# Confined run-and-tumble model with boundary aggregation: long time behavior
and convergence to the confined Fokker-Planck model

Jingyi Fu School of Mathematics, Institute of Natural Sciences, Shanghai Jiao
Tong University, 200240, Shanghai. Email<EMAIL_ADDRESS>Jiuyang Liang
School of Mathematics, Institute of Natural Sciences, Shanghai Jiao Tong
University, 200240, Shanghai. Email<EMAIL_ADDRESS>Benoit Perthame
Sorbonne Université, CNRS, Université de Paris, Inria, Laboratoire Jacques-
Louis Lions UMR7598, F-75005 Paris. Email : Benoit.Perthame@sorbonne-
universite.fr.  Min Tang School of Mathematics, Institute of Natural Sciences
and MOE-LSC, Shanghai Jiao Tong University, 200240, Shanghai. Email :
<EMAIL_ADDRESS>
###### Abstract

The motile micro-organisms such as E. coli, sperm, or some seaweed are usually
modelled by self-propelled particles that move with the run-and-tumble
process. Individual-based stochastic models are usually employed to model the
aggregation phenomenon at the boundary, which is an active research field that
has attracted a lot of biologists and biophysicists. Self-propelled particles
at the microscale have complex behaviors, while characteristics at the
population level are more important for practical applications but rely on
individual behaviors. Kinetic PDE models that describe the time evolution of
the probability density distribution of the motile micro-organisms are widely
used. However, how to impose the appropriate boundary conditions that take
into account the boundary aggregation phenomena is rarely studied. In this
paper, we propose the boundary conditions for a 2D confined run-and-tumble
model (CRTM) for self-propelled particle populations moving between two
parallel plates with a run-and-tumble process. The proposed model satisfies
the relative entropy inequality and thus long-time convergence. We establish
the relation between CRTM and the confined Fokker-Planck model (CFPM) studied
in [22]. We prove theoretically that when the tumble is highly forward peaked
and frequent enough, CRTM converges asymptotically to the CFPM. A numerical
comparison of the CRTM with aggregation and CFPM is given. The time evolution
of both the deterministic PDE model and individual-based stochastic
simulations are displayed, which match each other well.

##  1 Introduction

The run-and-tumble process is now standard to describe the movement of motile
organisms, in which the bacteria like E. coli move straight forward and take
reorientation after some time, then continue moving in a new direction [7].
The running time between two successive reorientations and the newly chosen
direction are two random variables, and the stochastic process describing this
behavior is also called the velocity jump process, which was first proposed in
[43].

When the self-propelled micro-organisms move in a confined environment, they
may aggregate at the boundary[8, 10, 29, 30]. This phenomenon has important
influences on several biological processes. For example, the aggregation of
sperm at the border affects mammalian reproduction[44, 46], and the
aggregation of bacteria near surfaces and their interaction with the external
flow affects the formation of biofilms [40, 25]. In 1963, Rothschild measured
the concentration of bull sperms swimming between glass plates and found that
bull sperms are non-uniformly distributed and the density peaks near the glass
wall [39]. Both sterical and hydrodynamical forces play important roles in
boundary aggregations. In [9], the authors measured the detailed density
distribution between two parallel plates and pointed out that non-tumbling
cells moving with hydrodynamical forces will reorient to the direction
parallel to the surfaces and hence be attracted by the wall. On the other
hand, the work of Guanglai Li et al.[30, 29] proposed a model of self-
propelled particles to explain the boundary aggregation by taking into account
the steric force between the swimming cells and the surface and the rotational
Brownian motion. More recently, observing by 3D holographic imaging, Bianchi
et al. found that reorientation of cells occurs when they are in contact with
the surface, and the orientations of cells always point into the surface, no
matter the boundary is no-slip or free-slip [12, 13].

One simple stochastic microscopic model that can take into account the
boundary aggregations is to assume that micro-organisms are confined in a
2-dimensional space between two horizontal boundaries $\\{y=\pm L\\}$. Cells
move with velocity $(V\cos\theta,V\sin\theta)$, where $V$ is the constant
speed and $\theta$ is the angle between its orientation and horizontal
direction. When the micro-organisms are away from the boundaries, they follow
the classical run-and-tumble movement; when moving toward the boundary, they
touch it, and then stay at the boundary until their orientations direct them
away from the boundary. For simplicity, we ignore any possible external fields
and the rotational/translational diffusion here. More precisely, for each
cell, let $Y_{t}$ be its vertical position, and $\Theta_{t}$ be the angle
between its orientation and horizontal direction. The evolution of
$(Y_{t},\Theta_{t})$ satisfies a Piecewise-Deterministic Markov Processes
[15], such that

$\begin{cases}dY_{t}=\begin{cases}V\sin\Theta_{t}dt,~{}~{}&\text{for}~{}|Y_{t}|<L,~{}~{}\text{or}~{}~{}|Y_{t}|=L~{}~{}\text{and}~{}~{}Y_{t}\Theta_{t}<0,\\\
0,~{}~{}&\text{for}~{}~{}|Y_{t}|=L~{}~{}\text{and}~{}~{}Y_{t}\Theta_{t}\geq
0,\end{cases}\\\\[12.0pt]
\Theta_{t}={\sum}\limits_{j=1}^{P_{t}}\Delta\Theta_{t}^{j},\end{cases}$ (1.1)

where $P_{t}$ is a non-homogeneous Poisson process with intensity
$\Gamma(Y_{t},\Theta_{t})>0$, and the jump size $\Delta\Theta^{j}_{t}$ are
i.i.d. random variables from the transitional distribution
$\tilde{K}(Y_{t},\Theta_{t},\Delta\Theta_{t})(|Y_{t}|<L)$ or
$\tilde{K}_{\pm}(\Theta_{t},\Delta\Theta_{t})(Y_{t}=\pm L)$ for given
$(Y_{t},\Theta_{t})$.

The computational costs of stochastic microscopic models are potentially huge
and for practical applications, characteristics at the macroscale are more
important. Stochastic simulations can only obtain the macroscale
characteristics after the whole simulation. It is desired to propose simpler
mesoscopic kinetic models for the probability density distribution of the
micro-organisms. The kinetic run-and-tumble model is developed and analyzed in
[2, 3, 33, 34, 35, 14, 21]. One interesting aspect of this model is that by
taking into account different scalings of difference processes, the kinetic
run-and-tumble model can converge to different macroscopic models at the
population level, including Patlak-Keller-Segel system [14, 23, 6] and
hyperbolic systems [21, 17]. However, most previous works of the kinetic run-
and-tumble model consider only the internal domain, and the boundary effects
are rarely considered. Boundary aggregations are not considered in the imposed
boundary conditions in some theoretical or numerical works. For example,
reflective boundary conditions are considered in [5, 47, 24], which assumes
that the cells collide with the boundary fully elastically. And no-flux
boundary conditions are used in [18, 20, 11, 24], which indicates that self-
propulsion and translational diffusion are balanced.

We propose a confined run-and-tumble model (CRTM) here that describes the time
evolution the probability density function of the stochastic process in (1.1).
Inside $|Y_{t}|<L$, the probability density function satisfies the classical
kinetic run-and-tumble model, but particles behave differently at the
boundaries and we propose appropriate new boundary conditions. Similar to
models in [19, 4, 22], the cells are classified into top boundary contacting
ones, bottom boundary contacting ones, and free-swimming ones, then one can
write down a coupled system of their probability density functions, whose
boundary conditions are deduced from the switching between these three
different phases.

A number of studies compute analytically the steady states of a 1D simplified
version of CRTM [19, 4, 1, 31, 37, 38]. The CRTM proposed here is more general
and we establish the relative entropy inequality. It is theoretically proved
that the CRTM has long-term convergence, i.e. the weak solution of the CRTM
will tend to the steady-state solution.

In [36], the authors proved the convergence from the run-and-tumble model to
the Fokker-Planck model for the radiative transport equation. Motivated by the
asymptotic analysis in [36], we prove theoretically that when the tumbling is
forward peaked and frequent enough, under proper scaling, the asymptotic limit
of the CRTM can recover the Confined Fokker-Planck model (CFPM) studied in
[22]. The CFPM has been proposed and considered by physicists in [28, 48], in
which cells move by rotational Brownian motion and accumulate at the boundary.
The boundary conditions for CFPM have quite different forms compared to the
boundary conditions for CRTM. We can establish their relationship of by
asymptotics.

The rest of this paper is organized as follows. In 2, we introduce the CRTM
for self-propelled particles moving between two parallel plates. We give the
proper boundary conditions and prove that the system is mass-conserving. In 3,
the long-time convergence is proved using relative entropy inequality. In 4,
proper scalings are introduced, and under some assumptions, one can derive the
CFPM from CRTM in the asymptotic limit. We give both stochastic and
deterministic numerical solvers in 5, the numerical results of both solvers
are compared and the analytical results of Section 4 are verified. Finally, we
conclude with some discussion in 6.

##  2 The confined run-and-tumble model (CRTM)

To simplify the problem, we consider self-propelled particles on a 2D plate,
they are confined between two parallel boundaries as in Fig. 2-1. The
direction perpendicular to the two parallel boundaries is denoted by $y$.
Assume that the magnitudes of the particle velocities are a constant $V$ while
their directions change according to the run-and-tumble process. When the
particles touch and move towards the boundaries, they do not move in the $y$
direction while their moving directions change by the run-and-tumble process.
They can only leave the boundaries when their moving directions are away from
the boundaries. Therefore, we consider the following CRTM:

$\displaystyle\left[\dfrac{\partial}{\partial
t}+V\sin\theta\dfrac{\partial}{\partial
y}+k(y,\theta)\right]n(t,y,\theta)-\\!\\!\int_{-\pi}^{\\!\pi}\\!K\left(y,\theta^{\prime},{\theta-\theta^{\prime}}\right)n(t,y,\theta^{\prime})d\theta^{\prime}=0,$
$\displaystyle\theta\in(-\pi,\pi),~{}y\in(-L,L),$ (2.1a)
$\displaystyle\left[\dfrac{\partial}{\partial
t}+k_{+}(\theta)\right]n_{+}(t,\theta)-\\!\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},{\theta-\theta^{\prime}}\right)n_{+}(t,\theta^{\prime})d\theta^{\prime}=V\sin\theta
n(t,L,\theta),$ $\displaystyle\theta\in(0,\pi),$ (2.1b)
$\displaystyle\left[\dfrac{\partial}{\partial
t}+k_{-}(\theta)\right]n_{-}(t,\theta)-\\!\\!\int_{-\pi}^{0}\\!K_{-}\left(\theta^{\prime},{\theta-\theta^{\prime}}\right)n_{-}(t,\theta^{\prime})d\theta^{\prime}=-V\sin\theta
n(t,-L,\theta),$ $\displaystyle\theta\in(-\pi,0).$ (2.1c)

Here $y$ is the position along the perpendicular direction of the two parallel
boundaries located at $y=\pm L$; $n(t,y,\theta)$ represents the probability
density distribution of the self-propelled particles moving in the direction
$\theta\in(-\pi,\pi)$ at time $t$ and position $y$; $n_{\pm}(t,\theta)$ is the
probability density distribution of the particles aggregated at $\\{y=\pm
L\\}$, at time $t$, and moving with direction $\pm\theta\in[0,\pi]$; for
particles moving with direction $\theta^{\prime}$,
$K(y,\theta^{\prime},\theta-\theta^{\prime})\geq 0$ and
$K_{\pm}(\theta^{\prime},\theta-\theta^{\prime})\geq 0$ give their jump rate
to the direction $\theta$. According to the experimental results in [12], $K$
can be different for different $y$, depending on how close the position is to
the boundary, and $K_{\pm}(\theta^{\prime},\theta-\theta^{\prime})$ can be
different from $K(\pm L,\theta^{\prime},\theta-\theta^{\prime})$ as well.
Here, $\theta^{\prime}$ and $\theta$ both take values from $[-\pi,\pi)$ in
$K(y,\theta^{\prime},\theta-\theta^{\prime})$; while for
$K_{+}(\theta^{\prime},\theta-\theta^{\prime})$
($K_{-}(\theta^{\prime},\theta-\theta^{\prime})$), $\theta^{\prime}$ and
$\theta$ take values respectively from $[0,\pi]$ ($[-\pi,0]$) and
$[-\pi,\pi]$. Moreover, we define the total tumbling rates

$k(y,\theta):=\int_{-\pi}^{\\!\pi}\\!K\left(y,\theta,\theta^{\prime}-\theta\right)d\theta^{\prime},\quad
k_{\pm}(\theta):=\int_{-\pi}^{\\!\pi}\\!K_{\pm}\left(\theta,\theta^{\prime}-\theta\right)d\theta^{\prime},$
(2.2)

which measure the frequency of particles moving in direction $\theta$ change
their direction.

Figure 2-1: Sketch map of the CRTM proposed in this paper.

#### Boundary conditions.

Boundary conditions play an important role in the coupling of the three
equations in (2.1). We consider $\theta\in[-\pi,\pi]$ and thus $n$ satisfies
the periodic boundary conditions in $\theta$

$n(t,y,-\pi)=n(t,y,\pi).$ (2.3)

When particles touch the boundary, we need to consider the transition between
$n(t,y,\theta)$ and $n_{\pm}(t,\theta)$. Self-propelled particles at the
boundary can leave the boundary only when $\theta\in[-\pi,0]$ at $y=L$, and
$\theta\in[0,\pi]$ at $y=-L$. Thus, we give the following boundary conditions:

$\displaystyle-V\sin\theta n(t,L,\theta)$
$\displaystyle=\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)n_{+}(t,\theta^{\prime})d\theta^{\prime},~{}\quad\theta\in(-\pi,0),$
(2.4) $\displaystyle V\sin\theta n(t,-L,\theta)$
$\displaystyle=\int_{-\pi}^{0}\\!K_{-}\left(\theta^{\prime},\theta-\theta^{\prime}\right)n_{-}(t,\theta^{\prime})d\theta^{\prime},~{}\quad\theta\in(0,\pi).$
(2.5)

#### Mass conservation.

Equation (2.1) with boundary conditions (2.3)–(2.5) and initial data such that

$n(0,y,\theta)=n_{0}(y,\theta)\geq 0,\qquad n_{+}(0,\theta)=n_{0+}(\theta)\geq
0,\qquad n_{-}(0,\theta)=n_{0-}(\theta)\geq 0$

form the CRTM. The following theorem gives the mass conservation.

######  Theorem 2.1.

The total mass $M(t)$ defined by

$M(t)=\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!n(t,y,\theta)d\theta{\rm
d}y+\int_{0}^{\\!\pi}\\!\\!n_{+}(t,\theta)d\theta+\int_{-\pi}^{0}\\!n_{-}(t,\theta)d\theta..$
(2.6)

is a constant in time.

###### Proof.

Integrating (2.1a) in $\theta$ from $-\pi$ to $\pi$, in $y$ from $-L$ to $L$
gives

$\displaystyle\dfrac{{\rm d}}{{\rm
d}t}\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!n(t,y,\theta)d\theta{\rm d}y$
$\displaystyle=$
$\displaystyle-\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta\frac{\partial
n}{\partial y}d\theta{\rm
d}y-\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!\left[k(y,\theta)n(t,y,\theta)-\\!\\!\int_{-\pi}^{\\!\pi}\\!K\left(y,\theta^{\prime},\theta-\theta^{\prime}\right)n(t,y,\theta^{\prime})d\theta^{\prime}\right]d\theta{\rm
d}y$ $\displaystyle=$
$\displaystyle-\\!\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta[n(t,L,\theta)-n(t,-L,\theta)]d\theta$
$\displaystyle-\\!\\!\int_{-L}^{L}\\!\left[\int_{-\pi}^{\\!\pi}\\!\int_{-\pi}^{\\!\pi}\\!\\!K(y,\theta,\theta^{\prime}-\theta)n(t,y,\theta)d\theta^{\prime}d\theta-\\!\\!\int_{-\pi}^{\\!\pi}\\!\int_{-\pi}^{\\!\pi}\\!\\!K\left(y,\theta^{\prime},\theta-\theta^{\prime}\right)n(t,y,\theta^{\prime})d\theta
d\theta^{\prime}\right]\\!{\rm d}y$ $\displaystyle=$
$\displaystyle-\\!\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta
n(t,L,\theta)d\theta+\int_{-\pi}^{\\!\pi}\\!V\sin\theta
n(t,-L,\theta)d\theta.$

Integrating (2.1b) with respect to $\theta$ from $0$ to $\pi$ and utilizing
the boundary condition in (2.4), we obtain

$\displaystyle\dfrac{{\rm d}}{{\rm
d}t}\int_{0}^{\\!\pi}\\!\\!n_{+}(t,\theta)d\theta=$
$\displaystyle\int_{0}^{\\!\pi}\\!\\!V\sin\theta
n(t,L,\theta)d\theta-\\!\\!\int_{0}^{\\!\pi}\\!\\!\left[k_{+}(\theta)n_{+}(t,\theta)-\\!\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)n_{+}(t,\theta^{\prime})d\theta^{\prime}\right]d\theta$
$\displaystyle=$ $\displaystyle\int_{0}^{\\!\pi}\\!\\!V\sin\theta
n(t,L,\theta)d\theta-\\!\\!\int_{-\pi}^{0}\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)n_{+}(t,\theta^{\prime})d\theta^{\prime}d\theta$
$\displaystyle=$ $\displaystyle\int_{-\pi}^{\\!\pi}\\!V\sin\theta
n(t,L,\theta)d\theta.$

Similarly, for $n_{-}(t,\theta)$ we have

$\dfrac{{\rm d}}{{\rm
d}t}\int_{-\pi}^{0}\\!n_{-}(t,\theta)d\theta=-\\!\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta
n(t,-L,\theta)d\theta.$

Summing up the above three equations yields ${\rm d}M(t)/{\rm d}t=0$, which
concludes the theorem. ∎

##  3 Convergence to steady state

Since the model constructed in (2.1)–(2.5) is linear, we take for granted that
the weak solution of the system exists and is non-negative. We prove that the
solutions of (2.1)–(2.5), converge to a unique stationary state $m(y,\theta)$
and $m_{\pm}(\theta)$ that satisfy

$\displaystyle V\sin\theta\dfrac{\partial m}{\partial
y}(y,\theta)+k(y,\theta)m(y,\theta)-\\!\\!\int_{-\pi}^{\\!\pi}\\!K\left(y,\theta^{\prime},\theta-\theta^{\prime}\right)m(y,\theta^{\prime})d\theta^{\prime}=0,$
$\displaystyle\theta\in(-\pi,\pi),~{}y\in(-L,L),$ (3.1a) $\displaystyle
k_{+}(\theta)m_{+}(\theta)-\\!\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}(\theta^{\prime})d\theta^{\prime}=V\sin\theta
m(L,\theta),$ $\displaystyle\theta\in(0,\pi),$ (3.1b) $\displaystyle
k_{-}(\theta)m_{-}(\theta)-\\!\\!\int_{-\pi}^{0}\\!K_{-}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{-}(\theta^{\prime})d\theta^{\prime}=-V\sin\theta
m(-L,\theta),$ $\displaystyle\theta\in(-\pi,0),$ (3.1c)

with the boundary conditions

$\displaystyle m(y,-\pi)=m(y,\pi),$ $\displaystyle y\in[-L,L],$ (3.2a)
$\displaystyle-V\sin\theta
m(L,\theta)=\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}(\theta^{\prime})d\theta^{\prime},$
$\displaystyle\theta\in(-\pi,0),$ (3.2b) $\displaystyle V\sin\theta
m(-L,\theta)=\int_{-\pi}^{0}\\!K_{-}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{-}(\theta^{\prime})d\theta^{\prime},$
$\displaystyle\theta\in(0,\pi),$ (3.2c)

and normalization condition

$\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!m(y,\theta)d\theta{\rm
d}y+\int_{0}^{\\!\pi}\\!\\!m_{+}(\theta)d\theta+\int_{-\pi}^{0}\\!m_{-}(\theta)d\theta=M(0).$
(3.3)

In the subsequent part, we first recall the relative entropy estimates,
following the general theory for positivity preserving equations [32], then we
show a priori $L^{\infty}$ bounds, and finally the long-time convergence.

###  3.1 Relative Entropy Inequality

Before establishing the relative entropy estimate, we define formally the
relative gaps $\omega$, $\omega_{\pm}$ by

$\displaystyle\omega(t,y,\theta)=\frac{n(t,y,\theta)}{m(y,\theta)},$
$\displaystyle\theta\in(-\pi,\pi),~{}y\in(-L,L),$ (3.4a)
$\displaystyle\omega_{\pm}(t,\theta)=\frac{n_{\pm}(t,\theta)}{m_{\pm}(\theta)},$
$\displaystyle\theta\in(0,\pi),$ (3.4b)

For simplicity, we hereafter use $\omega_{\pm L}$, $n_{\pm L}$, $m_{\pm L}$ to
denote $\omega(t,\pm L,\theta)$, $n(t,\pm L,\theta)$, $m(\pm L,\theta)$
respectively. It is important to note that $\omega_{\pm L}$, $n_{\pm L}$,
$m_{\pm L}$ are different from $\omega_{\pm}$, $n_{\pm}$, $m_{\pm}$. At the
same time, we use $\omega^{\prime}$, $n^{\prime}$, $m^{\prime}$ to represent
functions that depend on the variable $\theta^{\prime}$.

######  Theorem 3.1.

For any convex function $H\in C^{2}(\mathbb{R})$, the relative gaps $\omega$
and $\omega_{\pm}$ defined in (3.4) satisfy

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\left[\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!mH(\omega){\rm d}\theta{\rm
d}y+\int_{0}^{\\!\pi}\\!\\!m_{+}H\left(\omega_{+}\right){\rm
d}\theta+\int_{-\pi}^{0}\\!m_{-}H\left(\omega_{-}\right){\rm d}\theta\right]$
(3.5)
$\displaystyle\qquad\qquad=D_{H}^{(1)}(t)+D_{H,+}^{(1)}(t)+D_{H,-}^{(1)}(t)+D_{H,+}^{(2)}(t)+D_{H,-}^{(2)}(t)+D_{H,+}^{(3)}(t)+D_{H,-}^{(3)}(t)\leq
0,$

where the dissipations due to the run-and-tumble process in the interior, on
upper/lower boundaries, are respectively

$\displaystyle D_{H}^{(1)}(t)=$
$\displaystyle-\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!\int_{-\pi}^{\\!\pi}\\!K\left(y,\theta^{\prime},\theta-\theta^{\prime}\right)m^{\prime}\left[H(\omega^{\prime})-H(\omega)-H^{\prime}(\omega)(\omega^{\prime}-\omega)\right]d\theta^{\prime}d\theta{\rm
d}y,$ (3.6a) $\displaystyle D_{H,+}^{(1)}(t)=$
$\displaystyle-\\!\\!\int_{0}^{\\!\pi}\\!\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}[H(\omega_{+}^{\prime})-H(\omega_{+})-H^{\prime}\left(\omega_{+}\right)(\omega_{+}^{\prime}-\omega_{+})]d\theta^{\prime}d\theta,$
(3.6b) $\displaystyle D_{H,-}^{(1)}(t)=$
$\displaystyle-\\!\\!\int_{-\pi}^{0}\\!\int_{-\pi}^{0}\\!K_{-}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{-}^{\prime}[H(\omega_{-}^{\prime})-H(\omega_{-})-H^{\prime}\left(\omega_{-}\right)(\omega_{-}^{\prime}-\omega_{-})]d\theta^{\prime}d\theta,$
(3.6c)

the dissipations of state transition from interior to upper/lower boundary are

$\displaystyle D_{H,+}^{(2)}(t)=$
$\displaystyle-\\!\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta
m_{+L}\left[H\left(\omega_{+L}\right)-H\left(\omega_{+}\right)-H^{\prime}\left(\omega_{+}\right)\left(\omega_{+L}-\omega_{+}\right)\right]{\rm
d}\theta,$ (3.7a) $\displaystyle D_{H,-}^{(2)}(t)=-$
$\displaystyle\\!\\!\int_{-\pi}^{0}\\!V\sin\theta
m_{-L}\left[H\left(\omega_{-L}\right)-H\left(\omega_{-}\right)-H^{\prime}\left(\omega_{-}\right)\left(\omega_{-L}-\omega_{-}\right)\right]{\rm
d}\theta,$ (3.7b)

and the dissipations of state transition from the upper/lower boundary to the
interior are

$\displaystyle D_{H,+}^{(3)}(t)=$
$\displaystyle-\\!\\!\int_{-\pi}^{0}\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}[H(\omega_{+}^{\prime})-H(\omega_{+L})-H^{\prime}\left(\omega_{+L}\right)(\omega_{+}^{\prime}-\omega_{+L})]d\theta^{\prime}d\theta,$
(3.8a) $\displaystyle D_{H,-}^{(3)}(t)=$
$\displaystyle-\\!\\!\int_{0}^{\\!\pi}\\!\\!\int_{-\pi}^{0}\\!K_{-}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{-}^{\prime}[H(\omega_{-}^{\prime})-H(\omega_{-L})-H^{\prime}\left(\omega_{-L}\right)(\omega_{-}^{\prime}-\omega_{-L})]d\theta^{\prime}d\theta.$
(3.8b)

###### Proof.

We set

$E=\frac{{\rm d}}{{\rm
d}t}\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!mH(\omega){\rm d}\theta{\rm
d}y,\quad E_{+}=\frac{{\rm d}}{{\rm
d}t}\int_{0}^{\\!\pi}\\!\\!m_{+}H\left(\omega_{+}\right){\rm d}\theta,\quad
E_{-}=\frac{{\rm d}}{{\rm
d}t}\int_{-\pi}^{0}\\!m_{-}H\left(\omega_{-}\right){\rm d}\theta.$ (3.9)

We give an expression for each of those terms in the subsequent parts.

  * •

The expression for $E$. Plugging $n(t,y,\theta)=m(y,\theta)\omega(t,y,\theta)$
into (2.1a), we get

$\displaystyle m\frac{\partial\omega}{\partial t}=$
$\displaystyle-V\sin\theta\left[\frac{\partial m}{\partial
y}\omega+m\frac{\partial\omega}{\partial y}\right]-k(y,\theta)\omega
m+\int_{-\pi}^{\\!\pi}\\!K\left(y,\theta^{\prime},\theta-\theta^{\prime}\right)m^{\prime}\omega^{\prime}d\theta^{\prime}$
(3.10) $\displaystyle\xlongequal{\eqref{3.1a}}$ $\displaystyle-V\sin\theta
m\frac{\partial\omega}{\partial
y}-\\!\\!\int_{-\pi}^{\\!\pi}\\!K\left(y,\theta^{\prime},\theta-\theta^{\prime}\right)m^{\prime}(\omega-\omega^{\prime})d\theta^{\prime},$

Multiplying both sides of (3.10) by $H^{\prime}\left(\omega\right)$ gives

$\displaystyle m\frac{\partial H(\omega)}{\partial t}=$
$\displaystyle-V\sin\theta\left[mH^{\prime}(\omega)\frac{\partial\omega}{\partial
y}+\frac{\partial m}{\partial y}H(\omega)\right]+V\sin\theta\frac{\partial
m}{\partial y}H(\omega)$ (3.11)
$\displaystyle-\\!\\!\int_{-\pi}^{\\!\pi}\\!K\left(y,\theta^{\prime},\theta-\theta^{\prime}\right)m^{\prime}H^{\prime}(\omega)(\omega-\omega^{\prime})d\theta^{\prime}$
$\displaystyle\xlongequal{\eqref{3.1a}}$
$\displaystyle-V\sin\theta\frac{\partial[mH(\omega)]}{\partial
y}-k(y,\theta)mH(\omega)$
$\displaystyle+\int_{-\pi}^{\\!\pi}\\!K\left(y,\theta^{\prime},\theta-\theta^{\prime}\right)m^{\prime}[H(\omega)-H^{\prime}(\omega)(\omega-\omega^{\prime})]d\theta^{\prime}.$

Integrating both sides of (3.11) in $y$ from $-L$ to $L$, in $\theta$ from
$-\pi$ to $\pi$ yields (we use $\theta^{\prime}$ instead of $\theta$ in second
integral)

$\displaystyle E=$ $\displaystyle\frac{{\rm d}}{{\rm
d}t}\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!mH(\omega){\rm d}\theta{\rm d}y$
(3.12) $\displaystyle=$
$\displaystyle-\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta\frac{\partial[mH(\omega)]}{\partial
y}{\rm d}\theta{\rm
d}y-\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!k(y,\theta^{\prime})m^{\prime}H(\omega^{\prime}){\rm
d}\theta^{\prime}{\rm d}y$
$\displaystyle+\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!\int_{-\pi}^{\\!\pi}\\!K\left(y,\theta^{\prime},\theta-\theta^{\prime}\right)[H(\omega)-H^{\prime}(\omega)(\omega-\omega^{\prime})]m^{\prime}d\theta^{\prime}d\theta{\rm
d}y$ $\displaystyle\xlongequal{\eqref{2.2}}$
$\displaystyle-\\!\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta
m_{+L}H(\omega_{+L})d\theta+\int_{-\pi}^{\\!\pi}\\!V\sin\theta
m_{-L}H(\omega_{-L})d\theta$
$\displaystyle-\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!\int_{-\pi}^{\\!\pi}\\!K\left(y,\theta^{\prime},\theta-\theta^{\prime}\right)m^{\prime}\left[H(\omega^{\prime})-H(\omega)-H^{\prime}(\omega)(\omega^{\prime}-\omega)\right]d\theta^{\prime}d\theta{\rm
d}y$ $\displaystyle\xlongequal{\eqref{DH1}}$
$\displaystyle-\\!\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta
m_{+L}H(\omega_{+L})d\theta+\int_{-\pi}^{\\!\pi}\\!V\sin\theta
m_{-L}H(\omega_{-L})d\theta+D_{H}^{(1)}(t)$

  * •

Expression for $E_{+}$. Plugging
$n_{+}(t,\theta)=m_{+}(\theta)\omega_{+}(t,\theta)$ into (2.1b) gives

$\displaystyle m_{+}\frac{\partial\omega_{+}}{\partial t}=$ $\displaystyle-
k_{+}(\theta)m_{+}\omega_{+}+\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}\omega_{+}^{\prime}d\theta^{\prime}+V\sin\theta
m_{+L}\omega_{+L}$ (3.13) $\displaystyle\xlongequal{\eqref{3.1b}}$
$\displaystyle V\sin\theta
m_{+L}(\omega_{+L}-\omega_{+})-\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}(\omega_{+}-\omega_{+}^{\prime})d\theta^{\prime}.$

Multiplying both sides of (3.13) by $H^{\prime}(\omega_{+})$, and integrating
both sides in $\theta$ from $0$ to $\pi$, we obtain

$\displaystyle E_{+}=$ $\displaystyle\frac{{\rm d}}{{\rm
d}t}\int_{0}^{\\!\pi}\\!\\!m_{+}H\left(\omega_{+}\right){\rm d}\theta$
$\displaystyle=$ $\displaystyle\int_{0}^{\\!\pi}\\!\\!V\sin\theta
m_{+L}H^{\prime}(\omega_{+})(\omega_{+L}-\omega_{+}){\rm
d}\theta-\\!\\!\int_{0}^{\\!\pi}\\!\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}H^{\prime}(\omega_{+})(\omega_{+}-\omega_{+}^{\prime})d\theta^{\prime}{\rm
d}\theta^{\prime}$ $\displaystyle=$
$\displaystyle-\\!\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta
m_{+L}\left[H\left(\omega_{+L}\right)-H\left(\omega_{+}\right)-H^{\prime}\left(\omega_{+}\right)\left(\omega_{+L}-\omega_{+}\right)\right]{\rm
d}\theta$ $\displaystyle+\int_{0}^{\\!\pi}\\!\\!V\sin\theta
m_{+L}\left[H\left(\omega_{+L}\right)-H\left(\omega_{+}\right)\right]{\rm
d}\theta$
$\displaystyle-\\!\\!\int_{0}^{\\!\pi}\\!\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}\left[H(\omega_{+}^{\prime})-H(\omega_{+})-H^{\prime}\left(\omega_{+}\right)(\omega_{+}^{\prime}-\omega_{+})\right]d\theta^{\prime}d\theta$
$\displaystyle+\int_{0}^{\\!\pi}\\!\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}\left[H(\omega_{+}^{\prime})-H(\omega_{+})\right]d\theta^{\prime}d\theta$
$\displaystyle\xlongequal{\eqref{DH1p},\eqref{DH2p}}$ $\displaystyle
D_{H,+}^{(1)}(t)+D_{H,+}^{(2)}(t)-\\!\\!\int_{0}^{\\!\pi}\\!\\!\left[V\sin\theta
m_{+L}+\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}d\theta^{\prime}\right]H(\omega_{+})d\theta$
$\displaystyle+\int_{0}^{\\!\pi}\\!\\!V\sin\theta
m_{+L}H\left(\omega_{+L}\right){\rm
d}\theta+\int_{0}^{\\!\pi}\\!\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}H(\omega_{+}^{\prime})d\theta^{\prime}d\theta$

Using (3.1b), the expression of $E_{+}$ above can be further simplified, which
reads

$\displaystyle
E_{+}-D_{H,+}^{(1)}(t)-D_{H,+}^{(2)}(t)-\\!\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta
m_{+L}H\left(\omega_{+L}\right){\rm d}\theta$ (3.14)
$\displaystyle\xlongequal{\eqref{3.1b}}$
$\displaystyle\int_{0}^{\\!\pi}\\!\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}H(\omega_{+}^{\prime})d\theta^{\prime}d\theta-\\!\\!\int_{0}^{\\!\pi}\\!\\!k_{+}(\theta)m_{+}H(\omega_{+})d\theta$
$\displaystyle\xlongequal{\eqref{2.2}}$
$\displaystyle\int_{0}^{\\!\pi}\\!\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}H(\omega_{+}^{\prime})d\theta^{\prime}d\theta-\\!\\!\int_{0}^{\\!\pi}\\!\\!\left[\int_{-\pi}^{\\!\pi}\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)d\theta\right]m_{+}^{\prime}H(\omega_{+}^{\prime})d\theta^{\prime}$
$\displaystyle=$
$\displaystyle-\\!\\!\int_{-\pi}^{0}\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}H(\omega_{+}^{\prime})d\theta^{\prime}d\theta$

Inspired by the right hand side of (3.12), we subtract
$\int_{-\pi}^{0}\\!V\sin\theta m_{+L}H\left(\omega_{+L}\right){\rm d}\theta$
from both sides of (3.14) to obtain

$\displaystyle
E_{+}-D_{H,+}^{(1)}(t)-D_{H,+}^{(2)}(t)-\\!\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta
m_{+L}H\left(\omega_{+L}\right){\rm d}\theta$ (3.15) $\displaystyle=$
$\displaystyle-\\!\\!\int_{-\pi}^{0}\\!V\sin\theta
m_{+L}H\left(\omega_{+L}\right){\rm
d}\theta-\\!\\!\int_{-\pi}^{0}\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}H(\omega_{+}^{\prime})d\theta^{\prime}d\theta$
$\displaystyle\xlongequal{\eqref{3.2b}}$
$\displaystyle-\\!\\!\int_{-\pi}^{0}\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}\left[H(\omega_{+}^{\prime})-H(\omega_{+L})\right]d\theta^{\prime}d\theta$
$\displaystyle=$
$\displaystyle-\\!\\!\int_{-\pi}^{0}\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}[H(\omega_{+}^{\prime})-H(\omega_{+L})-H^{\prime}\left(\omega_{+L}\right)(\omega_{+}^{\prime}-\omega_{+L})]d\theta^{\prime}d\theta$
$\displaystyle-\\!\\!\int_{-\pi}^{0}\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}(\omega_{+}^{\prime}-\omega_{+L})d\theta^{\prime}H^{\prime}(\omega_{+L})d\theta$
$\displaystyle\xlongequal{\eqref{DH3p}}$ $\displaystyle
D_{H,+}^{(3)}(t)-\\!\\!\int_{-\pi}^{0}\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)n_{+}^{\prime}d\theta^{\prime}H^{\prime}(\omega_{+L})d\theta+\int_{-\pi}^{0}\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}d\theta^{\prime}H^{\prime}(\omega_{+L})\omega_{+L}d\theta.$

By boundary conditions (2.4) and (3.2b), the last two terms on the right hand
side of (3.15) cancel each other. More precisely, we have

$\displaystyle\int_{-\pi}^{0}\\!\int_{0}^{\\!\pi}$
$\displaystyle\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)n_{+}^{\prime}d\theta^{\prime}H^{\prime}(\omega_{+L})d\theta\xlongequal{\eqref{2.5}}\int_{-\pi}^{0}\\!-H^{\prime}(\omega_{+L})V\sin\theta
n_{+L}d\theta$ (3.16) $\displaystyle=$
$\displaystyle\int_{-\pi}^{0}\\!-H^{\prime}(\omega_{+L})V\sin\theta
m_{+L}\omega_{+L}d\theta\xlongequal{\eqref{3.2b}}\int_{-\pi}^{0}\\!\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)m_{+}^{\prime}d\theta^{\prime}H^{\prime}(\omega_{+L})\omega_{+L}d\theta.$

Hence $E_{+}$ can be expressed as

$E_{+}=D_{H,+}^{(1)}(t)+D_{H,+}^{(2)}(t)+D_{H,+}^{(3)}(t)+\int_{-\pi}^{\\!\pi}\\!V\sin\theta
m_{+L}H\left(\omega_{+L}\right){\rm d}\theta.$ (3.17)

  * •

Expression for $E_{-}$. Similarly, we can show that

$E_{-}=D_{H,-}^{(1)}(t)+D_{H,-}^{(2)}(t)+D_{H,-}^{(3)}(t)-\\!\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta
m_{-L}H\left(\omega_{-L}\right){\rm d}\theta.$ (3.18)

Summing up (3.12), (3.17), (3.18) yields the equality in (3.5). Then, since
the kernel functions $K$, $K_{\pm}$ and the stationary state distributions
$m$, $m_{\pm}$ are all non-negative, by the Jensen inequality, we conclude
that all dissipation terms on the right hand side of (3.5) are non-positive,
hence the theorem is concluded. ∎

###  3.2 L-infinity estimates and long time convergence

The relative entropy inequality (3.5) has an immediate consequence. When the
relative gaps are bounded initially, it stays bounded for all times and the
solution will weakly converge to its steady state solution.

######  Corollary 3.2 ($L^{\infty}$ estimates).

Assume there is a constant $\Gamma>0$ such that

$\|\omega(0,\cdot,\cdot)\|_{L^{\infty}([-L,L]\times[-\pi,\pi])}\leq\Gamma,\quad\left\|\omega_{\pm}(0,\cdot)\right\|_{L^{\infty}\left([0,\pm\pi]\right)}\leq\Gamma.$

Then for all $t\geq 0$, we have

$\|\omega(t,\cdot,\cdot)\|_{L^{\infty}([-L,L]\times[-\pi,\pi])}\leq\Gamma,\quad\left\|\omega_{\pm}(t,\cdot)\right\|_{L^{\infty}\left([0,\pm\pi]\right)}\leq\Gamma.$

###### Proof.

Choose $H(x)=\left[(x-\Gamma)_{+}\right]^{4}\in\mathcal{C}^{2}(\mathbb{R})$ in
(3.5), we get

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\left[\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!m(\omega-\Gamma)_{+}^{4}{\rm
d}\theta{\rm
d}y+\int_{0}^{\\!\pi}\\!\\!m_{+}\left(\omega_{+}-\Gamma\right)_{+}^{4}{\rm
d}\theta+\int_{-\pi}^{0}\\!m_{-}\left(\omega_{-}-\Gamma\right)_{+}^{4}{\rm
d}\theta\right]\leq 0.$

Moreover, we can obtain

$\displaystyle\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}$
$\displaystyle\\!m(\omega(T,y,\theta)-\Gamma)_{+}^{4}{\rm d}\theta{\rm
d}y+\int_{0}^{\\!\pi}\\!\\!m_{+}\left(\omega_{+}(T,\theta)-\Gamma\right)_{+}^{4}{\rm
d}\theta+\int_{-\pi}^{0}\\!m_{-}\left(\omega_{-}(T,\theta)-\Gamma\right)_{+}^{4}{\rm
d}\theta$ (3.19)
$\displaystyle\leq\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!m(\omega(0,y,\theta)-\Gamma)_{+}^{4}{\rm
d}\theta{\rm
d}y+\int_{0}^{\\!\pi}\\!\\!m_{+}\left(\omega_{+}(0,\theta)-\Gamma\right)_{+}^{4}{\rm
d}\theta+\int_{-\pi}^{0}\\!m_{-}\left(\omega_{-}(0,\theta)-\Gamma\right)_{+}^{4}{\rm
d}\theta=0.$

Since $m$ and $m_{\pm}$ are non-negative, we can conclude the result. ∎

To prove the long time convergence, we first give some definitions. For
$k\in\mathbb{N}$, we define the time shift

$\displaystyle\omega_{k}(t,y,\theta)=\omega(t+k,y,\theta)\quad\theta\in[-\pi,\pi],\
y\in[-L,L];\quad\omega_{k,\pm}(t,\theta)=\omega_{\pm}(t+k,\theta)\quad\theta\in[0,\pm\pi].$
(3.20)

From Corollary 3.2,
$\left\\{\omega_{k}\right\\},\left\\{\omega_{k,\pm}\right\\}$ are uniformly
bounded in $L^{\infty}$. By weak compactness of $L^{\infty}$ space, there
exists a subsequence $\left\\{n_{k}\right\\}_{k\in\mathbb{N}}$ and limit
function $\omega_{\infty}\in
L^{\infty}(\mathbb{R}\times[-L,L]\times[-\pi,\pi]),~{}\omega_{\infty,\pm}\in
L^{\infty}\left(\mathbb{R}\times[0,\pm\pi]\right)$, such that for all test
functions $\phi(t,y,\theta)\in L^{1}(\mathbb{R}\times[-L,L]\times[-\pi,\pi])$
and $\phi_{\pm}(t,\theta)\in L^{1}\left(\mathbb{R}\times[0,\pm\pi]\right)$ and
any $T\in\mathbb{R}$,

$\displaystyle\lim\limits_{n_{k}\rightarrow\infty}\int_{T}^{\infty}\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!\phi\omega_{n_{k}}{\rm
d}\theta{\rm d}y{\rm d}t$
$\displaystyle=\int_{T}^{\infty}\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!\phi\omega_{\infty}{\rm
d}\theta{\rm d}y{\rm d}t,$ (3.21)
$\displaystyle\lim\limits_{n_{k}\rightarrow\infty}\int_{T}^{\infty}\\!\int_{0}^{\pm\pi}\\!\phi_{\pm}\omega_{n_{k},\pm}{\rm
d}\theta{\rm d}t$
$\displaystyle=\int_{T}^{\infty}\\!\int_{0}^{\pm\pi}\\!\phi_{\pm}\omega_{\infty,\pm}{\rm
d}\theta{\rm d}t.$

We prove later that $\omega_{\infty},\omega_{\infty,\pm}$ are constant-valued
functions.

######  Theorem 3.3 (Long time convergence).

Assume initial data $\omega(0,\cdot,\cdot)$ and $\omega_{\pm}(0,\cdot)$ are
uniformly bounded in $L^{\infty}$, then

$\omega_{\infty}=1,\quad\omega_{\infty,\pm}=1,\quad\text{ a.e. }$ (3.22)

Furthermore, for any test functions $\phi(y,\theta)\in
L^{1}([-L,L]\times[-\pi,\pi])$, $\phi_{\pm}(\theta)\in L^{1}([0,\pm\pi])$, one
has

$\displaystyle\lim\limits_{t\rightarrow+\infty}$
$\displaystyle\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!\big{(}\omega(t,y,\theta)-1\big{)}\phi(y,\theta){\rm
d}\theta{\rm
d}y=\lim\limits_{t\rightarrow+\infty}\int_{0}^{\\!\pi}\\!\\!\big{(}\omega_{+}(t,\theta)-1\big{)}\phi_{+}(\theta){\rm
d}\theta=\lim\limits_{t\rightarrow+\infty}\int_{-\pi}^{0}\\!\big{(}\omega_{-}(t,\theta)-1\big{)}\phi_{-}(\theta){\rm
d}\theta=0.$ (3.23)

Since the proof of Theorem 3.3 is almost the same as the proof in Section 3 of
[22], we postpone it to the supplementary materials.

##  4 The Fokker-Planck limit

The CFPM has been studied in [22] and we explain now its connection with the
CRTM proposed here. Both models describe the movement of self-propelled
particles confined between two parallel plates, but one changes the velocity
by a tumble while the other by rotational diffusion. The model studied in [22]
writes

$\displaystyle\dfrac{\partial\tilde{n}}{\partial
t}+V\sin\theta\dfrac{\partial\tilde{n}}{\partial
y}-\dfrac{\partial^{2}}{\partial\theta^{2}}[D(y,\theta)\tilde{n}]=0,$
$\displaystyle\theta\in(-\pi,\pi),\quad y\in(-L,L),$ (4.1a)
$\displaystyle\dfrac{\partial\tilde{n}_{+}}{\partial
t}-\dfrac{\partial^{2}}{\partial\theta^{2}}[D_{+}(\theta)\tilde{n}_{+}]=V\sin\theta\tilde{n}(t,L,\theta),$
$\displaystyle\theta\in(0,\pi),$ (4.1b)
$\displaystyle\dfrac{\partial\tilde{n}_{-}}{\partial
t}-\dfrac{\partial^{2}}{\partial\theta^{2}}[D_{-}(\theta)\tilde{n}_{-}]=-V\sin\theta\tilde{n}(t,-L,\theta),$
$\displaystyle\theta\in(-\pi,0),$ (4.1c)

equipped with proper boundary conditions that take into account the switching
between particles within the domain and in contact to the boundaries

$\displaystyle\tilde{n}(t,y,-\pi)=\tilde{n}(t,y,\pi),\qquad\dfrac{\partial\tilde{n}}{\partial\theta}(t,y,-\pi)=\dfrac{\partial\tilde{n}}{\partial\theta}(t,y,\pi),\quad
y\in(-L,L),$ (4.2a)
$\displaystyle\tilde{n}(t,L,\theta)=0,\quad\theta\in(-\pi,0),\qquad\tilde{n}(t,-L,\theta)=0,\quad\theta\in(0,\pi),$
(4.2b)
$\displaystyle\tilde{n}_{+}(t,0)=\tilde{n}_{+}(t,\pi)=0,\qquad\tilde{n}_{-}(t,0)=\tilde{n}_{-}(t,-\pi)=0,$
(4.2c)
$\displaystyle\frac{\partial}{\partial\theta}\left[D(y,\theta)\tilde{n}(t,y,\theta)\right]\bigg{|}_{\theta=0_{-}}^{\theta=0_{+}}=-\frac{\partial}{\partial\theta}\left[D_{+}(0_{+})\tilde{n}_{+}(t,0_{+})\right]\delta_{y=L}+\frac{\partial}{\partial\theta}\left[D_{-}(0_{-})\tilde{n}_{-}(t,0_{-})\right]\delta_{y=-L},$
(4.2d)
$\displaystyle\frac{\partial}{\partial\theta}\left[D(y,\theta)\tilde{n}(t,y,\theta)\right]\bigg{|}_{\theta=\pi_{-}}^{\theta=-\pi_{+}}=-\frac{\partial}{\partial\theta}\left[D_{+}(\pi_{-})\tilde{n}_{+}(t,\pi_{-})\right]\delta_{y=L}+\frac{\partial}{\partial\theta}\left[D_{-}(-\pi_{+})\tilde{n}_{-}(t,-\pi_{+})\right]\delta_{y=-L}.$
(4.2e)

Inspired by the scaling of scattering equations proposed in [36], we consider
the forward peaked scattering kernel and obtain that the $L_{2}$-weak limit of
the scaled CRTM satisfies the CFPM, in the distributional sense.

Here we consider the jump sizes are scaled by $\varepsilon$, and the jump
frequency is rescaled by $\frac{1}{\varepsilon^{2}}$ with $\varepsilon$ being
very small, which indicates that the tumbles are quite frequent. Now the model
becomes

$\displaystyle\dfrac{\partial n^{\varepsilon}}{\partial
t}+V\sin\theta\dfrac{\partial n^{\varepsilon}}{\partial
y}+\dfrac{1}{\varepsilon^{2}}\left[k(y,\theta)n^{\varepsilon}-\\!\\!\int_{-\pi}^{\\!\pi}\\!\dfrac{1}{\varepsilon}K_{\varepsilon}\left(y,\theta^{\prime},\dfrac{\theta-\theta^{\prime}}{\varepsilon}\right)n^{\varepsilon\prime}d\theta^{\prime}\right]=0,$
$\displaystyle\theta\in(-\pi.\pi),\quad y\in(-L,L),$ (4.3a)
$\displaystyle\dfrac{\partial n_{+}^{\varepsilon}}{\partial
t}+\dfrac{1}{\varepsilon^{2}}\left[k_{+}(\theta)n_{+}^{\varepsilon}-\\!\\!\int_{0}^{\\!\pi}\\!\\!\dfrac{1}{\varepsilon}K_{\varepsilon+}\left(\theta^{\prime},\dfrac{\theta-\theta^{\prime}}{\varepsilon}\right)n_{+}^{\varepsilon\prime}d\theta^{\prime}\right]=V\sin\theta
n^{\varepsilon}(t,L,\theta),$ $\displaystyle\theta\in(0,\pi),$ (4.3b)
$\displaystyle\dfrac{\partial n_{-}^{\varepsilon}}{\partial
t}+\dfrac{1}{\varepsilon^{2}}\left[k_{-}(\theta)n_{-}^{\varepsilon}-\\!\\!\int_{-\pi}^{0}\\!\dfrac{1}{\varepsilon}K_{\varepsilon-}\left(\theta^{\prime},\dfrac{\theta-\theta^{\prime}}{\varepsilon}\right)n_{-}^{\varepsilon\prime}d\theta^{\prime}\right]=-V\sin\theta
n^{\varepsilon}(t,-L,\theta),$ $\displaystyle\theta\in(-\pi,0),$ (4.3c)

$K_{\varepsilon}\left(y,\theta,\dfrac{\theta^{\prime}-\theta}{\varepsilon}\right)\geq
0,\quad
k(y,\theta):=\int_{-\pi}^{\\!\pi}\\!\dfrac{1}{\varepsilon}K_{\varepsilon}\left(y,\theta,\dfrac{\theta^{\prime}-\theta}{\varepsilon}\right)d\theta^{\prime},\quad
K_{\varepsilon\pm}\left(\theta,\dfrac{\theta^{\prime}-\theta}{\varepsilon}\right)\geq
0,$ (4.4)
$k_{+}(\theta):=\int_{-\pi}^{\\!\pi}\\!\dfrac{1}{\varepsilon}K_{\varepsilon+}\left(\theta,\dfrac{\theta^{\prime}-\theta}{\varepsilon}\right)d\theta^{\prime},\quad
k_{-}(\theta):=\int_{-\pi}^{\\!\pi}\\!\dfrac{1}{\varepsilon}K_{\varepsilon-}\left(\theta,\dfrac{\theta^{\prime}-\theta}{\varepsilon}\right)d\theta^{\prime},$
(4.5)

with the boundary conditions

$\displaystyle n^{\varepsilon}(t,y,-\pi)=n^{\varepsilon}(t,y,\pi),$
$\displaystyle y\in(-L,L),$ (4.6a) $\displaystyle-V\sin\theta
n^{\varepsilon}(t,L,\theta)=\dfrac{1}{\varepsilon^{2}}\int_{0}^{\\!\pi}\\!\\!\dfrac{1}{\varepsilon}K_{\varepsilon+}\left(\theta^{\prime},\dfrac{\theta-\theta^{\prime}}{\varepsilon}\right)n_{+}^{\varepsilon}(t,\theta^{\prime})d\theta^{\prime},$
$\displaystyle\theta\in(-\pi,0),$ (4.6b) $\displaystyle V\sin\theta
n^{\varepsilon}(t,-L,\theta)=\dfrac{1}{\varepsilon^{2}}\int_{-\pi}^{0}\\!\dfrac{1}{\varepsilon}K_{\varepsilon-}\left(\theta^{\prime},\dfrac{\theta-\theta^{\prime}}{\varepsilon}\right)n_{-}^{\varepsilon}(t,\theta^{\prime})d\theta^{\prime},$
$\displaystyle\theta\in(0,\pi),$ (4.6c)

For simplicity, we only prove the case when the kernel functions satisfy the
following assumptions.

######  Assumption 4.1.

Assume that

  * (A1)

$K_{\varepsilon}\left(y,\theta,z\right)$ and
$K_{\varepsilon\pm}\left(\theta,z\right)$ are nonzero only if
$z\in\left[-1+\frac{2k\pi}{\varepsilon},1+\frac{2k\pi}{\varepsilon}\right]$
for some $k\in\mathbb{Z}$;

  * (A2)

$K_{\varepsilon}\left(y,\theta,z\right)$ and
$K_{\varepsilon\pm}\left(\theta,z\right)$ are periodic in $z$, i.e. for any
$y\in[-L,L]$, $\theta\in[-\pi,\pi]$, $k\in\mathbb{Z}$, we have

$K_{\varepsilon}\left(y,\theta,z\right)=K_{\varepsilon}\left(y,\theta,z+\frac{2k\pi}{\varepsilon}\right),\quad
K_{\varepsilon\pm}\left(\theta,z\right)=K_{\varepsilon\pm}\left(\theta,z+\frac{2k\pi}{\varepsilon}\right).$
(4.7)

  * (A3)

$K_{\varepsilon}\left(y,\theta,z\right)$ and
$K_{\varepsilon\pm}\left(\theta,z\right)$ are symmetric in $z$ when
$z\in[-1,1]$, i.e.

$K_{\varepsilon}\left(y,\theta,z\right)=K_{\varepsilon}\left(y,\theta,-z\right),\quad
K_{\varepsilon\pm}\left(\theta,z\right)=K_{\varepsilon\pm}\left(\theta,-z\right).$
(4.8)

  * (A4)

$K_{\varepsilon}\left(y,\theta,z\right)$ is periodic in $\theta$, i.e.

$K_{\varepsilon}\left(y,-\pi,z\right)=K_{\varepsilon}\left(y,\pi,z\right),\
\frac{\partial
K_{\varepsilon}}{\partial\theta}\left(y,-\pi,z\right)=\frac{\partial
K_{\varepsilon}}{\partial\theta}\left(y,\pi,z\right).$ (4.9)

A simple example of $K_{\varepsilon}\left(y,\theta,z\right)$ and
$K_{\varepsilon\pm}\left(\theta,z\right)$ that satisfy Assumption 4.1 is that

$K_{\varepsilon}\left(y,\theta,z\right)=k(y,\theta)f_{\varepsilon}(z),\qquad
K_{\varepsilon\pm}\left(\theta,z\right)=k_{\pm}(\theta)f_{\varepsilon}(z),$
(4.10)

where

$f_{\varepsilon}(z)=\left\\{\begin{aligned} &1-\lvert
z\rvert,&&z\in\left[\frac{2k\pi}{\varepsilon}-1,\frac{2k\pi}{\varepsilon}+1\right),\\\
&0,&&z\in\left[\frac{2k\pi-\pi}{\varepsilon},\frac{2k\pi}{\varepsilon}-1\right)\cup\left[\frac{2k\pi}{\varepsilon}+1,\frac{2k\pi+\pi}{\varepsilon}\right),\end{aligned}\right.$
(4.11)

as shown in (4-2), and $k(y,\theta)$ and $k_{\pm}(\theta)$ are nonzero,
$k(y,\theta)$ is periodic in $\theta$ with period $2\pi$.

$z$$-\frac{2\pi}{\varepsilon}$$-\frac{2\pi}{\varepsilon}+1$$-1$$1$$\frac{2\pi}{\varepsilon}-1$$\frac{2\pi}{\varepsilon}$$1$$f_{\varepsilon}(z)$
Figure 4-2: Diagram of $f_{\varepsilon}(z)$ for
$z\in\left[-\frac{2\pi}{\varepsilon},\frac{2\pi}{\varepsilon}\right)$

We define $\mathbb{M}^{1}([-L,L]\times[-\pi,\pi])$ the space of bounded
measure on $[-L,L]\times[-\pi,\pi]$ (see [27]). Then we have

######  Theorem 4.2.

Suppose that the initial data
$\left(n^{\varepsilon}(0,y,\theta),n_{\pm}^{\varepsilon}(0,\theta)\right)$
belong to $\mathbb{M}^{1}([-L,L]\times[-\pi,\pi])$ and are independent of
$\varepsilon$ and assume Assumption 4.1. Then, there exist subsequences of
solutions to the CRTM (4.3)–(4.6), still denoted by $n^{\varepsilon}$,
$n_{\pm}^{\varepsilon}$, such that for any $T>0$,

$\displaystyle n^{\varepsilon}(t,y,\theta)\xrightarrow[\varepsilon\to
0]{}\tilde{n}(t,y,\theta),\mbox{ in
}L^{\infty}\big{(}[0,T];\mathbb{M}^{1}([-L,L]\times[-\pi,\pi])\big{)}-weak-*,$
$\displaystyle n_{+}^{\varepsilon}(t,\theta)\xrightarrow[\varepsilon\to
0]{}\tilde{n}_{+}(t,\theta),\mbox{ in
}L^{\infty}\big{(}[0,T];\mathbb{M}^{1}([-\pi,0])\big{)}-weak-*,$
$\displaystyle n_{-}^{\varepsilon}(t,\theta)\xrightarrow[\varepsilon\to
0]{}\tilde{n}_{-}(t,\theta),\mbox{ in
}L^{\infty}\big{(}[0,T];\mathbb{M}^{1}([0,\pi])\big{)}-weak-*,$

with $\tilde{n}$, $\tilde{n}_{\pm}$ the solutions to the CFPM (4.1)–(4.2), and
the diffusion coefficients are determined by

$D(y,\theta)=\lim\limits_{\varepsilon\to
0}\frac{1}{2}\int_{-1}^{1}z^{2}K_{\varepsilon}(y,\theta,z)dz,\quad
D_{\pm}(\theta)=\lim\limits_{\varepsilon\to
0}\frac{1}{2}\int_{-1}^{1}z^{2}K_{\varepsilon\pm}(\theta,z)dz.$ (4.12)

###### Proof.

Since $n^{\varepsilon}(t,y,\theta)$ and $n_{+}^{\varepsilon}(t,\theta)$ are
non-negative and satisfy mass conservation, the family
$n^{\varepsilon}(t,y,\theta)$ is uniformly bounded in
$L^{\infty}([0,T];L^{1}([-L,L]\times[-\pi,\pi]))$ and hence in
$L^{\infty}([0,T];\mathbb{M}^{1}([-L,L]\times[-\pi,\pi]))$, the family
$n_{+}^{\varepsilon}(t,\theta)$ is uniformly bounded in
$L^{\infty}([0,T];L^{1}([0,\pi]))$ hence in
$L^{\infty}([0,T];\mathbb{M}^{1}([0,\pi]))$, the family
$n_{-}^{\varepsilon}(t,\theta)$ is uniformly bounded in
$L^{\infty}([0,T];L^{1}([-\pi,0]))$ hence in
$L^{\infty}([0,T];\mathbb{M}^{1}([-\pi,0]))$. Therefore, there exist
subsequences of $n^{\varepsilon}$, $n_{\pm}^{\varepsilon}$, still denoted by
$n^{\varepsilon}$, $n_{\pm}^{\varepsilon}$ which converge as stated in Theorem
4.2.
Next we prove that $\tilde{n}$, $\tilde{n}_{\pm}$ satisfy the CFPM (4.1)–(4.2)
in the distribution sense. Since the CFPM (4.1)–(4.2) is complex, we divide
our proof into three steps:

  1. I.

Prove that (4.3a) with boundary condition (4.6a) converges to (4.1a) with
boundary condition (4.2a) weakly.

  2. II.

Prove that (4.3b)–(4.3c) converge to (4.1b)–(4.1c) with boundary condition
(4.2c) weakly.

  3. III.

Prove that (4.6b)- (4.6c) converge to (4.2b), (4.2d), (4.2e) weakly.

Step I.  For any fixed $T>0$, consider a test function $\Phi(t,y,\theta)\in
C^{3}([0,T]\times[-L,L]\times[-\pi,\pi])$, which has compact support in $t$
and $y$, and satisfies periodic boundary condition in $\theta$

$\Phi(t,y,-\pi)=\Phi(t,y,\pi),\quad\frac{\partial}{\partial\theta}\Phi(t,y,-\pi)=\frac{\partial}{\partial\theta}\Phi(t,y,\pi),\quad
y\in[-L,L].$ (4.13)

Multiplying (4.3a) by such $\Phi(t,y,\theta)$ and integrating in $t$ from $0$
to $T$, in $y$ from $-L$ to $L$, in $\theta$ from $-\pi$ to $\pi$ yields

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!-n^{\varepsilon}(t,y,\theta)\left[\frac{\partial\Phi}{\partial
t}(t,y,\theta)+V\sin\theta\frac{\partial\Phi}{\partial
y}(t,y,\theta)\right]d\theta dydt$ $\displaystyle\xlongequal{\eqref{5.2}}$
$\displaystyle-\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!\Phi(t,y,\theta)n^{\varepsilon}(t,y,\theta)\int_{-\pi}^{\\!\pi}\\!\frac{1}{\varepsilon}K_{\varepsilon}\left(y,\theta,\frac{\theta^{\prime}-\theta}{\varepsilon}\right)d\theta^{\prime}d\theta
dydt$
$\displaystyle+\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!n^{\varepsilon}(t,y,\theta^{\prime})\int_{-\pi}^{\\!\pi}\\!\frac{1}{\varepsilon}K_{\varepsilon}\left(y,\theta^{\prime},\frac{\theta-\theta^{\prime}}{\varepsilon}\right)\Phi(t,y,\theta)d\theta
d\theta^{\prime}dydt$
$\displaystyle\xlongequal{z=\frac{\theta^{\prime}-\theta}{\varepsilon},z^{\prime}=\frac{\theta-\theta^{\prime}}{\varepsilon}}$
$\displaystyle-\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!\Phi(t,y,\theta)n^{\varepsilon}(t,y,\theta)\int_{\frac{-\theta-\pi}{\varepsilon}}^{\frac{-\theta+\pi}{\varepsilon}}\\!K_{\varepsilon}\left(y,\theta,z\right)dzd\theta
dydt$
$\displaystyle+\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!n^{\varepsilon}(t,y,\theta^{\prime})\int_{\frac{-\theta^{\prime}-\pi}{\varepsilon}}^{\frac{-\theta^{\prime}+\pi}{\varepsilon}}\\!K_{\varepsilon}\left(y,\theta^{\prime},z^{\prime}\right)\Phi(t,y,\theta^{\prime}+\varepsilon
z^{\prime})dz^{\prime}d\theta^{\prime}dydt$ $\displaystyle=$
$\displaystyle\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!n^{\varepsilon}(t,y,\theta)\int_{\frac{-\theta-\pi}{\varepsilon}}^{\frac{-\theta+\pi}{\varepsilon}}\\!K_{\varepsilon}\left(y,\theta,z\right)\left[\Phi(t,y,\theta+\varepsilon
z)-\Phi(t,y,\theta)\right]dzd\theta dydt.$

Next, we divide the right hand side of the above equation into three parts
$G_{1}^{\varepsilon}$, $G_{2}^{\varepsilon}$, $G_{3}^{\varepsilon}$, such that

$G_{1}^{\varepsilon}=\int_{-\pi+\varepsilon}^{\pi-\varepsilon}G(t,y,\theta)d\theta,\quad
G_{2}^{\varepsilon}=\int_{-\pi}^{-\pi+\varepsilon}G(t,y,\theta)d\theta,\quad
G_{3}^{\varepsilon}=\int_{\pi-\varepsilon}^{\\!\pi}G(t,y,\theta)d\theta,$

with

$G^{\varepsilon}(t,y,\theta)=\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!n^{\varepsilon}(t,y,\theta)\int_{\frac{-\theta-\pi}{\varepsilon}}^{\frac{-\theta+\pi}{\varepsilon}}\\!K_{\varepsilon}(y,\theta,z)\left[\Phi(t,y,\theta+\varepsilon
z)-\Phi(t,y,\theta)\right]dzdydt.$

When $\varepsilon\to 0$, the limits of $G_{1}^{\varepsilon}$,
$G_{2}^{\varepsilon}$, $G_{3}^{\varepsilon}$ are calculated in the subsequent
part.

  * •

The limit of $G_{1}^{\varepsilon}$

When $\theta\in[-\pi+\varepsilon,\pi-\varepsilon]$, we have

$supp_{z}\left\\{K_{\varepsilon}(y,\theta,z)\right\\}\cap\left[\dfrac{-\theta-\pi}{\varepsilon},\dfrac{-\theta+\pi}{\varepsilon}\right]\subset[-1,1].$
(4.14)

From (4.8), $zK_{\varepsilon}(y,\theta,z)$ is odd when $z\in[-1,1]$, we can
write

$\displaystyle G_{1}^{\varepsilon}=$
$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi+\varepsilon}^{\pi-\varepsilon}n^{\varepsilon}(t,y,\theta)\int_{\frac{-\theta-\pi}{\varepsilon}}^{\frac{-\theta+\pi}{\varepsilon}}\\!K_{\varepsilon}(y,\theta,z)\left[\frac{z}{\varepsilon}\frac{\partial\Phi}{\partial\theta}(t,y,\theta)+\frac{z^{2}}{2}\frac{\partial^{2}\Phi}{\partial\theta^{2}}(t,y,\theta)\right]dzd\theta
dydt+O(\varepsilon)$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi+\varepsilon}^{\pi-\varepsilon}n^{\varepsilon}(t,y,\theta)\frac{\partial\Phi}{\partial\theta}(t,y,\theta)\int_{-1}^{1}\frac{z}{\varepsilon}K_{\varepsilon}(y,\theta,z)dzd\theta
dydt$
$\displaystyle+\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi+\varepsilon}^{\pi-\varepsilon}\frac{\partial^{2}\Phi}{\partial\theta^{2}}(t,y,\theta)\int_{-1}^{1}\frac{z^{2}}{2}K_{\varepsilon}(y,\theta,z)dzd\theta
dydt+O(\varepsilon)$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi+\varepsilon}^{\pi-\varepsilon}D(y,\theta)n^{\varepsilon}(t,y,\theta)\frac{\partial^{2}\Phi}{\partial\theta^{2}}(t,y,\theta)d\theta
dydt+O(\varepsilon),$

where $D(y,\theta)$ is defined as in (4.12). We pass to the limit and get

$\lim\limits_{\varepsilon\to
0}G_{1}^{\varepsilon}=\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!D(y,\theta)\tilde{n}(t,y,\theta)\frac{\partial^{2}\Phi}{\partial\theta^{2}}(t,y,\theta)d\theta
dydt.$ (4.15)

  * •

Limits of $G_{2}^{\varepsilon}$, $G_{3}^{\varepsilon}$

Next we show that $\lim\limits_{\varepsilon\to 0}G_{2}^{\varepsilon}=0$. When
$\theta\in[-\pi,-\pi+\varepsilon]$, we have

$supp_{z}\left\\{K_{\varepsilon}(y,\theta,z)\right\\}\cap\left[\dfrac{-\theta-\pi}{\varepsilon},\dfrac{-\theta+\pi}{\varepsilon}\right]\subset\left[\frac{-\theta-\pi}{\varepsilon},1\right]\cap\left[\frac{2\pi}{\varepsilon}-1,\frac{-\theta+\pi}{\varepsilon}\right].$
(4.16)

Using assumption of periodicity and symmetry in (4.7), (4.8), we find

$\displaystyle\int_{\frac{2\pi}{\varepsilon}-1}^{\frac{-\theta+\pi}{\varepsilon}}\\!\\!K_{\varepsilon}(y,\theta,z)\left[\Phi(t,y,\theta+\varepsilon
z)-\Phi(t,y,\theta)\right]dz=$
$\displaystyle\int_{\frac{2\pi}{\varepsilon}-1}^{\frac{-\theta+\pi}{\varepsilon}}\\!\\!K_{\varepsilon}(y,\theta,\frac{2\pi}{\varepsilon}-z)\left[\Phi(t,y,\theta+\varepsilon
z)-\Phi(t,y,\theta)\right]dz$
$\displaystyle\xlongequal{z^{\prime}=\frac{2\pi}{\varepsilon}-z}$
$\displaystyle\int_{\frac{\theta+\pi}{\varepsilon}}^{1}\\!K_{\varepsilon}(y,\theta,z^{\prime})\left[\Phi(t,y,\theta+2\pi-\varepsilon
z^{\prime})-\Phi(t,y,\theta)\right]dz^{\prime},$

which gives us

$\displaystyle G_{2}^{\varepsilon}=$
$\displaystyle\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{-\pi+\varepsilon}n^{\varepsilon}(t,y,\theta)\int_{\frac{-\theta-\pi}{\varepsilon}}^{\frac{\theta+\pi}{\varepsilon}}K_{\varepsilon}(y,\theta,z)\left[\Phi(t,y,\theta+\varepsilon
z)-\Phi(t,y,\theta)\right]dzd\theta dydt$
$\displaystyle+\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{-\pi+\varepsilon}n^{\varepsilon}(t,y,\theta)\int_{\frac{\theta+\pi}{\varepsilon}}^{1}K_{\varepsilon}(y,\theta,z)\left[\Phi(t,y,\theta+\varepsilon
z)+\Phi(t,y,\theta+2\pi-\varepsilon z)-2\Phi(t,y,\theta)\right]dzd\theta
dydt.$

When $\theta\in[-\pi,-\pi+\varepsilon]$,
$\frac{\theta+\pi}{\varepsilon}\in[0,1]$. Hence by symmetry in (4.8), we have

$\int_{\frac{-\theta-\pi}{\varepsilon}}^{\frac{\theta+\pi}{\varepsilon}}K_{\varepsilon}(y,\theta,z)\left[\Phi(t,y,\theta+\varepsilon
z)-\Phi(t,y,\theta)\right]dzd\theta=\varepsilon\frac{\partial\Phi(t,y,\theta)}{\partial\theta}\int_{\frac{-\theta-\pi}{\varepsilon}}^{\frac{\theta+\pi}{\varepsilon}}zK_{\varepsilon}(y,\theta,z)dz+O(\varepsilon^{2})=O(\varepsilon^{2}).$

On the other hand, if we expand $\Phi(t,y,\theta+\varepsilon z)$ near
$\theta+\varepsilon z=-\pi$, and $\Phi(t,y,\theta+2\pi-\varepsilon z)$ near
$\theta+2\pi-\varepsilon z=\pi$, $\Phi(t,y,\theta)$ near $\theta=-\pi$, we
have

$\displaystyle\Phi(t,y,\theta+\varepsilon z)+\Phi(t,y,\theta+2\pi-\varepsilon
z)-2\Phi(t,y,\theta)$ $\displaystyle=$
$\displaystyle\Phi(t,y,\pi)-\Phi(t,y,-\pi)+\left(\theta+\pi-\varepsilon
z\right)\left[\frac{\partial\Phi}{\partial\theta}(t,y,\pi)-\frac{\partial\Phi}{\partial\theta}(t,y,-\pi)\right]+O(\varepsilon^{2}),$

which is just $O(\varepsilon^{2})$ for $\Phi(t,y,\theta)$ satisfying the
periodic boundary conditions in (4.13). Hence we have
$\lim\limits_{\varepsilon\to 0}G_{2}^{\varepsilon}=0$, and by similar
derivation, $\lim\limits_{\varepsilon\to 0}G_{3}^{\varepsilon}=0$ also holds.

We conclude that

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}$
$\displaystyle\\!-\tilde{n}\left[\frac{\partial\Phi}{\partial
t}(t,y,\theta)+V\sin\theta\frac{\partial\Phi}{\partial y}\right]d\theta dydt$
$\displaystyle=$ $\displaystyle\lim\limits_{\varepsilon\to
0}(G_{1}^{\varepsilon}+G_{2}^{\varepsilon}+G_{3}^{\varepsilon})=\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!D(y,\theta)\tilde{n}(t,y,\theta)\frac{\partial^{2}\Phi}{\partial\theta^{2}}(t,y,\theta)d\theta
dydt,$

which is the weak form of (4.1a) with periodic boundary condition (4.2a).

Step II. For any fixed $T>0$, consider a test function $\Phi_{+}(t,\theta)\in
C^{3}([0,T]\times[0,\pi])$ which satisfies

$\Phi_{+}(t,0)=\Phi_{+}(t,\pi)=0.$ (4.17)

We multiply (4.3b) with such a test function $\Phi_{+}(t,\theta)$ and
integrate in $t$ from $0$ to $T$, in $\theta$ from $0$ to $\pi$ to get

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!-n_{+}^{\varepsilon}\frac{\partial\Phi_{+}}{\partial
t}(t,\theta)-V\sin\theta n^{\varepsilon}(t,L,\theta)\Phi_{+}(t,\theta)d\theta
dt$ $\displaystyle\xlongequal{\eqref{5.2a}}$
$\displaystyle-\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!\Phi_{+}(t,\theta)n_{+}^{\varepsilon}(t,\theta)\int_{-\pi}^{\\!\pi}\\!\frac{1}{\varepsilon}K_{\varepsilon+}\left(\theta,\frac{\theta^{\prime}-\theta}{\varepsilon}\right)d\theta^{\prime}d\theta
dt$
$\displaystyle+\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!n_{+}^{\varepsilon}(t,\theta^{\prime})\int_{0}^{\\!\pi}\\!\\!\frac{1}{\varepsilon}K_{\varepsilon+}\left(\theta^{\prime},\frac{\theta-\theta^{\prime}}{\varepsilon}\right)\Phi_{+}(t,\theta)d\theta
d\theta^{\prime}dt$
$\displaystyle\xlongequal{z=\frac{\theta^{\prime}-\theta}{\varepsilon},z^{\prime}=\frac{\theta-\theta^{\prime}}{\varepsilon}}$
$\displaystyle-\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!\Phi_{+}(t,\theta)n_{+}^{\varepsilon}(t,\theta)\int_{\frac{-\theta-\pi}{\varepsilon}}^{\frac{-\theta+\pi}{\varepsilon}}\\!K_{\varepsilon+}\left(\theta,z\right)dzd\theta
dt$
$\displaystyle+\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!n_{+}^{\varepsilon}(t,\theta^{\prime})\int_{\frac{-\theta^{\prime}}{\varepsilon}}^{\frac{-\theta^{\prime}+\pi}{\varepsilon}}K_{\varepsilon+}\left(\theta^{\prime},z^{\prime}\right)\Phi_{+}(t,\theta^{\prime}+\varepsilon
z^{\prime})dz^{\prime}d\theta^{\prime}dt$ $\displaystyle=$
$\displaystyle\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!n_{+}^{\varepsilon}(t,\theta)\int_{\frac{-\theta}{\varepsilon}}^{\frac{-\theta+\pi}{\varepsilon}}K_{\varepsilon+}\left(\theta,z\right)\left[\Phi_{+}(t,\theta+\varepsilon
z)-\Phi_{+}(t,\theta)\right]dzd\theta dt$
$\displaystyle-\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!n_{+}^{\varepsilon}(t,\theta)\int_{\frac{-\theta-\pi}{\varepsilon}}^{\frac{-\theta}{\varepsilon}}K_{\varepsilon+}\left(\theta,z\right)\Phi_{+}(t,\theta)dzd\theta
dt.$ $\displaystyle\triangleq$ $\displaystyle
H_{1}^{\varepsilon}(t,\theta)+H_{2}^{\varepsilon}(t,\theta).$

In the subsequent part, we show that

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!-\tilde{n}_{+}\frac{\partial\Phi_{+}}{\partial
t}(t,\theta)-V\sin\theta\tilde{n}(t,L,\theta)\Phi_{+}(t,\theta)d\theta dt$
$\displaystyle=$ $\displaystyle\lim\limits_{\varepsilon\to
0}\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!-n_{+}^{\varepsilon}\frac{\partial\Phi_{+}}{\partial
t}(t,\theta)-V\sin\theta n^{\varepsilon}(t,L,\theta)\Phi_{+}(t,\theta)d\theta
dt$ (4.18) $\displaystyle=$ $\displaystyle\lim\limits_{\varepsilon\to
0}\left(H_{1}^{\varepsilon}(t,\theta)+H_{2}^{\varepsilon}(t,\theta)\right)=\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!\tilde{n}_{+}(t,\theta)D_{+}(\theta)\frac{\partial^{2}\Phi_{+}}{\partial\theta^{2}}(t,\theta)d\theta
dt-\frac{1}{2}D_{+}(\theta)\int_{0}^{T}\\!\\!\\!\tilde{n}_{+}(t,\theta)\frac{\partial\Phi_{+}}{\partial\theta}(t,\theta)dt\bigg{|}_{0}^{\\!\pi}.$

Note that

$supp_{z}\left\\{K_{\varepsilon+}(\theta,z)\right\\}\cup\left[\dfrac{-\theta}{\varepsilon},\dfrac{-\theta+\pi}{\varepsilon}\right]=\left\\{\begin{aligned}
&\left[\frac{-\theta}{\varepsilon},1\right],&&\theta\in[0,\varepsilon],\\\
&[-1,1],&&\theta\in[\varepsilon,\pi-\varepsilon],\\\
&\left[-1,\dfrac{-\theta+\pi}{\varepsilon}\right],&&\theta\in[\pi-\varepsilon,\pi],\\\
\end{aligned}\right.$ (4.19)
$supp_{z}\left\\{K_{\varepsilon+}(\theta,z)\right\\}\cup\left[\dfrac{-\theta-\pi}{\varepsilon},\dfrac{-\theta}{\varepsilon}\right]=\left\\{\begin{aligned}
&\left[-1,\frac{-\theta}{\varepsilon}\right],&&\theta\in[0,\varepsilon],\\\
&\emptyset,&&\theta\in[\varepsilon,\pi-\varepsilon],\\\
&\left[\frac{-\theta-\pi}{\varepsilon},-\frac{2\pi}{\varepsilon}+1\right],&&\theta\in[\pi-\varepsilon,\pi],\\\
\end{aligned}\right.$ (4.20)

Hence we can divide $H_{1}^{\varepsilon}(t,\theta)$ into
$H_{11}^{\varepsilon}(t,\theta)+H_{12}^{\varepsilon}(t,\theta)+H_{13}^{\varepsilon}(t,\theta)$,
where

$\displaystyle H_{11}^{\varepsilon}(t,\theta)=$
$\displaystyle\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{0}^{\varepsilon}n_{+}^{\varepsilon}(t,\theta)\int_{\frac{-\theta}{\varepsilon}}^{1}K_{\varepsilon+}\left(\theta,z\right)\left[\Phi_{+}(t,\theta+\varepsilon
z)-\Phi_{+}(t,\theta)\right]dzd\theta dt$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\\!\\!\\!n_{+}^{\varepsilon}(t,0)\frac{\partial\Phi_{+}}{\partial\theta}(t,0)\int_{-1}^{0}z^{2}K_{\varepsilon+}\left(0,z\right)dzdt+O(\varepsilon)$
$\displaystyle=$ $\displaystyle
D_{+}(0)\int_{0}^{T}\\!\\!\\!n_{+}^{\varepsilon}(t,0)\frac{\partial\Phi_{+}}{\partial\theta}(t,0)dt+O(\varepsilon),$
$\displaystyle H_{12}^{\varepsilon}(t,\theta)=$
$\displaystyle\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{\varepsilon}^{\pi-\varepsilon}n_{+}^{\varepsilon}(t,\theta)\int_{-1}^{1}K_{\varepsilon+}\left(\theta,z\right)\left[\Phi_{+}(t,\theta+\varepsilon
z)-\Phi_{+}(t,\theta)\right]dzd\theta dt$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!n_{+}^{\varepsilon}(t,\theta)D_{+}(\theta)\frac{\partial^{2}\Phi_{+}}{\partial\theta^{2}}(t,\theta)d\theta
dt+O(\varepsilon),$ $\displaystyle H_{13}^{\varepsilon}(t,\theta)=$
$\displaystyle\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{\pi-\varepsilon}^{\\!\pi}n_{+}^{\varepsilon}(t,\theta)\int_{-1}^{\frac{-\theta+\pi}{\varepsilon}}K_{\varepsilon+}\left(\theta,z\right)\left[\Phi_{+}(t,\theta+\varepsilon
z)-\Phi_{+}(t,\theta)\right]dzd\theta dt$ $\displaystyle=$ $\displaystyle-
D_{+}(\pi)\int_{0}^{T}\\!\\!\\!n_{+}^{\varepsilon}(t,\pi)\frac{\partial\Phi_{+}}{\partial\theta}(t,\pi)dt+O(\varepsilon).$

And $H_{2}^{\varepsilon}(t,\theta)$ can be divided into
$H_{21}^{\varepsilon}(t,\theta)+H_{22}^{\varepsilon}(t,\theta)$, where

$\displaystyle H_{21}^{\varepsilon}(t,\theta)=$
$\displaystyle-\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{0}^{\varepsilon}n_{+}^{\varepsilon}(t,\theta)\Phi_{+}(t,\theta)\int_{-1}^{\frac{-\theta}{\varepsilon}}K_{\varepsilon+}\left(\theta,z\right)dzd\theta
dt$ $\displaystyle=$
$\displaystyle-\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\Phi_{+}(0)n_{+}^{\varepsilon}(t,0)\int_{0}^{\varepsilon}\int_{-1}^{\frac{-\theta}{\varepsilon}}\left[K_{\varepsilon+}\left(0,z\right)+\theta\frac{\partial
K_{\varepsilon+}}{\partial\theta}\left(0,z\right)\right]dzd\theta dt$
$\displaystyle-\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\left[\frac{\partial\Phi_{+}}{\partial\theta}(0)n_{+}^{\varepsilon}(t,0)+\Phi_{+}(0)\frac{\partial
n_{+}^{\varepsilon}}{\partial\theta}(t,0)\right]\int_{0}^{\varepsilon}\int_{-1}^{\frac{-\theta}{\varepsilon}}\theta
K_{\varepsilon+}\left(0,z\right)dzd\theta dt+O(\varepsilon)$ $\displaystyle=$
$\displaystyle\left[-\frac{1}{\varepsilon}\int_{-1}^{0}-zK_{\varepsilon+}\left(0,z\right)dz-\frac{1}{2}z^{2}\frac{\partial
K_{\varepsilon+}}{\partial\theta}\left(0,z\right)dz\right]\int_{0}^{T}\\!\\!\\!\Phi_{+}(0)n_{+}^{\varepsilon}(t,0)dt$
$\displaystyle-\frac{1}{2}D_{+}(0)\int_{0}^{T}\\!\left[\frac{\partial\Phi_{+}}{\partial\theta}(0)n_{+}^{\varepsilon}(t,0)+\Phi_{+}(0)\frac{\partial
n_{+}^{\varepsilon}}{\partial\theta}(t,0)\right]dt+O(\varepsilon),$
$\displaystyle H_{22}^{\varepsilon}(t,\theta)=$
$\displaystyle-\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{\pi-\varepsilon}^{\\!\pi}n_{+}^{\varepsilon}(t,\theta)\Phi_{+}(t,\theta)\int_{\frac{-\theta-\pi}{\varepsilon}}^{-\frac{2\pi}{\varepsilon}+1}K_{\varepsilon+}\left(\theta,z\right)dzd\theta
dt$ $\displaystyle=$
$\displaystyle-\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\int_{\pi-\varepsilon}^{\\!\pi}n_{+}^{\varepsilon}(t,\theta)\Phi_{+}(t,\theta)\int_{-1}^{\frac{\theta-\pi}{\varepsilon}}K_{\varepsilon+}\left(\theta,z\right)dzd\theta
dt$ $\displaystyle=$
$\displaystyle\left[-\frac{1}{\varepsilon}\int_{-1}^{0}-zK_{\varepsilon+}\left(\pi,z\right)dz+\int_{-1}^{0}\frac{1}{2}z^{2}\frac{\partial
K_{\varepsilon+}}{\partial\theta}\left(\pi,z\right)dz\right]\int_{0}^{T}\\!\\!\\!\Phi_{+}(\pi)n_{+}^{\varepsilon}(t,\pi)dt$
$\displaystyle+\frac{1}{2}D_{+}(\pi)\int_{0}^{T}\\!\left[\frac{\partial\Phi_{+}}{\partial\theta}(\pi)n_{+}^{\varepsilon}(t,\pi)+\Phi_{+}(\pi)\frac{\partial
n_{+}^{\varepsilon}}{\partial\theta}(t,\pi)\right]dt+O(\varepsilon).$

Recall that the test function we use satisfies the Dirichlet boundary
conditions (4.17), hence we conclude (4).

If we choose $\Phi_{+}(t,\theta)\in C_{c}^{3}([0,T]\times[0,\pi]),$ it gives

$\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!-\tilde{n}_{+}^{\varepsilon}\frac{\partial\Phi_{+}}{\partial
t}(t,\theta)-V\sin\theta\tilde{n}^{\varepsilon}(t,L,\theta)\Phi_{+}(t,\theta)d\theta
dt=\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!\tilde{n}_{+}^{\varepsilon}(t,\theta)D_{+}(\theta)\frac{\partial^{2}\Phi_{+}}{\partial\theta^{2}}(t,\theta)d\theta
dt,$

which is the weak form of (4.1b).

On the other hand, if we choose $\Phi_{+}(t,\theta)\in
C^{3}([0,T]\times[0,\pi])$ such that $\Phi_{+}(t,\theta)$ satisfy (4.17) but
$\frac{\partial\Phi_{+}}{\partial\theta}(t,0)\neq 0$,
$\frac{\partial\Phi_{+}}{\partial\theta}(t,\pi)\neq 0$, then compared with the
weak form of (4.1b), we find

$\int_{0}^{T}\\!\\!\\!\tilde{n}_{+}(t,0)\frac{\partial\Phi_{+}}{\partial\theta}(t,0)dt=\int_{0}^{T}\\!\\!\\!\tilde{n}_{+}(t,\pi)\frac{\partial\Phi_{+}}{\partial\theta}(t,\pi)dt=0,$

which is the weak form of (4.2c).

Step III: If we choose $\Phi(t,0)\neq 0$, $\Phi(t,\pi)\neq 0$, according to
the proof in Step II, we have

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!-n_{+}^{\varepsilon}\frac{\partial\Phi_{+}}{\partial
t}(t,\theta)-V\sin\theta n^{\varepsilon}(t,L,\theta)\Phi_{+}(t,\theta)d\theta
dt$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!n_{+}D_{+}(\theta)\frac{\partial^{2}\Phi_{+}}{\partial\theta^{2}}(t,\theta)d\theta
dt+\frac{1}{2}D_{+}(\theta)\int_{0}^{T}\\!\\!\\!\Phi_{+}(t,\theta)\frac{\partial
n_{+}(t,\theta)}{\partial\theta}(t,\theta)dt\bigg{|}_{0}^{\\!\pi}$
$\displaystyle-\lim\limits_{\varepsilon\to
0}\frac{1}{\varepsilon}\left[\int_{-1}^{0}-zK_{\varepsilon}(0,z)dz\int_{0}^{T}\\!\\!\\!\Phi_{+}(t,0)n_{+}^{\varepsilon}(t,0)dt+\int_{-1}^{0}-zK_{\varepsilon}(\pi,z)dz\int_{0}^{T}\\!\\!\\!\Phi_{+}(t,\pi)n_{+}^{\varepsilon}(t,\pi)dt\right].$

Compared the weak form of (4.1b), it gives

$\displaystyle\lim\limits_{\varepsilon\to
0}\frac{1}{\varepsilon}\left[\int_{-1}^{0}-zK_{\varepsilon}(0,z)dz\int_{0}^{T}\\!\\!\\!\Phi_{+}(t,0)n_{+}^{\varepsilon}(t,0)dt+\int_{-1}^{0}-zK_{\varepsilon}(\pi,z)dz\int_{0}^{T}\\!\\!\\!\Phi_{+}(t,\pi)n_{+}^{\varepsilon}(t,\pi)dt\right]$
$\displaystyle=$
$\displaystyle-\frac{1}{2}D_{+}(\theta)\int_{0}^{T}\\!\\!\\!\Phi_{+}(t,\theta)\frac{\partial
n_{+}}{\partial\theta}(t,\theta)dt\bigg{|}_{0}^{\\!\pi}.$

Since $\Phi_{+}(t,0)$ and $\Phi_{+}(t,\pi)$ are chosen arbitrary, we conclude
that

$\lim\limits_{\varepsilon\to
0}\frac{1}{\varepsilon}n_{+}^{\varepsilon}(t,0)=\frac{D_{+}(0)\frac{\partial\tilde{n}_{+}}{\partial\theta}(t,0)}{2\lim\limits_{\varepsilon\to
0}\int_{-1}^{0}-zK_{\varepsilon}(0,z)dz},\quad\lim\limits_{\varepsilon\to
0}\frac{1}{\varepsilon}n_{+}^{\varepsilon}(t,\pi)=\frac{-D_{+}(\pi)\frac{\partial\tilde{n}_{+}}{\partial\theta}(t,\pi)}{2\lim\limits_{\varepsilon\to
0}\int_{-1}^{0}-zK_{\varepsilon}(\pi,z)dz}$ (4.21)

hold in the weak sense, these limits would be used later.

For any fixed $T>0$, consider a test function $\Phi(t,y,\theta)\in
C^{3}([0,T]\times[-L,L]\times[-\pi,\pi])$, which has compact support in $t$,
and satisfies periodic boundary condition in $\theta$ (4.13).

Following the similar procedures as in Step I, we obtain

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!-\Phi\left[\frac{\partial
n^{\varepsilon}}{\partial t}(t,y,\theta)+V\sin\theta\frac{\partial
n^{\varepsilon}}{\partial y}\right]d\theta
dydt=\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!D(y,\theta)n^{\varepsilon}(t,y,\theta)\frac{\partial^{2}\Phi}{\partial\theta^{2}}(t,y,\theta)d\theta
dydt+O(\varepsilon),$

which implies

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}$
$\displaystyle\\!-n^{\varepsilon}\left[\frac{\partial\Phi}{\partial
t}+V\sin\theta\frac{\partial\Phi}{\partial
y}-D(y,\theta)\frac{\partial^{2}\Phi}{\partial\theta^{2}}\right]d\theta dydt$
$\displaystyle=$
$\displaystyle-\\!\\!\int_{0}^{T}\\!\\!\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta\left[n^{\varepsilon}(t,L,\theta)\Phi(t,L,\theta)-n^{\varepsilon}(t,-L,\theta)\Phi(t,-L,\theta)\right]d\theta
dt.$

We first focus on the part $\int_{0}^{T}\\!\\!\\!\int_{-\pi}^{0}\\!V\sin\theta
n^{\varepsilon}(t,L,\theta)\Phi(t,L,\theta)d\theta dt$. Since
$K_{\varepsilon+}(\theta,z)$ is nonzero only if
$z\in\left[-1+\frac{2k\pi}{\varepsilon},\right.\\\
\left.1+\frac{2k\pi}{\varepsilon}\right]$ for some $k\in\mathbb{Z}$, we have

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{-\pi}^{0}\\!V\sin\theta
n^{\varepsilon}(t,L,\theta)\Phi(t,L,\theta)d\theta dt$
$\displaystyle\xlongequal{\eqref{5.3b}}$
$\displaystyle-\\!\\!\int_{0}^{T}\\!\\!\\!\int_{-\pi}^{0}\\!\Phi(t,L,\theta)\dfrac{1}{\varepsilon^{2}}\int_{0}^{\\!\pi}\\!\\!\dfrac{1}{\varepsilon}K_{\varepsilon+}\left(\theta^{\prime},\dfrac{\theta-\theta^{\prime}}{\varepsilon}\right)n_{+}^{\varepsilon}(t,\theta^{\prime})d\theta^{\prime}d\theta
dt$ $\displaystyle\xlongequal{Assumption(A1)}$
$\displaystyle-\\!\\!\int_{0}^{T}\\!\\!\\!\int_{-\pi}^{-\pi+\varepsilon}\Phi(t,L,\theta)\dfrac{1}{\varepsilon^{2}}\int_{\theta+2\pi-\varepsilon}^{\\!\pi}\dfrac{1}{\varepsilon}K_{\varepsilon+}\left(\theta^{\prime},\dfrac{\theta-\theta^{\prime}}{\varepsilon}\right)n_{+}^{\varepsilon}(t,\theta^{\prime})d\theta^{\prime}d\theta
dt$
$\displaystyle-\\!\\!\int_{0}^{T}\\!\\!\\!\int_{-\varepsilon}^{0}\Phi(t,L,\theta)\dfrac{1}{\varepsilon^{2}}\int_{0}^{\theta+\varepsilon}\dfrac{1}{\varepsilon}K_{\varepsilon+}\left(\theta^{\prime},\dfrac{\theta-\theta^{\prime}}{\varepsilon}\right)n_{+}^{\varepsilon}(t,\theta^{\prime})d\theta^{\prime}d\theta
dt$ $\displaystyle=$
$\displaystyle-\dfrac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\Phi(t,L,-\pi)\int_{-\pi}^{-\pi+\varepsilon}\\!\\!\int_{\theta+2\pi-\varepsilon}^{\\!\pi}\\!\\!\left[\frac{1}{\varepsilon}n_{+}^{\varepsilon}(t,\pi)+\dfrac{\theta^{\prime}-\pi}{\varepsilon}\frac{\partial
n_{+}^{\varepsilon}}{\partial\theta}(t,\pi)\right]K_{\varepsilon+}\left(\pi,\dfrac{\theta-\theta^{\prime}}{\varepsilon}\right)d\theta^{\prime}d\theta
dt$
$\displaystyle-\dfrac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\Phi(t,L,0)\int_{-\varepsilon}^{0}\int_{0}^{\theta+\varepsilon}\left[\frac{1}{\
\varepsilon}n_{+}^{\varepsilon}(t,0)+\dfrac{\theta^{\prime}}{\varepsilon}\frac{\partial
n_{+}^{\varepsilon}}{\partial\theta}(t,0)\right]K_{\varepsilon+}\left(0,\dfrac{\theta-\theta^{\prime}}{\varepsilon}\right)d\theta^{\prime}d\theta
dt+O(\varepsilon)$
$\displaystyle\xlongequal{z=\frac{\theta-\theta^{\prime}}{\varepsilon}}$
$\displaystyle-\dfrac{1}{\varepsilon}\int_{-\pi}^{-\pi+\varepsilon}\\!\\!\int_{\frac{\theta-\pi}{\varepsilon}}^{-\frac{2\pi}{\varepsilon}+1}\\!\\!-K_{\varepsilon+}\left(\pi,z\right)dzd\theta\int_{0}^{T}\\!\\!\\!\Phi(t,L,-\pi)\frac{1}{\varepsilon}n_{+}^{\varepsilon}(t,\pi)dt$
$\displaystyle-\dfrac{1}{\varepsilon}\int_{-\pi}^{-\pi+\varepsilon}\\!\\!\int_{\frac{\theta-\pi}{\varepsilon}}^{-\frac{2\pi}{\varepsilon}+1}\\!\\!-\left(\frac{\theta-\pi}{\varepsilon}-z\right)K_{\varepsilon+}\left(\pi,z\right)dzd\theta\int_{0}^{T}\\!\\!\\!\Phi(t,L,-\pi)\frac{\partial
n_{+}^{\varepsilon}(t,\pi)}{\partial\theta}dt$
$\displaystyle-\dfrac{1}{\varepsilon}\int_{-\varepsilon}^{0}\int_{-1}^{\frac{\theta}{\varepsilon}}-K_{\varepsilon+}\left(0,z\right)dzd\theta\int_{0}^{T}\\!\\!\\!\Phi(t,L,0)\frac{1}{\varepsilon}n_{+}^{\varepsilon}(t,0)dt$
$\displaystyle-\dfrac{1}{\varepsilon}\int_{-\varepsilon}^{0}\int_{-1}^{\frac{\theta}{\varepsilon}}-\left(\frac{\theta}{\varepsilon}-z\right)K_{\varepsilon+}\left(0,z\right)dzd\theta\int_{0}^{T}\\!\\!\\!\Phi(t,L,0)\frac{1}{\varepsilon}\frac{\partial
n_{+}^{\varepsilon}(t,0)}{\partial\theta}dt+O(\varepsilon)$ $\displaystyle=$
$\displaystyle-\\!\\!\int_{-1}^{0}-zK_{\varepsilon+}(\pi,z)dz\int_{0}^{T}\\!\\!\\!\Phi(t,L,-\pi)\frac{1}{\varepsilon}n_{+}^{\varepsilon}(t,\pi)dt-\\!\\!\int_{-1}^{0}-zK_{\varepsilon+}(0,z)dz\int_{0}^{T}\\!\\!\\!\Phi(t,L,0)\frac{1}{\varepsilon}n_{+}^{\varepsilon}(t,0)dt$
$\displaystyle+\frac{1}{2}\int_{-1}^{0}z^{2}K_{\varepsilon+}(\pi,z)dz\int_{0}^{T}\\!\\!\\!\Phi(t,L,-\pi)\frac{\partial
n_{+}^{\varepsilon}(t,\pi)}{\partial\theta}dt$
$\displaystyle-\frac{1}{2}\int_{-1}^{0}z^{2}K_{\varepsilon+}(0,z)dz\int_{0}^{T}\\!\\!\\!\Phi(t,L,0)\frac{\partial
n_{+}^{\varepsilon}(t,0)}{\partial\theta}dt+O(\varepsilon).$

Using (4.12), (4.21) and periodicity of $\Phi(t,L,\theta)$ in $\theta$, we
have

$\int_{0}^{T}\\!\\!\\!\int_{-\pi}^{0}\\!V\sin\theta
n^{\varepsilon}(t,L,\theta)\Phi(t,L,\theta)d\theta
dt=D_{+}(\theta)\int_{0}^{T}\\!\\!\\!\Phi(t,L,\theta)\frac{\partial
n_{+}^{\varepsilon}}{\partial\theta}(t,\theta)dt\bigg{|}_{0}^{\\!\pi}.$ (4.22)

Similarly, on $y=-L$ we have

$\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta
n^{\varepsilon}(t,-L,\theta)\Phi(t,-L,\theta)d\theta
dt=D_{-}(\theta)\int_{0}^{T}\\!\\!\\!\Phi(t,-L,\theta)\frac{\partial
n_{-}^{\varepsilon}}{\partial\theta}(t,\theta)dt\bigg{|}_{-\pi}^{0}.$ (4.23)

We can take $\Phi(t,L,0)=\Phi(t,L,\pi)=\Phi(t,-L,-\pi)=\Phi(t,-L,0)=0$, then
we pass to the limit and obtain the weak form of (4.2b).

On the other hand, when we take $\Phi(t,L,\theta)=0$ for $\theta\in(0,\pi)$
and $\Phi(t,-L,\theta)=0$ for $\theta\in(-\pi,0)$, we get

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}$
$\displaystyle\\!-n^{\varepsilon}\left[\frac{\partial\Phi}{\partial
t}+V\sin\theta\frac{\partial\Phi}{\partial
y}-D(y,\theta)\frac{\partial^{2}\Phi}{\partial\theta^{2}}\right]d\theta dydt$
$\displaystyle=$
$\displaystyle-\\!\\!\int_{0}^{T}\\!\\!\\!\int_{-\pi}^{0}\\!V\sin\theta
n^{\varepsilon}(t,L,\theta)\Phi(t,L,\theta)d\theta
dt+\int_{0}^{T}\\!\\!\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta
n^{\varepsilon}(t,-L,\theta)\Phi(t,-L,\theta)d\theta dt$
$\displaystyle\xlongequal{\eqref{eq3.21},\eqref{eq3.22}}$ $\displaystyle-
D_{+}(\theta)\int_{0}^{T}\\!\\!\\!\Phi(t,L,\theta)\frac{\partial
n_{+}^{\varepsilon}}{\partial\theta}(t,\theta)dt\bigg{|}_{0}^{\\!\pi}+D_{-}(\theta)\int_{0}^{T}\\!\\!\\!\Phi(t,-L,\theta)\frac{\partial
n_{-}^{\varepsilon}}{\partial\theta}(t,\theta)dt\bigg{|}_{-\pi}^{0},$

then we pass to the limit and get the weak form of (4.2d) and (4.2e). Hence we
have completed the proof. ∎

##  5 Numerical Simulations

We perform numerical simulations of the CRTM (2.1) in order to illustrate the
theoretical results and investigate some qualitative phenomena. Dimensionless
parameters are used in our simulations. The moving speed is fixed as $V=20$,
the distance between the two horizontal plates is $2L=20$, and the transition
kernel is set as in (4.10) and (4.11). If not being specified, we take
$k(y,\theta)=1$ in the following examples for convenience. All the simulations
were conducted by the method described in 5.1-5.3 and were performed on the
“Siyuan Mark-I” cluster at Shanghai Jiao Tong University, which comprises 2 ×
Intel Xeon ICX Platinum 8358 CPU (2.6 GHz, 32 cores) and 512 GB memory per
node.

$y$$\theta$$-L$$L$$\pi$$-\pi$$1$$2I$$1$$2J$ Figure 5-3: Stencil of the semi-
discretized scheme. Here nodes for $p(t,y_{i},\theta_{j})$ inside $\Omega$ are
black, while nodes for $p_{\pm}$ are blue and red respectively.

###  5.1 Description of numerical scheme

We consider an uniform mesh in both $y$ and $\theta$ in a rectangular domain
$\Omega=[-L,L]\times[-\pi,\pi]$. The two integers $N_{y}$ and $N_{\theta}$
define the mesh sizes of $\Omega$, $\Delta y=L/N_{y}$ and
$\Delta\theta=\pi/N_{\theta}$ along the $y$ and $\theta$ axes, respectively.
The index sets are defined as

$I_{y}=\\{1,2,\cdots,2N_{y}\\},~{}~{}I_{\theta+}=\\{N_{\theta+1},\cdots,2N_{\theta}\\},~{}~{}I_{\theta-}=\\{1,\cdots,N_{\theta}\\},~{}~{}I_{\theta}=I_{\theta+}\cup
I_{\theta-}.$ (5.1)

The grids inside the computational domain are

$(y_{i},\theta_{j})=((i-N_{y}-1/2)\Delta
y,~{}(j-N_{\theta}-1/2)\Delta\theta),~{}~{}i\in I_{y},~{}~{}j\in I_{\theta},$
(5.2)

and the nodes at the boundaries are

$(\pm L,~{}\theta_{j}),~{}~{}j\in I_{\theta}.$ (5.3)

The unknowns are $n_{\pm,j}\approx n_{\pm}^{\varepsilon}(t,\theta_{j})$ and

$n_{i,j}\approx\dfrac{1}{\Delta y}\int_{y_{i}-\Delta y/2}^{y_{i}+\Delta
y/2}n^{\varepsilon}(t,y_{i},\theta_{j})dy.$ (5.4)

We use upwind discretization for $\frac{\partial n}{\partial y}$ which has
first-order convergence w.r.t $y$, and forward Euler method for
$\frac{\partial n}{\partial t}$ which has first-order convergence w.r.t $t$.
The semi-discretized scheme for the scaled model (4.3) writes

$\begin{cases}\dfrac{dn_{i,j}}{dt}+V\sin\theta_{j}\dfrac{n_{i,j}-n_{i-1,j}}{\Delta
y}+\dfrac{1}{\varepsilon^{2}}\left(k_{i,j}n_{i,j}-\dfrac{1}{\varepsilon}\sum\limits_{j^{\prime}\in
I_{\theta}}K_{i,j^{\prime},j}n_{i,j^{\prime}}\right)=0,~{}~{}i\in
I_{y},~{}~{}j\in I_{\theta+},\\\\[12.0pt]
\dfrac{dn_{i,j}}{dt}+V\sin\theta_{j}\dfrac{n_{i+1,j}-n_{i,j}}{\Delta
y}+\dfrac{1}{\varepsilon^{2}}\left(k_{i,j}n_{i,j}-\dfrac{1}{\varepsilon}\sum\limits_{j^{\prime}\in
I_{\theta}}K_{i,j^{\prime},j}n_{i,j^{\prime}}\right)=0,~{}~{}i\in
I_{y},~{}~{}j\in I_{\theta-},\\\\[12.0pt]
\dfrac{dn_{+,j}}{dt}+\dfrac{1}{\varepsilon^{2}}\left(k_{2N_{y},j}n_{+,j}-\dfrac{1}{\varepsilon}\sum\limits_{j^{\prime}\in
I_{\theta+}}K_{+,j^{\prime},j}n_{+,j^{\prime}}\right)=V\sin\theta_{j}n_{2N_{y},j},~{}~{}j\in
I_{\theta+},\\\\[12.0pt]
\dfrac{dn_{-,j}}{dt}+\dfrac{1}{\varepsilon^{2}}\left(k_{1,j}n_{-,j}-\dfrac{1}{\varepsilon}\sum\limits_{j^{\prime}\in
I_{\theta-}}K_{-,j^{\prime},j}n_{-,j^{\prime}}\right)=-V\sin\theta_{j}n_{1,j},~{}~{}j\in
I_{\theta-},\end{cases}$ (5.5)

where

$\begin{split}&k_{i,j}=k(y_{i},\theta_{j}),\qquad
K_{i,j,j^{\prime}}=\int_{\theta_{j^{\prime}}-\frac{\Delta\theta}{2}}^{\theta_{j^{\prime}}+\frac{\Delta\theta}{2}}K\left(y_{i},\theta_{j},\dfrac{\theta^{\prime}-\theta_{j}}{\varepsilon}\right){\rm
d}\theta^{\prime},\\\ &k_{\pm,j}=k_{\pm}(\theta_{j}),\qquad
K_{\pm,j,j^{\prime}}=\int_{\theta_{j^{\prime}}-\frac{\Delta\theta}{2}}^{\theta_{j^{\prime}}+\frac{\Delta\theta}{2}}K_{\pm}\left(\theta_{j},\dfrac{\theta^{\prime}-\theta_{j}}{\varepsilon}\right){\rm
d}\theta^{\prime}.\end{split}$ (5.6)

Since both $K$ and $n^{\prime}$ have no singularity, the integrals in (5.6)
are discretized using either the adaptive Gauss-Legendre integral quadrature
[41] (function ”quadgk” in MATLAB) or the trapezoid rule [45] for their
efficiency. The boundary conditions in (4.6b) and (4.6c) are discretized as

$\begin{cases}-V\sin\theta_{j}n_{2N_{y}+1,j}=\dfrac{1}{\varepsilon^{2}}\sum\limits_{j^{\prime}\in
I_{\theta+}}\dfrac{1}{\varepsilon}K_{+,j^{\prime},j}n_{+,j^{\prime}},~{}\quad
j\in I_{\theta-},\\\\[12.0pt]
V\sin\theta_{j}n_{0,j}=\dfrac{1}{\varepsilon^{2}}\sum\limits_{j^{\prime}\in
I_{\theta-}}\dfrac{1}{\varepsilon}K_{-,j^{\prime},j}n_{-,j^{\prime}},~{}\quad
j\in I_{\theta+}.\end{cases}$ (5.7)

Non-negative initial data $n_{i,j}(0)$ and $n_{\pm,j}(0)$ are chosen such that
the condition

$\Delta y\Delta\theta\sum_{(i,j)\in I_{y}\times
I_{\theta}}n_{i,j}(0)+\Delta\theta\sum_{j\in
I_{\theta+}}n_{+,j}(0)+\Delta\theta\sum_{j\in I_{\theta-}}n_{-,j}(0)=1$ (5.8)

is satisfied and it is easy to verify that the numerical scheme conserves the
discrete total mass defined by

$\Delta y\Delta\theta\sum_{(i,j)\in I_{y}\times
I_{\theta}}n_{i,j}(t)+\Delta\theta\sum_{j\in
I_{\theta+}}n_{+,j}(t)+\Delta\theta\sum_{j\in I_{\theta-}}n_{-,j}(t).$ (5.9)

######  Remark 5.1.

Here we have used

$k_{i,j}=\frac{1}{\Delta\theta\Delta y}\int_{y_{i}-\frac{\Delta
y}{2}}^{y_{i}+\frac{\Delta
y}{2}}\int_{\theta_{j}-\frac{\Delta\theta}{2}}^{\theta_{j}+\frac{\Delta\theta}{2}}k(y,\theta){\rm
d}\theta{\rm d}y,$ (5.10) $K_{i,j,j^{\prime}}=\frac{1}{\Delta\theta^{2}\Delta
y}\int_{y_{i}-\frac{\Delta y}{2}}^{y_{i}+\frac{\Delta
y}{2}}\int_{\theta_{j}-\frac{\Delta\theta}{2}}^{\theta_{j}+\frac{\Delta\theta}{2}}\int_{\theta_{j^{\prime}}-\frac{\Delta\theta}{2}}^{\theta_{j^{\prime}}+\frac{\Delta\theta}{2}}K\left(y,\theta,\dfrac{\theta^{\prime}-\theta}{\varepsilon}\right){\rm
d}\theta^{\prime}{\rm d}\theta{\rm d}y,$ (5.11)

to approximate $k(y,\theta)$, $K(y,\theta,\theta^{\prime})$. The key point for
mass conservation is to keep

$k_{i,j}=\dfrac{1}{\varepsilon}\sum_{j^{\prime}\in
I_{\theta}}K_{i,j,j^{\prime}},\qquad
k_{\pm,j}=\dfrac{1}{\varepsilon}\sum_{j^{\prime}\in
I_{\theta}}K_{\pm,j,j^{\prime}}.$ (5.12)

When $t\rightarrow\infty$, the solution of the time-dependent model will
numerically converge to the solution of the following discretized steady-state
equation

$\begin{cases}V\sin\theta_{j}\dfrac{m_{i,j}-m_{i-1,j}}{\Delta
y}+\dfrac{1}{\varepsilon^{2}}\left(k_{i,j}m_{i,j}-\dfrac{1}{\varepsilon}\sum\limits_{j^{\prime}\in
I_{\theta}}K_{i,j^{\prime},j}m_{i,j^{\prime}}\right)=0,~{}~{}i\in
I_{y},~{}~{}j\in I_{\theta+},\\\
V\sin\theta_{j}\dfrac{m_{i+1,j}-m_{i,j}}{\Delta
y}+\dfrac{1}{\varepsilon^{2}}\left(k_{i,j}m_{i,j}-\dfrac{1}{\varepsilon}\sum\limits_{j^{\prime}\in
I_{\theta}}K_{i,j^{\prime},j}m_{i,j^{\prime}}\right)=0,~{}~{}i\in
I_{y},~{}~{}j\in I_{\theta-},\\\
\dfrac{1}{\varepsilon^{2}}\left(k_{2N_{y},j}m_{+,j}-\dfrac{1}{\varepsilon}\sum\limits_{j^{\prime}\in
I_{\theta+}}K_{+,j^{\prime},j}m_{+,j^{\prime}}\right)=V\sin\theta_{j}m_{2N_{y},j},~{}~{}j\in
I_{\theta+},\\\
\dfrac{1}{\varepsilon^{2}}\left(k_{1,j}m_{-,j}-\dfrac{1}{\varepsilon}\sum\limits_{j^{\prime}\in
I_{\theta-}}K_{-,j^{\prime},j}m_{-,j^{\prime}}\right)=-V\sin\theta_{j}m_{1,j},~{}~{}j\in
I_{\theta-},\end{cases}$ (5.13)

with boundary conditions

$\begin{cases}-V\sin\theta_{j}m_{2N_{y}+1,j}=\dfrac{1}{\varepsilon^{2}}\sum\limits_{j^{\prime}\in
I_{\theta+}}\dfrac{1}{\varepsilon}K_{+,j^{\prime},j}m_{+,j^{\prime}},~{}~{}j\in
I_{\theta-},\\\
V\sin\theta_{j}m_{0,j}=\dfrac{1}{\varepsilon^{2}}\sum\limits_{j^{\prime}\in
I_{\theta-}}\dfrac{1}{\varepsilon}K_{-,j^{\prime},j}m_{-,j^{\prime}},~{}~{}j\in
I_{\theta+},\end{cases}$ (5.14)

whose total mass satisfies

$\Delta y\Delta\theta\sum_{(i,j)\in I_{y}\times
I_{\theta}}m_{i,j}+\Delta\theta\sum_{j\in
I_{\theta+}}m_{+,j}+\Delta\theta\sum_{j\in I_{\theta-}}m_{-,j}=1.$ (5.15)

We define the following weighted $L^{2}$ norm to measure the numerical errors

$\begin{split}\left\|\left(e\right)\right\|_{L_{w}^{2}}^{2}&=\sum_{i\in
I_{y}}\sum_{j\in I_{\theta}}\Delta\theta^{2}\Delta y^{2}e_{i,j}^{2},\\\
\left\|\left(e_{+},e_{-}\right)\right\|_{L_{w}^{2}}^{2}&=\sum_{j\in
I_{\theta_{+}}}\Delta\theta^{2}e_{+,j}^{2}+\sum_{j\in
I_{\theta_{-}}}\Delta\theta^{2}e_{-,j}^{2}.\end{split}$ (5.16)

(5-1) gives the numerical errors of $n_{i,j}$ and $n_{\pm,j}$ at time $t=0.2$,
calculated with different mesh sizes, where the reference solution is computed
with a fine mesh $256\times 256$ and cubic interpolation implemented in
Matlab. Here we take $\varepsilon=0.05$ and $\Delta t=0.001$. From Fig. 5-4,
first-order convergence can be observed for both $y$ and $\theta$.

|  | $\Delta\theta$
---|---|---
|  | $\pi/8$ | $\pi/16$ | $\pi/32$ | $\pi/64$ | $\pi/128$
$\Delta y$ | $L/8$ | 5.45E-2 | 2.78E-2 | 1.38E-2 | 6.82E-3 | 3.42E-3
$L/16$ | 2.57E-2 | 1.36E-2 | 6.92E-3 | 3.35E-3 | 1.68E-3
$L/32$ | 1.49E-2 | 7.13E-3 | 3.65E-3 | 1.70E-3 | 8.54E-4
$L/64$ | 7.80E-3 | 4.09E-3 | 2.09E-3 | 9.03E-4 | 4.58E-4
$L/128$ | 5.02E-3 | 2.60E-3 | 1.33E-3 | 5.06E-4 | 2.59E-4
Table 5-1:
$\left\|\left(e\right)\right\|_{L_{w}^{2}}^{2}+\left\|\left(e_{+},e_{-}\right)\right\|_{L_{w}^{2}}^{2}.$
The weighted $L^{2}$ norm of the numerical errors as in (5.16). First order
convergence rate is observed with respect to both $\Delta\theta$ and $\Delta
y$.
Figure 5-4: Plots of
$\left\|\left(e\right)\right\|_{L_{w}^{2}}^{2}+\left\|\left(e_{+},e_{-}\right)\right\|_{L_{w}^{2}}^{2}$
in 5-1, as a function of (A) $\Delta y$ and (B) $\Delta\theta$. The scheme is
first order in both $y$ and $\theta$.

###  5.2 Numerical comparisons between the SDE and scattering models

The SDE (1.1) is discretized with the Euler-Maruyama-like scheme [26]. The
computational domain is $\Omega=\\{(y,\theta)|-L\leq y\leq
L,~{}-\pi\leq\theta\leq\pi\\}$ and $N_{\text{cell}}=5\times 10^{7}$
independent cells are employed for data statistics. Each cell is represented
by its position $Y_{i}$ along the $y$-axis and its orientation $\Theta_{i}$.
The initial positions for all cells are uniformly distributed on $(-L,L)$, and
the initial orientations are uniformly distributed on $(-\pi,\pi)$. Let
$\Delta t$ be the integration time step. The algorithm for evolving
$(Y_{i}^{n},\Theta_{i}^{n})(i=1,2,\cdots,N_{\text{cell}})$ at time
$t^{n}=n\Delta t$ is described in 1.

Algorithm 1 Monte Carlo simulation

1:For each cell, generate a number of jump $P_{i}^{n}$ from a Poisson process
with intensity  $\Gamma_{i}^{n}=\left\\{\begin{aligned}
&k(Y_{i}^{n-1},\Theta_{i}^{n-1}),&&\lvert Y_{i}^{n-1}\rvert<L,\\\
&k_{\pm}(\Theta_{i}^{n-1}),&&Y_{i}^{n-1}=\pm L,\end{aligned}\right.$ (5.17) at
time $\Delta t$, i.e.  $\mathbb{P}(P_{i}^{n}=J)=\frac{(\Gamma_{i}^{n}\Delta
t)^{J}}{J!}e^{-\Gamma_{i}^{n}\Delta t},\quad J\in\mathbb{N}.$ (5.18)

2:Sample $P_{i}^{n}$ random increments
$\\{\Delta\Theta_{i,j}^{n}\\}_{j=1,\cdots,P_{i}^{n}}$, which are independent
identically distributed from the distribution  $\left\\{\begin{aligned}
&\frac{K_{\varepsilon}\left(Y_{i}^{n-1},\Theta_{i}^{n-1},\dfrac{\Delta\Theta}{\varepsilon}\right)}{k(Y_{i}^{n-1},\Theta_{i}^{n-1})},&&\lvert
Y_{i}^{n-1}\rvert<L,\\\
&\frac{K_{\varepsilon\pm}\left(\Theta_{i}^{n-1},\dfrac{\Delta\Theta}{\varepsilon}\right)}{k_{\pm}(\Theta_{i}^{n-1})},&&Y_{i}^{n-1}=\pm
L.\end{aligned}\right.$ (5.19)

3:Sample $P_{i}^{n}$ i.i.d random numbers $\\{T_{j}\\}_{j=1,\cdots,P_{i}^{n}}$
which are uniformly distributed over $[0,1]$. Let $T_{0}=0$ and
$T_{P_{i}^{n}+1}=1$.

4:Initialize $Y_{i}^{n}=Y_{i}^{n-1}$, $\Theta_{i}^{n}=\Theta_{i}^{n-1}$.

5:for $j=1,2,\cdots P_{i}^{n}$ do

6: Compute
$\tilde{Y}_{i}^{n}=Y_{i}^{n}+V\sin(\Theta_{i}^{n})(T_{j}-T_{j-1})\Delta t$.
Evolve the position $Y_{i}^{n}$ by  $Y_{i}^{n}=\left\\{\begin{aligned}
&sign(\tilde{Y}_{i}^{n})L,&&\lvert\tilde{Y}_{i}^{n}\rvert>L,\\\
&\tilde{Y}_{i}^{n},&&\lvert\tilde{Y}_{i}^{n}\rvert\leq
L,.\end{aligned}\right.$ (5.20)

7: Evolve the orientation $\Theta_{i}^{n}$ by
$\Theta_{i}^{n}=\Theta_{i}^{n}+\Delta\Theta_{i,j}^{n}$.

8:end for

The above procedure is independent for different cells and thus is easy to
implement in parallel. We take $\Delta t=0.001$ and $\varepsilon=0.05$ for
both the PDE and the SDE. For the SDE, we count the number of particles inside
each cell of a $200\times 200$ uniform mesh and normalize them by the total
number of particles.

Figure 5-5: Density in the bulk $n_{i,j}$ at different times calculated by the
PDE model and the SDE model.

The discretization described in 5.1 is used to compute the numerical solution
of the PDE model. In practice, we take $N_{y}=N_{\theta}=100$ such that the
numerical results of the two models have the same resolution. The initial
condition is consistent with the SDE model, i.e. $n_{i,j}=1/4\pi L$ for all
$i\in I_{y},j\in I_{\theta}$. The time evolution of bulk density
$n(t,y,\theta)$ by numerical simulations based on both SDE and PDE models is
shown in Fig. 5-5. The results of the two models are comparable. There exist
two wells when $y=-L,\theta\in(0,\pi)$ and $y=L,\theta\in(-\pi,0)$, since
cells with positive (negative) $\theta$ tend to leave the boundary. When $t$
is larger, the well will stretch across the bulk region and the cell will
aggregate on the boundaries.

Figure 5-6: Density in the bulk near boundary $n_{1,j}$, $n_{2I_{y},j}$ and
boundary density $n_{-,j}$ and $n_{+,j}$ (from bottom to top) at different
times calculated by the PDE model (line) and SDE model (histogram). Notice
that $n_{1,j}$ and $n_{2N_{y},j}$ is close to $y=-L$ and $y=L$, but not at the
boundary, so the boundary condition (1.3) is not satisfied for $n_{1,j}$ and
$n_{2N_{y},j}$, respectively.

To verify the behavior of $n(t,y,\theta)$ near the boundaries, $n_{1,j}$ and
$n_{2I_{y},j}$ are plotted in Fig. 5-6. As $t$ becomes larger, the peaks at
$(\pm L,0)$, $(\pm L,\pm\pi)$ become sharper. The height of peaks stops
increasing when $t$ reaches $4$. The probability density distributions of
cells on both left and right boundaries are plotted in Fig. 5-6 as well, we
can see that their maximum increase with time until they reach a constant.

###  5.3 Asymptotic limit from CRTM towards CFPM

In 4, we have proved that the weak limit of the solution $n^{\varepsilon}$ of
the scaled CRTM (4.3)–(4.6) satisfies the Fokker-Planck system, as well as the
boundary conditions, in the distributional sense. In this subsection, we give
some numerical examples to illustrate this convergence. We take $\Delta
t=0.0001$, $N_{y}=N_{\theta}=400$, $t=4.0$, and vary the $\varepsilon$. The
results are shown in Fig. 5-7. When $\varepsilon$ decreases, more cells tend
to accumulate on the boundary.

Figure 5-7: Density in the bulk $n_{i,j}$, density in the bulk near the
boundary $n_{2I_{y},j}$, and boundary density $n_{+,j}$ at time $t=4.0$
calculated by the CRTM using different $\varepsilon$.

To further understand the behavior of the motion of cells, we examine the
ratio of the number of cells in the bulk, $M_{i}(t)$, to the number of cells
on the boundary, $M_{b}(t)$, where $M_{i}(t)$ and $M_{b}(t)$ are defined
respectively by

$M_{i}(t)=\Delta y\Delta\theta\sum_{(i,j)\in I_{y}\times
I_{\theta}}n_{i,j}(t),\quad M_{b}(t)=\Delta\theta\sum_{j\in
I_{\theta+}}n_{+,j}(t)+\Delta\theta\sum_{j\in I_{\theta-}}n_{-,j}(t).$ (5.21)

The results at $t=4.0$ at which the CRTM achieves a steady state are shown in
Fig. 5-8. It is observed that the ratio $M_{i}/M_{b}$ decays almost
exponentially with the increase of $\varepsilon$, indicating that the cells
tend to aggregate on the boundary when $\varepsilon$ is large. When
$\varepsilon\rightarrow 0$, the convergence of $M_{i}/M_{b}$ can be observed,
which verifies our theoretical results.

Figure 5-8: The ratio of the number of cells in the bulk to the number of
cells near the boundary, $M_{i}/M_{b}$, at steady state $t=4.0$ calculated by
the CRTM using different $\varepsilon$.

##  6 Conclusion

Motivated by recent biophysical experiments [30, 29, 12, 13], we study the
boundary aggregation phenomenon of run-and-tumble particles moving in a
confined environment. We build a run-and-tumble model with boundary conditions
that are more biologically relevant. The relative entropy inequality is
established and it is theoretically proved that the system will converge to a
steady state. Using similar scaling as in the derivation of the Fokker-Planck
limit from the radiative transport equation, we can recover the CFPM studied
in [22], in the strong scattering and forward peaked regime. In particular,
the same confinement boundary conditions can be obtained, which constitutes
the main difference with the classical Fokker-Planck limit derivation. The
asymptotics allows us to compute the coefficients of the CFPM based on the
individual rate of tumbling. Several numerical comparisons between the CRTM
and the CFPM are presented.

Physicists are particularly interested in the movement of active particles in
confined environments, some experiments are conducted [16, 42], and a lot of
theoretical works with simulations can be found in the physics literature [4,
1, 31, 37, 38, 49]. We intend to extend the model to more complex geometries.

##  7 Appendix: Proof of Theorem 3.1

###### Proof.

First Step: proof $\omega_{\infty},\omega_{\infty,\pm}$ are independent from
$\theta$ by relative entropy

At first, choose $H(x)=x^{2}$ in (3.5), we can obtain

$I(t)=\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!m\omega^{2}{\rm d}\theta{\rm
d}y+\int_{0}^{\\!\pi}\\!\\!m_{+}\omega_{+}^{2}{\rm
d}\theta+\int_{-\pi}^{0}\\!m_{-}\omega_{-}^{2}{\rm d}\theta\quad\text{
and}\quad\frac{{\rm d}I(t)}{{\rm d}t}\leq 0.$

Since initial data are bounded as in Corollary 3.2, we infer

$I(0)\leq(4\pi L+2\pi)\Gamma^{2}<\infty.$

Since $I(t)$ is non-negative, it converges to a limit as $t\rightarrow\infty$,
therefore

$\lim\limits_{\tau\rightarrow\infty}\int_{\tau}^{\infty}\left(D_{H}^{(1)}+D_{H}^{(2)}+D_{H}^{(3)}\right)\left[\omega,\omega_{\pm}\right]{\rm
d}t=\lim\limits_{\tau\rightarrow\infty}\int_{\tau}^{\infty}\frac{{\rm
d}I(t)}{{\rm d}t}{\rm d}t=0.$

For $\omega_{n_{k}}$ and $\omega_{n_{k},\pm}$, we have

$\displaystyle\lim\limits_{n_{k}\rightarrow\infty}$
$\displaystyle\int_{T}^{\infty}\\!\left(D_{H}^{(1)}+D_{H}^{(2)}+D_{H}^{(3)}\right)\left[\omega_{n_{k}},\omega_{n_{k},\pm}\right](t){\rm
d}t=\lim\limits_{n_{k}\rightarrow\infty}\int_{T+n_{k}}^{\infty}\left(D_{H}^{(1)}+D_{H}^{(2)}+D_{H}^{(3)}\right)\left[\omega,\omega_{\pm}\right](t){\rm
d}t=0.$

Due to the non-negativity of $D_{H}^{(1)},D_{H}^{(2)},D_{H}^{(3)}$ in (3.5),
we can extract a subsequence $n_{k}$ satisfying

$\displaystyle\lim\limits_{n_{k}\rightarrow\infty}\omega_{n_{k}}(t,y,\theta)-\omega_{n_{k}}(t,\theta^{\prime},y)=0,$
a.e. in $\displaystyle{[T,\infty)\times\Omega,}$ (A.1a)
$\displaystyle\lim\limits_{n_{k}\rightarrow\infty}\omega_{n_{k},\pm
L}(t,\theta)-\omega_{n_{k},\pm}(t,\theta)=0,$ a.e. in
$\displaystyle{[T,\infty)\times\Omega_{\pm},}$ (A.1b)
$\displaystyle\lim\limits_{n_{k}\rightarrow\infty}\omega_{n_{k},\pm}(t,\theta)-\omega_{n_{k},\pm}(t,\theta^{\prime})=0,$
a.e. in $\displaystyle{[T,\infty)\times\Omega_{\pm}}.$ (A.1c)

Second Step: proof $\omega_{\infty,\pm}$ are independent of time
From (3.4), it follows that $m_{\pm}$ are independent of time $t$ and thus we
know that for all $k\in\mathbb{N},\omega_{n_{k}}$ and $\omega_{n_{k},+}$
satisfy

$\displaystyle m_{+}\frac{\partial\omega_{n_{k},+}}{\partial t}+$
$\displaystyle
k_{+}(\theta)\omega_{n_{k},+}m_{+}-\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)\omega_{n_{k},+}^{\prime}m_{+}^{\prime}d\theta^{\prime}=V\sin\theta\omega_{n_{k},+L}m_{+L}.$
(A.2)

Next, we choose compact supported test function $\phi_{+}(t,\theta)\in
L^{1}\left([T,\infty)\times\Omega_{+}\right)\bigcap
C^{\infty}\left([T,\infty)\times\Omega_{+}\right)$, multiply both sides of
(A.2), integrate on $[T,\infty)\times(0,\pi)$, then

$\displaystyle\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!m_{+}\omega_{n_{k},+}\frac{\partial\phi_{+}}{\partial
t}d\theta d{t}=$
$\displaystyle\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!\phi_{+}[V\sin\theta\omega_{n_{k},+L}m_{+L}]d\theta
d{t}$
$\displaystyle-\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!\phi_{+}[k_{+}(\theta)\omega_{n_{k},+}m_{+}-\int_{0}^{\\!\pi}\\!\\!K_{+}\left(\theta^{\prime},\theta-\theta^{\prime}\right)\omega_{n_{k},+}^{\prime}m_{+}^{\prime}d\theta^{\prime}]d\theta
d{t}.$

Similar to the processing in the relative entropy inequality proof, we can
obtain

$\displaystyle\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!m_{+}\omega_{n_{k},+}\frac{\partial\phi_{+}}{\partial
t}d\theta
d{t}=\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!\phi_{+}[V\sin\theta(\omega_{n_{k},+L}-\omega_{n_{k},+})m_{+L}]d\theta
d{t}$
$\displaystyle+\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!\phi_{+}\int_{0}^{\\!\pi}\\!\\!K\left(y,\theta^{\prime},\theta-\theta^{\prime}\right)(\omega_{n_{k},+}^{\prime}-\omega_{n_{k},+})m_{+}^{\prime}d\theta^{\prime}d\theta
d{t}.$

Let $n_{k}\rightarrow\infty$ and pass to limit from (A.1) to get

$\displaystyle\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!m_{+}\omega_{n_{k},+}\frac{\partial\phi_{+}}{\partial
t}d\theta d{t}=0.$

Therefore, $\omega_{\infty,+}$ is independent of $t$. Similarly,
$\omega_{\infty,-}$ is independent of $t$.
Third Step : proof $\omega_{\infty}$ is independent of time $t$ and position
$y$.
Since $m$ is independent of $t$, for all $k\in\mathbb{N},$ we can obtain

$m\frac{\partial\omega_{n_{k}}}{\partial t}+V\sin\theta
m\frac{\partial\omega_{n_{k}}}{\partial
y}+\int_{-\pi}^{\\!\pi}\\!K\left(y,\theta^{\prime},\theta-\theta^{\prime}\right)(\omega_{n_{k}}-\omega_{n_{k}}^{\prime})m^{\prime}d\theta^{\prime}=0.$

Let $n_{k}\rightarrow\infty$, from (A.1), then

$m\frac{\partial\omega_{\infty}}{\partial t}=-V\sin\theta
m\frac{\partial\omega_{\infty}}{\partial y}.$

Since $m$ is strictly positive in $\Omega$ except for a zero-measure subset.
We divide both sides by $m$. At the same time, we multiply it by a test
function $\phi(t,y)\in C^{2}([0,\infty)\times(-L,L))$ and integrate within the
domain, then

$\int_{0}^{\infty}\int_{-L}^{L}\\!\phi\frac{\partial\omega_{\infty}}{\partial
t}{\rm d}y{\rm
d}t=-V\sin\theta\int_{0}^{\infty}\int_{-L}^{L}\\!\phi\frac{\partial\omega_{\infty}}{\partial
y}{\rm d}y{\rm d}t.$

The right-hand side of the above equation depends on $\theta$ while the left-
hand side does not, which indicates that both sides are zero.

Therefore, $\omega_{\infty}$ is independent of $t$ and $y$, Meanwhile,
$\omega_{\infty}$ and $\omega_{\infty,\pm}$ are both constant almost
everywhere in their domain.
Fourth Step: proof $\omega_{\infty}=\omega_{\infty,\pm}$.
Multiplying (7) by a test function $\phi\in C^{2}([0,+\infty)\times\Omega)$,
that is dependent of $t$ and $\theta$ but not in $y$. At the same time,
integrating on $[T,\infty)\times\Omega$

$\displaystyle\int_{T}^{\infty}\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!m\omega_{n_{k}}\frac{\partial\phi}{\partial
t}{\rm d}\theta{\rm d}y{\rm d}t=$
$\displaystyle-\int_{T}^{\infty}\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta\int_{-L}^{L}\\!m\phi\frac{\partial\omega_{n_{k}}}{\partial
y}{\rm d}y{\rm d}\theta{\rm d}t$
$\displaystyle\quad-\int_{T}^{\infty}\\!\int_{-\pi}^{\\!\pi}\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!K\left(y,\theta^{\prime},\theta-\theta^{\prime}\right)(\omega_{n_{k}}-\omega_{n_{k}}^{\prime})m^{\prime}d\theta^{\prime}{\rm
d}yd\theta d{t}.$

Combining the boundary conditions of the original equation and the steady
state equation and integrating by parts, we get

$\displaystyle-\int_{T}^{\infty}\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta\int_{-L}^{L}\\!m\phi\frac{\partial\omega_{n_{k}}}{\partial
y}{\rm d}y{\rm d}\theta{\rm d}t$
$\displaystyle=-\int_{T}^{\infty}\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta[m_{+L}\phi_{+L}\omega_{n_{k},+L}-m_{-L}\phi_{-L}\omega_{n_{k},-L}]d\theta
d{t}+\int_{T}^{\infty}\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta\omega_{n_{k}}[m_{+L}\phi_{+L}-m_{-L}\phi_{-L}]d\theta
d{t}$
$\displaystyle=\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta\phi_{+L}(\omega_{n_{k},+}-\omega_{n_{k},+L})m_{+L}d\theta
d{t}-\int_{T}^{\infty}\\!\int_{-\pi}^{0}\\!V\sin\theta\phi_{-L}(\omega_{n_{k},-}-\omega_{n_{k},-L})m_{-L}d\theta
d{t}.$
$\displaystyle\quad-\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta\phi_{+L}\omega_{n_{k},+}m_{+L}{\rm
d}\theta{\rm
d}t-\int_{T}^{\infty}\\!\int_{-\pi}^{0}\\!V\sin\theta\phi_{+L}\omega_{n_{k},+L}m_{+L}{\rm
d}\theta+\int_{T}^{\infty}\\!\int_{-\pi}^{0}\\!V\sin\theta\phi_{-L}\omega_{n_{k},-}m_{-L}{\rm
d}\theta{\rm d}t$
$\displaystyle\quad+\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta\phi_{-L}\omega_{n_{k},-L}m_{-L}{\rm
d}\theta+\int_{T}^{\infty}\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta\omega_{n_{k}}[m_{+L}\phi_{+L}-m_{-L}\phi_{-L}]d\theta
d{t}.$

Let $n_{k}\rightarrow\infty$ , by (A.1) we obtain

$\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta\phi_{L}(\omega_{n_{k},+}-\omega_{n_{k},+L})m_{+L}d\theta=-\int_{T}^{\infty}\\!\int_{-\pi}^{0}\\!V\sin\theta\phi_{-L}(\omega_{n_{k},-}-\omega_{n_{k},-L})m_{-L}{\rm
d}\theta{\rm d}=0,$

and thus

$\displaystyle\int_{T}^{\infty}\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}$
$\displaystyle\\!\omega_{\infty}\frac{\partial(m\phi)}{\partial t}{\rm
d}\theta{\rm d}y{\rm d}t$
$\displaystyle=-\omega_{\infty,+}\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta\phi_{+L}m_{+L}d\theta
d{t}-\omega_{\infty,+L}\int_{T}^{\infty}\\!\int_{-\pi}^{0}\\!V\sin\theta\phi_{+L}m_{+L}d\theta
d{t}$
$\displaystyle\quad+\omega_{\infty,-}\int_{T}^{\infty}\\!\int_{-\pi}^{0}\\!V\sin\theta\phi_{-L}m_{-L}d\theta
d{t}+\omega_{\infty,-L}\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta\phi_{-L}m_{-L}d\theta
d{t}$
$\displaystyle\quad+\omega_{\infty}\int_{T}^{\infty}\\!\int_{-\pi}^{\\!\pi}\\!V\sin\theta\omega_{n_{k}}[m_{+L}\phi_{+L}-m_{-L}\phi_{-L}]d\theta
d{t}$
$\displaystyle=\left(\omega_{\infty}-\omega_{\infty,+}\right)\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta\phi_{+L}m_{+L}d\theta
d{t}+\left(\omega_{\infty}-\omega_{\infty,+L}\right)\int_{T}^{\infty}\\!\int_{-\pi}^{0}\\!V\sin\theta\phi_{+L}m_{+L}d\theta
d{t}$
$\displaystyle+\left(\omega_{\infty,-}-\omega_{\infty}\right)\int_{T}^{\infty}\\!\int_{-\pi}^{0}\\!V\sin\theta\phi_{-L}m_{-L}d\theta
d{t}+\left(\omega_{\infty,-L}-\omega_{\infty}\right)\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta\phi_{-L}m_{-L}d\theta
d{t}.$

Since $\omega_{\infty}$ is independent on $y,$ we have
$\omega_{\infty,+L}=\omega_{\infty}=\omega_{\infty,-L},$ therefore

$\displaystyle\int_{T}^{\infty}\\!\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}$
$\displaystyle\\!\omega_{\infty}\frac{\partial(m\phi)}{\partial t}{\rm
d}\theta{\rm d}y{\rm d}t$
$\displaystyle=\left(\omega_{\infty}-\omega_{\infty,+}\right)\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta\phi_{+L}m_{+L}d\theta
d{t}+\left(\omega_{\infty,-}-\omega_{\infty}\right)\int_{T}^{\infty}\\!\int_{-\pi}^{0}\\!V\sin\theta\phi_{-L}m_{-L}d\theta
d{t}.$

Here we can choose a function $\phi(t,y,\theta)$ such that for $t$ in
$(0,+\infty)$ and $\theta$ in $(0,\pi)$ is compactly supported,and satisfies

$\int_{T}^{\infty}\\!\int_{-L}^{L}\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta\frac{\partial(m\phi)}{\partial
y}{\rm d}\theta{\rm d}y{\rm
d}t=\left.\int_{T}^{\infty}\\!\int_{0}^{\\!\pi}\\!\\!V\sin\theta(m\phi)\right|_{-L}^{L}{\rm
d}\theta{\rm d}y{\rm d}t\neq 0.$

Therefore, $\omega_{\infty}=\omega_{\infty,+}=C_{1}$. Likely, we have
$\omega_{\infty}=\omega_{\infty,-}=C_{2}$. So, there exist a constant $C$ that
satisfies

$\left.\omega_{\infty}\right|_{\Omega}=\left.\omega_{\infty,\pm}\right|_{\Omega_{\pm}}=C,\quad\text{
a.e. }$

By the conservation of mass,for any $n_{k}\in\mathbb{N},\omega_{n_{k}}$
satisfies

$\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!m\omega_{n_{k}}{\rm d}\theta{\rm
d}y+\int_{0}^{\\!\pi}\\!\\!m_{+}\omega_{n_{k},+}{\rm
d}\theta+\int_{-\pi}^{0}\\!m_{-}\omega_{n_{k},-}{\rm d}\theta=1.$

When $n_{k}\rightarrow\infty$, we obtain

$\int_{-L}^{L}\\!\int_{-\pi}^{\\!\pi}\\!m\omega_{\infty}{\rm d}\theta{\rm
d}y+\int_{0}^{\\!\pi}\\!\\!m_{+}\omega_{\infty,+}{\rm
d}\theta+\int_{-\pi}^{0}\\!m_{-}\omega_{\infty,-}{\rm d}\theta=1.$

Therefore, the constant $C$ is 1.
Fifth Step : proof the system is strongly convergent in time
For a smooth test function $\phi(y,\theta)\in C_{0}^{2}(\Omega)$, we define

$u(t)=\int_{\Omega}m\omega{\rm d}y{\rm d}\theta.$

Multiply both sides of the above formula by $\phi(y,\theta)$, and integrate
within $\Omega$, then

$\displaystyle\int_{\Omega}\frac{\partial(m\phi\omega)}{\partial t}{\rm
d}y{\rm d}\theta$ $\displaystyle=\int_{\Omega}\phi\frac{\partial n}{\partial
t}{\rm d}y{\rm d}\theta$
$\displaystyle=\int_{\Omega}\phi\left(-k(y,\theta)n+\int_{\Omega}K\left(y,\theta^{\prime},\theta-\theta^{\prime}\right)n^{\prime}d\theta^{\prime}-V\sin\theta\frac{\partial
n}{\partial y}\right){\rm d}y{\rm d}\theta\leq C^{\prime}.$

When the initial data meets the requirements of Corollary 3.2, $C^{\prime}$ is
a positive constant thanks to integration by parts. Therefore $u(t)$ is a
Lipschitz continuous function in time, and We have shown that it converges
weakly to $1$. From that we deduce that it is strongly convergent. Similarly,
strong convergence of $\omega_{\pm}$ in time cis obtained.

Therefore the long-term convergence Theorem 3.3 is proved. ∎

## Acknowledgment

Jingyi Fu and Min Tang are partially supported by NSFC 11871340, NSFC12031013,
Shanghai pilot innovation project 21JC1403500. B.P. has received funding from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No 740623).

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