega)}\leq\frac{1}{2}\|w^{\frac{1}{2}}\Delta\ln
w\|_{L^{2}(\Omega)}+\frac{1}{4}\|w^{-\frac{3}{2}}|\nabla
w|^{2}\|_{L^{2}(\Omega)}.$ (2.17)

(iv) There are $C>0$ and $\delta>0$ such that every positive $w\in
C^{2}(\bar{\Omega})$ fulfilling $\nabla w\cdot\nu=0$ on $\partial\Omega$
satisfies

$\begin{array}[]{rl}&\displaystyle{-2\int_{\Omega}\frac{|\Delta
w|^{2}}{w}+\int_{\Omega}\frac{|\nabla w|^{2}\Delta
w}{w^{2}}\leq-\delta\int_{\Omega}w|D^{2}\ln
w|^{2}-\delta\int_{\Omega}\frac{|\nabla w|^{4}}{w^{3}}+C\int_{\Omega}w.}\\\
\end{array}$ (2.18)

######  Lemma 2.4.

(Lemma 3.8 of [22]) Let $q\geq 1$,

$\lambda\in[2q+2,4q+1]$ (2.19)

and $\Omega\subset\mathbb{R}^{3}$ be a bounded convex domain with smooth
boundary. Then there exists $C>0$ such that for all $\varphi\in
C^{2}(\bar{\Omega})$ fulfilling
$\varphi\cdot\frac{\partial\varphi}{\partial\nu}=0$ on $\partial\Omega$ we
have

$\begin{array}[]{rl}&\|\nabla\varphi\|_{L^{\lambda}(\Omega)}\leq
C\||\nabla\varphi|^{q-1}D^{2}\varphi\|_{L^{2}(\Omega)}^{\frac{2(\lambda-3)}{(2q-1)\lambda}}\|\varphi\|_{L^{\infty}(\Omega)}^{\frac{6q-\lambda}{(2q-1)\lambda}}+C\|\varphi\|_{L^{\infty}(\Omega)}.\\\
\end{array}$ (2.20)

Let us state two well-known results of solution of (2.13).

######  Lemma 2.5.

The solution of (2.13) satisfies

$\begin{array}[]{rl}\|n_{\varepsilon}(\cdot,t)\|_{L^{1}(\Omega)}=\|n_{0}\|_{L^{1}(\Omega)}~{}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max})\end{array}$ (2.21)

and

$\begin{array}[]{rl}\|c_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}\leq\|c_{0}\|_{L^{\infty}(\Omega)}~{}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).\end{array}$ (2.22)

######  Lemma 2.6.

For any $l<\frac{3}{2}$, there exists $C:=C(l,\|n_{0}\|_{L^{1}(\Omega)})$ such
that

$\|Du{}(\cdot,t)\|_{L^{l}(\Omega)}\leq C~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).$ (2.23)

###### Proof.

Choosing $r=1$ in Lemma 2.7 and using (2.21), we can get the results. ∎

######  Lemma 2.7.

Let $m>\frac{10}{9}$. There exists $C>0$ independent of $\varepsilon$ such
that for every $\delta_{1}>0$, the solution of (2.13) satisfies

$\int_{\Omega}{|u_{\varepsilon}|^{2}}+\int_{\Omega}{|\nabla
u_{\varepsilon}|^{2}}\leq\delta_{1}\int_{\Omega}\frac{D_{\varepsilon}(n_{\varepsilon})|\nabla
n_{\varepsilon}|^{2}}{n_{\varepsilon}}+C~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),$ (2.24)

###### Proof.

Testing the third equation of (2.13) with $u_{\varepsilon}$, integrating by
parts and using $\nabla\cdot u_{\varepsilon}=0$

$\frac{1}{2}\int_{\Omega}{|u_{\varepsilon}|^{2}}+\int_{\Omega}{|\nabla
u_{\varepsilon}|^{2}}=\int_{\Omega}n_{\varepsilon}u_{\varepsilon}\cdot\nabla\phi~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),$ (2.25)

which together with the Hölder inequality, (2.6), the continuity of the
embedding $W^{1,2}(\Omega)\hookrightarrow L^{6}(\Omega)$, the
Gagliardo–Nirenberg inequality and(2.21) implies that there exists a positive
constants $C_{1},C_{2}$ and $C_{3}$ independent of $\varepsilon$ such that

$\begin{array}[]{rl}\displaystyle\int_{\Omega}n_{\varepsilon}u_{\varepsilon}\cdot\nabla\phi\leq&\displaystyle{\|\nabla\phi\|_{L^{\infty}(\Omega)}\|n_{\varepsilon}\|_{L^{\frac{6}{5}}(\Omega)}\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega)}}\\\
\leq&\displaystyle{C_{1}\|n_{\varepsilon}\|_{L^{\frac{6}{5}}(\Omega)}\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega)}}\\\ \leq&\displaystyle{C_{2}\|\nabla
n_{\varepsilon}^{\frac{m}{2}}\|_{L^{2}(\Omega)}^{\frac{1}{3m-1}}\|n_{\varepsilon}^{\frac{m}{2}}\|_{L^{\frac{2}{m}}(\Omega)}^{\frac{2}{m}-\frac{1}{3m-1}}\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega)}}\\\ \leq&\displaystyle{C_{3}(\|\nabla
n_{\varepsilon}^{\frac{m}{2}}\|_{L^{2}(\Omega)}^{\frac{1}{3m-1}}+1)\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega)}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).}\\\ \end{array}$ (2.26)

Next, with the help of the Young inequality and $m>\frac{10}{9}$, inserting
(2.26) into (2.25) and (1.4), we derive that

$\begin{array}[]{rl}\displaystyle\frac{1}{2}\int_{\Omega}{|u_{\varepsilon}|^{2}}+\frac{1}{2}\int_{\Omega}{|\nabla
u_{\varepsilon}|^{2}}\leq&\displaystyle{C_{4}(\|\nabla
n_{\varepsilon}^{\frac{m}{2}}\|_{L^{2}(\Omega)}^{\frac{2}{3m-1}}+1)}\\\
\leq&\displaystyle{\frac{\delta_{1}}{2C_{D}}\|\nabla
n_{\varepsilon}^{\frac{m}{2}}\|_{L^{2}(\Omega)}^{\frac{2}{3m-1}}+C_{5}}\\\
\leq&\displaystyle{\frac{\delta_{1}}{2}\int_{\Omega}\frac{D_{\varepsilon}(n_{\varepsilon})|\nabla
n_{\varepsilon}|^{2}}{n_{\varepsilon}}+C_{5}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),}\\\ \end{array}$ (2.27)

and some positive constants $C_{4}$ and $C_{5}.$ ∎

######  Lemma 2.8.

Let $\frac{10}{9}<m\leq 2$. There exist $\mu_{0}$ and $C>0$ independent of
$\varepsilon$ such that for every $\delta_{i}(i=2,3,4,5)>0$

$\begin{array}[]{rl}&\displaystyle{\frac{d}{dt}\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{2}}{c_{\varepsilon}}+\mu_{0}\int_{\Omega}c_{\varepsilon}|D^{2}\ln
c_{\varepsilon}|^{2}+(\mu_{0}-\frac{\delta_{2}}{4}-\frac{\delta_{3}}{4})\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{4}}{c_{\varepsilon}^{3}}}\\\
\leq&\displaystyle{(\frac{\delta_{4}}{4}+\frac{\delta_{5}}{4})\int_{\Omega}\frac{D_{\varepsilon}(n_{\varepsilon})|\nabla
n_{\varepsilon}|^{2}}{n_{\varepsilon}}+\frac{4}{\delta_{2}}\|c_{0}\|_{L^{\infty}(\Omega)}\int_{\Omega}|\nabla
u_{\varepsilon}|^{2}+C~{}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max})}\\\
\end{array}$ (2.28)

###### Proof.

Firstly, by calculation, we derive that

$\begin{array}[]{rl}\displaystyle\frac{d}{dt}\displaystyle\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{2}}{c_{\varepsilon}}=&\displaystyle{2\int_{\Omega}\frac{\nabla
c_{\varepsilon}\cdot\nabla c_{\varepsilon
t}}{c_{\varepsilon}}-\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{2}c_{\varepsilon t}}{c_{\varepsilon}^{2}}}\\\
=&\displaystyle{-2\int_{\Omega}\frac{\Delta c_{\varepsilon}c_{\varepsilon
t}}{c_{\varepsilon}}+\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{2}c_{\varepsilon t}}{c_{\varepsilon}^{2}}}\\\
=&\displaystyle{-2\int_{\Omega}\frac{|\Delta
c_{\varepsilon}|^{2}}{c_{\varepsilon}}+2\int_{\Omega}\frac{\Delta
c_{\varepsilon}n_{\varepsilon}c_{\varepsilon}}{c_{\varepsilon}}+2\int_{\Omega}\frac{\Delta
c_{\varepsilon}}{c_{\varepsilon}}u_{\varepsilon}\cdot\nabla
c_{\varepsilon}}\\\ &+\displaystyle{\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{2}\Delta
c_{\varepsilon}}{c_{\varepsilon}^{2}}-\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{2}n_{\varepsilon}c_{\varepsilon}}{c_{\varepsilon}^{2}}-\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{2}u_{\varepsilon}\cdot\nabla
c_{\varepsilon}}{c_{\varepsilon}^{2}}~{}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).}\\\ \end{array}$ (2.29)

Due to (vi) of Lemma 2.3 and the Young inequality, there exist $\mu_{0}>0$ and
$C(\mu_{0})>0$ such that

$\begin{array}[]{rl}&\displaystyle{-2\int_{\Omega}\frac{|\Delta
c_{\varepsilon}|^{2}}{c_{\varepsilon}}+\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{2}\Delta
c_{\varepsilon}}{c_{\varepsilon}^{2}}\leq-\mu_{0}\int_{\Omega}c_{\varepsilon}|D^{2}\ln
c_{\varepsilon}|^{2}-\mu_{0}\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{4}}{c_{\varepsilon}^{3}}+C(\mu_{0})\int_{\Omega}c_{\varepsilon}}\\\
\end{array}$ (2.30)

for all $t\in(0,T_{max})$. On the other hand, for all $t\in(0,T_{max})$, with
the help of the computing, the Young inequality and Lemma 2.5 implies that for
any $\delta_{2}>0$

$\begin{array}[]{rl}\displaystyle\hfil&2\int_{\Omega}\frac{\Delta
c_{\varepsilon}}{c_{\varepsilon}}(u_{\varepsilon}\cdot\nabla
c_{\varepsilon})-\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{2}}{c_{\varepsilon}^{2}}u_{\varepsilon}\cdot\nabla
c_{\varepsilon}\\\ =&\displaystyle{2\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{2}}{c_{\varepsilon}^{2}}u_{\varepsilon}\cdot\nabla
c_{\varepsilon}-2\int_{\Omega}\frac{1}{c_{\varepsilon}}\nabla
c_{\varepsilon}\cdot(\nabla u_{\varepsilon}\nabla
c_{\varepsilon})-2\int_{\Omega}\frac{1}{c_{\varepsilon}}u_{\varepsilon}\cdot
D^{2}c_{\varepsilon}\nabla
c_{\varepsilon}+2\int_{\Omega}\frac{1}{c_{\varepsilon}}u_{\varepsilon}\cdot
D^{2}c_{\varepsilon}\nabla c_{\varepsilon}}\\\
=&\displaystyle{-2\int_{\Omega}\frac{1}{c_{\varepsilon}}\nabla
c_{\varepsilon}\cdot(\nabla u_{\varepsilon}\nabla c_{\varepsilon})}\\\
\leq&\displaystyle{\frac{\delta_{2}}{4}\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{4}}{c_{\varepsilon}^{3}}+\frac{4}{\delta_{2}}\int_{\Omega}c_{\varepsilon}|\nabla
u_{\varepsilon}|^{2}}\\\
\leq&\displaystyle{\frac{\delta_{2}}{4}\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{4}}{c_{\varepsilon}^{3}}+C_{1}\int_{\Omega}|\nabla
u_{\varepsilon}|^{2}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max}),}\\\
\end{array}$ (2.31)

where $C_{1}:=\frac{4}{\delta_{2}}\|c_{0}\|_{L^{\infty}(\Omega)}$. In view of
integration by parts, the Young inequality, (1.4) and (2.22), we also derive
that

$\begin{array}[]{rl}2\displaystyle\int_{\Omega}\frac{\Delta
c_{\varepsilon}n_{\varepsilon}c_{\varepsilon}}{c_{\varepsilon}}=&\displaystyle{-2\int_{\Omega}\nabla
n_{\varepsilon}\cdot\nabla c_{\varepsilon}}\\\
\leq&\displaystyle{\frac{\delta_{3}}{4}\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{4}}{c_{\varepsilon}^{3}}+2^{\frac{4}{3}}\delta^{-\frac{1}{3}}_{3}\int_{\Omega}c_{\varepsilon}|\nabla
n_{\varepsilon}|^{\frac{4}{3}}}\\\
\leq&\displaystyle{\frac{\delta_{3}}{4}\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{4}}{c_{\varepsilon}^{3}}+\frac{\delta_{4}}{4C_{D}}\int_{\Omega}n_{\varepsilon}^{m-2}|\nabla
n_{\varepsilon}|^{2}+C_{2}\int_{\Omega}n_{\varepsilon}^{4-2m}c_{\varepsilon}^{3}}\\\
\leq&\displaystyle{\frac{\delta_{3}}{4}\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{4}}{c_{\varepsilon}^{3}}+\frac{\delta_{4}}{4}\int_{\Omega}\frac{D_{\varepsilon}(n_{\varepsilon})|\nabla
n_{\varepsilon}|^{2}}{n_{\varepsilon}}+C_{3}\int_{\Omega}n_{\varepsilon}^{4-2m}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),}\\\ \end{array}$ (2.32)

where
$\delta_{3},\delta_{4},C_{2}:=C_{2}(\delta_{3},\delta_{4}),C_{3}:=C_{3}(\delta_{3},\delta_{4},\|c_{0}\|_{L^{\infty}(\Omega)})$
are positive constants.

Case $\frac{10}{9}<m<\frac{3}{2}$: Due to the Gagliardo–Nirenberg inequality
and (2.21), we conclude that

$\begin{array}[]{rl}C_{3}\displaystyle\int_{\Omega}n_{\varepsilon}^{4-2m}=&\displaystyle{C_{3}\|n_{\varepsilon}^{\frac{m}{2}}\|_{L^{\frac{2(4-2m)}{m}}(\Omega)}^{\frac{2(4-2m)}{m}}}\\\
\leq&\displaystyle{C_{4}\|\nabla
n_{\varepsilon}^{\frac{m}{2}}\|_{L^{2}(\Omega)}^{\frac{2(4-2m)\mu_{1}}{m}}\|n_{\varepsilon}^{\frac{m}{2}}\|_{L^{\frac{2}{m}}(\Omega)}^{\frac{2(4-2m)(1-\mu_{1})}{m}}+\|n_{\varepsilon}^{\frac{m}{2}}\|_{L^{\frac{2}{m}}(\Omega)}^{\frac{2(4-2m)}{m}}}\\\
\leq&\displaystyle{C_{5}(\|\nabla
n_{\varepsilon}^{\frac{m}{2}}\|_{L^{2}(\Omega)}^{\frac{2(4-2m)\mu_{1}}{m}}+1)}\\\
=&\displaystyle{C_{5}(\|\nabla
n_{\varepsilon}^{\frac{m}{2}}\|_{L^{2}(\Omega)}^{\frac{6(3-2m)}{3m-1}}+1)~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),}\\\ \end{array}$ (2.33)

where $C_{4}$ and $C_{5}$ are positive constants,

$\mu_{1}=\frac{\frac{3m}{2}-\frac{3m}{2(4-2m)}}{\frac{3m-1}{2}}\in(0,1).$

Now, in view of $m>\frac{10}{9},$ with the help of the Young inequality and
(2.33), for any $\delta_{5}>0$, we have

$\begin{array}[]{rl}&\displaystyle{C_{3}\int_{\Omega}n_{\varepsilon}^{4-2m}\leq\frac{\delta_{5}}{4C_{D}}\|\nabla
n_{\varepsilon}^{\frac{m}{2}}\|_{L^{2}(\Omega)}^{2}+C_{6}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max})}\\\ \end{array}$ (2.34)

with some $C_{6}>0$.

Case $\frac{3}{2}\leq m\leq 2$: With the help of the Young inequality and
(2.21), we derive that

$\begin{array}[]{rl}C_{3}\displaystyle\int_{\Omega}n_{\varepsilon}^{4-2m}\leq&\displaystyle{\int_{\Omega}n_{\varepsilon}+C_{7}}\\\
\leq&\displaystyle{C_{8}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max})}\\\
\end{array}$ (2.35)

where $C_{7}$ and $C_{8}$ are positive constants independent of $\varepsilon.$
Finally, collecting (2.30)–(2.35) and (2.29), we can get the results. ∎

######  Lemma 2.9.

Let $\frac{10}{9}<m\leq 2$ and $\delta>0$. There is $C>0$ such that for any
$\delta_{6}$ and $\delta_{7}$

$\begin{array}[]{rl}\displaystyle\frac{d}{dt}\int_{\Omega}n_{\varepsilon}\ln
n_{\varepsilon}+(1-\frac{\delta_{7}}{4})\int_{\Omega}\frac{D_{\varepsilon}(n_{\varepsilon})|\nabla
n_{\varepsilon}|^{2}}{n_{\varepsilon}}\leq\frac{\delta_{6}}{4}\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{4}}{c_{\varepsilon}^{3}}+C~{}\mbox{for
all}~{}t\in(0,T_{max}).\end{array}$ (2.36)

###### Proof.

Using these estimates and the first equation of (2.13), from integration by
parts we obtain

$\begin{array}[]{rl}\frac{d}{dt}\displaystyle\int_{\Omega}n_{\varepsilon}\ln
n_{\varepsilon}=&\displaystyle{\int_{\Omega}n_{\varepsilon t}\ln
n_{\varepsilon}+\int_{\Omega}n_{\varepsilon t}}\\\
=&\displaystyle{\int_{\Omega}\nabla\cdot(D_{\varepsilon}(n_{\varepsilon})\nabla
n_{\varepsilon})\ln n_{\varepsilon}-\int_{\Omega}\ln
n_{\varepsilon}\nabla\cdot(n_{\varepsilon}S_{\varepsilon}(x,n_{\varepsilon},c_{\varepsilon})\nabla
c_{\varepsilon})-\int_{\Omega}\ln n_{\varepsilon}u_{\varepsilon}\cdot\nabla
n_{\varepsilon}}\\\
\leq&\displaystyle{-\int_{\Omega}\frac{D_{\varepsilon}(n_{\varepsilon})|\nabla
n_{\varepsilon}|^{2}}{n_{\varepsilon}}+\int_{\Omega}S_{0}(c_{\varepsilon})|\nabla
n_{\varepsilon}||\nabla c_{\varepsilon}|}\\\ \end{array}$ (2.37)

for all $t\in(0,T_{max})$. Now, in view of (2.22), employing the same argument
of (2.32)–(2.35), for any $\delta_{6}>0$ and $\delta_{7}>0$, we conclude that

$\begin{array}[]{rl}&\displaystyle{\int_{\Omega}S_{0}(c_{\varepsilon})|\nabla
n_{\varepsilon}||\nabla c_{\varepsilon}|}\\\
\leq&\displaystyle{S_{0}(\|c_{0}\|_{L^{\infty}(\Omega)})\int_{\Omega}|\nabla
n_{\varepsilon}||\nabla c_{\varepsilon}|}\\\
\leq&\displaystyle{\frac{\delta_{6}}{4}\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{4}}{c_{\varepsilon}^{3}}+\frac{\delta_{7}}{4}\int_{\Omega}\frac{D_{\varepsilon}(n_{\varepsilon})|\nabla
n_{\varepsilon}|^{2}}{n_{\varepsilon}}+C_{9}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).}\\\ \end{array}$ (2.38)

with $C_{9}>0$ independent of $\varepsilon.$ Now, in conjunction with (2.37)
and (2.38), we get the results. This completes the proof of Lemma 2.9. ∎

Properly combining Lemmata 2.7–2.9, we arrive at the following Lemma, which
plays a key rule in obtaining the existence of solutions to (2.13).

######  Lemma 2.10.

Let $\frac{10}{9}<m\leq 2$ and $S$ satisfy (1.2)–(1.3). Suppose that (1.4) and
(2.6)–(2.7) holds. Then there exists $C>0$ such that the solution of (2.13)
satisfies

$\begin{array}[]{rl}&\displaystyle{\int_{\Omega}n_{\varepsilon}(\cdot,t)\ln
n_{\varepsilon}(\cdot,t)+\int_{\Omega}|\nabla\sqrt{c_{\varepsilon}}(\cdot,t)|^{2}+\int_{\Omega}|u_{\varepsilon}(\cdot,t)|^{2}\leq
C}\\\ \end{array}$ (2.39)

for all $t\in(0,T_{max})$. Moreover, for each $T\in(0,T_{max})$, one can find
a constant $C>0$ such that

$\begin{array}[]{rl}&\displaystyle{\int_{0}^{T}\int_{\Omega}n_{\varepsilon}^{m-2}|\nabla{n_{\varepsilon}}|^{2}\leq
C}\\\ \end{array}$ (2.40)

and

$\begin{array}[]{rl}&\displaystyle{\int_{0}^{T}\int_{\Omega}|\nabla{c_{\varepsilon}}|^{4}\leq
C}\\\ \end{array}$ (2.41)

as well as

$\begin{array}[]{rl}&\displaystyle{\int_{0}^{T}\int_{\Omega}c_{\varepsilon}|D^{2}\ln
c_{\varepsilon}|^{2}\leq C.}\\\ \end{array}$ (2.42)

###### Proof.

Take an evident linear combination of the inequalities provided by Lemmata
2.7–2.9, we conclude that there exists a positive constant $C_{1}>0$ such that

$\begin{array}[]{rl}&\displaystyle{\frac{d}{dt}\left(\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{2}}{c_{\varepsilon}}+L\int_{\Omega}n_{\varepsilon}\ln
n_{\varepsilon}+K|u_{\varepsilon}|^{2}\right)+(K-\frac{4}{\delta_{2}}\|c_{0}\|_{L^{\infty}(\Omega)})\int_{\Omega}|\nabla
u_{\varepsilon}|^{2}+\mu_{0}\int_{\Omega}c_{\varepsilon}|D^{2}\ln
c_{\varepsilon}|^{2}}\\\
&\displaystyle{+[(\mu_{0}-\frac{\delta_{2}}{4}-\frac{\delta_{3}}{4})-L\frac{\delta_{6}}{4}]\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{4}}{c_{\varepsilon}^{3}}+[L(1-\frac{\delta_{7}}{4})-\frac{\delta_{4}}{4}-\frac{\delta_{5}}{4}-K\delta_{1}]\int_{\Omega}\frac{D_{\varepsilon}(n_{\varepsilon})|\nabla
n_{\varepsilon}|^{2}}{n_{\varepsilon}}}\\\
\leq&\displaystyle{C_{1}~{}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max,\varepsilon}),}\\\ \end{array}$ (2.43)

where $K,L$ are positive constants. Now, choosing $\delta_{7}=1,$
$\delta_{6}=\frac{\mu_{0}}{L},\delta_{3}=\mu_{0},\delta_{4}=\delta_{5}=L,\delta_{1}=\frac{L}{8K}$
and $\delta_{2}=\frac{8}{K}\|c_{0}\|_{L^{\infty}(\Omega)}$ and $K$ large
enough such that $\frac{8}{K}\|c_{0}\|_{L^{\infty}(\Omega)}<\mu_{0}$ in
(2.43), we derive that

$\begin{array}[]{rl}&\displaystyle{\frac{d}{dt}\left(\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{2}}{c_{\varepsilon}}+L\int_{\Omega}n_{\varepsilon}\ln
n_{\varepsilon}+K|u_{\varepsilon}|^{2}\right)+\frac{K}{2}\int_{\Omega}|\nabla
u_{\varepsilon}|^{2}+\mu_{0}\int_{\Omega}c_{\varepsilon}|D^{2}\ln
c_{\varepsilon}|^{2}}\\\
&\displaystyle{+\frac{\mu_{0}}{4}\int_{\Omega}\frac{|\nabla
c_{\varepsilon}|^{4}}{c_{\varepsilon}^{3}}+\frac{L}{8}\int_{\Omega}\frac{D_{\varepsilon}(n_{\varepsilon})|\nabla
n_{\varepsilon}|^{2}}{n_{\varepsilon}}}\\\
\leq&\displaystyle{C_{2}~{}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max,\varepsilon}).}\\\ \end{array}$ (2.44)

and some positive constant $C_{2}.$ Hence, by some basic calculation, we can
conclude (2.39)–(2.42). ∎

######  Remark 2.2.

From Lemma 2.10, $m>\frac{10}{9}$ yields the (2.39), which is the natural
energy functional of $3D$ chemotaxis-stokes system with nonlinear diffusion
and rotation. Furthermore, Lemma 2.10 expands the contents of [22] (see also
the introduction of [2], Wang et. al [18, 19]). Moreover, rely on Lemma 2.10,
we also conclude that the large diffusion exponent $m$ ($>\frac{10}{9}$)
benefits the existence of solutions to (2.13).

Employing almost exactly the same arguments as in the proof of Lemma 3.5 and
Lemma 3.6 in [15] (the minor necessary changes are left as an easy exercise to
the reader), we conclude the following Lemmata:

######  Lemma 2.11.

Let $p>1$. Then the solution of (2.13) from Lemma 2.2 satisfies

$\begin{array}[]{rl}&\displaystyle{\frac{1}{{p}}\frac{d}{dt}\|n_{\varepsilon}\|^{{{p}}}_{L^{{p}}(\Omega)}+\frac{C_{D}(p-1)}{2}\int_{\Omega}n_{\varepsilon}^{m+p-3}|\nabla
n_{\varepsilon}|^{2}\leq C\int_{\Omega}n_{\varepsilon}^{p+1-m}|\nabla
c_{\varepsilon}|^{2},}\\\ \end{array}$ (2.45)

where $C>0$ is a positive constant depends on $p$ and $C_{D}$.

######  Lemma 2.12.

Let $q>1.$ Then the solution of (2.13) from Lemma 2.2 satisfies

$\begin{array}[]{rl}&\displaystyle{\frac{1}{{2{q}}}\frac{d}{dt}\|\nabla
c_{\varepsilon}{}\|^{{{2{q}}}}_{L^{{2{q}}}(\Omega)}+\frac{({q}-1)}{{{q}^{2}}}\int_{\Omega}\left|\nabla|\nabla
c_{\varepsilon}{}|^{{q}}\right|^{2}+\frac{1}{2}\int_{\Omega}|\nabla
c_{\varepsilon}{}|^{2{q}-2}|D^{2}c_{\varepsilon}{}|^{2}}\\\
\leq&\displaystyle{C(\int_{\Omega}n_{\varepsilon}^{2}{}|\nabla
c_{\varepsilon}{}|^{2{q}-2}+\int_{\Omega}|Du_{\varepsilon}{}||\nabla
c_{\varepsilon}{}|^{2{q}}),}\\\ \end{array}$ (2.46)

where $C>0$ is a positive constant depends on $q$ and
$\|c_{0}\|_{L^{\infty}(\Omega)}$.

In the following, we will estimate the integrals on the right-hand sides of
(2.45) and (2.46). Indeed, we first give the following Lemma which plays an
important rule in estimating the boundedness of $|Du_{\varepsilon}{}||\nabla
c_{\varepsilon}{}|^{2{q}}$.

######  Lemma 2.13.

(Lemma 3.7 of [22]) For any $\varepsilon\in(0,1),$ we have

$\begin{array}[]{rl}\displaystyle{\frac{d}{dt}\int_{\Omega}|A^{\frac{1}{2}}u_{\varepsilon}|^{2}+\int_{\Omega}|Au_{\varepsilon}|^{2}\leq\|\nabla\phi\|_{L^{\infty}(\Omega)}^{2}\int_{\Omega}n^{2}_{\varepsilon}~{}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).}\end{array}$ (2.47)

Now, in order to estimate $\int_{\Omega}n_{\varepsilon}^{2}{}|\nabla
c_{\varepsilon}{}|^{2{q}-2},$ we recall the following Lemma which comes from
Lemma 2.7 of [15] (see also [28])

######  Lemma 2.14.

(Lemma 2.7 of [15]) Assume that the initial data $(n_{0},c_{0},u_{0})$
fulfills (2.7). Let $p_{0}\in(0,9(m-1)),m>1$ and $T_{0}>0$, then one can find
a constant $C(p_{0},T)>0$ such that

$\begin{array}[]{rl}&\displaystyle{\|{n}_{\varepsilon}(\cdot,t)\|_{L^{p_{0}}(\Omega)}\leq
C(p_{0},T)~{}~{}~{}\mbox{for all}~{}~{}t\in(0,T)}\\\ \end{array}$ (2.48)

holds with

$T:=\min\\{T_{0},T_{max}\\}.$ (2.49)

######  Lemma 2.15.

Assuming that $p_{0}\in[1,9(m-1)),m>1$ and ${q}>1$. If

$\max\\{1,\frac{p_{0}}{3}+1-m,m-1+p_{0}\frac{q}{q+1}\\}<p<[2(m-1)+\frac{2p_{0}}{3}]q+m-1,$
(2.50)

then for all small $\delta>0$, we can find a constant $C:=C(p,{q},\delta)>0$
such that

$\begin{array}[]{rl}&\displaystyle{\int_{\Omega}n^{p+1-m}_{\varepsilon}|\nabla{c}_{\varepsilon}|^{2}\leq\delta\int_{\Omega}|n_{\varepsilon}^{\frac{p+m-1}{2}}|^{2}+\delta\||\nabla{c}_{\varepsilon}|^{{q}-1}D^{2}{c}_{\varepsilon}\|_{L^{2}(\Omega)}^{2}+C}\\\
\end{array}$ (2.51)

for all $t\in(0,T)$, where $T$ is given by (2.49).

###### Proof.

We apply the Hölder inequality with exponents $q+1$ and $\frac{q+1}{q}$ to
obtain

$\begin{array}[]{rl}J_{1}&:=\displaystyle{\int_{\Omega}n^{p+1-m}_{\varepsilon}|\nabla{c}_{\varepsilon}|^{2}}\\\
&\leq\displaystyle{\left(\int_{\Omega}n^{\frac{q+1}{q}(p+1-m)}_{\varepsilon}\right)^{\frac{1}{\frac{q+1}{q}}}\left(\int_{\Omega}|\nabla{c}_{\varepsilon}|^{2(q+1)}\right)^{\frac{1}{\kappa}}}\\\
&=\displaystyle{\|n^{\frac{m+p-1}{2}}_{\varepsilon}\|^{\frac{2(p+1-m)}{m+p-1}}_{L^{\frac{2\frac{q+1}{q}(p+1-m)}{m+p-1}}(\Omega)}\|\nabla{c}_{\varepsilon}\|_{L^{2(q+1)}(\Omega)}^{2}.}\\\
\end{array}$ (2.52)

Due to $p>m-1+p_{0}\frac{q}{q+1},m>1$ and $q>1$, we have

$\frac{p_{0}}{m+p-1}\leq\frac{\frac{q+1}{q}(p+1-m)}{m+p-1}\leq 3,$

which together with Lemma 2.14 and the Gagliardo–Nirenberg inequality (see
e.g. [26]) implies that there exist positive constants $C_{1}$ and $C_{2}$
such that

$\begin{array}[]{rl}&\displaystyle{\|n^{\frac{m+p-1}{2}}_{\varepsilon}\|^{\frac{2(p+1-m)}{m+p-1}}_{L^{\frac{2\frac{q+1}{q}(p+1-m)}{m+p-1}}(\Omega)}}\\\
\leq&\displaystyle{C_{1}(\|n_{\varepsilon}^{\frac{p+m-1}{2}}\|_{L^{2}(\Omega)}^{\mu_{1}}\|n^{\frac{m+p-1}{2}}_{\varepsilon}\|_{L^{\frac{2p_{0}}{m+p-1}}(\Omega)}^{1-\mu_{1}}+\|n^{\frac{m+p-1}{2}}_{\varepsilon}\|_{L^{\frac{2p_{0}}{m+p-1}}(\Omega)})^{\frac{2(p+1-m)}{m+p-1}}}\\\
\leq&\displaystyle{C_{2}(\|n_{\varepsilon}^{\frac{p+m-1}{2}}\|_{L^{2}(\Omega)}^{\frac{2(p+1-m)\mu_{1}}{m+p-1}}+1)}\\\
=&\displaystyle{C_{2}(\|n_{\varepsilon}^{\frac{p+m-1}{2}}\|_{L^{2}(\Omega)}^{\frac{3(\frac{p+1-m}{p_{0}}-\frac{q}{q+1})}{-\frac{1}{2}+\frac{3({m+p-1})}{2p_{0}}}}+1),}\\\
\end{array}$ (2.53)

where

$\mu_{1}=\frac{\frac{3({m+p-1})}{2p_{0}}-\frac{3({m+p-1})}{2\frac{q+1}{q}(p+1-m)}}{-\frac{1}{2}+\frac{3({m+p-1})}{2p_{0}}}=({m+p-1})\frac{\frac{3}{2p_{0}}-\frac{3}{2\frac{q+1}{q}(p+1-m)}}{-\frac{1}{2}+\frac{3({m+p-1})}{2p_{0}}}\in(0,1).$

On the other hand, with the help of Lemma 2.5 and Lemma 2.4, we conclude that
there exist some positive constants $C_{3}$ and $C_{4}$ such that

$\begin{array}[]{rl}\|\nabla{c}_{\varepsilon}\|_{L^{2(q+1)}(\Omega)}^{2}\leq&\displaystyle{C_{3}\||\nabla{c}_{\varepsilon}|^{q-1}D^{2}{c}_{\varepsilon}\|_{L^{2}(\Omega)}^{\frac{2}{q+1}}\|{c}_{\varepsilon}\|_{L^{\infty}(\Omega)}^{\frac{2(q-1)}{(2q-1)(q+1)}}+C_{3}\|{c}_{\varepsilon}\|_{L^{\infty}(\Omega)}^{2}}\\\
\leq&\displaystyle{C_{4}(\||\nabla{c}_{\varepsilon}|^{q-1}D^{2}{c}_{\varepsilon}\|_{L^{2}(\Omega)}^{\frac{2}{q+1}}+1).}\\\
\end{array}$ (2.54)

Inserting (2.53)–(2.54) into (2.52) and using the Young inequality and (2.50),
we derive that there exist positive constants $C_{5}$ and $C_{6}$ such that
for every $\delta>0,$

$\begin{array}[]{rl}J_{1}=&\displaystyle{C_{5}(\|n_{\varepsilon}^{\frac{p+m-1}{2}}\|_{L^{2}(\Omega)}^{\frac{3[{\frac{p+1-m}{p_{0}}-\frac{1}{\frac{q+1}{q}}}]}{-\frac{1}{2}+\frac{3(m+{p}-1)}{2p_{0}}}}+1)(\||\nabla{c}_{\varepsilon}|^{q-1}D^{2}{c}_{\varepsilon}\|_{L^{2}(\Omega)}^{\frac{2}{q+1}}+1)}\\\
\leq&\displaystyle{\delta\int_{\Omega}|n_{\varepsilon}^{\frac{p+m-1}{2}}|^{2}+\delta\||\nabla{c}_{\varepsilon}|^{q-1}D^{2}{c}_{\varepsilon}\|_{L^{2}(\Omega)}^{2}+C_{6}~{}~{}\mbox{for
all}~{}~{}t\in(0,T).}\\\ \end{array}$ (2.55)

Here we have use the fact that

$0<{\frac{3[{\frac{p+1-m}{p_{0}}-\frac{1}{\frac{q+1}{q}}}]}{-\frac{1}{2}+\frac{3(m+{p}-1)}{2p_{0}}}}+{\frac{2}{q+1}}<2$

and

${\frac{3[{\frac{p+1-m}{p_{0}}-\frac{1}{\frac{q+1}{q}}}]}{-\frac{1}{2}\frac{3(m+{p}-1)}{2p_{0}}}}>0,{\frac{2}{q+1}}>0.$

∎

######  Lemma 2.16.

Assuming that ${q}>\max\\{1,p_{0}-1\\}$ If

$p>\max\\{1,1-m+\frac{q+1}{3},q+2-\frac{2p_{0}}{3}-m\\},$ (2.56)

then for all small $\delta>0$, we can find a constant
$C:=C(p,q,p_{0},\delta)>0$ such that

$\begin{array}[]{rl}&\displaystyle{\int_{\Omega}n^{2}_{\varepsilon}|\nabla{c}_{\varepsilon}|^{2{q}-2}+\int_{\Omega}n^{2}_{\varepsilon}\leq\delta\int_{\Omega}|n_{\varepsilon}^{\frac{p+m-1}{2}}|^{2}+\delta\||\nabla{c}_{\varepsilon}|^{q-1}D^{2}{c}_{\varepsilon}\|_{L^{2}(\Omega)}^{2}+C}\\\
\end{array}$ (2.57)

for all $t\in(0,T)$, where $T$ is given by (2.49).

###### Proof.

Firstly, in light of Hölder inequality with exponents $\frac{q+1}{{q}-1}$ and
$\frac{q+1}{2}$, we obtain

$\begin{array}[]{rl}J_{2}&:=\displaystyle{\int_{\Omega}n^{2}_{\varepsilon}|\nabla{c}_{\varepsilon}|^{2{q}-2}}\\\
&\leq\displaystyle{\left(\int_{\Omega}n^{q+1}_{\varepsilon}\right)^{\frac{2}{q+1}}\left(\int_{\Omega}|\nabla{c}_{\varepsilon}|^{2(q+1)}\right)^{\frac{1}{\frac{q+1}{q-1}}}}\\\
&=\displaystyle{\|n^{\frac{m+p-1}{2}}_{\varepsilon}\|^{\frac{4}{m+p-1}}_{L^{\frac{2(q+1)}{m+p-1}}(\Omega)}\|\nabla{c}_{\varepsilon}\|_{L^{2(q+1)}(\Omega)}^{(2{q}-2)}.}\\\
\end{array}$ (2.58)

On the other hand, in view of $p>1-m+\frac{q+1}{3}$ and Lemma 2.14 and the
Gagliardo–Nirenberg inequality we conclude that there exist positive constants
$C_{1}$ and $C_{2}$ such that

$\begin{array}[]{rl}&\displaystyle{\|n^{\frac{m+p-1}{2}}_{\varepsilon}\|^{\frac{4}{m+p-1}}_{L^{\frac{2(q+1)}{m+p-1}}(\Omega)}}\\\
\leq&\displaystyle{C_{1}(\|\nabla
n_{\varepsilon}^{\frac{p+m-1}{2}}\|_{L^{2}(\Omega)}^{\mu_{3}}\|n^{\frac{m+p-1}{2}}_{\varepsilon}\|_{L^{\frac{2p_{0}}{m+p-1}}(\Omega)}^{(1-\mu_{3})}+\|n^{\frac{m+p-1}{2}}_{\varepsilon}\|_{L^{\frac{2p_{0}}{m+p-1}}(\Omega)})^{\frac{4}{m+p-1}}}\\\
\leq&\displaystyle{C_{2}(\|\nabla
n_{\varepsilon}^{\frac{p+m-1}{2}}\|_{L^{2}(\Omega)}^{\frac{4\mu_{3}}{m+p-1}}+1).}\\\
\end{array}$ (2.59)

Here

$\mu_{3}=\frac{\frac{3{(m+p-1)}}{2p_{0}}-\frac{3{(m+p-1)}}{2(q+1)}}{-\frac{1}{2}+\frac{3{(m+p-1)}}{2p_{0}}}={(m+p-1)}\frac{\frac{3}{2p_{0}}-\frac{3}{2(q+1)}}{-\frac{1}{2}+\frac{3{(m+p-1)}}{2p_{0}}}\in(0,1).$

On the other hand, according to Lemma 2.5 and the Gagliardo–Nirenberg
inequality, we can find some positive constants $C_{3}$ and $C_{4}$ such that

$\begin{array}[]{rl}\|\nabla{c}_{\varepsilon}\|_{L^{2(q+1)}(\Omega)}^{(2{q}-2)}\leq&\displaystyle{C_{3}\||\nabla{c}_{\varepsilon}|^{q-1}D^{2}{c}_{\varepsilon}\|_{L^{2}(\Omega)}^{{\frac{2(q-1)}{q+1}}}\|{c}_{\varepsilon}\|_{L^{\infty}(\Omega)}^{\frac{2(q-1)}{q+1}}+C_{3}\|{c}_{\varepsilon}\|_{L^{\infty}(\Omega)}^{2q-2}}\\\
\leq&\displaystyle{C_{4}(\||\nabla{c}_{\varepsilon}|^{q-1}D^{2}{c}_{\varepsilon}\|_{L^{2}(\Omega)}^{{\frac{2(q-1)}{q+1}}}+1).}\\\
\end{array}$ (2.60)

Now, observing that

$0<{\frac{3(\frac{2}{p_{0}}-\frac{2}{q+1})}{-\frac{1}{2}+\frac{3{(m+p-1)}}{2p_{0}}}}+{{\frac{2(q-1)}{q+1}}}<2$

and

${\frac{3(\frac{2}{p_{0}}-\frac{2}{q+1})}{-\frac{1}{2}+\frac{3{(m+p-1)}}{2p_{0}}}}>0,{{\frac{2(q-1)}{q+1}}}>0,$

hence, inserting (2.59)–(2.60) into (2.58) and using (2.56) and the
Gagliardo–Nirenberg inequality, we derive that for any $\delta>0,$

$\begin{array}[]{rl}J_{2}\leq&\displaystyle{C_{5}(\|\nabla
n_{\varepsilon}^{\frac{p+m-1}{2}}\|_{L^{2}(\Omega)}^{\frac{3(\frac{2}{p_{0}}-\frac{2}{q+1})}{-\frac{1}{2}+\frac{3{(m+p-1)}}{2p_{0}}}}+1)(\||\nabla{c}_{\varepsilon}|^{q-1}D^{2}{c}_{\varepsilon}\|_{L^{2}(\Omega)}^{{\frac{2(q-1)}{q+1}}}+1)}\\\
\leq&\displaystyle{\delta\int_{\Omega}|\nabla
n_{\varepsilon}^{\frac{p+m-1}{2}}|^{2}+\delta\||\nabla{c}_{\varepsilon}|^{q-1}D^{2}{c}_{\varepsilon}\|_{L^{2}(\Omega)}^{2}+C_{6}~{}~{}\mbox{for
all}~{}~{}t\in(0,T).}\\\ \end{array}$ (2.61)

with some positive constants $C_{5}$ and $C_{6}.$

Since the integral $\int_{\Omega}n^{2}_{\varepsilon}$ can be estimated
similarly upon a straightforward simplification of the above argument, this
establishes (2.57). ∎

######  Lemma 2.17.

For any $1\leq l<\frac{3}{2}$, if

$1<q<\frac{2l+3}{3},$

then for all small $\delta>0$, the solution of (2.13) from Lemma 2.2 satisfies

$\begin{array}[]{rl}&\displaystyle{\int_{\Omega}|Du{}_{\varepsilon}||\nabla
c_{\varepsilon}{}|^{2{q}}\leq\delta\int_{\Omega}|Au_{\varepsilon}|^{2}+\delta\|\nabla|\nabla
c_{\varepsilon}|^{q}\|_{L^{2}(\Omega)}^{2}+C~{}~{}\mbox{for
all}~{}~{}t\in(0,T),}\\\ \end{array}$ (2.62)

where a positive constant $C$ depends on $p$, ${q}$ and $\delta$.

###### Proof.

Firstly, applying the Hölder inequality leads

$\begin{array}[]{rl}J_{3}&:=\displaystyle{\int_{\Omega}|Du{}_{\varepsilon}||\nabla
c_{\varepsilon}|^{2{q}}}\\\
&\leq\displaystyle{\|Du_{\varepsilon}\|_{L^{q+1}(\Omega)}\|\nabla
c_{\varepsilon}\|_{L^{2(q+1)}(\Omega)}^{2{q}}.}\\\ \end{array}$ (2.63)

Due to Lemma 2.7, there exists a positive constant $C_{1}$ such that

$\|Du_{\varepsilon}\|_{L^{l}(\Omega)}\leq C_{1},$ (2.64)

where $l<\frac{3}{2}$ is the same as Lemma 2.7. Hence, using the
Gagliardo–Nirenberg inequality and (2.64), since $q+1\leq\frac{2l+3}{3}+1<6$
we can find $C_{1}$, $C_{2}$ and $C_{3}$ such that

$\begin{array}[]{rl}&\displaystyle{\|Du_{\varepsilon}\|_{L^{q+1}(\Omega)}}\\\
\leq&\displaystyle{C_{1}\|u_{\varepsilon}\|_{W^{2,2}(\Omega)}^{\frac{6(q+1-l)}{(q+1)(6-l)}}\|u_{\varepsilon}\|_{W^{1,l}(\Omega)}^{\frac{(5-q)l}{(q+1)(6-l)}}}\\\
\leq&\displaystyle{C_{2}\|Au_{\varepsilon}\|_{L^{2}(\Omega)}^{\frac{6(q+1-l)}{(q+1)(6-l)}}\|Du_{\varepsilon}\|_{L^{l}(\Omega)}^{\frac{(5-q)l}{(q+1)(6-l)}}}\\\
\leq&\displaystyle{C_{3}\|Au_{\varepsilon}\|_{L^{2}(\Omega)}^{\frac{6(q+1-l)}{(q+1)(6-l)}}.}\\\
\end{array}$ (2.65)

Now, in light of Lemma 2.5 and the Gagliardo–Nirenberg inequality that we have

$\begin{array}[]{rl}\|\nabla{c_{\varepsilon}}\|_{L^{2(q+1)}(\Omega)}^{2{q}}\leq&\displaystyle{C_{4}(\||\nabla{c_{\varepsilon}}|^{q-1}D^{2}{c{}}\|_{L^{2}(\Omega)}^{\frac{2q[2(2q+2)-6]}{(2q-1)(2q+2)}}\|{c_{\varepsilon}}\|_{L^{\infty}(\Omega)}^{\frac{2q[6q-2(q+1)]}{(2q-1)(2q+2)}}+\|{c_{\varepsilon}}\|_{L^{\infty}(\Omega)}^{2q})}\\\
\leq&\displaystyle{C_{5}(\||\nabla{c_{\varepsilon}}|^{q-1}D^{2}{c_{\varepsilon}}\|_{L^{2}(\Omega)}^{\frac{2q}{q+1}}+1),}\\\
\end{array}$ (2.66)

where $C_{4}$ and $C_{5}$ are positive constants independent of $\varepsilon$.
Inserting (2.59)–(2.60) into (2.58) and using $q<\frac{2l+3}{3}$ and the Young
inequality, we have

$\begin{array}[]{rl}J_{3}\leq&\displaystyle{C_{6}\|Au_{\varepsilon}\|_{L^{2}(\Omega)}^{\frac{6(q+1-l)}{(q+1)(6-l)}}(\||\nabla{c_{\varepsilon}}|^{q-1}D^{2}{c_{\varepsilon}}\|_{L^{2}(\Omega)}^{\frac{2q}{q+1}}+1)}\\\
\leq&\displaystyle{\delta\int_{\Omega}|Au_{\varepsilon}|^{2}+\delta\||\nabla{c_{\varepsilon}}|^{q-1}D^{2}{c_{\varepsilon}}\|_{L^{2}(\Omega)}^{2}+C_{7}~{}~{}\mbox{for
all}~{}~{}t\in(0,T).}\\\ \end{array}$ (2.67)

with some positive constants $C_{6}$ and $C_{7}.$ ∎

######  Lemma 2.18.

Assuming that ${q}>1$. If

$\|Du_{\varepsilon}(\cdot,t)\|_{L^{2}(\Omega)}\leq K~{}~{}\mbox{for
all}~{}~{}t\in(0,T),$ (2.68)

then for all small $\delta>0$, the solution of (2.13) from Lemma 2.2 satisfies

$\begin{array}[]{rl}&\displaystyle{\int_{\Omega}|Du{}_{\varepsilon}||\nabla
c_{\varepsilon}{}|^{2{q}}\leq\delta\|\nabla|\nabla
c_{\varepsilon}|^{q}\|_{L^{2}(\Omega)}^{2}+C~{}~{}\mbox{for
all}~{}~{}t\in(0,T),}\\\ \end{array}$ (2.69)

where a positive constant $C$ depends on ${q}$ and $\delta$.

###### Proof.

Firstly, using the Hölder inequality, we find

$\begin{array}[]{rl}J_{3}&:=\displaystyle{\int_{\Omega}|Du{}_{\varepsilon}||\nabla
c_{\varepsilon}|^{2{q}}}\\\
&\leq\displaystyle{\|Du_{\varepsilon}\|_{L^{2}(\Omega)}\|\nabla
c_{\varepsilon}\|_{L^{4q}(\Omega)}^{2{q}}}\\\ &\leq\displaystyle{K\|\nabla
c_{\varepsilon}\|_{L^{4q}(\Omega)}^{2{q}}~{}~{}\mbox{for
all}~{}~{}t\in(0,T).}\\\ \end{array}$ (2.70)

Now, it then follows from Lemma 2.5 and the Gagliardo–Nirenberg inequality
that we have

$\begin{array}[]{rl}\|\nabla{c_{\varepsilon}}\|_{L^{4q}(\Omega)}^{2{q}}\leq&\displaystyle{C_{1}(\||\nabla{c_{\varepsilon}}|^{q-1}D^{2}{c{}}\|_{L^{2}(\Omega)}^{\frac{4q[4q-3]}{(2q-1)4q}}\|{c_{\varepsilon}}\|_{L^{\infty}(\Omega)}^{\frac{2q[6q-4q]}{(2q-1)4q}}+\|{c_{\varepsilon}}\|_{L^{\infty}(\Omega)}^{2q})}\\\
\leq&\displaystyle{C_{2}(\||\nabla{c_{\varepsilon}}|^{q-1}D^{2}{c_{\varepsilon}}\|_{L^{2}(\Omega)}^{\frac{2q-3}{2q-1}}+1),}\\\
\end{array}$ (2.71)

where some positive constants $C_{1},C_{2}$. Inserting (2.71) into (2.70) and
employing the Young inequality, we can get (2.69). ∎

######  Lemma 2.19.

Assuming that $m>\frac{9}{8}$. Then for all $p>1$ and ${q}>1$,

$\|{n}_{\varepsilon}(\cdot,t)\|_{L^{p}(\Omega)}+\|\nabla{c}_{\varepsilon}(\cdot,t)\|_{L^{2{q}}(\Omega)}\leq
C~{}~{}\mbox{for all}~{}~{}t\in(0,T),$ (2.72)

where $T$ is given by (2.49).

###### Proof.

We only need to prove case $2\geq m>\frac{9}{8}$. Since $m>2$, employing
almost exactly the same arguments as in the proof of Lemma 4.1 in [27] (see
also [22]) (the minor necessary changes are left as an easy exercise to the
reader), we conclude the estimate (2.72).

Case $2\geq m>\frac{9}{8}$. We divide the proof into two steps.

Step 1. We first make sure that there exists $C_{1}>0$ such that

$\|Du_{\varepsilon}(\cdot,t)\|_{L^{2}(\Omega)}\leq C_{1}~{}~{}\mbox{for
all}~{}~{}t\in(0,T).$ (2.73)

To achieve this, in light of $m>\frac{9}{8},$ $p_{0}\geq 1$ and $q>1$, then

$\begin{array}[]{rl}[2(m-1)+\frac{2p_{0}}{3}]q+m-1>&3(m-1)+\frac{2p_{0}}{3}\\\
>&3\times\frac{1}{9}+\frac{2}{3}=1,\end{array}$ (2.74)
$\begin{array}[]{rl}[2(m-1)+\frac{2p_{0}}{3}]q+m-1>&m-1+\frac{2p_{0}}{3}q\\\
>&m-1+p_{0}\frac{q}{q+1}\end{array}$ (2.75)

and

$\begin{array}[]{rl}\left\\{[2(m-1)+\frac{2p_{0}}{3}]q+m-1\right\\}-\left\\{\frac{q+1}{3}+1-m\right\\}=&2(m-1)(q-1)+(\frac{2p_{0}}{3}-\frac{1}{3})q\\\
>&\frac{1}{3}q>0.\end{array}$ (2.76)

Now, since $p_{0}\in[1,9(m-1))$ is arbitrary, with the help of
$m>\frac{9}{8}$, we can finally pick $p_{0}<9(m-1)$ sufficiently close to
$9(m-1)$ such that

$2(m-1)+\frac{2p_{0}}{3}-1>0,$

which implies that for all $q>1$ and $p_{0}<9(m-1)$ sufficiently close
$9(m-1)$ to

$\begin{array}[]{rl}q+2-\frac{2p_{0}}{3}-m<[2(m-1)+\frac{2p_{0}}{3}]q+m-1,\end{array}$
(2.77)

which along with (2.74)–(2.76) yields that

$\begin{array}[]{rl}&\max\\{1,m-1+p_{0}\frac{q}{q+1},\frac{q+1}{3}+1-m,q+2-\frac{2p_{0}}{3}-m\\}\\\
<&[2(m-1)+\frac{2p_{0}}{3}]q+m-1~{}~{}\mbox{for all}~{}~{}q>1.\end{array}$
(2.78)

Now, in view of $l<\frac{3}{2}$ is arbitrary, we can finally pick $q_{0}<2$
sufficiently close to $2$ such that $q_{0}<\frac{2l+3}{3}$. Now, we choose
$q=q_{0}$ in (2.78),

$p\in(\max\\{1,m-1+p_{0}\frac{q_{0}}{q_{0}+1},\frac{q_{0}+1}{3}+1-m,q_{0}+2-\frac{2p_{0}}{3}-m\\},[2(m-1)+\frac{2p_{0}}{3}]q_{0}+m-1)$

and $\delta$ small enough in Lemma 2.13–Lemma 2.17, then by Lemma 2.11, we
conclude that

$\begin{array}[]{rl}&\displaystyle{\frac{d}{dt}\left(\int_{\Omega}n^{p}_{\varepsilon}+\int_{\Omega}|\nabla{c}_{\varepsilon}|^{2{q_{0}}}+\int_{\Omega}|A^{\frac{1}{2}}u_{\varepsilon}|^{2}\right)+\frac{(p-1)}{4C_{D}}\int_{\Omega}n_{\varepsilon}^{m+p-3}|\nabla
n_{\varepsilon}|^{2}}\\\
&+\displaystyle{\frac{({q_{0}}-1)}{{{q_{0}}^{2}}}\int_{\Omega}\left|\nabla|\nabla{c}_{\varepsilon}|^{{q_{0}}}\right|^{2}+\frac{1}{4}\int_{\Omega}|\nabla{c}_{\varepsilon}|^{2{q_{0}}-2}|D^{2}c_{\varepsilon}|^{2}+\int_{\Omega}|Au_{\varepsilon}|^{2}}\\\
\leq&\displaystyle{C_{2}}\\\ \end{array}$ (2.79)

with some positive constant $C_{2}.$ Assuming that
$y:=\int_{\Omega}n^{p}_{\varepsilon}+\int_{\Omega}|\nabla{c}_{\varepsilon}|^{2{q_{0}}}+\int_{\Omega}|A^{\frac{1}{2}}u_{\varepsilon}|^{2}$,
then (2.79) implies that there exist positive constants $C_{3}$ and $C_{4}$
such that

$\frac{d}{dt}y(t)+C_{3}y^{h}(t)\leq C_{4},$

where $h$ is a positive constant. Then an ODE comparison argument yields that
there exists a positive constant $C_{5}$ independent of $\varepsilon$ such
that

$\|n_{\varepsilon}(\cdot,t)\|_{L^{p}(\Omega)}+\|\nabla{c}_{\varepsilon}(\cdot,t)\|_{L^{2{q_{0}}}(\Omega)}+\|A^{\frac{1}{2}}u_{\varepsilon}(\cdot,t)\|_{L^{2}(\Omega)}\leq
C_{5}~{}~{}\mbox{for all}~{}~{}t\in(0,T),$ (2.80)

which together with $q_{0}<2$, $p<[2(m-1)+\frac{2p_{0}}{3}]q_{0}+m-1$,
$p_{0}<9(m-1)$ sufficiently close to $9(m-1)$ and $2\geq m>\frac{9}{8}$ imply
that

$\|n_{\varepsilon}(\cdot,t)\|_{L^{\frac{7}{4}}(\Omega)}\leq
C_{6}~{}~{}\mbox{for all}~{}~{}t\in(0,T).$ (2.81)

with some positive constant $C_{6}$. Therefore, by Lemma 2.1, we can get
(2.73).

Step 2. We proceed to prove the statement of the lemma. To this end, by
(2.73), we can get (2.68), hence, (2.69) holds. Now, inserting (2.51), (2.57),
(2.62) and (2.69) into (2.45) and using (2.78) yields to

$\begin{array}[]{rl}&\displaystyle{\frac{d}{dt}\left(\int_{\Omega}n^{p}_{\varepsilon}+\int_{\Omega}|\nabla{c}_{\varepsilon}|^{2{q}}\right)+\frac{(p-1)}{4C_{D}}\int_{\Omega}n_{\varepsilon}^{m+p-3}|\nabla
n_{\varepsilon}|^{2}}\\\
&+\displaystyle{\frac{({q}-1)}{{{q}^{2}}}\int_{\Omega}\left|\nabla|\nabla{c}_{\varepsilon}|^{{q}}\right|^{2}+\frac{1}{4}\int_{\Omega}|\nabla{c}_{\varepsilon}|^{2{q}-2}|D^{2}c_{\varepsilon}|^{2}}\\\
\leq&\displaystyle{C_{7},}\\\ \end{array}$ (2.82)

where $C_{7}$ is a positive constant independent of $\varepsilon$. Here we
have picked the $\delta$ small enough. Employing the same arguments as in the
proof of step 1, we conclude (2.72). The proof of Lemma 2.19 is complete. ∎

##  3 The proof of main results

In preparation of an Aubin-Lions type compactness argument, we intend to
supplement Lemma 2.19 with bounds on time-derivatives. With these estimates,
we can construct weak solutions by means of a standard extraction procedure,
therefore, we only give general idea, one can see [27, 22] for more details.

The proof of Theorem 2.1 Firstly, choosing $p=2$ in (2.72), we derive that

$\|n_{\varepsilon}(\cdot,t)\|_{L^{2}(\Omega)}\leq C_{1}~{}~{}\mbox{for
all}~{}~{}t\in(0,T)$ (3.1)

with some positive constant $C_{1}.$ Now, in view of (3.1), employing the
variation-of-constants formula for $u_{\varepsilon}$ and by the properties of
the Stokes semigroup, we conclude that there exists a positive constant
$C_{2}$ such that

$\|A^{\gamma}u_{\varepsilon}(\cdot,t)\|_{L^{2}(\Omega)}\leq
C_{2}~{}~{}\mbox{for all}~{}~{}t\in(0,T),$ (3.2)

where $\gamma\in(\frac{3}{4},1)$ and $T$ are the same as (2.7) and (2.49),
respectively. Since, $D(A^{\gamma})$ is continuously embedded into
$L^{\infty}(\Omega)$, hence, in view of (3.2), we may find that

$\begin{array}[]{rl}\|u_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}\leq
C_{3}~{}~{}\mbox{for all}~{}~{}t\in(0,T).\\\ \end{array}$ (3.3)

for some positive constant $C_{3}.$ Next, choosing $q=2$ in (2.72), we get
that

$\|\nabla c_{\varepsilon}(\cdot,t)\|_{L^{4}(\Omega)}\leq C_{4}~{}~{}\mbox{for
all}~{}~{}t\in(0,T),$ (3.4)

where $C_{4}$ is positive constant independent of $\varepsilon.$ Hence, in
light of (3.3)–(3.4) and the $L^{p}$-$L^{q}$ estimates associated heat
semigroup, we derive that there exists a positive constant $C_{5}$ such that

$\|c_{\varepsilon}(\cdot,t)\|_{W^{1,\infty}(\Omega)}\leq C_{5}~{}~{}\mbox{for
all}~{}~{}t\in(0,T).$ (3.5)

Now, due to (3.3) and (3.5), we may use the standard Moser-type iteration to
conclude that there exists a positive constant $C_{6}$ such that

$\|n_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}\leq C_{6}~{}~{}\mbox{for
all}~{}~{}t\in(0,T).$ (3.6)

Next, with the help of (3.3), (3.5) and (3.6), for all $\varepsilon\in(0,1),$
we can fix a positive constants $C_{7}$ such that

$n_{\varepsilon}\leq C_{7},|\nabla c_{\varepsilon}|\leq
C_{7}~{}~{}\mbox{and}~{}~{}|u_{\varepsilon}|\leq
C_{7}~{}~{}\mbox{in}~{}~{}\Omega\times(0,T).$ (3.7)

Now, employing the same argument of Lemma 5.1 of [27] (see also Lemmata 3.22
and 3.23 of [22]), we may derive that there exists a positive constant $C_{8}$
such that

$\|\partial_{t}n_{\varepsilon}(\cdot,t)\|_{(W^{2,2}_{0}(\Omega))^{*}}\leq
C_{8}~{}~{}\mbox{for all}~{}~{}t>0$ (3.8)

as well as

$\int_{0}^{T}\|\partial_{t}n_{\varepsilon}^{\varsigma}(\cdot,t)\|_{(W^{3,2}_{0}(\Omega))^{*}}dt\leq
C_{8}(T+1)$ (3.9)

and

$\int_{0}^{T}\int_{\Omega}n_{\varepsilon}^{m+p-3}|\nabla
n_{\varepsilon}|^{2}\leq C_{8}T,$ (3.10)

where $p:=2\zeta-m+1$ and $\varsigma>m$ satisfying $\varsigma\geq 2(m-1)$.

Now, in conjunction with (3.2), (3.3), (3.5) and (3.6) and the Aubin–Lions
compactness lemma ([11]), we thus infer that there exists a sequence of
numbers $\varepsilon:=\varepsilon_{j}\searrow 0$ such that

$n_{\varepsilon}\rightharpoonup n~{}~{}\mbox{weakly star
in}~{}~{}L^{\infty}_{loc}(\Omega\times(0,\infty)),$ (3.11)
$n_{\varepsilon}\rightarrow
n~{}~{}\mbox{in}~{}~{}L^{\infty}_{loc}([0,\infty);(W^{3,2}_{0}(\Omega))^{*}),$
(3.12) $c_{\varepsilon}\rightarrow
c~{}~{}~{}~{}~{}\mbox{a.e.}~{}~{}\mbox{in}~{}~{}\Omega\times(0,\infty),$
(3.13) $\nabla c_{\varepsilon}\rightarrow\nabla
c~{}~{}\mbox{in}~{}~{}L^{\infty}_{loc}(\bar{\Omega}\times[0,\infty)),$ (3.14)
$u_{\varepsilon}\rightarrow
u~{}~{}\mbox{a.e.}~{}~{}\mbox{in}~{}~{}\Omega\times(0,\infty),$ (3.15)

and

$Du_{\varepsilon}\rightharpoonup Du~{}~{}\mbox{weakly
in}~{}~{}L^{\infty}_{loc}(\Omega\times[0,\infty))$ (3.16)

holds for some triple $(n,c,u)$.

Since $p=2\zeta-m+1$, then by (3.10) implies that for each $T>0,$
$(n_{\varepsilon}^{\varsigma})_{\varepsilon\in(0,1)}$ is bounded in
$L^{2}((0,T);W^{1,2}(\Omega))$. With the help of (3.9), we also show that

$(\partial_{t}n_{\varepsilon}^{\varsigma})_{\varepsilon\in(0,1)}~{}~{}\mbox{is
bounded in}~{}~{}L^{1}((0,T);(W^{3,2}_{0}(\Omega))^{*})~{}~{}\mbox{for
each}~{}~{}T>0.$

Hence, an Aubin-Lions lemma (see e.g. [11]) applies to the above inequality we
have the strong precompactness of
$(n_{\varepsilon}^{\varsigma})_{\varepsilon\in(0,1)}$ in
$L^{2}(\Omega\times(0,T))$. Therefore, we can pick a suitable subsequence such
that $n_{\varepsilon}^{\varsigma}\rightarrow z^{\varsigma}$ for some
nonnegative measurable $z:\Omega\times(0,\Omega)\rightarrow\mathbb{R}$. In
light of (3.11) and the Egorov theorem, we have $z=n$ necessarily, so that

$n_{\varepsilon}\rightarrow
n~{}~{}\mbox{a.e.}~{}~{}\mbox{in}~{}~{}\Omega\times(0,\infty)$ (3.17)

is valid. Now, in light of (3.11), (3.14), and (3.16), we derive that (2.9) is
hold.

Now, we will prove that $(n,c,u)$ is the global weak solution of (1.1). To
this end, by (3.17), (3.11), (3.13), (3.14) and (3.15), we have $n$, $c$ are
nonnegative and (2.1) and (2.2) are valid. Next, with the help of the standard
arguments, letting $\varepsilon=\varepsilon_{j}\searrow 0$ in the approximate
system (2.13) and using (3.17)–(3.11) and (3.13)–(3.16), we can get
(2.3)–(2.5).

Acknowledgement: This work is partially supported by the National Natural
Science Foundation of China (No. 11601215), the Natural Science Foundation of
Shandong Province of China (No. ZR2016AQ17) and the Doctor Start-up Funding of
Ludong University (No. LA2016006).

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[a,b]B. Betancourt Kamenetskaia

# A theoretical model of an off-axis GRB jet

N. Fraija  M. Dainotti  A. Gálvan-Gámez  R. Barniol Duran  S. Dichiara

###### Abstract

In light of the most recent observations of late afterglows produced by the
merger of compact objects or by the core-collapse of massive dying stars, we
research the evolution of the afterglow produced by an off-axis top-hat jet
and its interaction with a surrounding medium. The medium is parametrized by a
power law distribution of the form $n(r)\propto r^{-k}$is the stratification
parameter and contains the development when the surrounding density is
constant ($k=0$) or wind-like ($k=2$). We develop an analytical synchrotron
forward-shock model when the outflow is viewed off-axis, and it is decelerated
by a stratified medium. Using the X-ray data points collected by a large
campaign of orbiting satellites and ground telescopes, we have managed to
apply our model and fit the X-ray spectrum of the GRB afterglow associated to
SN 2020bvc with conventional parameters. Our model predicts that its
circumburst medium is parametrized by a power law with stratification
parameter $k=1.5$

##  1 Introduction

On February 4, 2020, SN 2020bvc was first detected by the All Sky Automated
Survey for SuperNovae (ASAS-SN) survey. It was associated to the host galaxy
UGC 09379, with a redshift of $z=0.025235$ ($D=120\ \mathrm{Mpc}$)[1]. A later
report confirmed this association and redshift and based on the blue
featureless continuum and the absolute magnitude at the discovery of -18.1
classified this event as a young core-collapse supernova [2].

Twelve days after its discovery, on February 16, the Very Large Array (VLA)
observed the position of SN 2020bvc and detected a point source with a flux
density of $66\ \mu\textrm{Jy}$ in the X-band and a luminosity of $1.3\times
10^{27}\ {\rm erg\,s^{-1}Hz^{-1}}$ [3]. Such X-ray emission can be seen as
another argument in favor of the off-axis GRB scenario, as the radiation can
be seen as the afterglow component of a GRB.

Chevalier [4] explored how an adiabatic flow interacts in a circumstellar
density profile of the form $n(r)\propto r^{-2}$ for Type II supernovae. Since
then, such a power law has become standard and it has been applied in
subsequent publications for other sorts of supernovae, including those by [5],
[6], [7] and [8], amongst others. Nevertheless, in a more recent study by
Moriya and Tominaga [9], the authors demonstrated that the density slope of
the surrounding dense wind can explain the variety in the spectrum of Type II
luminous supernovae. To this end, they put forward a wind density power law
distribution in the form $n(r)\propto r^{-k}$, where $k$ is known as the
stratification parameter. They discovered that the ratio of the diffusion
timescale in the optically thick region of the wind and the shock propagation
timescale after the shock breakout is strongly influenced by the
stratification parameter, resulting in variations in the supernova’s spectral
development.

In this proceedings, we base ourselves on external forward shocks to introduce
an analytic model for calculating the synchrotron emission from an off-axis
jet, which is decelerated in an arbitrary orientation with respect to the
observer. We analyze the behaviour of our model and compare it with the latest
observations of the afterglow of SN2020bvc.

##  2 Theoretical Model

###  2.1 Relativistic Phase: Before the Jet Break

Once the outflow launched by the merger of NSs sweeps up enough cirumburst
material, electrons are cooled down by synchrotron radiation. We consider that
the jet concentrated within an opening angle $\theta_{j}$ “top-hat jet"
producing the afterglow emission is not aligned with the observer’s line of
sight and it differs by an angle $\Delta\theta$.

Considering the adiabatic evolution of the forward shock [10, 11], the bulk
Lorentz factor evolves as

$\Gamma\propto(1+z)^{-\frac{k-3}{2}}\xi^{k-3}\zeta_{e}^{2}A_{k}^{-\frac{1}{2}}\theta_{j}^{-1}\Delta\theta^{-(k-3)}\tilde{E}^{\frac{1}{2}}t^{\frac{k-3}{2}}\,,$
(1)

where $\tilde{E}$ is the fiducial energy, $\zeta_{e}$ denotes the fraction of
electrons that were accelerated by the shock front [12] and $\xi\sim 1$ is a
parameter introduced when the Lorentz factor is approximated as a power law in
the radius [13], as is the case in this model. We also denote the redshift as
$z$ and the density parameter as $A_{k}$ from the stratification law
$n(r)=A_{k}r^{-k}$.

We assume an electron distribution described as
$dN/d\gamma_{e}\propto\gamma_{e}^{-p}$ for $\gamma_{e}\geq\gamma_{m}$, where
$p$ is the index of the electron distribution and $\gamma_{m}$ is the Lorentz
factor of the lowest-energy electrons. Using the bulk Lorentz factor (eq. 1),
the evolution of the minimum and cooling electron Lorentz factors
$\gamma_{m}\propto t^{\frac{k-3}{2}}$ and $\gamma_{c}$, respectively, and the
comoving magnetic field $B^{\prime}\propto t^{-\frac{3}{2}}$, the synchrotron
spectral breaks and the maximum flux can be written as

$\displaystyle\nu^{\rm syn}_{\rm m}$
$\displaystyle\propto(1+z)^{\frac{4-k}{2}}\xi^{k-6}\zeta_{e}^{-2}A_{k}^{-\frac{1}{2}}\epsilon_{e}^{2}\epsilon_{B}^{\frac{1}{2}}\theta_{j}^{-2}\Delta\theta^{4-k}\tilde{E}t^{\frac{k-6}{2}}$
(2) $\displaystyle\nu^{\rm syn}_{\rm c}$
$\displaystyle\propto(1+z)^{-\frac{k+4}{2}}\xi^{k+2}A_{k}^{-\frac{1}{2}}(1+Y)^{-2}\epsilon_{B}^{-\frac{3}{2}}\theta_{j}^{2}\Delta\theta^{-(k+4)}\tilde{E}^{-1}t^{\frac{k+2}{2}}$
$\displaystyle F^{\rm syn}_{\rm max}$
$\displaystyle\propto(1+z)^{\frac{5k-8}{2}}\xi^{12-5k}\zeta_{e}A_{k}^{\frac{5}{2}}\epsilon_{B}^{\frac{1}{2}}D_{z}^{-2}\theta_{j}^{2}\Delta\theta^{5k-18}\tilde{E}^{-1}t^{\frac{12-5k}{2}}\,,$

where where $Y$ is the Compton parameter, $D_{z}$ is the luminosity distance,
$\epsilon_{e}$ is the fraction of the shock’s thermal energy density that is
transmitted to the electrons and $\epsilon_{B}$ is the fraction turned into
magnetic energy density [14].

Using the synchrotron spectral breaks, the maximum flux (eqs. 2), the
synchrotron light curves in the fast- and slow-cooling regimes can be written
as:

$F^{\rm
syn}_{\nu}\propto\begin{cases}t^{\frac{17-8k}{3}}\,\nu^{\frac{1}{3}},\hskip
48.36958pt\nu<\nu^{\rm syn}_{\rm c},\cr
t^{\frac{26-9k}{4}}\,\nu^{-\frac{1}{2}}\,\hskip 45.52458pt\nu^{\rm syn}_{\rm
c}<\nu<\nu^{\rm syn}_{\rm m},\,\,\,\,\,\cr
t^{\frac{32-10k-6p+kp}{4}}\,\nu^{-\frac{p}{2}},\hskip 14.22636pt\nu^{\rm
syn}_{\rm m}<\nu\,.\cr\end{cases}$ (3)

and

$F^{\rm
syn}_{\nu}\propto\begin{cases}t^{\frac{21-8k}{3}}\,\nu^{\frac{1}{3}},\hskip
68.28644pt\nu<\nu^{\rm syn}_{\rm m},\cr
t^{\frac{(p-11)k-6(p-5)}{4}}\nu^{-\frac{p-1}{2}},\hskip 22.76228pt\nu^{\rm
syn}_{\rm m}<\nu<\nu^{\rm syn}_{\rm c},\,\,\,\,\,\cr
t^{\frac{32-10k-6p+kp}{4}}\,\nu^{-\frac{p}{2}},\hskip 31.2982pt\nu^{\rm
syn}_{\rm c}<\nu\,.\cr\end{cases}$ (4)

It is worth noting that when $k=0$, the observable quantities derived in [15]
and the light curves of the synchrotron forward-shock emission are recovered
[16, 17, e.g., see].

###  2.2 Lateral expansion: After the jet break

During the lateral expansion phase, the beaming cone of the radiation emitted
off-axis, $\Delta\theta$, broaden increasingly until this cone reaches our
field of view $\Gamma_{\rm j}\sim\Delta\theta^{-1}$ [18, 19]. Considering the
adiabatic evolution in this phase, the bulk Lorentz factor evolves as:

$\Gamma\propto(1+z)^{\frac{1}{2}}\xi^{-1}A_{k}^{\frac{1}{2(k-3)}}\tilde{E}^{-\frac{1}{2(k-3)}}t^{-\frac{1}{2}}\,.$
(5)

Using the bulk Lorentz factor (eq. 5) and the synchrotron afterglow theory
introduced in [20], the synchrotron spectral breaks and the maximum flux
evolve as:

$\displaystyle\nu^{\rm syn}_{\rm m}$
$\displaystyle\propto(1+z)\xi^{-4}\zeta_{e}^{-2}A_{k}^{\frac{1}{2(k-3)}}\epsilon_{e}^{2}\epsilon_{B}^{\frac{1}{2}}\tilde{E}^{\frac{k-4}{2(k-3)}}t^{-2}$
(6) $\displaystyle\nu^{\rm syn}_{\rm c}$
$\displaystyle\propto(1+z)^{-1}\xi^{0}A_{k}^{\frac{5}{2(k-3)}}\epsilon_{B}^{-\frac{3}{2}}(1+Y)^{-2}\tilde{E}^{\frac{4-3k}{2(k-3)}}t^{0}$
$\displaystyle F^{\rm syn}_{\rm max}$
$\displaystyle\propto(1+z)^{3}\xi^{-2}\zeta_{e}A_{k}^{\frac{1}{2(3-k)}}\epsilon_{B}^{\frac{1}{2}}D_{z}^{-2}\tilde{E}^{\frac{8-3k}{2(3-k)}}t^{-1}\,.$

Using the observed synchrotron spectrum in the slow-cooling regime with eq.
(6), the synchrotron light curves in the slow-cooling regimes can be written
as:

$F^{\rm syn}_{\nu}\propto\begin{cases}t\,\nu^{\frac{1}{3}},\hskip
51.21504pt\nu<\nu^{\rm syn}_{\rm m},\cr t^{-p}\nu^{-\frac{p-1}{2}},\hskip
28.45274pt\nu^{\rm syn}_{\rm m}<\nu<\nu^{\rm syn}_{\rm c},\,\,\,\,\,\cr
t^{-p}\,\nu^{-\frac{p}{2}},\hskip 34.14322pt\nu^{\rm syn}_{\rm
c}<\nu\,,\cr\end{cases}$ (7)

##  3 Results and discussion

In Figure 1 the light curves in the X-ray band (1 keV) are plotted. Different
stratification parameters were used; the purple curve corresponds to $k=0$,
the green one to $k=1$, the blue one to $k=1.5$ and the yellow one to $k=2$.
The rising behavior of the flux is represented by the relativistic phase,
while the decrease is due to the lateral expansion. The Figure shows that the
behavior after the jet breaks is independent of the stratification of the
medium, while an increase of $k$ in the relativistic phase leads to flatter
profiles. The parameters used for this figure are: $\tilde{E}=10^{51}\,{\rm
erg}$, $\epsilon_{\rm B}=10^{-2}$, $\epsilon_{\rm e}=10^{-1}$, $p=2.6$,
$\zeta_{e}=1$, $\xi=1$, $\Delta\theta=15^{\circ}$, $\theta_{j}=5^{\circ}$ and
$D_{z}=26.5\,{\rm Mpc}$. The density parameters were: $A_{0}=1\
\mathrm{cm}^{-3}$, $A_{1}=1.5\times 10^{19}\ \mathrm{cm}^{-2}$,
$A_{1.5}=2.7\times 10^{28}\ \mathrm{cm}^{-\frac{3}{2}}$ and $A_{2}=3\times
10^{36}\ \mathrm{cm}^{-1}$.

Figure 1: The X-ray light curves for different stratification parameters.

Figure 2 shows the X-ray observations of SN 2020bvc with the best fit
synchrotron light curve generated by the deceleration of an off-axis jet in a
medium with stratification parameter $k=1.5$. The synchrotron light curve is
plotted for the following parameter values: $\tilde{E}=5.3\times 10^{49}\,{\rm
erg}$, $\epsilon_{\rm B}=2\times 10^{-2}$ $\epsilon_{\rm e}=3.5\times
10^{-3}$, $A_{k}=8.47\times 10^{25}\ \mathrm{cm}^{-\frac{3}{2}}$, $p=2.2$,
$\Delta\theta=23^{\circ}$, $\theta_{j}=5^{\circ}$ and $D_{z}=120\,{\rm Mpc}$.

Figure 2: The X-ray data points of SN 2020bvc with the best-fit curve obtained
with the model presented in this proceedings for a stratification parameter of
$k=1.5$.

##  4 Conclusion

Using external forward shocks as our starting point, we have derived a
theoretical model to describe the afterglow emission of an off-axis
relativistic jet. Influenced by earlier work on supernova emission, we have
assumed that the jet interacts with a medium parametrized by a power law
number density distribution $\propto R^{-k}$. This general approach is
advantageous, as it allows us to not only consider a homogeneous medium
($k=0$) and a wind-like medium ($k=2$), but regions with non-standard
stratification parameters, in particular $k=1$, $1.5$ or $2.5$.

Our model has considered the jet’s evolution during the relativistic phase and
after the jet break. In both stages, we have calculated the synchrotron light
curves in the fast- and slow-cooling regimes, which agree with previous
literature on the subject for the particular cases of a homogeneous and a
wind-like medium.

We have analyzed the behaviour of the light curves for different sets of
parameters. In the case of variation of the stratification parameter, we have
noticed that during the relativistic phase, an increase of this parameter
leads to flatter profiles. As for the case of the flux after the jet break, we
have shown that its time evolution is independent of $k$. Therefore, an
advantage of this model is the freedom with which one can explain the early-
time evolution of the radiation, without spoiling the long-time results.

We have applied our model to the radiation from SN 2020bvc and have been able
to successfully fit all reported values with the parameters specified in
Section 3. A stratification parameter of $k=1.5$ was used.

## Acknowledgments

We acknowledge the support from Consejo Nacional de Ciencia y Tecnología
(CONACyT), México, grants IN106521.

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