# Well-posedness and stationary states for a crowded active Brownian system
with size-exclusion

Martin Burger  Computational Imaging Group and Helmholtz Imaging, Deutsches
Elektronen-Synchrotron (DESY), Notkestr. 85, 22607 Hamburg, Germany, and
Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg,
Germany<EMAIL_ADDRESS>and  Simon Schulz  Scuola Normale Superiore,
Centro di Ricerca Matematica Ennio De Giorgi, P.zza dei Cavalieri, 3, 56126
Pisa, Italy<EMAIL_ADDRESS>
###### Abstract.

We prove the existence of solutions to a non-linear, non-local, degenerate
equation which was previously derived as the formal hydrodynamic limit of an
active Brownian particle system, where the particles are endowed with a
position and an orientation. This equation incorporates diffusion in both the
spatial and angular coordinates, as well as a non-linear non-local drift term,
which depends on the angle-independent density. The spatial diffusion is non-
linear degenerate and also comprises diffusion of the angle-independent
density, which one may interpret as cross-diffusion with infinitely many
species. Our proof relies on interpreting the equation as the perturbation of
a gradient flow in a Wasserstein-type space. It generalizes the boundedness-
by-entropy method to this setting and makes use of a gain of integrability due
to the angular diffusion. For this latter step, we adapt a classical
interpolation lemma for function spaces depending on time. We also prove
uniqueness in the particular case where the non-local drift term is null, and
provide existence and uniqueness results for stationary equilibrium solutions.

Keywords: Boundedness-by-entropy; active particles; cross-diffusion; gradient
flow.

AMS Subject Classification:  35K65; 35K55; 76M30; 35Q92.

###### Contents

  1. 1 Introduction
  2. 2 Entropy Variable and Transformed Problem
  3. 3 Galerkin Approximation for the Transformed Regularised System
  4. 4 Proof of the Main Existence Theorem
  5. 5 Uniqueness for Null Péclet Number
  6. 6 Existence and Uniqueness of Stationary States
  7. A Interpolation Lemma
  8. B Regularised Initial Data

##  1\. Introduction

We study the following non-local non-linear degenerate equation, which was
derived in [6] as the formal hydrodynamic limit of a many-particle system
modeling the interaction of active Brownian particles endowed with a two-
dimensional position $x\in\Omega=(0,2\pi)^{2}$ and orientation
$\theta\in(0,2\pi)$:

(1.1)
$\displaystyle\partial_{t}f+\mathrm{Pe}\operatorname{div}\big{(}f(1-\rho){\bf
e}(\theta)\big{)}=D_{e}\operatorname{div}\big{(}(1-\rho)\nabla
f+f\nabla\rho\big{)}+\partial^{2}_{\theta}f,$

where $\rho(t,x)=\int_{0}^{2\pi}f(t,x,\theta)\,\mathrm{d}\theta$ is the
_angle-independent density_ , ${\bf e}(\theta)=(\cos\theta,\sin\theta)$, the
real constant parameters $\mathrm{Pe}$ and $D_{e}>0$ are respectively the
Péclet number and spatial diffusion coefficient, supplemented with initial
data $f_{0}$ and periodic boundary conditions in both the angle and space
variables; the periodic cell for the space-angle coordinates is denoted by
$\Upsilon:=\Omega\times(0,2\pi)$. We prove the existence of weak solutions to
(1.1), show uniqueness of such solutions in the case of zero Péclet number,
and provide existence and uniqueness results for stationary states under
various assumptions on the Péclet number.

Active Brownian systems model the motion of self-propelled particles and have
an abundance of applications; for instance in biology, pedestrian and traffic
flows, and the modeling of collective behaviour in general (cf. e.g. [24, 10,
23, 11, 26, 27, 25]). Much like kinetic models, active Brownian systems depict
the _mesoscopic_ dynamics of a group of interacting agents and, indeed,
equation (1.1) presents similarities with kinetic Fokker–Planck equations as
well as non-linear degenerate parabolic systems. At first glance, (1.1)
appears similar to the cross-diffusion system studied in [8]:

(1.2) $\displaystyle\partial_{t}r=\operatorname{div}\big{(}(1-\sigma)\nabla
r+r\nabla\sigma+r(1-\sigma)\nabla V\big{)},$
$\displaystyle\partial_{t}b=\operatorname{div}\big{(}(1-\sigma)\nabla
b+b\nabla\sigma+b(1-\sigma)\nabla W\big{)},$

where $\sigma=r+b$ and $V,W$ are given external potentials. Equation (1.1) may
be interpreted as the analog of this system with infinitely-many species; the
diffusive term $\operatorname{div}f\nabla\rho$ encodes the diffusion of the
uncountable family of angles in the interval $(0,2\pi)$. However, unlike the
system in [8], the equation (1.1) is non-local; both in its diffusive and
drift terms. This non-locality is a feature shared by certain kinetic models,
in particular the Vlasov–Benney equation [4, 16] and the recent contribution
[5]. Moreover, the angle-independent density, which formally satisfies the
strongly parabolic drift-diffusion equation

$\partial_{t}\rho+\operatorname{div}\big{(}(1-\rho){\bf p}\big{)}=\Delta\rho,$

where ${\bf p}=\int_{0}^{2\pi}f{\bf e}(\theta)\,\mathrm{d}\theta$, is expected
to be more regular than $f$ as its governing equation is no longer degenerate.
This mechanism is evocative of velocity averaging lemmas (see, e.g., [22]),
illustrating another parallel between kinetic theory and active Brownian
systems. The zones of degeneracy $\\{\rho=1\\}$ and $\\{f=0\\}$ comprised in
(1.1) make its analysis delicate as one expects the formation of
discontinuities and other losses of regularity along these interfaces; this
first degenerate region is all-the-more troublesome because one can only infer
information about the local behaviour of $\rho$, not $f$. In turn, there
exists relatively few known results in the existing literature for such
equations.

To the authors’ knowledge, there exists only one prior contribution studying
equations similar to (1.1) (except the authors’ own work [7], which has a
different diffusion model and thus less degeneracy), namely the result of
Erignoux [13, Theorem 2.3.3] . Therein, the author relies solely on
probabilistic techniques to compute the hydrodynamic limit of the underlying
particle system, thereby establishing the existence of a very weak solution;
defined only as a measure. The present contribution provides an entirely
deterministic PDE-theoretic proof, and we show the existence of a Sobolev weak
solution with stronger regularity properties by means of the boundedness-by-
entropy principle and a vanishing diffusivity method. Our strategy relies on
using the linear diffusion in angle to upgrade the integrability of $f$ and
obtain improved compactness properties; for this we adapt a classical
interpolation lemma (cf. Appendix A) . We also prove uniqueness of our
solution within the stronger class of admissible weak solutions when the
Péclet number is assumed to be null, and provide existence and uniqueness
results for stationary solutions.

At the core of the aforementioned boundedness-by-entropy method (originally
developed in [8]) is the notion that equation (1.1) may be interpreted as a
perturbation of a Wasserstein gradient flow associated to the entropy
functional

$E[f]=\int_{\Upsilon}\big{(}f\log
f+(1-\rho)\log(1-\rho)\big{)}\,\mathrm{d}\bm{\xi};$

see §2 for further details. One is able to obtain additional _a priori_
estimates on the equation by considering the natural dual problem from the
point of view of the underlying variational evolution. That is to say, we
consider the first variation $u=E^{\prime}[f]$ and rewrite (1.1) as an
equation on $u$, which reads as a perturbation of a gradient flow associated
to the convex conjugate functional $E^{*}[u]$. By rewriting the original
unknown $f$ in terms of $u$, one is able to immediately determine the non-
negativity of $f$ and the _a priori_ bounds $0\leq\rho\leq 1$. This
generalizes the boundedness-by-entropy method developed for the related cross-
diffusion system (1.2) in [8] and later generalized in [17]. The underlying
philosophy is analogous to the reformulation of Lagrangian to Hamiltonian
mechanics; now written in the context of Wasserstein gradient flows with
nontrivial mobility.

The rest of the paper is organised as follows. In §1.1 we state our main
results precisely, and outline our strategy in §1.2. §2 introduces the
boundedness-by-entropy method; which we use to construct a sequence of
suitable transformed regularised problems related to (1.1). In §3, we prove
the existence of the Galerkin approximation to the aforementioned transformed
regularised problems and obtain uniform estimates. In §4, we give the proof of
our main existence result. In §5, we prove uniqueness of the solution obtained
in §4 under the assumption that the Péclet number is null. Finally, in §6, we
prove the existence and uniqueness of stationary states under particular
assumptions on the Péclet number. Appendix A contains an interpolation lemma
pertaining to the uniform estimates of §3.2, while Appendix B contains
technical results related to the sequence of regularised initial data used for
the existence proof.

Notation:  Throughout this work, the letter $C$ will denote a positive
constant independent of the parameters $\varepsilon,n$, which may change from
line to line. Unless stated explicitly otherwise, the symbols
$\operatorname{div},\nabla,\Delta$ denote divergences, gradients, and
Laplacians taken with respect to the space variable $x$, while the symbols
$\nabla_{\bm{\xi}},\Delta_{\bm{\xi}}$ denote gradients and Laplacians with
respect to the concatenated space-angle variable $\bm{\xi}=(x,\theta)$. The
domain on which the equation is posed will be denoted by
$\Upsilon_{T}:=(0,T)\times\Upsilon$, the closure of which we denote by
$\overline{\Upsilon}_{T}:=[0,T]\times\overline{\Upsilon}$; the sets
$\Omega_{T}$ and $\overline{\Omega}_{T}$ are defined analogously. In later
estimates and weak formulations, we shall make use of the families of
admissible test functions $\mathcal{A},\mathcal{A}_{s}$, defined by

(1.3) $\displaystyle\mathcal{A}:=\big{\\{}\psi\in
C^{\infty}([0,T]\times\mathbb{R}^{3}):\,\psi(t,\cdot)\text{ triply
}2\pi\text{-periodic in }x,\theta,\,\psi(0,\cdot)=\psi(T,\cdot)=0\big{\\}},$
$\displaystyle\mathcal{A}_{s}:=\big{\\{}\psi\in
C^{\infty}([0,T]\times\mathbb{R}^{2}):\,\psi(t,\cdot)\text{ doubly
}2\pi\text{-periodic in }x,\,\psi(0,\cdot)=\psi(T,\cdot)=0\big{\\}},$

and test functions will always be denoted by the letter $\psi$. We define the
function spaces

$\displaystyle X:=L^{6}(0,T;W^{1,6}_{\mathrm{per}}(\Upsilon)),\quad
Y:=L^{2}(0,T;H^{1}_{\mathrm{per}}(\Omega)),\quad
Z:=W^{1,6}_{\mathrm{per}}(\Upsilon),$ $\displaystyle\begin{aligned}
\mathcal{X}:=\Big{\\{}f\in&L^{3}(0,T;L^{3}_{\mathrm{per}}(\Upsilon)):f\geq
0\text{ a.e.},\partial_{\theta}\sqrt{f}\in
L^{2}(\Upsilon_{T}),\partial_{t}f\in X^{\prime},\\\
&\rho={\int_{0}^{2\pi}}f\,\mathrm{d}\theta\in[0,1]\text{
a.e.},\partial_{t}\rho\in Y^{\prime},\nabla\sqrt{1-\rho}\in
L^{2}(\Omega_{T}),\sqrt{1-\rho}\nabla\sqrt{f}\in
L^{2}(\Upsilon_{T})\Big{\\}},\end{aligned}$

and, in what follows, the notation $\langle\cdot,\cdot\rangle$ will always
refer to the obvious underlying duality product where there is no confusion,
unless stated explicitly otherwise. In the previous definitions and in what
follows, the functions spaces denoted by $Z_{\mathrm{per}}(S)$ for
$S\in\\{(0,2\pi)^{d}\\}_{d=1}^{3}$ are understood to mean

$Z_{\mathrm{per}}(S):=\big{\\{}g:\mathbb{R}^{d}\to\mathbb{R}:\,\|g\|_{Z(S)}<\infty,\text{
and }g(y+2\pi{\bf e}_{i})=g(y)\,\forall
y\in\mathbb{R}^{d},i\in\\{1,\dots,d\\}\big{\\}},$

where $\\{{\bf e}_{i}\\}_{i=1}^{d}$ is the standard basis of $\mathbb{R}^{d}$.
The case $d=2$ corresponds to functions depending on $x$ and $d=3$ to
functions depending on $x,\theta$.

###  1.1. Main theorems

In order to state our main existence result, we begin by introducing our
notion of solution.

######  Definition 1.1 (Weak solution).

We say that $f\in\mathcal{X}$ is a _weak solution_ of (1.1) with periodic
boundary conditions if, for all $\psi\in X$, there holds

(1.4)
$\displaystyle-\langle\partial_{t}f,\psi\rangle+\mathrm{Pe}\int_{\Upsilon_{T}}(1-\rho)$
$\displaystyle f{\bf
e}(\theta)\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t$
$\displaystyle=\int_{\Upsilon_{T}}\Big{(}D_{e}\big{(}(1-\rho)\nabla
f+f\nabla\rho\big{)}\cdot\nabla\psi+\partial_{\theta}f\partial_{\theta}\psi\Big{)}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t.$

We note that all integrals in the above weak formulation are well-defined
since, using the Jensen and Hölder inequalities,

(1.5)
$\displaystyle\partial_{\theta}f\partial_{\theta}\psi=2\underbrace{\sqrt{f}}_{\in
L^{6}}\underbrace{\partial_{\theta}\sqrt{f}}_{\in
L^{2}}\underbrace{\partial_{\theta}\psi}_{\in L^{6}}\in
L^{\frac{6}{5}}(\Upsilon_{T})\subset L^{1}(\Upsilon_{T})$ $\displaystyle
f\nabla\rho\cdot\nabla\psi=-2\underbrace{f}_{\in
L^{3}}\underbrace{\sqrt{1-\rho}}_{\in
L^{\infty}}\underbrace{\nabla\sqrt{1-\rho}}_{L^{2}}\cdot\underbrace{\nabla\psi}_{L^{6}}\in
L^{1}(\Upsilon_{T})$ $\displaystyle(1-\rho)\nabla
f\cdot\nabla\psi=2\underbrace{\sqrt{1-\rho}}_{\in
L^{\infty}}\underbrace{\sqrt{f}}_{\in
L^{6}}\underbrace{\sqrt{1-\rho}\nabla\sqrt{f}}_{\in
L^{2}}\cdot\underbrace{\nabla\psi}_{\in L^{6}}\in
L^{\frac{6}{5}}(\Upsilon_{T})\subset L^{1}(\Upsilon_{T}).$

Note that one may identify the duality product with

$\langle\partial_{t}f,\psi\rangle=\int_{0}^{T}\langle\partial_{t}f(t,\cdot),\psi(t,\cdot)\rangle_{Z^{\prime}\times
Z}\,\mathrm{d}t,$

whence a standard argument (cf. [3]) shows that the weak formulation of
Definition 1.1 is equivalent to requiring that, for all $\psi\in\mathcal{A}$,
there holds

(1.6)
$\displaystyle\int_{\Upsilon_{T}}f\partial_{t}\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t+\mathrm{Pe}\int_{\Upsilon_{T}}(1-\rho)$
$\displaystyle f{\bf
e}(\theta)\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t$
$\displaystyle=\int_{\Upsilon_{T}}\Big{(}D_{e}\big{(}(1-\rho)\nabla
f+f\nabla\rho\big{)}\cdot\nabla\psi+\partial_{\theta}f\partial_{\theta}\psi\Big{)}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t.$

Our main existence result is the following.

######  Theorem 1 (Existence of weak solution).

Let $T>0$ be arbitrary. Let $f_{0}$ be measurable non-negative initial data
such that $f_{0}\in L^{p}(\Upsilon)$ for some $p>1$ and
$\rho_{0}=\int_{0}^{2\pi}f_{0}\,\mathrm{d}\theta\in[0,1]$ a.e. in $\Omega$.
Then, there exists $f\in\mathcal{X}$ a weak solution of (1.1) in the sense of
Definition 1.1. Furthermore, the initial data is attained in the sense
$\lim_{t\to 0^{+}}\|f(t,\cdot)-f_{0}\|_{Z^{\prime}}=0$.

The essential requirement on $f_{0}$ in the above theorem is the finiteness of
the entropy functional $E[f_{0}]$. This is of course satisfied by imposing the
stronger condition $f_{0}\in L^{p}(\Upsilon)$ for some $p>1$. For clarity of
exposition, we choose this latter requirement in order to frame the problem in
Lebesgue spaces instead of Orlicz spaces. We note in passing that the weak
solution obtained in Theorem 1 also satisfies an alternative notion of weak
solution which is more convenient when studying regularity (cf. [1]); this
alternative weak formulation is written at the end of §4.

######  Remark 1.2 (Attainment of initial data).

Note that $\mathcal{X}\subset W^{1,\frac{6}{5}}(0,T;Z^{\prime})$, whence (see,
e.g., [14, Theorem 2, §5.9.2]) $f\in C([0,T];Z^{\prime})$ and
$\|f(t,\cdot)-f(s,\cdot)\|_{Z^{\prime}}\leq|t-s|^{1/6}C(\|\partial_{t}f\|_{Y^{\prime}})$
for a.e. $0<s<t<T$. Hence, the attainment of the initial data is phrased in
the sense of $Z^{\prime}$.

Our main uniqueness result is the following, and focuses on the case of zero
Péclet number.

######  Theorem 2 (Uniqueness for null Péclet number).

If $\mathrm{Pe}=0$, then the weak solution provided by Theorem 1 is unique in
$\mathcal{X}$.

Finally, we study the existence and uniqueness of stationary states of (1.1)
in §6. We mostly concentrate on uniqueness, which we divide into two parts:
uniqueness for zero Péclet number, and uniqueness of strong solutions near
_constant_ stationary states for general Péclet number. We delay the
statements of these results to §6.

###  1.2. Strategy and set-up

In this section, we describe our strategy of proof for the main existence
result Theorem 1 in §1.2.1, and then describe our mollification procedure for
regularising the initial data in §1.2.2.

####  1.2.1. Strategy

We briefly outline our strategy of proof. As already mentioned, the equation
(1.1) may be interpreted as a perturbation of a Wasserstein gradient flow
associated to the entropy functional $E$. It is therefore natural to reframe
the problem in terms of the first variation of $E[f]$, which we denote by
$u=E^{\prime}[f]$; see §2 for more details. The resulting equation for $u$ is
again non-linear, non-local, and degenerate-parabolic. We therefore add the
regularisation term $\varepsilon\Delta_{\bm{\xi}}u$ to this equation, and
compute Galerkin approximations of the solution of this equation, which we
denote by $u^{\varepsilon}_{n}$; where $n$ denotes the $n$-th Galerkin
approximation. These approximations give rise to approximations
$f^{\varepsilon}_{n}$ for the solution of (1.1) by a simple explicit formula.
After proving the required uniform estimates for the families of functions
$\\{u^{\varepsilon}_{n},f^{\varepsilon}_{n},\rho^{\varepsilon}_{n}\\}_{n,\varepsilon}$,
which are independent of $n$ and $\varepsilon$, we perform a diagonal
procedure so as to simultaneously take the limit $n\to\infty$ and
$\varepsilon\to 0$, and we obtain a weak solution of (1.1) in the sense of
Definition 1.1.

####  1.2.2. Set-up for initial data

We shall approximate the initial data with smooth strictly positive functions.
We consider the product mollifier sequence

$\eta_{\varepsilon}(x,\theta):=\alpha_{\varepsilon}(x)\beta_{\varepsilon}(\theta),$

where $\alpha,\beta$ are compactly supported smooth non-negative bump
functions with unit integral;
$\alpha_{\varepsilon}(x):=\varepsilon^{-2}\alpha(x/\varepsilon)$,
$\beta_{\varepsilon}(\theta)=\varepsilon^{-\gamma}\beta(\theta/\varepsilon^{\gamma})$
for $\gamma>2$. Define, for $\varepsilon\in(0,1)$, the strictly positive
periodic functions

$\displaystyle
f_{0}^{\varepsilon}:=\frac{\varepsilon}{4\pi}+(1-\varepsilon)\eta_{\varepsilon}*f_{0}\in
C^{\infty}_{\mathrm{per}}(\overline{\Upsilon}),$
$\displaystyle\rho_{0}^{\varepsilon}:=\int_{0}^{2\pi}f_{0}^{\varepsilon}(\cdot,\theta)\,\mathrm{d}\theta=\frac{\varepsilon}{2}+(1-\varepsilon)\alpha_{\varepsilon}*\rho_{0}\in
C^{\infty}_{\mathrm{per}}(\overline{\Omega}),$

where the convolutions are understood in the sense
$\eta_{\varepsilon}*f_{0}=\int_{\mathbb{R}^{3}}\eta_{\varepsilon}(\cdot-\bm{\xi})f_{0}(\bm{\xi})\,\mathrm{d}\bm{\xi}$
and
$\alpha_{\varepsilon}*\rho_{0}=\int_{\mathbb{R}^{2}}\alpha_{\varepsilon}(\cdot-x)\rho_{0}(x)\,\mathrm{d}x$.
A straightforward computation (cf. Appendix B) shows that
$0<f^{\varepsilon}_{0}$ and $0<\rho^{\varepsilon}_{0}<1$, whence the function

$\displaystyle
u_{0}^{\varepsilon}:=\log\Big{(}\frac{f^{\varepsilon}_{0}}{1-\rho^{\varepsilon}_{0}}\Big{)}\in
C^{\infty}_{\mathrm{per}}(\overline{\Upsilon})$

is also well-defined. The next lemma summarises the key properties of the
regularised initial data. The proof is delayed to Appendix B.

######  Lemma 1.3 (Properties of the regularised initial data).

There exists a subsequence of the regularised initial data sequence
$\\{f^{\varepsilon}_{0},\rho^{\varepsilon}_{0},u^{\varepsilon}_{0}\\}_{\varepsilon}$,
which we do not relabel, such that

$f^{\varepsilon}_{0}\to f_{0}\text{ strongly in
}L^{p}(\Upsilon),\quad\rho^{\varepsilon}_{0}\to\rho_{0}\text{ strongly in
}L^{q}(\Omega)\text{ for all }q\in[1,\infty),$

and for which there exists a positive constant $C=C(p,\Upsilon)$, independent
of $\varepsilon$, such that

$\displaystyle
E[f^{\varepsilon}_{0}]+\varepsilon\int_{\Upsilon}|u^{\varepsilon}_{0}|^{2}\,\mathrm{d}\bm{\xi}\leq
C\big{(}1+\|f_{0}\|_{L^{p}(\Upsilon)}\big{)}.$

##  2\. Entropy Variable and Transformed Problem

In the following we will perform a transformation of the problem (1.1) into
entropy variables, which diagonalizes the diffusion part at the cost of
coupling the time derivatives of $f$ and $\rho$. This transformation allows
for a suitable approximation via the Galerkin method, as it did for the cross-
diffusion system (1.2) considered in [8], and an application of the
boundedness-by-entropy principle; developed again in [8] and subsequently
generalised in [17].

For this sake we consider the aforementioned _entropy functional_ of (1),
i.e.,

$E[f]=\int_{\Upsilon}\big{(}f\log
f+(1-\rho)\log(1-\rho)\big{)}\,\mathrm{d}x\,\mathrm{d}\theta,$

where $\rho=\int_{0}^{2\pi}f\,\mathrm{d}\theta$. Note that $E[f]$ is bounded
from below by a fixed constant for all $f$, provided $0\leq\rho\leq 1$ and
$f\geq 0$ a.e. We can define $E$ on a dense subspace of $L^{p}(\Upsilon)$ for
any $p>1$, e.g., smooth functions, and extend it uniquely to the whole space.
Note that $E$ is a convex functional, and we may define its convex conjugate,
or equivalently its Legendre transform,
$E^{*}:L^{p^{\prime}}(\Upsilon)\to\mathbb{R}$ via

$E^{*}[u]:=\sup\big{\\{}\langle u,f\rangle-E[f]:f\in
L^{p}(\Upsilon)\big{\\}},$

which is again a convex functional. In fact, we may compute $E^{*}$ explicitly
as

$E^{*}[u]=\int_{\Omega}\log\left(1+\int_{0}^{2\pi}e^{u}\,\mathrm{d}\theta\right)\,\mathrm{d}x.$

As usual we define the _entropy variable_ (or dual variable) related to $f$
via the functional derivative

(2.1) $u:=E^{\prime}[f]=\log f-\log(1-\rho).$

We therefore write $f$ and $\rho$ as functions of
$u:\Upsilon_{T}\to\mathbb{R}$ via the expressions:

$\displaystyle
f=(1-\rho)e^{u},\qquad\rho=(1-\rho)\int_{0}^{2\pi}e^{u}\,\mathrm{d}\theta.$

It follows that

(2.2)
$\rho(t,x)=\frac{\int_{0}^{2\pi}e^{u(t,x,\theta)}\,\mathrm{d}\theta}{1+\int_{0}^{2\pi}e^{u(t,x,\theta)}\,\mathrm{d}\theta}$

automatically satisfies the bound $0\leq\rho\leq 1$. In turn, we obtain

(2.3)
$f(t,x,\theta)=\frac{e^{u(t,x,\theta)}}{1+\int_{0}^{2\pi}e^{u(t,x,\theta)}\,\mathrm{d}\theta}.$

Note that this latter formula makes the relation
$(E^{*})^{\prime}=(E^{\prime})^{-1}$ explicit, i.e.,

$u=E^{\prime}[f],\quad f=(E^{*})^{\prime}[u].$

The main idea to analyze the system is as follows: the original problem (1.1)
is considered as a perturbation of the gradient flow

$\partial_{t}f=\nabla_{\bm{\xi}}\cdot\left(M(f,\rho)(\nabla_{\bm{\xi}}E^{\prime}[f]+V)\right)$

with mobility $M$ and perturbative vector field $V$, independent of $f$, given
by

$M(f,\rho)=\left(\begin{array}[]{cc}D_{e}f(1-\rho)I&0\\\
0&f\end{array}\right),\qquad
V(x,\theta)=\left(\begin{array}[]{c}-\mathrm{Pe}~{}{\bf e}(\theta)\\\
0\end{array}\right).$

Transforming to the entropy variable we arrive at

(2.4)
$\partial_{t}((E^{*})^{\prime}[u])=\nabla_{\bm{\xi}}\cdot\left(\tilde{M}(u)(\nabla_{\bm{\xi}}u+V)\right),$

where $\tilde{M}(u)=M(f[u],\rho[u])$ is given by

(2.5)
$\tilde{M}(u)=\left(\begin{matrix}\frac{e^{u}}{(1+\int_{0}^{2\pi}e^{u}\,\mathrm{d}\theta)^{2}}D_{e}I&0\\\
0&\frac{e^{u}}{1+\int_{0}^{2\pi}e^{u}\,\mathrm{d}\theta}\end{matrix}\right).$

This dual formulation has several advantages. First of all, the boundedness-
by-entropy principle applies, i.e. if we find a weak solution $u$ of the
above, it is apparent that after transformation $f\geq 0$ and $0\leq\rho\leq
1$ hold almost everywhere. Moreover, the transformed system can be
approximated more easily, e.g. by regularising with a multiple of the heat
operator. This yields a problem of the form

(2.6)
$(\varepsilon\mathrm{id}+A(u^{\varepsilon}))[\partial_{t}u^{\varepsilon}]=\nabla_{\bm{\xi}}\cdot\left((\varepsilon
I+\tilde{M}(u^{\varepsilon}))\nabla_{\bm{\xi}}u^{\varepsilon}+\tilde{M}(u^{\varepsilon})V\right),$

where, for all $u\in L^{p^{\prime}}(\Upsilon)$, the quantity $A(u)$ is a
linear operator which corresponds to the second functional derivative of
$E^{*}$ evaluated on the element $u$; furthermore, since $E^{*}$ is convex,
this quantity is a positive semi-definite linear operator. In fact, one can
explicitly write

(2.7) $\displaystyle
A(u^{\varepsilon})[\partial_{t}u^{\varepsilon}]=\frac{e^{u^{\varepsilon}}}{1+\int_{0}^{2\pi}e^{u^{\varepsilon}}\,\mathrm{d}\theta}\partial_{t}u^{\varepsilon}-\frac{e^{u^{\varepsilon}}}{\big{(}1+\int_{0}^{2\pi}e^{u^{\varepsilon}}\,\mathrm{d}\theta\big{)}^{2}}\int_{0}^{2\pi}e^{u^{\varepsilon}}\partial_{t}u^{\varepsilon}\,\mathrm{d}\theta.$

In principle, one can now write a proof of existence and uniqueness including
some regularity for the regularised problem (2.7); this is the content of §3.
Moreover, testing the equation with $u$ preserves _a priori_ estimates for the
entropy, noticing that

(2.8) $\frac{\mathrm{d}}{\mathrm{d}t}E[f]=\langle
E^{\prime}(f),\partial_{t}f\rangle_{\Upsilon}=\langle
u,\partial_{t}((E^{*})^{\prime}[u])\rangle_{\Upsilon}.$

##  3\. Galerkin Approximation for the Transformed Regularised System

The main result of this section is as follows. We shall prove it in several
steps: first, we obtain the existence of the Galerkin approximations to the
regularised problem (2.6) in §3.1, and then we obtain the desired uniform
estimates in §3.2.

######  Proposition 3.1 (Existence and uniform estimates for Galerkin
approximations).

Let $\varepsilon\in(0,1)$ and $n\in\mathbb{N}$. There exists functions
$u^{\varepsilon}_{n},f^{\varepsilon}_{n},\rho^{\varepsilon}_{n}\in
C^{1}(\overline{\Upsilon}_{T})$ satisfying the relations

(3.1)
$f^{\varepsilon}_{n}(t,x,\theta)=\frac{e^{u^{\varepsilon}_{n}(t,x,\theta)}}{1+\int_{0}^{2\pi}e^{u^{\varepsilon}_{n}(t,x,\theta)}\,\mathrm{d}\theta},\qquad\rho^{\varepsilon}_{n}(t,x)=\int_{0}^{2\pi}f^{\varepsilon}_{n}(t,x,\theta)\,\mathrm{d}\theta\in[0,1]$

in $\Upsilon_{T}$, such that, for all $\psi\in\mathcal{A}$, there holds

(3.2) $\displaystyle-\langle\partial_{t}(\varepsilon u^{\varepsilon}_{n}+$
$\displaystyle
f^{\varepsilon}_{n}),\psi\rangle+\mathrm{Pe}\int_{\Upsilon_{T}}(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}{\bf
e}(\theta)\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t$ $\displaystyle=$
$\displaystyle\int_{\Upsilon_{T}}\Big{(}\varepsilon\nabla_{\bm{\xi}}u^{\varepsilon}_{n}\cdot\nabla_{\bm{\xi}}\psi+D_{e}\big{(}(1-\rho^{\varepsilon}_{n})\nabla
f^{\varepsilon}_{n}+f^{\varepsilon}_{n}\nabla\rho^{\varepsilon}_{n}\big{)}\cdot\nabla\psi+\partial_{\theta}f^{\varepsilon}_{n}\partial_{\theta}\psi\Big{)}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t,$

Furthermore there exists a positive constant $C$, independent of $\varepsilon$
and $n$, such that

(3.3)
$\displaystyle\operatorname{ess\,sup}_{t\in[0,T]}E[f^{\varepsilon}_{n}](t)+\operatorname{ess\,sup}_{t\in[0,T]}\int_{\Upsilon}|\sqrt{\varepsilon}u^{\varepsilon}_{n}(t)|^{2}\,\mathrm{d}\bm{\xi}+\int_{\Upsilon_{T}}|\nabla_{\bm{\xi}}\sqrt{\varepsilon}u^{\varepsilon}_{n}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t$
$\displaystyle+\int_{\Upsilon_{T}}(1-\rho^{\varepsilon}_{n})|\nabla_{\bm{\xi}}\sqrt{f^{\varepsilon}_{n}}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t+\int_{\Omega_{T}}|\nabla\sqrt{1-\rho^{\varepsilon}_{n}}|^{2}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle+\int_{\Upsilon_{T}}|\partial_{\theta}\sqrt{f^{\varepsilon}_{n}}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t+\int_{\Omega_{T}}|\nabla\rho^{\varepsilon}_{n}|^{2}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\leq C,$

as well as

(3.4) $\displaystyle\|f^{\varepsilon}_{n}\|_{L^{3}(\Upsilon_{T})}\leq C$ (3.5)
$\displaystyle\|\sqrt{(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}}\|_{L^{\frac{10}{3}}(\Upsilon_{T})}\leq
C$ (3.6)
$\displaystyle\|f^{\varepsilon}_{n}\sqrt{1-\rho^{\varepsilon}_{n}}\|_{L^{\frac{6}{5}}(0,T;W^{1,\frac{6}{5}}(\Upsilon))}\leq
C,$

and the time-derivative bounds

(3.7) $\displaystyle\|\partial_{t}(\varepsilon
u^{\varepsilon}_{n}+f^{\varepsilon}_{n})\|_{X^{\prime}}\leq C,$ (3.8)
$\displaystyle\|\partial_{t}(\varepsilon
U^{\varepsilon}_{n}+\rho^{\varepsilon})\|_{Y^{\prime}}\leq C,$

where
$U^{\varepsilon}_{n}(t,x):=\int_{0}^{2\pi}u^{\varepsilon}_{n}(t,x,\theta)\,\mathrm{d}\theta$
also satisfies

(3.9)
$\|\sqrt{\varepsilon}U^{\varepsilon}_{n}\|_{L^{\infty}(0,T;L^{2}(\Omega))}+\|\sqrt{\varepsilon}U^{\varepsilon}_{n}\|_{L^{2}(0,T;H^{1}(\Omega))}\leq
C.$

###  3.1. Existence of Galerkin approximants

In what follows, we use that the biharmonic operator $\Delta_{\bm{\xi}}^{2}$,
considered with periodic boundary conditions in space and angle on $\Upsilon$,
is a coercive, bounded, linear self-adjoint operator with respect to the inner
product of $H^{2}_{\mathrm{per}}(\Upsilon)$, where

$\langle
f,g\rangle_{H^{2}_{\mathrm{per}}(\Upsilon)}=\int_{\Upsilon}\Delta_{\bm{\xi}}f\Delta_{\bm{\xi}}g\,\mathrm{d}\bm{\xi}.$

It follows from the classical spectral theorem on Hilbert spaces that
$\Delta_{\bm{\xi}}^{2}$ admits a discrete spectrum
$0<\lambda_{1}<\lambda_{2}<\dots$ with eigenfunctions
$\\{\varphi_{j}\\}_{j=1}^{\infty}$ such that there holds
$\Delta_{\bm{\xi}}^{2}\varphi_{j}=\lambda_{j}\varphi_{j}$ for all $j$; a
classical bootstrapping procedure ensures that $\varphi_{j}\in
C^{\infty}_{\mathrm{per}}(\overline{\Upsilon})$ for all $j$, cf. e.g. [2,
Chapitre 7]. Furthermore, $\\{\varphi_{j}\\}_{j=1}^{\infty}$ is an orthonormal
basis of $H^{2}_{\mathrm{per}}(\Upsilon)$.

We note in passing that our choice of functional setting
$H^{2}_{\mathrm{per}}(\Upsilon)$ is motivated by the fact that we must control
the $L^{\infty}$ norm of the sequence of Galerkin approximations
$\\{u_{n}^{\varepsilon}(0,\cdot)\\}_{n}$ in order to estimate the
approximations $\\{f_{n}^{\varepsilon}(0,\cdot)\\}_{n}$ via the entropy
variable formula (3.1); no $L^{p}$ control for $p<\infty$ suffices. This is
made possible by approximating $u^{\varepsilon}(0,\cdot)$ in
$H^{2}_{\mathrm{per}}(\Upsilon)$ and using Morrey’s inequality over the three-
dimensional open domain $\Upsilon$; see Step 1 of the proof of Lemma 3.3 for
details.

For each fixed $n\in\mathbb{N}$ we define the closed subspace
$\mathcal{V}_{n}:=\overline{\mathrm{span}}\\{\varphi_{j}\\}_{j=1}^{n}$. Note
from our previous discussion that $\bigcup_{n}\mathcal{V}_{n}$ is dense in
$H^{2}_{\mathrm{per}}(\Upsilon)$, $\mathcal{V}_{n}\subset\mathcal{V}_{n+1}$
for all $n$, and, for each $n$, the finite subset
$\\{\varphi_{j}\\}_{j=1}^{n}$ is an orthonormal basis of $\mathcal{V}_{n}$. We
now define the $n$-th approximant to the entropy variable $u^{\varepsilon}$,
to be given by the explicit formula

(3.10)
$u^{\varepsilon}_{n}(t,x,\theta):=\sum_{j=1}^{n}\alpha_{j}^{\varepsilon,n}(t)\varphi_{j}(x,\theta),$

where the coefficients $\alpha^{\varepsilon,n}_{j}$ are to be determined. This
construction is such that, for large $n$, the function $u^{\varepsilon}_{n}$
ought to be a suitable approximation of $u^{\varepsilon}$; a weak solution in
$H^{1}_{\mathrm{per}}(\Upsilon)$ of (2.6).

One can see, by plugging in $u^{\varepsilon}_{n}$ in place of
$u^{\varepsilon}$ in (2.6), that the coefficients $\alpha^{\varepsilon,n}_{j}$
will be determined from requiring

(3.11)
$(\varepsilon\mathrm{id}+A(u^{\varepsilon}_{n}))\Big{[}\sum_{j=1}^{n}\frac{\mathrm{d}\alpha^{\varepsilon,n}_{j}}{\mathrm{d}t}\varphi_{j}\Big{]}=\nabla_{\bm{\xi}}\cdot\Big{(}(\varepsilon
I+\tilde{M}(u^{\varepsilon}_{n}))\sum_{j=1}^{n}\alpha^{\varepsilon,n}_{j}\nabla_{\bm{\xi}}\varphi_{j}+\tilde{M}(u^{\varepsilon}_{n})V\Big{)}$

in the weak sense in $\mathcal{V}_{n}$, i.e., for all
$\phi\in\mathcal{V}_{n}$, there holds

$\displaystyle\sum_{j=1}^{n}\frac{\mathrm{d}\alpha^{\varepsilon,n}_{j}}{\mathrm{d}t}\int_{\Upsilon}(\varepsilon\mathrm{id}+$
$\displaystyle A(u^{\varepsilon}_{n}))[\varphi_{j}]\phi\,\mathrm{d}\bm{\xi}$
$\displaystyle=-\sum_{j=1}^{n}\alpha^{\varepsilon,n}_{j}\int_{\Upsilon}\nabla_{\bm{\xi}}\phi\cdot\left((\varepsilon
I+\tilde{M}(u^{\varepsilon}_{n}))\nabla_{\bm{\xi}}\varphi_{j}+\tilde{M}(u^{\varepsilon}_{n})V\right)\,\mathrm{d}\bm{\xi},$

for all $t\in(0,T)$, where we used linearity and
$A(u^{\varepsilon}_{n})[\frac{\mathrm{d}}{\mathrm{d}t}\alpha_{j}^{\varepsilon,n}(t)\varphi_{j}]=\frac{\mathrm{d}}{\mathrm{d}t}\alpha_{j}^{\varepsilon,n}(t)A(u^{\varepsilon}_{n})[\varphi_{j}]$,
which can be seen directly from the formula (2.7). Using the decomposition of
$\phi$ into the basis functions $\\{\varphi_{k}\\}_{k=1}^{n}$, the previous
relation is equivalent to requiring: for all $k\in\\{1,\dots,n\\}$, there
holds

$\displaystyle\sum_{j=1}^{n}\frac{\mathrm{d}\alpha^{\varepsilon,n}_{j}}{\mathrm{d}t}\int_{\Upsilon}(\varepsilon\mathrm{id}+$
$\displaystyle
A(u^{\varepsilon}_{n}))[\varphi_{j}]\varphi_{k}\,\mathrm{d}\bm{\xi}$
$\displaystyle=-\sum_{j=1}^{n}\alpha^{\varepsilon,n}_{j}\int_{\Upsilon}\nabla_{\bm{\xi}}\varphi_{k}\cdot\left((\varepsilon
I+\tilde{M}(u^{\varepsilon}_{n}))\nabla_{\bm{\xi}}\varphi_{j}+\tilde{M}(u^{\varepsilon}_{n})V\right)\,\mathrm{d}\bm{\xi},$

for all $t\in(0,T)$. Note that, provided the coefficients
$\alpha^{\varepsilon,n}_{j}$ belong to $C^{1}([0,T])$, all terms in the
previous equality are well-defined; in particular, the integrals are finite
due to the boundedness of the functions $\varphi_{j}\in
C^{\infty}_{\mathrm{per}}(\overline{\Upsilon})$. The above can be recast as a
quasilinear system of ODEs of the form

(3.12) $\left\\{\begin{aligned}
&\mathcal{M}(\gamma^{\varepsilon,n}(t))\frac{\mathrm{d}\gamma^{\varepsilon,n}}{\mathrm{d}t}(t)=\mathcal{R}(\gamma^{\varepsilon,n}(t)),\\\
&\gamma^{\varepsilon,n}(0)=\gamma^{\varepsilon,n}_{0},\end{aligned}\right.$

where we have defined the vector function
$\gamma^{\varepsilon,n}=(\alpha^{\varepsilon,n}_{1},\dots,\alpha^{\varepsilon,n}_{n})$
and the entries of the matrices are given explicitly by

(3.13)
$\displaystyle\mathcal{M}(\gamma^{\varepsilon,n}(t))_{kj}=\int_{\Upsilon}(\varepsilon\mathrm{id}+A(u^{\varepsilon}_{n}))[\varphi_{j}]\varphi_{k}\,\mathrm{d}\bm{\xi},$
$\displaystyle\mathcal{R}(\gamma^{\varepsilon,n}(t))_{kj}=-\int_{\Upsilon}\nabla_{\bm{\xi}}\varphi_{k}\cdot\left((\varepsilon
I+\tilde{M}(u^{\varepsilon}_{n}))\nabla_{\bm{\xi}}\varphi_{j}+\tilde{M}(u^{\varepsilon}_{n})V\right)\,\mathrm{d}\bm{\xi},$

where $k,j$ range over all possible values in $\\{1,\dots,n\\}$, and the
initial data is given explicitly by

(3.14) $\alpha_{j}^{\varepsilon,n}(0)=\langle
u^{\varepsilon}_{0},\varphi_{j}\rangle_{H^{2}_{\mathrm{per}}(\Upsilon)},$

such that, with
$u^{\varepsilon}_{n}(0)=u^{\varepsilon}_{n,0}\equiv\sum_{j=1}^{n}\alpha^{\varepsilon,n}_{j}(0)\varphi_{j}$,
there holds the strong convergence

(3.15) $u^{\varepsilon}_{n}(0)\to u^{\varepsilon}_{0}\quad\text{in
}H^{2}_{\mathrm{per}}(\Upsilon).$

It follows from the positivity $\varepsilon>0$ and the positive
semidefiniteness of the operator $A$ that the matrix $\mathcal{M}$ is
invertible, and furthermore the product $\mathcal{M}^{-1}\mathcal{R}$ is
locally Lipschitz; indeed, this quantity and its derivatives involve only
exponential functions of the $\alpha^{\varepsilon,n}_{j}$ and their products,
weighted by the reciprocal of a determinant which is bounded strictly away
from zero for each $\varepsilon>0$. In turn, the Cauchy–Lipschitz Theorem for
ODE systems implies the existence and uniqueness of $\gamma^{\varepsilon,n}\in
C^{1}([0,T])$ solving (3.12).

It follows that we have proved the following result, which concludes this
subsection.

######  Lemma 3.2.

Let $\varepsilon\in(0,1)$, $n\in\mathbb{N}$, and $u^{\varepsilon}_{n,0}\in
H^{2}_{\mathrm{per}}(\Upsilon)$ be given by

$u^{\varepsilon}_{n,0}=\sum_{j=1}^{n}\varphi_{j}\langle
u^{\varepsilon}_{0},\varphi_{j}\rangle_{H^{2}_{\mathrm{per}}(\Upsilon)}.$

There exists $u^{\varepsilon}_{n}\in C^{1}(\overline{\Upsilon}_{T})$ such that
$u^{\varepsilon}_{n}(t,\cdot)\in\mathcal{V}_{n}$ for all $t\in[0,T]$,
$u^{\varepsilon}_{n}(0)=u^{\varepsilon}_{n,0}$ and, for all
$\psi\in\mathcal{V}_{n}$, there holds for all $t\in(0,T)$

(3.16)
$\int_{\Upsilon}(\varepsilon\mathrm{id}+A(u^{\varepsilon}_{n}))[\partial_{t}u^{\varepsilon}_{n}]\psi\,\mathrm{d}\bm{\xi}=-\int_{\Upsilon}\nabla_{\bm{\xi}}\psi\cdot\left((\varepsilon
I+\tilde{M}(u^{\varepsilon}_{n}))\nabla_{\bm{\xi}}u^{\varepsilon}_{n}+\tilde{M}(u^{\varepsilon}_{n})V\right)\,\mathrm{d}\bm{\xi},$

i.e., in the weak sense with respect to $\mathcal{V}_{n}$,

(3.17)
$(\varepsilon\mathrm{id}+A(u^{\varepsilon}_{n}))[\partial_{t}u^{\varepsilon}_{n}]=\nabla_{\bm{\xi}}\cdot\left((\varepsilon
I+\tilde{M}(u^{\varepsilon}_{n}))\nabla_{\bm{\xi}}u^{\varepsilon}_{n}+\tilde{M}(u^{\varepsilon}_{n})V\right).$

We then define $f^{\varepsilon}_{n}\in C^{1}([0,T]\times\overline{\Upsilon})$
and its angle-independent density $\rho^{\varepsilon}_{n}$ by

(3.18)
$f^{\varepsilon}_{n}:=(E^{*})^{\prime}[u^{\varepsilon}_{n}]=\frac{e^{u^{\varepsilon}_{n}}}{1+\int_{0}^{2\pi}e^{u^{\varepsilon}_{n}}\,\mathrm{d}\theta},\quad\rho^{\varepsilon}_{n}:=\int_{0}^{2\pi}f^{\varepsilon}_{n}\,\mathrm{d}\theta=\frac{\int_{0}^{2\pi}e^{u^{\varepsilon}_{n}}\,\mathrm{d}\theta}{1+\int_{0}^{2\pi}e^{u^{\varepsilon}_{n}}\,\mathrm{d}\theta},$

whence

$(\varepsilon\mathrm{id}+A(u^{\varepsilon}_{n}))[\partial_{t}u^{\varepsilon}_{n}]=\varepsilon\partial_{t}u^{\varepsilon}_{n}+\partial_{t}((E^{*})^{\prime}[u^{\varepsilon}_{n}])=\varepsilon\partial_{t}u^{\varepsilon}_{n}+\partial_{t}f^{\varepsilon}_{n},$

and from which we obtain, using also (2.8),

(3.19)
$\int_{\Upsilon}(\varepsilon\mathrm{id}+A(u^{\varepsilon}_{n}))[\partial_{t}u^{\varepsilon}_{n}]u^{\varepsilon}_{n}\,\mathrm{d}\bm{\xi}=\frac{\varepsilon}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Upsilon}|u^{\varepsilon}_{n}|^{2}\,\mathrm{d}\bm{\xi}+\frac{\mathrm{d}}{\mathrm{d}t}E[f^{\varepsilon}_{n}].$

At this point, we record the relations

(3.20) $\displaystyle
f^{\varepsilon}_{n}\partial_{\theta}u^{\varepsilon}_{n}=\partial_{\theta}f^{\varepsilon}_{n}=2\sqrt{f^{\varepsilon}_{n}}\partial_{\theta}\sqrt{f^{\varepsilon}_{n}},$
$\displaystyle(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}\nabla
u^{\varepsilon}_{n}=(1-\rho^{\varepsilon}_{n})\nabla
f^{\varepsilon}_{n}+f^{\varepsilon}_{n}\nabla\rho^{\varepsilon}_{n}=2\sqrt{(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}}\nabla\sqrt{(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}}+2f^{\varepsilon}_{n}\nabla\rho^{\varepsilon}_{n},$

whence the equation (3.17) may be rewritten as

(3.21)
$\varepsilon\partial_{t}u^{\varepsilon}_{n}+\partial_{t}f^{\varepsilon}_{n}+\mathrm{Pe}\operatorname{div}((1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}{\bf
e}(\theta))=\varepsilon\Delta_{\bm{\xi}}u^{\varepsilon}_{n}+D_{e}\operatorname{div}((1-\rho^{\varepsilon}_{n})\nabla
f^{\varepsilon}_{n}+f^{\varepsilon}_{n}\nabla\rho^{\varepsilon}_{n})+\partial^{2}_{\theta}f^{\varepsilon}_{n}.$

It follows from an integration by parts that (3.2) is verified. We have
therefore proved the existence part of Proposition 3.1. It remains to obtain
the uniform estimates, which is the focus of the next section.

###  3.2. Uniform estimates for the Galerkin approximations

In this section, we perform the main entropy-dissipation estimate, and obtain
bounds uniform in $n$ and $\varepsilon$ for both the entropy variable and the
original unknown. After this, we establish similar estimates for their
respective time derivatives.

We first remark that, for all fixed $t,x$ there holds
$\|f^{\varepsilon}_{n}(t,x,\cdot)\|_{L^{1}((0,2\pi))}\leq 1$ as well as
$f^{\varepsilon}_{n}\geq 0$ and $0\leq\rho^{\varepsilon}_{n}\leq 1$, from
which we get

(3.22) $\|f^{\varepsilon}_{n}\|_{L^{\infty}(\Omega_{T};L^{1}(0,2\pi))}\leq 1.$

This bound will be used in the proof of the main result of this subsection,
which we now state.

######  Lemma 3.3 (Uniform estimates for Galerkin approximation).

Let $\varepsilon,n,u^{\varepsilon}_{n}$ be as in Lemma 3.2 and
$f^{\varepsilon}_{n},\rho^{\varepsilon}_{n}$ be as in (3.18). There exists a
positive constant $C$, independent of $\varepsilon,n$ such that there holds
the uniform estimates:

(3.23)
$\displaystyle\sup_{t\in[0,T]}E[f^{\varepsilon}_{n}](t)+\sup_{t\in[0,T]}\int_{\Upsilon}|\sqrt{\varepsilon}u^{\varepsilon}_{n}(t)|^{2}\,\mathrm{d}\bm{\xi}+\int_{\Upsilon_{T}}|\nabla_{\bm{\xi}}\sqrt{\varepsilon}u^{\varepsilon}_{n}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t$
$\displaystyle+\int_{\Upsilon_{T}}(1-\rho^{\varepsilon}_{n})|\nabla_{\bm{\xi}}\sqrt{f^{\varepsilon}_{n}}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t+\int_{\Upsilon_{T}}|\nabla\sqrt{1-\rho^{\varepsilon}_{n}}|^{2}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle+\int_{\Upsilon_{T}}|\partial_{\theta}\sqrt{f^{\varepsilon}_{n}}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t+\int_{\Omega_{T}}|\nabla\rho^{\varepsilon}_{n}|^{2}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\leq C,$

and

(3.24) $\displaystyle\|f^{\varepsilon}_{n}\|_{L^{3}(\Upsilon_{T})}\leq C$
(3.25)
$\displaystyle\|\sqrt{(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}}\|_{L^{\frac{10}{3}}(\Upsilon_{T})}\leq
C$ (3.26)
$\displaystyle\|f^{\varepsilon}_{n}\sqrt{1-\rho^{\varepsilon}_{n}}\|_{L^{\frac{6}{5}}(0,T;W^{1,\frac{6}{5}}(\Upsilon))}\leq
C.$

###### Proof.

1\. Entropy dissipation: By testing (3.17) with $u^{\varepsilon}_{n}$, which
is an admissible test function since one can directly verify
$u^{\varepsilon}_{n},\partial_{t}u^{\varepsilon}_{n}\in\mathcal{V}_{n}$, we
compute, using (3.19), the entropy dissipation

$\displaystyle\frac{\varepsilon}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Upsilon}|u^{\varepsilon}_{n}|^{2}\,\mathrm{d}x\,\mathrm{d}\theta+\frac{\mathrm{d}}{\mathrm{d}t}E[f^{\varepsilon}_{n}]+\int_{\Upsilon}(\varepsilon
I+\tilde{M}(u^{\varepsilon}_{n}))\nabla_{\bm{\xi}}u^{\varepsilon}_{n}\cdot\nabla_{\bm{\xi}}u^{\varepsilon}_{n}\,\mathrm{d}\bm{\xi}=-\int_{\Upsilon}\tilde{M}(u^{\varepsilon}_{n})V\cdot\nabla_{\bm{\xi}}u^{\varepsilon}_{n}\,\mathrm{d}\bm{\xi}.$

Again, we emphasise that, since $\varphi_{j}\in
C^{\infty}_{\mathrm{per}}(\overline{\Upsilon})$ for all $j$, the integrands in
the previous line are composed of functions belonging to
$C^{1}(\overline{\Upsilon}_{T})$, whence the integrals are well-defined.
Recall that the matrix $\tilde{M}$ is positive semi-definite and symmetric. It
follows that there exists a unique square-root matrix $\tilde{N}$ of the same
dimension such that $\tilde{N}^{T}\tilde{N}=\tilde{M}$, namely

$\tilde{N}(u)=N(f[u])=\left(\begin{matrix}\sqrt{D_{e}f(1-\rho)}I&0\\\
0&\sqrt{f}\end{matrix}\right).$

In turn, there holds, for all $t\in[0,T]$,

(3.27) $\displaystyle
E[f^{\varepsilon}_{n}](t)+\frac{\varepsilon}{2}\int_{\Upsilon}$
$\displaystyle|u^{\varepsilon}_{n}(t)|^{2}\,\mathrm{d}\bm{\xi}+\varepsilon\int_{0}^{t}\int_{\Upsilon}|\nabla_{\bm{\xi}}u^{\varepsilon}_{n}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau+\int_{0}^{t}\int_{\Upsilon}\underbrace{\tilde{M}(u^{\varepsilon}_{n})\nabla_{\bm{\xi}}u^{\varepsilon}_{n}\cdot\nabla_{\bm{\xi}}u^{\varepsilon}_{n}}_{\geq
0}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau$
$\displaystyle=E[f^{\varepsilon}_{n}](0)+\frac{\varepsilon}{2}\int_{\Upsilon}|u^{\varepsilon}_{n}(0)|^{2}\,\mathrm{d}\bm{\xi}-\int_{0}^{t}\int_{\Upsilon}\tilde{N}(u^{\varepsilon}_{n})\nabla_{\bm{\xi}}u^{\varepsilon}_{n}\cdot\tilde{N}(u^{\varepsilon}_{n})V\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau,$

and observe that the final term on the right-hand side may be bounded by means
of the Young inequality as follows

(3.28)
$\displaystyle|\tilde{N}(u^{\varepsilon}_{n})\nabla_{\bm{\xi}}u^{\varepsilon}_{n}\cdot\tilde{N}(u^{\varepsilon}_{n})V|\leq$
$\displaystyle\frac{1}{2}|\tilde{N}(u^{\varepsilon}_{n})\nabla_{\bm{\xi}}u^{\varepsilon}_{n}|^{2}+\frac{1}{2}|\tilde{N}(u^{\varepsilon}_{n})V|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\tilde{M}(u^{\varepsilon}_{n})\nabla_{\bm{\xi}}u^{\varepsilon}_{n}\cdot\nabla_{\bm{\xi}}u^{\varepsilon}_{n}+\frac{1}{2}\tilde{M}(u^{\varepsilon}_{n})V\cdot
V$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\tilde{M}(u^{\varepsilon}_{n})\nabla_{\bm{\xi}}u^{\varepsilon}_{n}\cdot\nabla_{\bm{\xi}}u^{\varepsilon}_{n}+\frac{1}{2}\|V\|_{L^{\infty}(\Upsilon_{T})}^{2}|f^{\varepsilon}_{n}|,$

where we used the boundedness of $\rho^{\varepsilon}_{n}$; cf. (3.22). Recall
from (3.15) that $u^{\varepsilon}_{n}(0)\to u^{\varepsilon}_{0}$ strongly in
$H^{2}_{\mathrm{per}}(\Upsilon)$, and hence almost everywhere for a
subsequence; which, with slight abuse of notation, we still label as
$\\{u^{\varepsilon}_{n}\\}_{n}$. In view of the Morrey embedding
$H^{2}(\Upsilon)\hookrightarrow C^{0,1/2}(\Upsilon)$, we deduce

$\|u^{\varepsilon}_{n}(0)-u^{\varepsilon}_{0}\|_{L^{\infty}(\Upsilon)}\to 0,$

and hence

$\sup_{n}\|u^{\varepsilon}_{n}(0)\|_{L^{\infty}(\Upsilon)}<\infty.$

It follows from the continuity of the exponential and the relations (3.18)
that $f^{\varepsilon}_{n}(0)\to f^{\varepsilon}_{0}$ almost everywhere, and

$0<f^{\varepsilon}_{n}(0)\leq\exp\big{(}\sup_{n}{\|u^{\varepsilon}_{n}(0)\|_{L^{\infty}(\Upsilon)}}\big{)}<\infty,$

and we remark that
$\exp(\sup_{n}{\|u^{\varepsilon}_{n}(0)\|_{L^{\infty}(\Upsilon)}})$ is
constant and is therefore an integrable dominating function on the bounded
domain $\Upsilon$. Using the previous estimate, an application of the
Dominated Convergence Theorem yields, for all $q\in[1,\infty)$,

(3.29) $f^{\varepsilon}_{n}(0)\to f^{\varepsilon}_{0}\quad\text{strongly in
}L^{q}(\Upsilon)\text{ as }n\to\infty,$

and a similar argument using also (3.22), cf. e.g. Appendix B, yields
$\lim_{n\to\infty}E[f^{\varepsilon}_{n}(0)]=E[f^{\varepsilon}_{0}]$ in the
sense of real numbers. It therefore follows, using also (3.22), (3.27), and
(3.28), that

$\displaystyle\sup_{t\in[0,T]}E[f^{\varepsilon}_{n}](t)+$
$\displaystyle\frac{\varepsilon}{2}\bigg{(}\sup_{t\in[0,T]}\int_{\Upsilon}|u^{\varepsilon}_{n}(t)|^{2}\,\mathrm{d}\bm{\xi}+\int_{\Upsilon_{T}}|\nabla_{\bm{\xi}}u^{\varepsilon}_{n}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t\bigg{)}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\int_{\Upsilon_{T}}\tilde{M}(u^{\varepsilon}_{n})\nabla_{\bm{\xi}}u^{\varepsilon}_{n}\cdot\nabla_{\bm{\xi}}u^{\varepsilon}_{n}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t\leq
C\bigg{(}1+E[f^{\varepsilon}_{0}]+\varepsilon\int_{\Upsilon}|u^{\varepsilon}_{0}|^{2}\,\mathrm{d}\bm{\xi}\bigg{)},$

for some positive constant $C=C(\mathrm{Pe},D_{e},T,\Upsilon)$ independent of
$\varepsilon,n$. Our choice of regularised initial data (cf. Lemma 1.3)
implies that the entire right-hand side of the above is bounded independently
of $\varepsilon,n$. By rewriting the final term on the left-hand side in terms
of $f^{\varepsilon}_{n},\rho^{\varepsilon}_{n}$ using (3.20), we get

$\displaystyle\sup_{t\in[0,T]}E[f^{\varepsilon}_{n}](t)+$
$\displaystyle\frac{\varepsilon}{2}\bigg{(}\sup_{t\in[0,T]}\int_{\Upsilon}|u^{\varepsilon}_{n}(t)|^{2}\,\mathrm{d}\bm{\xi}+\int_{\Upsilon_{T}}|\nabla_{\bm{\xi}}u^{\varepsilon}_{n}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t\bigg{)}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\int_{\Upsilon_{T}}\Big{(}D_{e}f^{\varepsilon}_{n}(1-\rho^{\varepsilon}_{n})\big{|}\nabla\log\big{(}\frac{f^{\varepsilon}_{n}}{1-\rho^{\varepsilon}_{n}}\big{)}\big{|}^{2}+f^{\varepsilon}_{n}\big{|}\partial_{\theta}\log\big{(}\frac{f^{\varepsilon}_{n}}{1-\rho^{\varepsilon}_{n}}\big{)}\big{|}^{2}\Big{)}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t\leq
C,$

i.e.,

(3.30)
$\displaystyle\sup_{t\in[0,T]}E[f^{\varepsilon}_{n}](t)+\sup_{t\in[0,T]}\int_{\Upsilon}|\sqrt{\varepsilon}u^{\varepsilon}_{n}(t)|^{2}\,\mathrm{d}\bm{\xi}+\int_{\Upsilon_{T}}|\nabla_{\bm{\xi}}\sqrt{\varepsilon}u^{\varepsilon}_{n}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t$
$\displaystyle+\int_{\Upsilon_{T}}(1-\rho^{\varepsilon}_{n})|\nabla\sqrt{f^{\varepsilon}_{n}}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t+\int_{\Omega_{T}}\rho^{\varepsilon}_{n}|\nabla\sqrt{1-\rho^{\varepsilon}_{n}}|^{2}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle+\int_{\Upsilon_{T}}|\partial_{\theta}\sqrt{f^{\varepsilon}_{n}}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t+\int_{\Omega_{T}}|\nabla\rho^{\varepsilon}_{n}|^{2}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\leq C,$

where we used the Fubini–Tonelli Theorem and the commutativity of the
derivative in $x$ and integral in $\theta$ to rewrite
$\int_{\Upsilon_{T}}\nabla
f^{\varepsilon}_{n}\cdot\nabla\rho^{\varepsilon}_{n}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t=\int_{\Omega_{T}}|\nabla\rho^{\varepsilon}_{n}|^{2}\,\mathrm{d}x\,\mathrm{d}t$
for the final term on the left-hand side of (3.30). Observe that, using the
bounds in the previous estimate,

$\displaystyle\int_{\Omega_{T}}|\nabla\sqrt{1-\rho^{\varepsilon}_{n}}|^{2}\,\mathrm{d}x\,\mathrm{d}t=\frac{1}{4}\int_{\Omega_{T}}|\nabla\rho^{\varepsilon}_{n}|^{2}\,\mathrm{d}x\,\mathrm{d}t+\int_{\Omega_{T}}\rho^{\varepsilon}_{n}|\nabla\sqrt{1-\rho^{\varepsilon}_{n}}|^{2}\,\mathrm{d}x\,\mathrm{d}t\leq
C,$

and similarly

(3.31)
$\displaystyle\int_{\Upsilon_{T}}(1-\rho^{\varepsilon}_{n})|\nabla_{\bm{\xi}}\sqrt{f^{\varepsilon}_{n}}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t\leq\int_{\Upsilon_{T}}(1-\rho^{\varepsilon}_{n})|\nabla\sqrt{f^{\varepsilon}_{n}}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t+\int_{\Upsilon_{T}}|\partial_{\theta}\sqrt{f^{\varepsilon}_{n}}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t\leq
C.$

Collating these latter two bounds with (3.30) yields the estimate (3.23).

2\. Interpolated bounds on $\sqrt{f}$ and $\sqrt{(1-\rho)f}$: Using the
product rule and the estimates (3.31) and (3.30), we find

$\displaystyle\int_{\Upsilon_{T}}|\nabla_{\bm{\xi}}\sqrt{(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t$
$\displaystyle\leq
C\bigg{(}\int_{\Upsilon_{T}}(1-\rho^{\varepsilon}_{n})|\nabla_{\bm{\xi}}\sqrt{f^{\varepsilon}_{n}}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t+\int_{\Omega_{T}}\rho^{\varepsilon}_{n}|\nabla\sqrt{1-\rho^{\varepsilon}_{n}}|^{2}\,\mathrm{d}x\,\mathrm{d}t\bigg{)}$
$\displaystyle\leq C.$

Meanwhile, the estimate (3.22) implies
$\|\sqrt{(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}}\|_{L^{\infty}(0,T;L^{2}(\Upsilon))}\leq
C$, whence we deduce

$\|\sqrt{(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}}\|_{L^{\infty}(0,T;L^{2}(\Upsilon))}+\|\sqrt{(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}}\|_{L^{2}(0,T;H^{1}(\Upsilon))}\leq
C.$

In turn, the classical interpolation lemma [12, §1, Proposition 3.2] implies

$\|\sqrt{(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}}\|_{L^{\frac{10}{3}}(\Upsilon_{T})}\leq
C,$

and thus (3.25) is proved.

Our objective is then to make use of the _linear_ diffusion in angle to
upgrade the integrability of $f^{\varepsilon}_{n}$; to do so, we use an
adapted version of the aforementioned interpolation lemma to the space
$L^{\infty}(\Omega_{T};L^{2}(0,2\pi))\cap L^{2}(\Omega_{T};H^{1}(0,2\pi))$,
cf. Appendix A. Using both (3.22) and the estimate on
$\partial_{\theta}\sqrt{f^{\varepsilon}_{n}}$ from (3.30), we remark that

$\|\sqrt{f^{\varepsilon}_{n}}\|_{L^{\infty}(\Omega_{T};L^{2}(0,2\pi))}+\|\sqrt{f^{\varepsilon}_{n}}\|_{L^{2}(\Omega_{T};H^{1}(0,2\pi))}\leq
C.$

A direct application of Lemma A.1 then implies

$\|\sqrt{f^{\varepsilon}_{n}}\|_{L^{6}(\Upsilon_{T})}\leq C,$

and (3.24) follows.

3\. Estimate on $f\sqrt{1-\rho}$: We write

$\displaystyle\sqrt{1-\rho^{\varepsilon}_{n}}\nabla_{\bm{\xi}}f^{\varepsilon}_{n}=2\sqrt{f^{\varepsilon}_{n}}\sqrt{1-\rho^{\varepsilon}_{n}}\nabla_{\bm{\xi}}\sqrt{f^{\varepsilon}_{n}},$

whence Hölder’s inequality yields

$\|\sqrt{1-\rho^{\varepsilon}_{n}}\nabla_{\bm{\xi}}f^{\varepsilon}_{n}\|_{L^{\frac{3}{2}}(\Upsilon_{T})}\leq
2\|\sqrt{f^{\varepsilon}_{n}}\|_{L^{6}(\Upsilon_{T})}\|\sqrt{1-\rho^{\varepsilon}_{n}}\nabla_{\bm{\xi}}\sqrt{f^{\varepsilon}_{n}}\|_{L^{2}(\Upsilon_{T})}\leq
C,$

where we used (3.23) and (3.24). In turn, using the Hölder, Minkowski, and
Jensen inequalities, we get

$\displaystyle\|\nabla_{\bm{\xi}}\big{(}f^{\varepsilon}_{n}\sqrt{1-\rho^{\varepsilon}_{n}}\big{)}\|_{L^{\frac{6}{5}}(\Upsilon_{T})}$
$\displaystyle\leq\|\sqrt{1-\rho^{\varepsilon}_{n}}\nabla_{\bm{\xi}}f^{\varepsilon}_{n}\|_{L^{\frac{6}{5}}(\Upsilon_{T})}+\|f^{\varepsilon}_{n}\nabla\sqrt{1-\rho^{\varepsilon}_{n}}\|_{L^{\frac{6}{5}}(\Upsilon_{T})}$
$\displaystyle\leq
C\|\sqrt{1-\rho^{\varepsilon}_{n}}\nabla_{\bm{\xi}}f^{\varepsilon}_{n}\|_{L^{\frac{3}{2}}(\Upsilon_{T})}+\|f^{\varepsilon}_{n}\|_{L^{3}(\Upsilon_{T})}\|\nabla\sqrt{1-\rho^{\varepsilon}_{n}}\|_{L^{2}(\Omega_{T})}$
$\displaystyle\leq C,$

where we again used (3.23) and (3.24). Therefore, using again Jensen’s
inequality and (3.24) to bound
$\|f^{\varepsilon}_{n}\sqrt{1-\rho^{\varepsilon}_{n}}\|_{L^{\frac{6}{5}}(\Upsilon_{T})}$
uniformly in $n$ and $\varepsilon$, we obtain the estimate (3.25). ∎

In what follows, we estimate the time derivatives of the transformed variable
and original unknowns; these bounds will then permit us to apply the
Aubin–Lions Lemma in the proof of Theorem 1, cf. §4. To this end, we recall
the angle-independent integrated entropy variable

$U^{\varepsilon}_{n}(t,x)=\int_{0}^{2\pi}u^{\varepsilon}_{n}(t,x,\theta)\,\mathrm{d}\theta,$

which was defined at the end of the statement of Proposition 3.1. Observe
that, by virtue of Jensen’s inequality, the entropy estimate (3.23) yields

(3.32)
$\sup_{t\in[0,T]}\int_{\Omega}|\sqrt{\varepsilon}U^{\varepsilon}_{n}(t)|^{2}\,\mathrm{d}x+\int_{\Omega_{T}}|\nabla\sqrt{\varepsilon}U^{\varepsilon}_{n}|^{2}\,\mathrm{d}x\,\mathrm{d}t\leq
C.$

Furthermore, integrating (3.21) with respect to the angle variable, yields the
evolution equation on the angle-independent quantities:

(3.33) $\partial_{t}(\varepsilon
U^{\varepsilon}_{n}+\rho^{\varepsilon}_{n})+\mathrm{Pe}\operatorname{div}((1-\rho^{\varepsilon}_{n}){\bf
p}^{\varepsilon}_{n})=\varepsilon\Delta
U^{\varepsilon}_{n}+D_{e}\Delta\rho^{\varepsilon}_{n}$

in the weak sense, where

${\bf
p}^{\varepsilon}_{n}(t,x)=\int_{0}^{2\pi}f^{\varepsilon}_{n}(t,x,\theta){\bf
e}(\theta)\,\mathrm{d}\theta$

satisfies, using the non-negativity of $f^{\varepsilon}_{n}$ and the
boundedness of $\rho^{\varepsilon}_{n}$, the uniform estimate

$\|{\bf p}^{\varepsilon}_{n}\|_{L^{\infty}(\Omega_{T})}\leq 1.$

######  Corollary 3.4 (Uniform estimates on time derivatives).

Let $\varepsilon,n,u^{\varepsilon}_{n}$ be as in Lemma 3.2 and
$f^{\varepsilon}_{n},\rho^{\varepsilon}_{n}$ be as in (3.18). There exists a
positive constant $C$, independent of $\varepsilon,n$, such that there holds
the uniform estimates:

(3.34) $\displaystyle\|\partial_{t}(\varepsilon
u^{\varepsilon}_{n}+f^{\varepsilon}_{n})\|_{X^{\prime}}\leq C,$ (3.35)
$\displaystyle\|\partial_{t}(\varepsilon
U^{\varepsilon}_{n}+\rho^{\varepsilon}_{n})\|_{Y^{\prime}}\leq C.$

###### Proof.

In view of the estimates of Lemma 3.3, a standard density argument shows that
the weak formulation (3.16), cf. (3.21), generalises to the space of test
functions $\mathcal{A}$ defined in (1.3), i.e., for all $\psi\in\mathcal{A}$,
there holds

(3.36) $\displaystyle-\langle\partial_{t}(\varepsilon u^{\varepsilon}_{n}$
$\displaystyle+f^{\varepsilon}_{n}),\psi\rangle+\mathrm{Pe}\int_{\Upsilon_{T}}(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}{\bf
e}(\theta)\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t$ $\displaystyle=$
$\displaystyle\int_{\Upsilon_{T}}\Big{(}\varepsilon\nabla_{\bm{\xi}}u^{\varepsilon}_{n}\cdot\nabla_{\bm{\xi}}\psi+D_{e}\big{(}(1-\rho^{\varepsilon}_{n})\nabla
f^{\varepsilon}_{n}+f^{\varepsilon}_{n}\nabla\rho^{\varepsilon}_{n}\big{)}\cdot\nabla\psi+\partial_{\theta}f^{\varepsilon}_{n}\partial_{\theta}\psi\Big{)}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t.$

1\. Estimate on $\partial_{t}(\varepsilon u+f)$: The weak formulation (3.36)
implies that, for all $\psi\in\mathcal{A}$, there holds

(3.37) $\displaystyle|\langle\partial_{t}(\varepsilon
u^{\varepsilon}_{n}+f^{\varepsilon}_{n}),\psi\rangle|\leq$ $\displaystyle
D_{e}\bigg{|}\int_{\Upsilon_{T}}(1-\rho^{\varepsilon}_{n})\nabla
f^{\varepsilon}_{n}\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t\bigg{|}+D_{e}\bigg{|}\int_{\Upsilon_{T}}f^{\varepsilon}_{n}\nabla\rho^{\varepsilon}_{n}\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t\bigg{|}$
$\displaystyle+\sqrt{\varepsilon}\bigg{|}\int_{\Upsilon_{T}}\nabla_{\bm{\xi}}\sqrt{\varepsilon}u^{\varepsilon}_{n}\cdot\nabla_{\bm{\xi}}\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t\bigg{|}+\bigg{|}\int_{\Upsilon_{T}}\partial_{\theta}f^{\varepsilon}_{n}\partial_{\theta}\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t\bigg{|}$
$\displaystyle+\mathrm{Pe}\bigg{|}\int_{\Upsilon_{T}}(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}{\bf
e}(\theta)\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t\bigg{|}$
$\displaystyle=:$ $\displaystyle\sum_{j=1}^{5}I_{j}.$

Using the estimates of Lemma 3.3, we bound each term on the right-hand side of
(3.37) individually. To begin with, Hölder’s inequality yields

$I_{2}\leq\|f^{\varepsilon}_{n}\|_{L^{3}(\Upsilon_{T})}\|\nabla\rho^{\varepsilon}_{n}\|_{L^{2}(\Upsilon_{T})}\|\nabla_{\bm{\xi}}\psi\|_{L^{6}(\Upsilon_{T})}\leq
C\|\nabla_{\bm{\xi}}\psi\|_{L^{6}(\Upsilon_{T})}.$

Next, we use the second relation in (3.20), the previous bound, and Jensen’s
inequality to write

$\displaystyle I_{1}\leq
C\Big{(}\|\sqrt{(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}}\|_{L^{\frac{10}{3}}(\Upsilon_{T})}\|\nabla_{\bm{\xi}}\sqrt{(1-\rho^{\varepsilon}_{n})f^{\varepsilon}_{n}}\|_{L^{2}(\Upsilon_{T})}\|\nabla_{\bm{\xi}}\psi\|_{L^{5}(\Upsilon_{T})}\\!+\\!I_{2}\Big{)}\\!\leq
C\|\nabla_{\bm{\xi}}\psi\|_{L^{6}(\Upsilon_{T})}.$

Similarly, using the first relation in (3.20),

$\displaystyle
I_{4}\leq\|\sqrt{f^{\varepsilon}_{n}}\|_{L^{6}(\Upsilon_{T})}\|\partial_{\theta}\sqrt{f^{\varepsilon}_{n}}\|_{L^{2}(\Upsilon_{T})}\|\partial_{\theta}\psi\|_{L^{3}(\Upsilon_{T})}\leq
C\|\partial_{\theta}\psi\|_{L^{3}(\Upsilon_{T})}.$

For the third term, we have

$I_{3}\leq\sqrt{\varepsilon}\|\nabla_{\bm{\xi}}\sqrt{\varepsilon}u^{\varepsilon}_{n}\|_{L^{2}(\Upsilon_{T})}\|\nabla_{\bm{\xi}}\psi\|_{L^{2}(\Upsilon)}\leq
C\sqrt{\varepsilon}\|\nabla_{\bm{\xi}}\psi\|_{L^{2}(\Upsilon)}.$

For the final integral, the uniform boundedness of $\rho^{\varepsilon}_{n}$
and Jensen’s inequality implies

$\displaystyle I_{5}\leq$
$\displaystyle\|f^{\varepsilon}_{n}\|_{L^{3}(\Upsilon_{T})}\|\nabla_{\bm{\xi}}\psi\|_{L^{\frac{3}{2}}(\Upsilon_{T})}\leq
C\|\nabla_{\bm{\xi}}\psi\|_{L^{\frac{3}{2}}(\Upsilon_{T})}.$

By combining the previous bounds and using Jensen’s inequality, we deduce

$|\langle\partial_{t}(\varepsilon
u^{\varepsilon}_{n}+f^{\varepsilon}_{n}),\psi\rangle|\leq
C\|\nabla_{\bm{\xi}}\psi\|_{L^{6}(\Upsilon_{T})},$

and (3.34) follows.

2\. Estimate on $\partial_{t}(\varepsilon U+\rho)$: Testing (3.33) against
$\psi\in\mathcal{A}_{s}$, we obtain

$\displaystyle\langle\partial_{t}(\varepsilon
U^{\varepsilon}_{n}+\rho^{\varepsilon}_{n}),\psi\rangle=$
$\displaystyle\mathrm{Pe}\int_{\Omega_{T}}(1-\rho^{\varepsilon}_{n}){\bf
p}^{\varepsilon}_{n}\cdot\nabla\psi\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle-\sqrt{\varepsilon}\int_{\Omega_{T}}\nabla\sqrt{\varepsilon}U^{\varepsilon}_{n}\cdot\nabla\psi\,\mathrm{d}x\,\mathrm{d}t-D_{e}\int_{\Omega_{T}}\nabla\rho^{\varepsilon}_{n}\cdot\nabla\psi\,\mathrm{d}x\,\mathrm{d}t,$

from which we deduce, using (3.32), the uniform boundedness of ${\bf
p}^{\varepsilon}_{n}$, and Jensen’s inequality,

$\displaystyle|\langle\partial_{t}(\varepsilon U^{\varepsilon}_{n}$
$\displaystyle+\rho^{\varepsilon}_{n}),\psi\rangle|$ $\displaystyle\leq$
$\displaystyle
C\Big{(}\|\nabla\psi\|_{L^{1}(\Omega_{T})}+\sqrt{\varepsilon}\|\nabla\sqrt{\varepsilon}U^{\varepsilon}_{n}\|_{L^{2}(\Omega_{T})}\|\nabla\psi\|_{L^{2}(\Omega_{T})}+\|\nabla\rho^{\varepsilon}_{n}\|_{L^{2}(\Omega_{T})}\|\nabla\psi\|_{L^{2}(\Omega_{T})}\Big{)}$
$\displaystyle\leq$ $\displaystyle C\|\nabla\psi\|_{L^{2}(\Omega_{T})},$

whence we deduce (3.35). ∎

The results of Lemmas 3.2-3.3 and Corollary 3.4 prove Proposition 3.1.

##  4\. Proof of the Main Existence Theorem

In this section, we study the convergence of the approximate solutions and
obtain the existence of a weak solution to (1.1). We shall essentially take
the double-limit as $\varepsilon\to 0$ in the regularisation parameter and
$n\to\infty$ in the Galerkin approximation simultaneously, which will entail
the use of a diagonal argument. We explain why we proceed in this manner in
the next paragraph.

The reason for employing a double-limit strategy is that the time-derivative
estimates of Corollary 3.4 do not decouple into separate estimates on
$\partial_{t}\varepsilon U^{\varepsilon}_{n}$ and
$\partial_{t}\rho^{\varepsilon}_{n}$ individually. This means that the
Aubin–Lions Lemma yields the strong convergence of only the sum $\varepsilon
U^{\varepsilon}_{n}+\rho^{\varepsilon}_{n}$ as $n\to\infty$ for fixed
$\varepsilon$. It is then, to the authors’ knowledge, not possible to obtain
the strong convergence $\rho^{\varepsilon}_{n}\to\rho^{\varepsilon}$ for fixed
$\varepsilon$ without having already determined that $\varepsilon
U^{\varepsilon}_{n}$ converges strongly. While the classical theory of [20]
implies that the linear heat term
$\varepsilon(\partial_{t}u^{\varepsilon}_{n}-\Delta_{\bm{\xi}}u^{\varepsilon}_{n})$
ought to yield $H^{2}$-regularity of $u^{\varepsilon}_{n}$ and hence
$L^{2}$-boundedness of $\partial_{t}u^{\varepsilon}_{n}$ (from which one could
infer strong convergence of $u^{\varepsilon}_{n}$ by another application of
Aubin–Lions), the presence of the non-local operator $A$ and the non-local
matrix $\tilde{M}$ in the equation (3.17) makes _quantitative estimates_
rather intractable. In turn, we are not able to deduce that
$U^{\varepsilon}_{n}$ converges strongly as $n\to\infty$ while keeping
$\varepsilon$ fixed. Instead, we rely on the uniform bound on
$\|\sqrt{\varepsilon}U^{\varepsilon}_{n}\|_{L^{2}(\Omega_{T})}$ to deduce that
$\varepsilon U^{\varepsilon}_{n}\to 0$ strongly as $\varepsilon\to 0$; whence
the aforementioned strong convergence of the sum $\varepsilon
U^{\varepsilon}_{n}+\rho^{\varepsilon}_{n}$ yields the desired strong
convergence of $\rho^{\varepsilon}_{n}$.

The main difficulty in implementing this strategy lies in making sure that the
sequence of regularised initial data converges appropriately. This is all-the-
more subtle since the initial functions
$\\{f^{\varepsilon}_{n}(0),\rho^{\varepsilon}_{n}(0)\\}_{\varepsilon,n}$ are
determined via the usual non-linear transformation from
$u^{\varepsilon}_{n}(0)$, the $n$-th Galerkin approximation of the initial
data $u^{\varepsilon}_{0}$, i.e.,

(4.1) $u^{\varepsilon}_{n}(0)=\sum_{j=1}^{n}\langle
u^{\varepsilon}_{0},\varphi_{j}\rangle_{H^{2}_{\mathrm{per}}(\Upsilon)}\varphi_{j},\quad
f^{\varepsilon}_{n}(0)=\frac{e^{u^{\varepsilon}_{n}(0)}}{1+\int_{0}^{2\pi}e^{u^{\varepsilon}_{n}(0)}\,\mathrm{d}\theta},\quad\rho^{\varepsilon}_{n}(0)=\int_{0}^{2\pi}f^{\varepsilon}_{n}(0)\,\mathrm{d}\theta,$

where we recall that $\\{\varphi_{j}\\}_{j}$ is the orthonormal basis of
$H^{2}_{\mathrm{per}}(\Upsilon)$ used in §3.1.

The next lemma shows that there exists a choice of diagonal subsequence from
the array $\\{f^{\varepsilon}_{n}\\}_{\varepsilon,n}$ which preserves the
desired convergence to the initial data $f_{0}$. The proof is delayed to
Appendix B.

######  Lemma 4.1 (Diagonal convergence of initial data).

There exists a diagonal subsequence of the array
$\\{f^{\varepsilon}_{n}\\}_{\varepsilon,n}$, which, with slight abuse of
notation, we denote by $\\{f^{\varepsilon}\\}_{\varepsilon}$, such that

$\lim_{\varepsilon\to 0}\|f^{\varepsilon}(0)-f_{0}\|_{L^{p}(\Upsilon)}=0.$

We are now ready to give the proof of the main existence theorem; Theorem 1.
We will show that the sequence $\\{f^{\varepsilon}\\}_{\varepsilon}$
constructed in the previous lemma admits a subsequential limit which is indeed
a weak solution of (1.1) in the sense of Definition 1.1. We emphasise that
$\\{f^{\varepsilon}\\}_{\varepsilon}$ satisfies the uniform estimates of
Proposition 3.1.

###### Proof of Theorem 1.

We assume for the time being that the test function $\psi$ belongs to the
stronger space $\mathcal{A}$. We shall relax this assumption at the end of the
proof.

1\. Convergence of time-derivative: The uniform bound (3.7) implies

$\partial_{t}(\varepsilon
u^{\varepsilon}+f^{\varepsilon})\overset{*}{\rightharpoonup}\zeta\quad\text{
weakly-* in }X^{\prime},$

for some unkown $\zeta\in X^{\prime}$. The estimate (3.4) implies
$f^{\varepsilon}\rightharpoonup f$ weakly in $L^{3}(\Upsilon_{T})$ whence, for
all $\psi\in\mathcal{A}$,

$\displaystyle\langle\zeta,\psi\rangle=\lim_{\varepsilon\to
0}\langle\partial_{t}(\varepsilon
u^{\varepsilon}+f^{\varepsilon}),\psi\rangle$
$\displaystyle=-\lim_{\varepsilon\to
0}\sqrt{\varepsilon}\int_{\Upsilon_{T}}\sqrt{\varepsilon}u^{\varepsilon}\partial_{t}\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t-\lim_{\varepsilon\to
0}\int_{\Upsilon_{T}}f^{\varepsilon}\partial_{t}\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t$
$\displaystyle=-\int_{\Upsilon_{T}}f\partial_{t}\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t,$

where we used that
$\|\sqrt{\varepsilon}u^{\varepsilon}\|_{L^{2}(\Upsilon_{T})}\leq C$ from (3.3)
to obtain the final line. It therefore follows that $\zeta=\partial_{t}f$ and
$\partial_{t}f^{\varepsilon}\overset{*}{\rightharpoonup}\partial_{t}f$
weakly-* in $X^{\prime}$. An analogous argument shows
$\partial_{t}\rho^{\varepsilon}\overset{*}{\rightharpoonup}\partial_{t}\rho$
weakly-* in $Y^{\prime}$.

2\. Strong convergence of $\rho^{\varepsilon}$: To begin with, (3.3) implies
$\|\rho^{\varepsilon}\|_{L^{2}(0,T;H^{1}(\Omega))}\leq C$, whence it follows
that $\rho^{\varepsilon}\rightharpoonup\rho$ weakly
$L^{2}(0,T;H^{1}(\Omega))$. Meanwhile, the uniform estimates from Proposition
3.1 on $\sqrt{\varepsilon}U^{\varepsilon}$ and $\rho^{\varepsilon}$ imply that

$\|\varepsilon
U^{\varepsilon}+\rho^{\varepsilon}\|_{L^{\infty}(0,T;L^{2}(\Omega))}+\|\varepsilon
U^{\varepsilon}+\rho^{\varepsilon}\|_{L^{2}(0,T;H^{1}(\Omega))}\leq C.$

In turn, the time-derivative bound (3.8) and the Aubin–Lions Lemma imply that,
up to a subsequence which we do not relabel,

$\varepsilon U^{\varepsilon}+\rho^{\varepsilon}\to A\quad\text{strongly in
}L^{2}(\Omega_{T}).$

for some unknown $A\in L^{2}(\Omega_{T})$. Note that the triangle inequality
and the uniform estimate on $\sqrt{\varepsilon}U^{\varepsilon}$ from
Proposition 3.1 imply

$\displaystyle\|\rho^{\varepsilon}-A\|_{L^{2}(\Omega_{T})}\leq$
$\displaystyle\|\varepsilon
U^{\varepsilon}+\rho^{\varepsilon}-A\|_{L^{2}(\Omega_{T})}+\sqrt{\varepsilon}\underbrace{\|\sqrt{\varepsilon}U^{\varepsilon}\|_{L^{2}(\Omega_{T})}}_{\leq
C}\to 0,$

and thus by the uniqueness of limits, $A=\rho$, i.e.,

(4.2) $\rho^{\varepsilon}\to\rho\quad\text{strongly in }L^{2}(\Omega_{T}),$

and $\rho\in[0,1]$ a.e. in $\Omega_{T}$. Moreover, for all $q\in(2,\infty)$,
the boundedness of $\rho,\rho^{\varepsilon}$ implies

$\displaystyle\int_{\Omega_{T}}|\rho^{\varepsilon}-\rho|^{q}\,\mathrm{d}x\,\mathrm{d}t=\int_{\Omega_{T}}|\rho^{\varepsilon}-\rho|^{q-2}|\rho^{\varepsilon}-\rho|^{2}\,\mathrm{d}x\,\mathrm{d}t\leq\int_{\Omega_{T}}|\rho^{\varepsilon}-\rho|^{2}\,\mathrm{d}x\,\mathrm{d}t\to
0.$

Using Jensen’s inequality to cover the indices $q\in[1,2)$, it follows that,
for all $q\in[1,\infty)$,

(4.3) $\rho^{\varepsilon}\to\rho\quad\text{strongly in }L^{q}(\Omega_{T}).$

The previous convergence implies that $\rho^{\varepsilon}\to\rho$ a.e. in
$\Omega_{T}$; up to a subsequence which we do not relabel. In turn, given any
$q\in[1,\infty)$, an application of the Dominated Convergence Theorem shows,
using again $0\leq\rho,\rho^{\varepsilon}_{n}\leq 1$ to establish an
admissible constant majorant function,

$\displaystyle\int_{\Omega_{T}}\big{|}\sqrt{1-\rho^{\varepsilon}}-\sqrt{1-\rho}\big{|}^{q}\,\mathrm{d}x\,\mathrm{d}t\to
0,$

i.e., for all $q\in[1,\infty)$,

(4.4) $\sqrt{1-\rho^{\varepsilon}}\to\sqrt{1-\rho}\quad\text{strongly in
}L^{q}(\Omega_{T}).$

3\. Weak convergences and bounds: The uniform estimate (3.6) implies that
there exists $B\in L^{\frac{6}{5}}(0,T;W^{1,\frac{6}{5}}(\Upsilon))$ such
that, up to a subsequence which we do not relabel, there holds

$f^{\varepsilon}\sqrt{1-\rho^{\varepsilon}}\rightharpoonup B\quad\text{weakly
in }L^{\frac{6}{5}}(0,T;W^{1,\frac{6}{5}}(\Upsilon)).$

Meanwhile, the uniform estimate (3.4) implies that
$f^{\varepsilon}\rightharpoonup f$ weakly in $L^{3}(\Upsilon_{T})$, so that,
recalling the strong convergence (4.4), there holds

$f^{\varepsilon}\sqrt{1-\rho^{\varepsilon}}\rightharpoonup
f\sqrt{1-\rho}\quad\text{weakly in }L^{3}(\Upsilon_{T}),$

and the uniqueness of weak limits implies $B=f\sqrt{1-\rho}$ a.e. in
$\Upsilon_{T}$. We therefore have

(4.5) $f^{\varepsilon}\sqrt{1-\rho^{\varepsilon}}\rightharpoonup
f\sqrt{1-\rho}\quad\text{weakly in
}L^{\frac{6}{5}}(0,T;W^{1,\frac{6}{5}}(\Upsilon)).$

In turn, recalling the strong convergence (4.4), we have that

(4.6)
$\sqrt{1-\rho^{\varepsilon}}\nabla_{\bm{\xi}}(f^{\varepsilon}\sqrt{1-\rho^{\varepsilon}})\rightharpoonup\sqrt{1-\rho}\nabla_{\bm{\xi}}(f\sqrt{1-\rho})\quad\text{weakly
in }L^{\frac{6}{5}}(\Upsilon_{T}).$

Analogous arguments show
$\sqrt{1-\rho^{\varepsilon}}\nabla\sqrt{f^{\varepsilon}}\rightharpoonup\sqrt{1-\rho}\nabla\sqrt{f}$
and $\nabla\sqrt{1-\rho^{\varepsilon}}\rightharpoonup\nabla\sqrt{1-\rho}$
weakly in $L^{2}(\Omega_{T})$.

Similarly, we deduce from (3.3) and (3.4) that

$\|\partial_{\theta}f^{\varepsilon}\|_{L^{\frac{3}{2}}(\Upsilon_{T})}\leq
C\|\sqrt{f^{\varepsilon}}\|_{L^{6}(\Upsilon_{T})}\|\partial_{\theta}\sqrt{f^{\varepsilon}}\|_{L^{2}(\Upsilon_{T})}\leq
C,$

whence $f^{\varepsilon}$ is uniformly bounded in
$L^{\frac{3}{2}}(\Omega_{T};W^{1,\frac{3}{2}}(0,2\pi))$. Using the uniqueness
of limits and $f^{\varepsilon}\rightharpoonup f$ weakly in
$L^{3}(\Upsilon_{T})$, we deduce as before that

(4.7)
$\partial_{\theta}f^{\varepsilon}\rightharpoonup\partial_{\theta}f\quad\text{weakly
in }L^{\frac{3}{2}}(\Upsilon_{T}).$

An analogous approach shows
$\partial_{\theta}\sqrt{f^{\varepsilon}}\rightharpoonup\partial_{\theta}\sqrt{f}$
weakly in $L^{2}(\Upsilon_{T})$.

In view of the weak (resp. weak-*) lower semicontinuity of the norms, the
limits satisfy

(4.8) $\displaystyle f\in L^{3}(\Upsilon_{T}),\quad\partial_{t}f\in
X^{\prime},\quad\partial_{\theta}\sqrt{f}\in
L^{2}(\Upsilon_{T}),\quad\nabla\sqrt{1-\rho}\in L^{2}(\Omega_{T}),$
$\displaystyle\partial_{t}\rho\in
Y^{\prime},\quad\sqrt{1-\rho}\nabla\sqrt{f}\in L^{2}(\Upsilon_{T}),\quad
f\sqrt{1-\rho}\in L^{\frac{6}{5}}(0,T;W^{1,\frac{6}{5}}(\Upsilon)).$

4\. Convergence of spatial diffusions: We rewrite the diffusive terms as
follows

(4.9) $\displaystyle(1-\rho^{\varepsilon})\nabla
f^{\varepsilon}+f^{\varepsilon}\nabla\rho^{\varepsilon}=\sqrt{1-\rho^{\varepsilon}}\nabla(f^{\varepsilon}\sqrt{1-\rho^{\varepsilon}})+\frac{3}{2}f^{\varepsilon}\nabla\rho^{\varepsilon}.$

The weak convergence (4.6) implies

(4.10) $\lim_{\varepsilon\to
0}\int_{\Upsilon_{T}}\sqrt{1-\rho^{\varepsilon}}\nabla(f^{\varepsilon}\sqrt{1-\rho^{\varepsilon}})\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t=\int_{\Upsilon_{T}}\sqrt{1-\rho}\nabla(f\sqrt{1-\rho})\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t,$

and the desired convergence for the first term on the right-hand side of (4.9)
is verified.

The second term on the right-hand side of (4.9) is dealt with as follows; we
essentially use an integration by parts to place two derivatives on the test
function prior to passing to the limit. We write

$f^{\varepsilon}\nabla\rho^{\varepsilon}=-2f^{\varepsilon}\sqrt{1-\rho^{\varepsilon}}\nabla\sqrt{1-\rho^{\varepsilon}},$

and observe that an integration by parts yields

(4.11)
$\displaystyle\int_{\Upsilon_{T}}f^{\varepsilon}\nabla\rho^{\varepsilon}\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t=$
$\displaystyle-2\int_{\Upsilon_{T}}f^{\varepsilon}\sqrt{1-\rho^{\varepsilon}}\nabla\sqrt{1-\rho^{\varepsilon}}\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t$
$\displaystyle=$ $\displaystyle
2\int_{\Upsilon_{T}}\sqrt{1-\rho^{\varepsilon}}\operatorname{div}\big{(}f^{\varepsilon}\sqrt{1-\rho^{\varepsilon}}\nabla\psi\big{)}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t$
$\displaystyle=$ $\displaystyle I_{1}^{\varepsilon}+I_{2}^{\varepsilon},$

where

$\displaystyle I_{1}^{\varepsilon}:=$ $\displaystyle
2\int_{0}^{T}\int_{\Upsilon}\sqrt{1-\rho^{\varepsilon}}\nabla\big{(}f^{\varepsilon}\sqrt{1-\rho^{\varepsilon}}\big{)}\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t,$
$\displaystyle I_{2}^{\varepsilon}:=$ $\displaystyle
2\int_{0}^{T}\int_{\Upsilon}\sqrt{1-\rho^{\varepsilon}}\big{(}f^{\varepsilon}\sqrt{1-\rho^{\varepsilon}}\big{)}\Delta\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t.$

The weak convergence from (4.5) along with the strong convergence from (4.4)
implies that there holds the weak convergences in $L^{1}(\Upsilon_{T})$:

$\displaystyle\sqrt{1-\rho^{\varepsilon}}\nabla\big{(}f^{\varepsilon}\sqrt{1-\rho^{\varepsilon}}\big{)}\rightharpoonup\sqrt{1-\rho}\nabla\big{(}f\sqrt{1-\rho}\big{)},$
$\displaystyle\sqrt{1-\rho^{\varepsilon}}\big{(}f^{\varepsilon}\sqrt{1-\rho^{\varepsilon}}\big{)}\rightharpoonup\sqrt{1-\rho}\big{(}f\sqrt{1-\rho}\big{)},$

from which we deduce

$\displaystyle\lim_{\varepsilon\to 0}I_{1}^{\varepsilon}=$ $\displaystyle
2\int_{\Upsilon_{T}}\sqrt{1-\rho}\nabla\big{(}f\sqrt{1-\rho}\big{)}\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t,$
$\displaystyle\lim_{\varepsilon\to 0}I_{2}^{\varepsilon}=$ $\displaystyle
2\int_{\Upsilon_{T}}\sqrt{1-\rho}\big{(}f\sqrt{1-\rho}\big{)}\Delta\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t.$

Using the computation in (4.11) again, we obtain

$\displaystyle\lim_{\varepsilon\to
0}\int_{\Upsilon_{T}}f^{\varepsilon}\nabla\rho^{\varepsilon}\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t=$
$\displaystyle
2\int_{\Upsilon_{T}}\sqrt{1-\rho}\operatorname{div}\big{(}f\sqrt{1-\rho}\nabla\psi\big{)}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t$
$\displaystyle=$
$\displaystyle\int_{\Upsilon_{T}}f\nabla\rho\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t,$

where we used that the above quantities are regular enough to perform the
integration by parts; cf. (4.8). By collating the previous limit with (4.10)
and using (4.9), we get

$\lim_{\varepsilon\to 0}\int_{\Upsilon_{T}}\big{(}(1-\rho^{\varepsilon})\nabla
f^{\varepsilon}+f^{\varepsilon}\nabla\rho^{\varepsilon}\big{)}\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t=\int_{\Upsilon_{T}}\big{(}(1-\rho)\nabla
f+f\nabla\rho\big{)}\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t.$

5\. Convergence of angular diffusion, non-local drift, and $\varepsilon$-error
term: For the angular diffusion, the weak convergence of
$\partial_{\theta}f^{\varepsilon}$ in (4.7) implies directly

$\displaystyle\lim_{\varepsilon\to
0}\int_{\Upsilon_{T}}\partial_{\theta}f^{\varepsilon}\partial_{\theta}\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t=\int_{\Upsilon_{T}}\partial_{\theta}f\partial_{\theta}\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t.$

For the non-local drift, the strong convergence of $\rho^{\varepsilon}$ from
(4.3) and the weak convergence of $f^{\varepsilon}$ implies
$(1-\rho^{\varepsilon})f^{\varepsilon}\rightharpoonup(1-\rho)f$ weakly in
$L^{1}(\Upsilon_{T})$, whence

$\lim_{\varepsilon\to
0}\int_{\Upsilon_{T}}(1-\rho^{\varepsilon})f^{\varepsilon}{\bf
e}(\theta)\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t=\int_{\Upsilon_{T}}(1-\rho^{\varepsilon})f^{\varepsilon}{\bf
e}(\theta)\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t.$

Similarly, we have

$\lim_{\varepsilon\to
0}\int_{\Upsilon}\varepsilon\nabla_{\bm{\xi}}u^{\varepsilon}\cdot\nabla_{\bm{\xi}}\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t\leq\lim_{\varepsilon\to
0}\sqrt{\varepsilon}\underbrace{\|\nabla_{\bm{\xi}}\sqrt{\varepsilon}u^{\varepsilon}\|_{L^{2}(\Upsilon)}}_{\leq
C}\|\nabla_{\bm{\xi}}\psi\|_{L^{2}(\Upsilon)}=0,$

where we used the uniform estimate (3.3).

6\. Admissible test functions and attainment of initial data: It therefore
follows that the weak formulation of Definition 1.1 holds for all test
functions $\psi\in\mathcal{A}$. Using the density of this latter space in $X$
and the fact that $f,\rho$ and their derivatives are bounded in suitable
spaces, cf. equation (1.5), the weak formulation extends to test functions
belonging to $X$.

Finally, the attainment of the initial data in the dual space $Z^{\prime}$ is
clear from Remark 1.2 and Lemma 4.1. The proof is complete. ∎

We end this section by recording that the solution obtained in Theorem 1 also
satisfies an alternative notion of weak solution, which does not require the
test functions of the weak formulation to be periodic with respect to the
space-angle variable $\bm{\xi}$; this is the content of the next lemma. This
formulation is convenient when implementing the method of De Giorgi to study
the regularity of the solution (cf. [1]), which will be the subject of future
works. The proof is identical to that of Theorem 1, using that integrating by
parts over all of $\mathbb{R}^{3}$ against an arbitrary non-periodic test
function $\psi\in C^{\infty}_{c}((0,T)\times\mathbb{R}^{3})$ yields no
boundary terms, as was also the case when testing against periodic test
functions $\psi\in X$ and integrating over one periodic cell.

######  Lemma 4.2.

Let $f\in\mathcal{X}$ be the weak solution obtained in Theorem 1. Then, for
all $\psi\in C^{\infty}_{c}((0,T)\times\mathbb{R}^{3})$, there holds

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{3}}f\partial_{t}\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t+$
$\displaystyle\mathrm{Pe}\int_{0}^{T}\int_{\mathbb{R}^{3}}(1-\rho)f{\bf
e}(\theta)\cdot\nabla\psi\,\mathrm{d}\bm{\xi}\,\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\int_{\mathbb{R}^{3}}\Big{(}D_{e}\big{(}(1-\rho)\nabla
f+f\nabla\rho\big{)}\cdot\nabla\psi+\partial_{\theta}f\partial_{\theta}\psi\Big{)}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t.$

##  5\. Uniqueness for Null Péclet Number

In the case $\mathrm{Pe}=0$, the equation reads

(5.1) $\left\\{\begin{aligned}
&\partial_{t}f=D_{e}\operatorname{div}((1-\rho)\nabla
f+f\nabla\rho)+\partial^{2}_{\theta}f,\\\
&\partial_{t}\rho=D_{e}\Delta\rho.\end{aligned}\right.$

The main observation is that, as $\rho$ satisfies the heat equation,
uniqueness for $\rho$ is trivial. With this information, we employ the method
of Gajewski [15] to deduce that the solution of (5.1) is unique within the
class $\mathcal{X}$; this approach had been used for a similar uniqueness
argument by Jüngel and Zamponi in [19].

###### Proof of Theorem 2.

Let $f_{1},f_{2}\in\mathcal{X}$ be two weak solutions of (5.1) with identical
admissible initial data, and define $\rho_{1},\rho_{2}$ accordingly. As
$\rho_{1}-\rho_{2}$ satisfies the heat equation with zero initial data, the
standard theory implies $\rho_{1}=\rho_{2}$; we therefore denote this common
function by $\rho$, which satisfies the usual estimate $0\leq\rho\leq 1$. It
therefore follows that

(5.2) $\partial_{t}f_{i}=D_{e}\operatorname{div}((1-\rho)\nabla
f_{i}+f_{i}\nabla\rho)+\partial^{2}_{\theta}f_{i}$

for $i\in\\{1,2\\}$.

Consider the distance

(5.3)
$d(u,v):=\int_{\Upsilon}\Big{(}\zeta(u)+\zeta(v)-2\zeta\big{(}\frac{u+v}{2}\big{)}\Big{)}\,\mathrm{d}\bm{\xi},$

where $\zeta(s)=s(\log s-1)+1$ for $s\geq 0$. Note that the convexity of
$\zeta$ implies that $d(u,v)\geq 0$. In fact we shall consider the regularised
variant for $\delta>0$:

(5.4)
$d_{\delta}(u,v):=\int_{\Upsilon}\Big{(}\zeta_{\delta}(u)+\zeta_{\delta}(v)-2\zeta_{\delta}\big{(}\frac{u+v}{2}\big{)}\Big{)}\,\mathrm{d}\bm{\xi},$

where $\zeta_{\delta}(s)=\zeta(s+\delta)$, and again $d_{\delta}$ is non-
negative because $\zeta_{\delta}$ is convex. Note that
$\zeta^{\prime}(s)=\log(s)$ for $s>0$ and that, by a standard argument using
Taylor’s Theorem, there holds for all $\delta>0$:

(5.5)
$\zeta_{\delta}(u)+\zeta_{\delta}(v)-2\zeta_{\delta}\big{(}\frac{u+v}{2}\big{)}\geq\frac{1}{8}|u-v|^{2};$

we refer the reader to [19, §6] for further details.

Our goal is to show that $f_{1}\equiv f_{2}$ a.e. in $\Upsilon_{T}$. With this
in mind, we compute

(5.6) $\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}d_{\delta}(f_{1},f_{2})=$
$\displaystyle\langle\partial_{t}f_{1},\log(f_{1}+\delta)\rangle+\langle\partial_{t}f_{2},\log(f_{2}+\delta)\rangle-\langle\partial_{t}f_{1}+\partial_{t}f_{2},\log\big{(}\frac{f_{1}+f_{2}}{2}+\delta\big{)}\rangle$
$\displaystyle=$
$\displaystyle-4D_{e}\int_{\Upsilon}\Big{(}\sum_{i=1}^{2}|\nabla\sqrt{f_{i}+\delta}|^{2}-|\nabla\sqrt{f_{1}+f_{2}+2\delta}|^{2}\Big{)}(1-\rho)\,\mathrm{d}\bm{\xi}$
$\displaystyle-4\int_{\Upsilon}\Big{(}\sum_{i=1}^{2}|\partial_{\theta}\sqrt{f_{i}+\delta}|^{2}-|\partial_{\theta}\sqrt{f_{1}+f_{2}+2\delta}|^{2}\Big{)}\,\mathrm{d}\bm{\xi}$
$\displaystyle-
D_{e}\sum_{i=1}^{2}\int_{\Upsilon}\Big{(}\frac{f_{i}}{f_{1}+\delta}-\frac{f_{1}+f_{2}}{f_{1}+f_{2}+2\delta}\Big{)}\nabla
f_{i}\cdot\nabla\rho\,\mathrm{d}\bm{\xi}.$

The first two integrals above are non-negative by virtue of the subadditivity
of the Fisher information [19, Lemma 9]; cf. [21, §3.6]. Our strategy will be
to show that the final term on the right-hand side of the previous equation
vanishes in the limit as $\delta\to 0$. It will suffice to first show that the
term $\nabla f_{i}\cdot\nabla\rho$ is integrable, and to then apply the
Dominated Convergence Theorem. We rewrite this product as

$\nabla
f_{i}\cdot\nabla\rho=-4\sqrt{1-\rho}\nabla\sqrt{f_{i}}\cdot\sqrt{f_{i}}\nabla\sqrt{1-\rho},$

from which it follows, using the Young inequality, that

$\displaystyle\int_{\Upsilon_{T}}|\nabla f_{i}\cdot\nabla\rho|$
$\displaystyle\leq
2\int_{\Upsilon_{T}}(1-\rho)|\nabla_{\bm{\xi}}\sqrt{f_{i}}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t+2\int_{0}^{T}\Big{(}\int_{\Omega}|\nabla\sqrt{1-\rho}|^{2}\underbrace{\int_{0}^{2\pi}f_{i}\,\mathrm{d}\theta}_{=\rho}\Big{)}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\leq
2\int_{\Upsilon_{T}}(1-\rho)|\nabla_{\bm{\xi}}\sqrt{f_{i}}|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t+2\int_{\Omega_{T}}|\nabla\sqrt{1-\rho}|^{2}\,\mathrm{d}x\,\mathrm{d}t,$

and thus $\nabla f_{i}\cdot\nabla\rho\in L^{1}(\Upsilon_{T})$; the boundedness
of the right-hand side follows from the definition of $\mathcal{X}$. It
follows that, by integrating (5.6) in time, there holds, for a.e.$t\in(0,T)$,

$d_{\delta}(f_{1},f_{2})(t)-\underbrace{d_{\delta}(f_{1},f_{2})(0)}_{=0}\leq
D_{e}\sum_{i=1}^{2}\int_{0}^{t}\int_{\Upsilon}\Big{(}\frac{f_{i}}{f_{1}+\delta}-\frac{f_{1}+f_{2}}{f_{1}+f_{2}+2\delta}\Big{)}\nabla
f_{i}\cdot\nabla\rho\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau,$

whence, by the Dominated Convergence Theorem and using $d_{\delta}\geq 0$, we
deduce that, for a.e. $t\in(0,T)$,

$\lim_{\delta\to 0^{+}}d_{\delta}(f_{1},f_{2})(t)=0.$

It follows from Fatou’s Lemma that

$\lim_{\delta\to
0^{+}}\Big{(}\zeta_{\delta}(f_{1})+\zeta_{\delta}(f_{1})-2\zeta_{\delta}\big{(}\frac{f_{1}+f_{2}}{2}\big{)}\Big{)}=0\quad\text{a.e.~{}in
}\Upsilon_{T},$

whence the estimate (5.5) implies that $f_{1}\equiv f_{2}$ a.e. in
$\Upsilon_{T}$, as required. ∎

##  6\. Existence and Uniqueness of Stationary States

We consider the existence and uniqueness of stationary solutions
$f_{\infty}:\Upsilon\to\mathbb{R}$ which satisfy, with
$\rho_{\infty}(x)=\int_{0}^{2\pi}f_{\infty}(x,\theta)\,\mathrm{d}\theta$,

(6.1) $\mathrm{Pe}\operatorname{div}\big{(}(1-\rho_{\infty})f_{\infty}{\bf
e}(\theta)\big{)}=D_{e}\operatorname{div}\big{(}(1-\rho_{\infty})\nabla
f_{\infty}+f_{\infty}\nabla\rho_{\infty}\big{)}+\partial^{2}_{\theta}f_{\infty}.$

The question of existence of stationary states is straightforward, with no
restriction on the Péclet number. Indeed, given any
$\mathrm{Pe}\in\mathbb{R}$, there exists at least one non-trivial constant
stationary solution; namely $f_{\infty}=1/2\pi$, which gives rise to
$\rho_{\infty}=1$. In fact, by writing the drift term as

$\operatorname{div}\big{(}(1-\rho_{\infty})f_{\infty}{\bf
e}(\theta)\big{)}={\bf
e}(\theta)\cdot\nabla\big{(}(1-\rho_{\infty})f_{\infty}\big{)},$

for sufficiently regular $f_{\infty}$, we deduce that all constant solutions
satisfy (6.1).

We therefore focus entirely on uniqueness henceforth, and divide our results
into two subsections; §6.1 focused exclusively on the case $\mathrm{Pe}=0$ by
means of a simple variational approach, and §6.2 concerned with uniqueness of
strong solutions near constant stationary states for general $\mathrm{Pe}$,
which requires a more involved analysis.

###  6.1. Uniqueness of stationary solutions for null Péclet number

The case $\mathrm{Pe}=0$ is particularly simple since, in this case, the
equation (1.1) admits an exact gradient flow structure. We shall show that
stationary states are unique up to a mass constraint via a variational
approach. The proof does not readily generalise to $\mathrm{Pe}\neq 0$; this
is essentially because the perturbative term $V$ in (2.4) does not admit a
primitive, whence $\nabla_{\bm{\xi}}\cdot(\tilde{M}(u)(\nabla_{\bm{\xi}}u+V))$
cannot be rewritten as the dissipation of a suitable energy (cf. [8, §3.3]).

######  Theorem 3.

Let $\mathrm{Pe}=0$ and $m>0$. Then, the stationary solution $f_{\infty}$ of
(6.1) is unique within the class
$\int_{\Upsilon}f_{\infty}\,\mathrm{d}\bm{\xi}=m$.

Further results regarding the convergence to stationary states can be obtained
using the theory of Bakry–Emery or log-Sobolev inequalities applied to
equations endowed with an exact Wasserstein gradient flow structure (cf. e.g.
[3, 9, 18]). This is well-known, and we do not include the details here. The
study of the convergence to stationary states by means of hypocoercivity
methods for the case of non-zero Péclet number, for which one does not have an
exact gradient flow structure, will be the subject of future investigation.

###### Proof.

1\. Variational consequence of stationary equation: As per §2, we introduce
the transformed variable $u_{\infty}$

(6.2) $u_{\infty}:=\log\big{(}\frac{f_{\infty}}{1-\rho_{\infty}}\big{)}.$

For clarity of presentation, we assume that the above is well-defined and
perform formal computations; these may easily be rendered rigorous by standard
approximation arguments. We show that any stationary solution satisfies

(6.3)
$\int_{\Upsilon}f_{\infty}(1-\rho_{\infty})|\nabla_{\bm{\xi}}u_{\infty}|^{2}\,\mathrm{d}\bm{\xi}=0.$

Indeed, define the functional

$F[f]:=\int_{\Upsilon}\Big{(}D_{e}(1-\rho)f|\nabla
u|^{2}+f|\partial_{\theta}u|^{2}\Big{)}\,\mathrm{d}\bm{\xi}=\int_{\Upsilon}\tilde{M}(u)\nabla_{\bm{\xi}}u\cdot\nabla_{\bm{\xi}}u\,\mathrm{d}\bm{\xi}\geq
0,$

where we used that $\tilde{M}(u)$ is positive semidefinite, and observe that
testing the equation (6.1) with $u_{\infty}$ yields $F[f_{\infty}]=0$. The
equality (6.3) follows from the non-negativity of the integrand in the
functional $F$ and basic manipulations.

2\. Uniqueness from mass constraint: The relation (6.3) implies that all
stationary solutions with $\mathrm{Pe}=0$ are constant. The mass constraint
$\int_{\Upsilon}f_{\infty}\,\mathrm{d}\bm{\xi}=m$ then uniquely determines the
stationary solution, as required. ∎

###  6.2. Uniqueness of strong solutions near constant stationary states

Our aim in this section is to prove that, provided a strong solution is
sufficiently near a constant stationary state (in a sense to be made precise),
then it is the only such solution. We proceed by means of a fixed-point
argument employing the Contraction Mapping Theorem, similar to [8, §3.5], for
which we define the spaces

(6.4) $\displaystyle\Xi_{t}:=L^{\infty}(t,T;H^{2}(\Upsilon))\cap
L^{2}(t,T;H^{3}(\Upsilon))\cap H^{1}(t,T;H^{1}(\Upsilon)),$
$\displaystyle\Lambda_{t}:=L^{\infty}(t,T;H^{1}(\Upsilon))\cap
L^{2}(t,T;H^{2}(\Upsilon))\cap H^{1}(t,T;L^{2}(\Upsilon)),$
$\displaystyle\Theta_{t}:=L^{\infty}(t,T;L^{2}(\Upsilon))\cap
L^{2}(t,T;H^{1}(\Upsilon)).$

Where no confusion arises, we shall omit the $t$ subscript and simply write
$\Xi,\Lambda,\Theta$. The main result of this section is as follows.

######  Theorem 4.

Let $D_{e}>0$, $\mathrm{Pe}\in\mathbb{R}$, $f_{\infty}\in(0,1/2\pi)$ be a
given constant stationary state, and $f$ be a solution of (1.1). Assume there
exists $t\in(0,T)$ such that $f(t,\cdot)\in H^{2}(\Upsilon)$ with

$\|f(t,\cdot)-f_{\infty}\|_{H^{2}(\Upsilon)}\leq R^{2}$

for $R$ sufficiently small, depending only on $f_{\infty},D_{e}$ and
$\mathrm{Pe}$. Then, there exists a unique solution of (1.1) in

$B_{R,t}:=\big{\\{}f\in\Xi:\,\|f-f_{\infty}\|_{\Xi_{t}}\leq R\big{\\}}.$

The proof of Theorem 4 entails studying the time-evolution of the error term
$w:=f-f_{\infty}$, with $W:=\rho-\rho_{\infty}$, which satisfies

$\displaystyle\partial_{t}w-\partial^{2}_{\theta}w-D_{e}\big{[}(1-\rho_{\infty})\Delta
w$ $\displaystyle+f_{\infty}\Delta
W\big{]}+\mathrm{Pe}\big{[}(1-\rho_{\infty})\nabla w-f_{\infty}\nabla
W\big{]}\cdot{\bf e}(\theta)$ $\displaystyle=$ $\displaystyle
D_{e}\big{[}w\Delta W-W\Delta w\big{]}+\mathrm{Pe}\big{[}w\nabla W+W\nabla
w\big{]}\cdot{\bf e}(\theta).$

We therefore construct the operator $\Gamma$ to be the solution map
$\Gamma:w\mapsto z$, where $z$ solves the linear problem

(6.5) $\left\\{\begin{aligned}
&\partial_{t}z-\partial^{2}_{\theta}z-D_{e}\big{[}(1-\rho_{\infty})&&\Delta
z+f_{\infty}\Delta Z\big{]}+\mathrm{Pe}\big{[}(1-\rho_{\infty})\nabla
z-f_{\infty}\nabla Z\big{]}\cdot{\bf e}(\theta)\\\
&&&=\underbrace{D_{e}\big{(}w\Delta W-W\Delta
w\big{)}+\mathrm{Pe}\big{(}w\nabla W+W\nabla w\big{)}\cdot{\bf
e}(\theta)}_{=:G(w)},\\\ &z(0,\cdot)=z_{0},&&\end{aligned}\right.$

and $Z:=\int_{0}^{2\pi}z\,\mathrm{d}\theta$; we write $\Gamma=S\circ G$, where
$S$ is the solution mapping to the above equation for given right-hand side
and initial data $z_{0}$, and $G$ maps precisely to this desired right-hand
source term. Our goal is to show that $\Gamma$ admits a fixed point.

Note that, provided the above is supplemented with adequate initial data and
the right-hand term $G(w)$ is sufficiently regular, the solution map $\Gamma$
is well-defined in a suitable space using a standard application of the
Galerkin method/Fourier series. Indeed, writing the ansatz

$z(t,\bm{\xi})=\sum_{\mathbf{n}\in\mathbb{Z}^{3}}\alpha_{\mathbf{n}}(t)e^{\mathrm{i}\mathbf{n}\cdot\bm{\xi}},\quad
Z(t,x)=2\pi\sum_{\mathbf{n}^{\prime}\in\mathbb{Z}^{2}}\alpha_{(\mathbf{n}^{\prime},0)}(t)e^{\mathrm{i}\mathbf{n}^{\prime}\cdot
x},$

where $\mathbf{n}=(n_{1},n_{2},n_{3})$, $\mathbf{n}^{\prime}=(n_{1},n_{2})$,
the equation (6.5) reads

(6.6)
$\displaystyle\alpha_{\mathbf{n}}^{\prime}(t)+n_{3}^{2}\alpha_{\mathbf{n}}(t)+D_{e}(n_{1}^{2}+n_{2}^{2})\big{[}(1-\rho_{\infty})\alpha_{\mathbf{n}}(t)$
$\displaystyle+\mathds{1}_{\\{n_{3}=0\\}}f_{\infty}2\pi\alpha_{(\mathbf{n}^{\prime},0)}(t)\big{]}$
$\displaystyle+\mathrm{Pe}(1-\rho_{\infty})2\pi\mathrm{i}n_{1}\alpha_{\mathbf{n}}(t)=\hat{G}_{\mathbf{n}}(t),$

where

$\hat{G}_{\mathbf{n}}(t)=\frac{1}{(2\pi)^{2}}\int_{0}^{2\pi}G(w)(t,\bm{\xi})e^{-\mathrm{i}\mathbf{n}\cdot\bm{\xi}}\,\mathrm{d}\bm{\xi}.$

It follows that, provided $\hat{G}_{\mathbf{n}}(w)$ is sufficiently well-
behaved, the ODE system (6.6) is solvable by standard methods for all values
of $f_{\infty}\geq 0$ and $0\leq\rho_{\infty}\leq 1$. This existence
construction is standard; we therefore deliberately omit further details for
clarity of presentation, and henceforth focus exclusively on the _a priori_
estimates required for the fixed-point strategy. Our results are encapsulated
in the following two lemmas, which will be used to prove that $\Gamma$ admits
a unique fixed point.

######  Lemma 6.1 (Estimates for the operator $S$).

Assume $f_{\infty}\in(0,1/2\pi)$, i.e., $\rho_{\infty}\in(0,1)$. Then, there
exists a positive constant
$C_{\infty,D_{e}}=C_{\infty,D_{e}}(f_{\infty},D_{e})$ such that

(6.7) $\displaystyle\|Sv\|_{\Theta}\leq
C_{\infty,D_{e}}C_{0}^{1/2}\Big{(}\|z_{0}\|_{L^{2}(\Upsilon)}+\|v\|_{L^{2}(\Upsilon_{T})}\Big{)},$
$\displaystyle\|\nabla Sv\|_{\Theta}\leq C_{\infty,D_{e}}C_{0}\Big{(}\|\nabla
z_{0}\|_{L^{2}(\Upsilon)}+\|v\|_{L^{2}(\Upsilon_{T})}\Big{)},$
$\displaystyle\|\partial_{\theta}Sv\|_{\Theta}\leq
C_{\infty,D_{e}}C_{0}^{3/2}\Big{(}\|z_{0}\|_{L^{2}(\Upsilon)}+\|\partial_{\theta}z_{0}\|_{L^{2}(\Upsilon)}+\|v\|_{L^{2}(\Upsilon_{T})}\Big{)},$
$\displaystyle\|\partial_{t}Sv\|_{L^{2}(\Upsilon_{T})}\leq
C_{\infty,D_{e}}C_{1}\Big{(}\|z_{0}\|_{H^{1}(\Upsilon)}+\|v\|_{L^{2}(\Upsilon_{T})}\Big{)},$

where

(6.8) $\displaystyle C_{0}=e^{T(1+\mathrm{Pe}^{2}{D_{e}^{-1}})},\qquad
C_{1}=(1+\mathrm{Pe}^{2}D_{e}^{-1})^{1/2}C_{0}^{3/2},$

and

(6.9) $\|Sv\|_{\Xi}\leq
C_{\infty,D_{e}}C_{1}\Big{(}\|z_{0}\|_{H^{2}(\Upsilon)}+\|v\|_{\Theta}\Big{)}.$

######  Lemma 6.2 (Estimates for the operator $G$).

There exists a positive constant $C_{D_{e}}$, depending solely on $D_{e}$ and
$\Upsilon$, such that, for all $w_{1},w_{2}\in\Xi$, there holds

(6.10) $\|G(w_{1})-G(w_{2})\|_{\Theta}\leq
C_{D_{e}}(1+\mathrm{Pe}D_{e}^{-1})\|w_{1}-w_{2}\|_{\Xi}\big{(}\|w_{1}\|_{\Xi}+\|w_{2}\|_{\Xi}\big{)},$

and, for all $w\in\Xi$,

(6.11) $\|G(w)\|_{\Theta}\leq
C_{D_{e}}(1+\mathrm{Pe}D_{e}^{-1})\|w\|_{\Xi}^{2}.$

Before proving Lemmas 6.1 and 6.2, we provide the proof of Theorem 4.

###### Proof of Theorem 4.

Suppose for the time being that $T\leq 1$. This assumption will be removed in
the third step of the proof.

1\. $\Gamma$ is self-mapping: Suppose $f\in B_{R}$. Using the estimates of
Lemmas 6.1 and 6.2, we find that

$\|\Gamma w\|_{\Xi}\leq
C_{\infty,D_{e}}C_{1}\Big{(}1+C_{D_{e}}(1+\mathrm{Pe}D_{e}^{-1})\Big{)}R^{2}.$

Since we have restricted $T\leq 1$, it follows that there exists $R$
sufficiently small, depending only on $D_{e},f_{\infty},\mathrm{Pe}$ such that
there holds

$\|\Gamma w\|_{\Xi}\leq R,$

whence $\Gamma$ is self-mapping from $B_{R}$ to itself.

2\. $\Gamma$ is contractive: Let $w_{1},w_{2}\in B_{R}$. Observe that the
linearity of $S$ implies $\Gamma w_{1}-\Gamma w_{2}=S(Gw_{1}-Gw_{2})$, where
the initial data $z_{0}$ in (6.5) is null. It follows from Lemmas 6.1 and 6.2
that there holds

$\|\Gamma w_{1}-\Gamma w_{2}\|_{\Xi}\leq
C_{\infty,D_{e}}C_{1}C_{D_{e}}(1+\mathrm{Pe}D_{e}^{-1})R\|w_{1}-w_{2}\|_{\Xi}.$

As per the first part of the proof, it follows that one may select $R$
sufficiently small, depending only on $f_{\infty},D_{e},\mathrm{Pe}$ such that
$\Gamma$ is a contraction. An application of Banach’s Contraction Mapping
Theorem implies that there exists a unique $f$ solving (1.1) and such that
$\|f-f_{\infty}\|_{\Xi}$ over the time interval $t\in(0,1)$.

3\. Extension to arbitrary time intervals: We apply Step 2 of the proof
repeatedly over overlapping time intervals $(\alpha_{j},\beta_{j})$ for
Lebesgue points $\alpha_{j},\beta_{j}$ chosen such that
$\beta_{j-1}>\alpha_{j}$ for all $j$ and such that
$\bigcup_{j}(\alpha_{j},\beta_{j})\supset(t,T)$. In turn, obtain that there
exists a unique $f\in B_{R}$ over the entire time interval $(0,T)$. ∎

###### Proof of Lemma 6.1.

By testing (6.5) with $z$, we obtain the classical parabolic estimate

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Upsilon}|z|^{2}\,\mathrm{d}\bm{\xi}+\int_{\Upsilon}$
$\displaystyle|\partial_{\theta}z|^{2}\,\mathrm{d}\bm{\xi}+D_{e}\Big{(}(1-\rho_{\infty})\int_{\Upsilon}|\nabla
z|^{2}\,\mathrm{d}\bm{\xi}+f_{\infty}\int_{\Omega}|\nabla
Z|^{2}\,\mathrm{d}x\Big{)}$ $\displaystyle\leq$
$\displaystyle\mathrm{Pe}\int_{\Upsilon}\big{[}(1-\rho_{\infty})|z||\nabla
z|+f_{\infty}|z||\nabla
Z|\big{]}\,\mathrm{d}\bm{\xi}+\int_{\Upsilon}G(w)z\,\mathrm{d}\bm{\xi}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}D_{e}(1-\rho_{\infty})\int_{\Upsilon}|\nabla
z|^{2}\,\mathrm{d}\bm{\xi}+\frac{1}{2}D_{e}f_{\infty}\int_{\Omega}|\nabla
Z|^{2}\,\mathrm{d}x$ $\displaystyle+\frac{1}{2}\big{(}(1-\rho_{\infty}+2\pi
f_{\infty})\mathrm{Pe}^{2}{D_{e}^{-1}}+1\big{)}\int_{\Upsilon}|z|^{2}\,\mathrm{d}\bm{\xi}+\frac{1}{2}\int_{\Upsilon}|G(w)|^{2}\,\mathrm{d}\bm{\xi}$

for some positive universal constant $C$, from which we get, for a.e.
$t\in(0,T)$,

$\displaystyle\int_{\Upsilon}|z(t)|^{2}\,\mathrm{d}\bm{\xi}+$
$\displaystyle\int_{0}^{t}\int_{\Upsilon}|\partial_{\theta}z|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau+D_{e}\Big{(}(1-\rho_{\infty})\int_{0}^{t}\int_{\Upsilon}|\nabla
z|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau+f_{\infty}\int_{0}^{t}\int_{\Omega}|\nabla
Z|^{2}\,\mathrm{d}x\,\mathrm{d}\tau\Big{)}$ $\displaystyle\leq$
$\displaystyle(1+\mathrm{Pe}^{2}{D_{e}^{-1}})\int_{0}^{t}\int_{\Upsilon}|z|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau+\Big{(}\int_{\Upsilon}|z_{0}|^{2}\,\mathrm{d}\bm{\xi}+\int_{0}^{t}\int_{\Upsilon}|G(w)|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau\Big{)},$

and Grönwall’s Lemma yields

$\displaystyle\int_{\Upsilon}|z(t)|^{2}\,\mathrm{d}\bm{\xi}\leq$
$\displaystyle\Big{(}\int_{\Upsilon}|z_{0}|^{2}\,\mathrm{d}\bm{\xi}+\int_{0}^{t}\int_{\Upsilon}|G(w)|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau\Big{)}e^{t(1+\mathrm{Pe}^{2}{D_{e}^{-1}})},$

so that

$\displaystyle\int_{0}^{T}\int_{\Upsilon}|z(t)|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}t\leq$
$\displaystyle\Big{(}\|z_{0}\|_{L^{2}(\Upsilon)}^{2}+\|G(w)\|_{L^{2}(\Upsilon_{T})}^{2}\Big{)}\frac{1}{(1+\mathrm{Pe}^{2}{D_{e}^{-1}})}\big{(}e^{T(1+\mathrm{Pe}^{2}{D_{e}^{-1}})}-1\big{)},$

whence

(6.12)
$\displaystyle\|z\|_{L^{\infty}(0,T;L^{2}(\Upsilon))}^{2}+\|\partial_{\theta}z\|_{L^{2}(\Upsilon_{T})}^{2}+D_{e}\Big{(}(1-\rho_{\infty})\|$
$\displaystyle\nabla z\|_{L^{2}(\Upsilon_{T})}^{2}+f_{\infty}\|\nabla
Z\|_{L^{2}(\Omega_{T})}^{2}\Big{)}$ $\displaystyle\leq$ $\displaystyle
C_{0}\Big{(}\|z_{0}\|_{L^{2}(\Upsilon)}^{2}+\|G(w)\|_{L^{2}(\Upsilon_{T})}^{2}\Big{)},$

and the first estimate in (6.7) follows, where
$C_{0}=C_{0}(\mathrm{Pe},D_{e},T)$ is independent of $f_{\infty},z_{0}$ and is
given by (6.8).

Similarly, by testing (6.5) with $(1-\rho_{\infty})\Delta z+f_{\infty}\Delta
Z$, we get

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}(1-\rho_{\infty})\int_{\Upsilon}$
$\displaystyle|\nabla
z|^{2}\,\mathrm{d}\bm{\xi}+f_{\infty}\int_{\Omega}|\nabla
Z|^{2}\,\mathrm{d}x\Big{)}$ $\displaystyle+$
$\displaystyle\frac{1}{2}D_{e}\int_{\Upsilon}\big{|}(1-\rho_{\infty})\Delta
z+f_{\infty}\Delta
Z\big{|}^{2}\,\mathrm{d}\bm{\xi}+(1-\rho_{\infty})\int_{\Upsilon}|\nabla\partial_{\theta}z|^{2}\,\mathrm{d}\bm{\xi}$
$\displaystyle\leq\frac{1}{2}\mathrm{Pe}^{2}D_{e}^{-1}\int_{\Upsilon}|(1-\rho_{\infty})\nabla
z-f_{\infty}\nabla
Z|^{2}\,\mathrm{d}\bm{\xi}+\frac{1}{2}D_{e}^{-1}\int_{\Upsilon}|G(w)|^{2}\,\mathrm{d}\bm{\xi}.$

It follows that

$\displaystyle\Big{(}(1-$ $\displaystyle\rho_{\infty})\int_{\Upsilon}|\nabla
z|^{2}\,\mathrm{d}\bm{\xi}+f_{\infty}\int_{\Omega}|\nabla
Z|^{2}\,\mathrm{d}x\Big{)}(t)$
$\displaystyle+D_{e}\int_{0}^{t}\int_{\Upsilon}\big{|}(1-\rho_{\infty})\Delta
z+f_{\infty}\Delta
Z\big{|}^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau+(1-\rho_{\infty})\int_{0}^{t}\int_{\Upsilon}|\nabla\partial_{\theta}z|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\Big{(}(1-\rho_{\infty})\int_{\Upsilon}|\nabla
z_{0}|^{2}\,\mathrm{d}\bm{\xi}+f_{\infty}\int_{\Omega}|\nabla
Z_{0}|^{2}\,\mathrm{d}x\Big{)}+D_{e}^{-1}\int_{0}^{t}\int_{\Upsilon}|G(w)|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau$
$\displaystyle+\mathrm{Pe}^{2}D_{e}^{-1}\Big{(}(1-\rho_{\infty})^{2}\int_{0}^{t}\int_{\Upsilon}|\nabla
z|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau+(2(1-\rho_{\infty})f_{\infty}+2\pi
f_{\infty}^{2})\int_{0}^{t}\int_{\Omega}|\nabla
Z|^{2}\,\mathrm{d}x\,\mathrm{d}\tau\Big{)},$

whence, since $2\pi f_{\infty}=\rho_{\infty}\in[0,1]$, we get

$\displaystyle\Big{(}(1-\rho_{\infty})$ $\displaystyle\int_{\Upsilon}|\nabla
z|^{2}\,\mathrm{d}\bm{\xi}+f_{\infty}\int_{\Omega}|\nabla
Z|^{2}\,\mathrm{d}x\Big{)}(t)$
$\displaystyle+D_{e}\int_{0}^{t}\int_{\Upsilon}\big{|}(1-\rho_{\infty})\Delta
z+f_{\infty}\Delta
Z\big{|}^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau+(1-\rho_{\infty})\int_{0}^{t}\int_{\Upsilon}|\nabla\partial_{\theta}z|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\Big{(}(1-\rho_{\infty})\int_{\Upsilon}|\nabla
z_{0}|^{2}\,\mathrm{d}\bm{\xi}+f_{\infty}\int_{\Omega}|\nabla
Z_{0}|^{2}\,\mathrm{d}x\Big{)}+D_{e}^{-1}\int_{0}^{t}\int_{\Upsilon}|G(w)|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau$
$\displaystyle+2\mathrm{Pe}^{2}D_{e}^{-1}\Big{(}(1-\rho_{\infty})\int_{0}^{t}\int_{\Upsilon}|\nabla
z|^{2}\,\mathrm{d}\bm{\xi}\,\mathrm{d}\tau+f_{\infty}\int_{0}^{t}\int_{\Omega}|\nabla
Z|^{2}\,\mathrm{d}x\,\mathrm{d}\tau\Big{)}.$

Grönwall’s Lemma yields, for a.e. $t\in(0,T)$,

$\displaystyle\Big{(}$ $\displaystyle(1-\rho_{\infty})\int_{\Upsilon}|\nabla
z|^{2}\,\mathrm{d}\bm{\xi}+f_{\infty}\int_{\Omega}|\nabla
Z|^{2}\,\mathrm{d}x\Big{)}(t)$
$\displaystyle\leq\Big{[}(1-\rho_{\infty})\|\nabla
z_{0}\|_{L^{2}(\Upsilon)}^{2}+f_{\infty}\|\nabla
Z_{0}\|_{L^{2}(\Omega)}^{2}+D_{e}^{-1}\|G(w)\|_{L^{2}((0,t)\times\Upsilon)}^{2}\Big{]}e^{t2\mathrm{Pe}^{2}D_{e}^{-1}}.$

We therefore obtain

$\displaystyle(1-\rho_{\infty})\|\nabla
z\|_{L^{\infty}(0,T;L^{2}(\Upsilon))}^{2}$ $\displaystyle+f_{\infty}\|\nabla
Z\|_{L^{\infty}(0,T;L^{2}(\Omega))}^{2}$
$\displaystyle+D_{e}\|(1-\rho_{\infty})\Delta z$
$\displaystyle+f_{\infty}\Delta
Z\|_{L^{2}(\Upsilon_{T})}^{2}+(1-\rho_{\infty})\|\nabla\partial_{\theta}z\|^{2}_{L^{2}(\Upsilon_{T})}$
$\displaystyle\leq C_{0}^{2}\Big{(}(1-\rho_{\infty})$ $\displaystyle\|\nabla
z_{0}\|_{L^{2}(\Upsilon)}^{2}+f_{\infty}\|\nabla
Z_{0}\|_{L^{2}(\Omega)}^{2}+D_{e}^{-1}\|G(w)\|_{L^{2}(\Upsilon_{T})}^{2}\Big{)}.$

Note in particular that the Fubini–Tonelli Theorem implies

$\displaystyle\|(1-\rho_{\infty})\Delta z+f_{\infty}\Delta
Z\|_{L^{2}(\Upsilon_{T})}^{2}=(1-\rho_{\infty})^{2}\|\Delta
z\|^{2}_{L^{2}(\Upsilon_{T})}+f_{\infty}(2(1-\rho_{\infty})+2\pi
f_{\infty})\|\Delta Z\|^{2}_{L^{2}(\Omega_{T})},$

whence the previous estimate yields

(6.13) $\displaystyle(1-\rho_{\infty})\|\nabla
z\|_{L^{\infty}(0,T;L^{2}(\Upsilon))}^{2}$ $\displaystyle+f_{\infty}\|\nabla
Z\|_{L^{\infty}(0,T;L^{2}(\Omega))}^{2}+(1-\rho_{\infty})\|\nabla\partial_{\theta}z\|^{2}_{L^{2}(\Upsilon_{T})}$
$\displaystyle+D_{e}\Big{(}(1-\rho_{\infty})^{2}$ $\displaystyle\|\Delta
z\|^{2}_{L^{2}(\Upsilon_{T})}+f_{\infty}(2-\rho_{\infty})\|\Delta
Z\|^{2}_{L^{2}(\Omega_{T})}\Big{)}$ $\displaystyle\leq C_{0}^{2}\Big{(}$
$\displaystyle(1-\rho_{\infty})\|\nabla
z_{0}\|_{L^{2}(\Upsilon)}^{2}+f_{\infty}\|\nabla
Z_{0}\|_{L^{2}(\Omega)}^{2}+D_{e}^{-1}\|G(w)\|_{L^{2}(\Upsilon_{T})}^{2}\Big{)},$

and the second estimate in (6.7) follows. Similarly, we test (6.5) against
$\partial^{2}_{\theta}z$ to obtain

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Upsilon}|\partial_{\theta}z|^{2}\,\mathrm{d}\bm{\xi}+$
$\displaystyle\int_{\Upsilon}|\partial^{2}_{\theta}z|\,\mathrm{d}\bm{\xi}+D_{e}(1-\rho_{\infty})\int_{\Upsilon}|\nabla\partial_{\theta}z|^{2}\,\mathrm{d}\bm{\xi}$
$\displaystyle=$
$\displaystyle\mathrm{Pe}\int_{\Upsilon}\big{[}(1-\rho_{\infty})\nabla
z-f_{\infty}\nabla Z]\cdot{\bf
e}(\theta)\partial^{2}_{\theta}z\,\mathrm{d}\bm{\xi}-\int_{\Upsilon}G(w)\partial^{2}_{\theta}z\,\mathrm{d}\bm{\xi}$
$\displaystyle=$
$\displaystyle-\mathrm{Pe}(1-\rho_{\infty})\int_{\Upsilon}\nabla\partial_{\theta}z\cdot{\bf
e}(\theta)\partial_{\theta}z\,\mathrm{d}\bm{\xi}$
$\displaystyle-\mathrm{Pe}\int_{\Upsilon}\big{[}(1-\rho_{\infty})\nabla
z-f_{\infty}\nabla Z\big{]}\cdot{\bf
e}^{\prime}(\theta)\partial_{\theta}z\,\mathrm{d}\bm{\xi}-\int_{\Upsilon}G(w)\partial^{2}_{\theta}z\,\mathrm{d}\bm{\xi}.$

Using ${\bf e}^{\prime}(\theta)=(-\sin\theta,\cos\theta)$ and the Young
inequality, we get

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Upsilon}|\partial_{\theta}z|^{2}\,\mathrm{d}\bm{\xi}$
$\displaystyle+\int_{\Upsilon}|\partial^{2}_{\theta}z|\,\mathrm{d}\bm{\xi}+D_{e}(1-\rho_{\infty})\int_{\Upsilon}|\nabla\partial_{\theta}z|^{2}\,\mathrm{d}\bm{\xi}$
$\displaystyle\leq$
$\displaystyle\mathrm{Pe}(1-\rho_{\infty})\int_{\Upsilon}|\nabla\partial_{\theta}z||\partial_{\theta}z|\,\mathrm{d}\bm{\xi}+\mathrm{Pe}(1-\rho_{\infty})\int_{\Upsilon}|\nabla
z||\partial_{\theta}z|\,\mathrm{d}\bm{\xi}$
$\displaystyle+\mathrm{Pe}f_{\infty}\int_{\Upsilon}|\nabla
Z||\partial_{\theta}z|\,\mathrm{d}\bm{\xi}+\int_{\Upsilon}|G(w)||\partial^{2}_{\theta}z|\,\mathrm{d}\bm{\xi}.$

We obtain, for a.e. $t\in(0,T)$,

$\displaystyle\|\partial_{\theta}z\|_{L^{\infty}(0,t;L^{2}(\Upsilon))}^{2}+\|\partial^{2}_{\theta}z$
$\displaystyle\|^{2}_{L^{2}(\Upsilon_{t})}+D_{e}(1-\rho_{\infty})\int_{\Upsilon_{t}}|\nabla\partial_{\theta}z|^{2}\,\mathrm{d}\bm{\xi}$
$\displaystyle\leq$ $\displaystyle
2\mathrm{Pe}^{2}D_{e}^{-1}(1-\rho_{\infty})\|\partial_{\theta}z\|^{2}_{L^{2}(\Upsilon_{t})}+D_{e}(1-\rho_{\infty})\|\nabla
z\|^{2}_{L^{2}(\Upsilon_{T})}$
$\displaystyle+\|\partial_{\theta}z_{0}\|^{2}_{L^{2}(\Upsilon)}+\|G(w)\|^{2}_{L^{2}(\Upsilon_{t})}$
$\displaystyle\leq$ $\displaystyle
2\mathrm{Pe}^{2}D_{e}^{-1}\|\partial_{\theta}z\|^{2}_{L^{2}(\Upsilon_{t})}$
$\displaystyle+C_{0}\Big{(}\|z_{0}\|_{L^{2}(\Upsilon)}^{2}+\|\partial_{\theta}z_{0}\|^{2}_{L^{2}(\Upsilon)}+2\|G(w)\|^{2}_{L^{2}(\Upsilon_{t})}\Big{)},$

where we used $\rho_{\infty}\in[0,1]$ and estimate (6.12) to obtain the final
line. Grönwall’s Lemma yields

(6.14)
$\displaystyle\|\partial_{\theta}z\|_{L^{\infty}(0,T;L^{2}(\Upsilon))}^{2}+\|\partial^{2}_{\theta}z\|^{2}_{L^{2}(\Upsilon_{T})}+$
$\displaystyle
D_{e}(1-\rho_{\infty})\|\nabla\partial_{\theta}z\|_{L^{2}(\Upsilon_{T})}^{2}$
$\displaystyle\leq$ $\displaystyle
C_{0}^{3}\Big{(}\|z_{0}\|_{L^{2}(\Upsilon)}^{2}+\|\partial_{\theta}z_{0}\|^{2}_{L^{2}(\Upsilon)}+2\|G(w)\|_{L^{2}(\Upsilon_{T})}^{2}\Big{)},$

and the third estimate in (6.7) follows. It follows, using directly the
equation (6.5), that there holds, for some positive universal constant $C$,
the estimate on the time derivative:

$\displaystyle\|\partial_{t}z\|_{L^{2}(\Upsilon_{T})}\leq C\Big{(}$
$\displaystyle\|\partial^{2}_{\theta}z\|_{L^{2}(\Upsilon_{T})}^{2}+D_{e}^{2}(1-\rho_{\infty})^{2}\|\Delta
z\|_{L^{2}(\Upsilon_{T})}^{2}+D_{e}^{2}f_{\infty}^{2}\|\Delta
Z\|_{L^{2}(\Upsilon_{T})}^{2}$
$\displaystyle+\mathrm{Pe}^{2}(1-\rho_{\infty})^{2}\|\nabla
z\|_{L^{2}(\Upsilon_{T})}^{2}+\mathrm{Pe}^{2}f_{\infty}^{2}\|\nabla
Z\|_{L^{2}(\Upsilon_{T})}^{2}+\|G(w)\|_{L^{2}(\Upsilon_{T})}^{2}\Big{)}$

and thus, using (6.12), (6.13), (6.14),

$\displaystyle\|\partial_{t}z\|_{L^{2}(\Upsilon_{T})}^{2}\leq$ $\displaystyle
C_{\infty,D_{e}}(1+\mathrm{Pe}^{2}D_{e}^{-1})C_{0}^{3}\Big{(}\|z_{0}\|_{H^{1}(\Upsilon)}^{2}+\|G(w)\|_{L^{2}(\Upsilon_{T})}^{2}\Big{)}.$

Thus, there holds

$\|z\|_{\Lambda}\leq
C_{\infty,D_{e}}(1+\mathrm{Pe}^{2}D_{e}^{-1})^{1/2}C_{0}^{3/2}\Big{(}\|z_{0}\|_{H^{1}(\Upsilon)}+\|G(w)\|_{L^{2}(\Upsilon_{T})}\Big{)}.$

Finally, using the linearity of the equation to introduce difference quotients
with respect to the space-angle variable $\bm{\xi}$, we reproduce the strategy
required to obtain (6.7) and get

$\|\nabla_{\bm{\xi}}z\|_{\Lambda}\leq
C_{\infty,D_{e}}C_{0}^{3/2}\Big{(}\|\nabla_{\bm{\xi}}z_{0}\|_{H^{1}(\Upsilon)}+\|\nabla_{\bm{\xi}}G(w)\|_{L^{2}(\Upsilon_{T})}\Big{)}.$

It follows that $\|z\|_{\Xi}$ is controlled by
$\|z_{0}\|_{H^{2}(\Upsilon)}+\|G(w)\|_{L^{2}(0,T;H^{1}(\Upsilon))}$ as per the
previous inequality, and the final estimate (6.9) follows. ∎

###### Proof of Lemma 6.2.

We focus on the contractivity of $G$. Direct computation yields

(6.15) $\displaystyle\|G(w_{1})-G(w_{2})\|_{L^{2}(\Upsilon_{T})}=D_{e}\Big{(}$
$\displaystyle\|(w_{1}-w_{2})\Delta
W_{1}\|_{L^{2}(\Upsilon_{T})}+\|w_{2}\Delta(W_{1}-W_{2})\|_{L^{2}(\Upsilon_{T})}$
$\displaystyle+\|W_{1}\Delta(w_{1}-w_{2})\|_{L^{2}(\Upsilon_{T})}+\|(W_{1}-W_{2})\Delta
w_{2}\|_{L^{2}(\Upsilon_{T})}$ $\displaystyle+\mathrm{Pe}\Big{(}$
$\displaystyle\|(w_{1}-w_{2})\nabla
W_{1}\|_{L^{2}(\Upsilon_{T})}+\|w_{2}\nabla(W_{1}-W_{2})\|_{L^{2}(\Upsilon_{T})}$
$\displaystyle+\|W_{1}\nabla(w_{1}-w_{2})\|_{L^{2}(\Upsilon_{T})}+\|(W_{1}-W_{2})\nabla
w_{2}\|_{L^{2}(\Upsilon_{T})}\Big{)}.$

Using Morrey’s embedding $H^{2}(\Upsilon)\hookrightarrow L^{\infty}(\Upsilon)$
for the bounded open domain $\Upsilon\subset\mathbb{R}^{3}$, there exists a
positive constant $C_{M}$ depending only on $\Upsilon$ such that

(6.16) $\|w_{i}(t,\cdot)\|_{L^{\infty}(\Upsilon)}\leq
C_{M}\|w_{i}(t,\cdot)\|_{H^{2}(\Upsilon)},$

whence

$\displaystyle\|w_{i}\|_{L^{\infty}(\Upsilon_{T})}\leq C_{M}\|w_{i}\|_{\Xi}.$

It follows that, for instance,

$\displaystyle\|(w_{i}-w_{j})\Delta W_{i}\|_{L^{2}(\Upsilon_{T})}\leq 2\pi
C_{M}\|w_{i}-w_{j}\|_{\Xi}\|w_{i}\|_{L^{2}(0,T;H^{2}(\Upsilon))}\leq 2\pi
C_{M}\|w_{i}-w_{j}\|_{\Xi}\|w_{i}\|_{\Xi},$

and thus, arguing similarly for the other terms, it follows from (6.15) that

$\|G(w_{1})-G(w_{2})\|_{L^{2}(\Upsilon_{T})}\leq 4\pi
C_{M}D_{e}(1+\mathrm{Pe}D_{e}^{-1})\|w_{1}-w_{2}\|_{\Xi}\big{(}\|w_{1}\|_{\Xi}+\|w_{2}\|_{\Xi}\big{)}.$

The same strategy yields an analogous estimate for
$\|G(w_{1})-G(w_{2})\|_{L^{\infty}(0,T;L^{2}(\Upsilon))}$.

Observe that

$\displaystyle\nabla_{\bm{\xi}}G(w)=$ $\displaystyle
D_{e}\big{(}w\nabla_{\bm{\xi}}\Delta W+\nabla_{\bm{\xi}}w\Delta
W-\nabla_{\bm{\xi}}W\Delta w-W\nabla_{\bm{\xi}}\Delta w\big{)}$
$\displaystyle+\mathrm{Pe}\big{(}w\nabla_{\bm{\xi}}\Delta
W+\nabla_{\bm{\xi}}w\Delta W+\nabla_{\bm{\xi}}W\Delta
w+W\nabla_{\bm{\xi}}\Delta w\big{)}\cdot{\bf e}(\theta)$
$\displaystyle+\mathrm{Pe}\big{(}w\Delta W+W\Delta w\big{)}\cdot{\bf
e}^{\prime}(\theta),$

and, for $w\in\Xi$, arguing as per (6.16), Morrey’s embedding implies $\nabla
w\in L^{2}(0,T;L^{\infty}(\Upsilon))$; furthermore this latter norm is
controlled by the norm of $\Xi$. We also have $\Delta w\in
L^{\infty}(0,T;L^{2}(\Upsilon))$ and $\nabla\Delta w\in L^{2}(\Upsilon_{T})$,
and analogously for the quantities involving $W$; furthermore, all of these
norms are all controlled by $\|w\|_{\Xi}$. Thus, by arguing as per (6.15) we
obtain

$\|\nabla_{\bm{\xi}}G(w_{1})-\nabla_{\bm{\xi}}G(w_{2})\|_{L^{2}(\Upsilon_{T})}\leq
8\pi
C_{M}D_{e}(1+\mathrm{Pe}D_{e}^{-1})\|w_{1}-w_{2}\|_{\Xi}\big{(}\|w_{1}\|_{\Xi}+\|w_{2}\|_{\Xi}\big{)},$

and (6.10) follows. Finally, since $G(0)=0$, the previous estimate yields
(6.11). ∎

##  Appendix A Interpolation Lemma

In the present appendix, we derive a variant of a classical interpolation
estimate for spaces of functions depending on time and space; our function
spaces include dependence on time, space, and angle.

######  Lemma A.1.

Let $m,p\geq 1$ and define the Banach spaces

$\displaystyle V^{m,p}:=L^{\infty}(\Omega_{T};L^{m}(0,2\pi)))\cap
L^{p}(\Omega_{T};W^{1,p}(0,2\pi)),$ $\displaystyle
V^{m,p}_{0}:=L^{\infty}(\Omega_{T};L^{m}(0,2\pi))\cap
L^{p}(\Omega_{T};W^{1,p}_{0}(0,2\pi)),$

both equipped with the norm

$\|v\|_{V^{m,p}}:=\operatorname{ess\,sup}_{(t,x)\in\Omega_{T}}\|v(t,x,\cdot)\|_{L^{m}(0,2\pi)}+\|\partial_{\theta}v\|_{L^{p}(\Upsilon_{T})}.$

There exists a positive constant $C$ depending only on $p,m,\Upsilon,T$ such
that, for all $v\in V^{m,p}$, there holds

$\|v\|_{L^{q}(\Upsilon_{T})}\leq C\|v\|_{V^{m,p}},$

where $q=p(m+1)$.

###### Proof.

1\. Inequality for $V_{0}^{m,p}$: Assume for the time being that $v\in
V^{m,p}_{0}$. Observe that $p\geq 1>\frac{m}{m+1}$, i.e. $p-m+mp>0$. The
Gagliardo–Nirenberg inequality [12, §1, Theorem 2.1, (2.2-i)] yields, for a.e.
$(t,x)\in\Omega_{T}$,

$\|v(t,x,\cdot)\|_{L^{q}(0,2\pi)}\leq
C\|\partial_{\theta}v(t,x,\cdot)\|_{L^{p}(0,2\pi)}^{\frac{p}{q}}\|v(t,x,\cdot)\|^{1-\frac{p}{q}}_{L^{m}(0,2\pi)},$

where $q=p(m+1)$. In turn, raising the previous inequality to the power $q$
and integrating over $\Omega_{T}$ yields

$\displaystyle\|v\|_{L^{q}(\Upsilon_{T})}^{q}\leq
C\bigg{(}\int_{\Omega_{T}}\|\partial_{\theta}v(t,x,\cdot)\|^{p}_{L^{p}(0,2\pi)}\,\mathrm{d}x\,\mathrm{d}t\bigg{)}\operatorname{ess\,sup}_{(t,x)\in\Omega_{T}}\|v(t,x,\cdot)\|^{q-p}_{L^{m}(0,2\pi)},$

and thus

$\displaystyle\|v\|_{L^{q}(\Upsilon_{T})}\leq
C\|\partial_{\theta}v\|^{\frac{p}{q}}_{L^{p}(\Upsilon_{T})}\|v(t,x,\cdot)\|^{1-\frac{p}{q}}_{L^{\infty}(\Omega_{T};L^{m}(0,2\pi))}.$

Applying the Young inequality to the previous right-hand side with exponent
$q/p>1$ and its Hölder conjugate yields

$\displaystyle\|v\|_{L^{q}(\Upsilon_{T})}\leq C\|v\|_{V^{m,p}}.$

A classical argument shows that the bound above still holds for functions
$v\in V^{m,p}$ such that $\int_{0}^{2\pi}v\,\mathrm{d}\theta=0$ a.e. in
$\Omega_{T}$; cf. [12, §1 Remark 2.1].

2\. Relaxation to $V^{m,p}$: Let $v\in V^{m,p}$ and define, for a.e.
$(t,x,\theta)$, its mean-zero counterpart

$w(t,x,\theta):=v(t,x,\theta)-\frac{1}{2\pi}\int_{0}^{2\pi}v(t,x,\theta)\,\mathrm{d}\theta.$

In view of the remark at the end of Step 1, we deduce that there holds

$\|w\|_{L^{q}(\Upsilon_{T})}\leq C\|w\|_{V^{m,p}}.$

Notice that $\partial_{\theta}w=\partial_{\theta}v$ and
$\|w(t,x,\cdot)\|_{L^{m}(0,2\pi)}\leq 2\|v(t,x,\cdot)\|_{L^{m}(0,2\pi)}$ a.e.
in $\Omega_{T}$, and that Jensen’s inequality yields

$\displaystyle\Big{\|}\int_{0}^{2\pi}v(\cdot,\cdot,\theta)\,\mathrm{d}\theta\Big{\|}_{L^{q}(\Upsilon_{T})}\leq$
$\displaystyle
C\bigg{(}\int_{0}^{2\pi}\int_{\Omega_{T}}\bigg{(}\int_{0}^{2\pi}|v(t,x,\theta^{\prime})|^{m}\,\mathrm{d}\theta^{\prime}\bigg{)}^{\frac{q}{m}}\,\mathrm{d}x\,\mathrm{d}t\,\mathrm{d}\theta\bigg{)}^{\frac{1}{q}}$
$\displaystyle\leq$ $\displaystyle
C\operatorname{ess\,sup}_{(t,x)\in\Omega_{T}}\|v(t,x,\cdot)\|_{L^{m}(0,2\pi)}.$

It therefore follows from the Minkowski inequality that

$\displaystyle\|v\|_{L^{q}(\Upsilon_{T})}\leq$
$\displaystyle\|w\|_{L^{q}(\Upsilon_{T})}+\frac{1}{2\pi}\Big{\|}\int_{0}^{2\pi}v\,\mathrm{d}\theta\Big{\|}_{L^{q}(\Upsilon_{T})}$
$\displaystyle\leq$ $\displaystyle
C\bigg{(}\|\underbrace{\partial_{\theta}w}_{=\partial_{\theta}v}\|_{L^{p}(\Upsilon_{T})}+\operatorname{ess\,sup}_{(t,x)\in\Omega_{T}}\|v(t,x,\cdot)\|_{L^{m}(0,2\pi)}\bigg{)},$

and the result follows. ∎

##  Appendix B Regularised Initial Data

.

###### Proof of Lemma 1.3.

The strong convergences in the statement of the lemma follow directly from
recalling the mollification technique of [7, Appendix C]. It remains to verify
the estimates; both those depending on $\varepsilon$ and those independent of
$\varepsilon$.

1\. Bounds depending on $\varepsilon$ : using the non-negativity of
$\rho_{0}$, there holds

$0<\frac{\varepsilon}{2}\leq\rho^{\varepsilon}_{0}(x)\leq\frac{\varepsilon}{2}+(1-\varepsilon)\int_{\mathbb{R}^{2}}\alpha_{\varepsilon}\,\mathrm{d}x=1-\frac{\varepsilon}{2}<1,$

whence, for all $\varepsilon\in(0,1)$ and a.e. $x\in\Omega$,

(B.1) $1-\frac{\varepsilon}{2}\geq
1-\rho^{\varepsilon}_{0}(x)\geq\frac{\varepsilon}{2}.$

Similarly, we note the bound

(B.2) $f^{0}_{\varepsilon}\geq\frac{\varepsilon}{4\pi},$

which holds due to the non-negativity of $f_{0}$.

2\. Uniform $L^{2}$-bound on $u^{\varepsilon}_{0}$ : we estimate the quantity

$\displaystyle\varepsilon\int_{\Upsilon}|u^{\varepsilon}_{0}|^{2}\,\mathrm{d}\bm{\xi}=\varepsilon\int_{\\{0<f^{\varepsilon}_{0}/(1-\rho^{\varepsilon}_{0})<1\\}}|u^{\varepsilon}_{0}|^{2}\,\mathrm{d}\bm{\xi}+\varepsilon\int_{\\{f^{\varepsilon}_{0}/(1-\rho^{\varepsilon}_{0})\geq
1\\}}|u^{\varepsilon}_{0}|^{2}\,\mathrm{d}\bm{\xi}=:I^{\varepsilon}_{1}+I^{\varepsilon}_{2}.$

Using that there exists $C>0$ such that $|\log s|\mathds{1}_{\\{0<s<1\\}}\leq
C|s|^{-1/2}$, the monotonicity of $s\mapsto s^{1/2}$ for $s>0$, and the
estimates (B.1)-(B.2), the first integral is controlled by

$\displaystyle I^{\varepsilon}_{1}\leq\varepsilon
C\int_{\Upsilon}(1-\rho^{\varepsilon}_{0})|{f^{\varepsilon}_{0}}|^{-1}\,\mathrm{d}\bm{\xi}\leq\varepsilon
C\int_{\Upsilon}\varepsilon^{-1}\,\mathrm{d}\bm{\xi}\leq C,$

and the right-hand side is uniformly bounded independently of $\varepsilon$.
Similarly, using that $|\log s|\mathds{1}_{\\{s\geq
1\\}}\leq|s|^{\frac{1}{2}}$, there holds

$I^{\varepsilon}_{2}\leq
C\varepsilon\int_{\Upsilon}\varepsilon^{-1}f^{\varepsilon}_{0}\,\mathrm{d}\bm{\xi}\leq
C\int_{\Omega}\rho^{\varepsilon}_{0}\,\mathrm{d}x\leq C,$

and the right-hand side is again uniformly bounded.

3\. Uniform bound on entropy functional: the aforementioned strong
convergences $f^{\varepsilon}_{0}\to f_{0}$ and
$\rho^{\varepsilon}_{0}\to\rho_{0}$ imply that, up to a subsequence which we
do not relabel, there holds $f^{\varepsilon}_{0}\to f_{0}$ a.e. in $\Upsilon$
and $\rho^{\varepsilon}_{0}\to\rho_{0}$ a.e. in $\Omega$.

Since $s\mapsto s\log s$ is locally bounded, the bound
$0<\rho^{\varepsilon}_{0}<1$ implies the uniform boundedness of
$(1-\rho^{\varepsilon}_{0})\log(1-\rho^{\varepsilon}_{0})$, whence the
Dominated Convergence Theorem implies

$\lim_{\varepsilon\to
0}\int_{\Omega}(1-\rho^{\varepsilon}_{0})\log(1-\rho^{\varepsilon}_{0})\,\mathrm{d}x=\int_{\Omega}(1-\rho_{0})\log(1-\rho_{0})\,\mathrm{d}x\leq
C,$

for $C$ depending only on the maximum value of $s\log s$ over the interval
$[0,1]$ and $\Omega$.

Similarly, we decompose

$\int_{\Upsilon}f^{\varepsilon}_{0}\log
f^{\varepsilon}_{0}\,\mathrm{d}\bm{\xi}=\int_{\\{0<f^{\varepsilon}_{0}<1\\}}f^{\varepsilon}_{0}\log
f^{\varepsilon}_{0}\,\mathrm{d}\bm{\xi}+\int_{\\{f^{\varepsilon}_{0}\geq
1\\}}f^{\varepsilon}_{0}\log f^{\varepsilon}_{0}\,\mathrm{d}\bm{\xi},$

and the local boundedness of $s\log s$ and the Dominated Convergence Theorem
that

$\lim_{\varepsilon\to
0}\int_{\\{0<f^{\varepsilon}_{0}<1\\}}f^{\varepsilon}_{0}\log
f^{\varepsilon}_{0}\,\mathrm{d}\bm{\xi}=\int_{\\{0<f_{0}<1\\}}f_{0}\log
f_{0}\,\mathrm{d}\bm{\xi}\leq C,$

where the right-hand side of the above is bounded by virtue of the function
$s\log s$ being bounded over the interval $s\in(0,1)$. Meanwhile, the
assumption $f_{0}\in L^{p}(\Upsilon)$ for $p>1$ and the estimate $|\log
s|\mathds{1}_{\\{s\geq 1\\}}\leq C_{p}|s|^{p-1}$ imply

$\int_{\\{f^{\varepsilon}_{0}>1\\}}f^{\varepsilon}_{0}\log
f^{\varepsilon}_{0}\,\mathrm{d}\bm{\xi}\leq
C_{p}\int_{\\{f^{\varepsilon}_{0}>1\\}}|f^{\varepsilon}_{0}|^{p}\,\mathrm{d}\bm{\xi}\leq
C_{p}(1+\|f_{0}\|_{L^{p}(\Upsilon)}^{p}),$

and the right-hand side is bounded independently of $\varepsilon$. ∎

The next proof consists in a quantitative diagonal approach; the argument is
classical, but it is nevertheless useful as it helps identify exactly which
diagonal subsequence we select. We recall that for
$\\{\sigma^{\prime}(n)\\}_{n}$ to be a subsequence of
$\\{\sigma(n)\\}_{n\in\mathbb{N}}$, it is sufficient that there holds:
$\\{\sigma^{\prime}(n)\\}_{n}\subset\\{\sigma(n)\\}_{n}$,
$\sigma^{\prime}(n)\geq\sigma(n)$, and
$\sigma^{\prime}(n+1)>\sigma^{\prime}(n)$ for all $n$.

###### Proof of Lemma 4.1.

For the purposes of the argument that follows, it is crucial that we replace
the continuum $\\{\varepsilon\\}_{\varepsilon\in(0,1)}$ with a countable set.
For this reason, we select $1>\varepsilon_{1}>\varepsilon_{2}>\dots>0$ and
define

$f^{m}_{n}(0):=f^{\varepsilon_{m}}_{n}(0),\qquad
f^{m}_{0}:=f^{\varepsilon_{m}}_{0},$

so that the limit $m\to\infty$ morally corresponds to $\varepsilon\to 0$.

Recall from equation (3.29) of §3.1 that for each fixed $m$, we have

$\|f^{m}_{n}(0)-f^{m}_{0}\|_{L^{p}(\Upsilon)}\to 0\quad\text{as }n\to\infty.$

In turn, for $m=1$, there exists a subsequence $\\{\sigma_{1}(n)\\}_{n}$ of
$\mathbb{N}$ such that, for all $n$,

(B.3)
$\|f^{1}_{\sigma_{1}(n)}(0)-f^{1}_{0}\|_{L^{p}(\Upsilon)}\leq\frac{1}{n}.$

Note in particular that the above implies, for all $\ell\geq n$,

(B.4)
$\|f^{1}_{\sigma_{1}(\ell)}(0)-f^{1}_{0}\|_{L^{p}(\Upsilon)}\leq\frac{1}{n}.$

Then, for $m=2$, since we also know that
$\|f^{2}_{n}(0)-f^{2}_{0}\|_{L^{p}(\Upsilon)}\to 0$ as $n\to\infty$, we may
construct a subsequence $\\{\sigma_{2}(n)\\}_{n}$ of $\\{\sigma_{1}(n)\\}_{n}$
such that, for all $n$,

$\|f^{2}_{\sigma_{2}(n)}(0)-f^{2}_{0}\|_{L^{p}(\Upsilon)}\leq\frac{1}{n}.$

Furthermore, since $\\{\sigma_{2}(n)\\}_{n}$ is a subsequence of
$\\{\sigma_{1}(n)\\}_{n}$, the bound (B.4) implies, for all $n$,

$\|f^{1}_{\sigma_{2}(n)}(0)-f^{1}_{0}\|_{L^{p}(\Upsilon)}\leq\frac{1}{n}.$

We continue iteratively, and construct for each $m\in\mathbb{N}$ a subsequence
$\\{\sigma_{m}(n)\\}_{n}$ of all previous $m-1$ subsequences, such that there
holds, for all $j\in\\{1,\dots,m\\}$ and all $n$,

(B.5)
$\|f^{j}_{\sigma_{m}(n)}(0)-f^{j}_{0}\|_{L^{p}(\Upsilon)}\leq\frac{1}{n}.$

In turn, we define the diagonal sequence

(B.6) $f^{m}:=f^{m}_{\sigma_{m}(m)},$

and note that (B.5) implies, for all $m$,

$\|f^{m}(0)-f^{m}_{0}\|_{L^{p}(\Upsilon)}\leq\frac{1}{m}.$

Using the above and the strong convergence of $f^{m}_{0}\to f_{0}$ in
$L^{p}(\Upsilon)$ of Lemma 1.3, the triangle inequality yields

$\|f^{m}(0)-f_{0}\|_{L^{p}(\Upsilon)}\leq\underbrace{\|f^{m}(0)-f_{0}^{m}\|_{L^{p}(\Upsilon)}}_{\leq
1/m}+\underbrace{\|f_{0}^{m}-f_{0}\|_{L^{p}(\Upsilon)}}_{\to 0}\to 0$

as $m\to\infty$, and the proof is complete. ∎

Acknowledgements:

MB acknowledges support from DESY (Hamburg, Germany), a member of the
Helmholtz Association HGF, and the German Science Foundation (DFG) through CRC
TR 154, subproject C06. SMS acknowledges support from Centro di Ricerca
Matematica Ennio De Giorgi.

The authors report there are no competing interests to declare.

## References

  * [1] L. C. B. Alasio, J. Guerand, and S. M. Schulz, Regularity and trend to equilibrium for a non-local advection-diffusion model of active particles, In preparation, (2023).
  * [2] G. Allaire, Analyse numérique et optimisation, Éditions de l’École Polytechnique, 91128 Palaiseau Cedex, 2nd ed., 2012.
  * [3] L. Ambrosio, N. Gigli, and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, second ed., 2008.
  * [4] C. Bardos and N. Besse, The cauchy problem for the vlasov–dirac–benney equation and related issues in fluid mechanics and semi-classical limits, Kinetic and Related Models, 6 (2013), pp. 893–917.
  * [5] M. Briant and N. Meunier, Well-posedness for systems of self-propelled particles, arXiv:2305.05336, (2023).
  * [6] M. Bruna, M. Burger, A. Esposito, and S. M. Schulz, Phase separation in systems of interacting active brownian particles, SIAM J. Appl. Math., 82 (2022), pp. 1635–1660.
  * [7] , Well-posedness of an integro-differential model for active brownian particles, SIAM J. Math. Anal., 54 (2022), pp. 5662–5697.
  * [8] M. Burger, M. Di Francesco, J.-F. Pietschmann, and B. Schlake, Nonlinear cross-diffusion with size exclusion, SIAM J. Math. Anal., 42 (2010), pp. 2842–2871.
  * [9] J. A. Carrillo, R. J. McCann, and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), pp. 971–1018.
  * [10] M. E. Cates and J. Tailleur, When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-induced phase separation, EPL, 101 (2013), p. 20010.
  * [11] M. E. Cates and J. Tailleur, Motility-Induced Phase Separation, Annu. Rev. Cond. Ma. P., 6 (2015), pp. 219–244.
  * [12] E. DiBenedetto, Degenerate Parabolic Equations, Springer Science & Business Media, 2012.
  * [13] C. Erignoux, Hydrodynamic Limit For An Active Exclusion Process, 2016\.
  * [14] L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second ed., 2010.
  * [15] H. Gajewski, On a variant of monotonicity and its applications to differential equations, Nonlinear Analysis, 22 (1994), pp. 73–80.
  * [16] P.-E. Jabin and A. Nouri, Analytic solutions to a strongly nonlinear vlasov equation, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), pp. 541–546.
  * [17] A. Jüngel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), p. 1963.
  * [18] , Entropy methods for diffusive partial differential equations, vol. 804, Springer, 2016.
  * [19] A. Jüngel and N. Zamponi, Analysis of degenerate cross-diffusion population models with volume filling, Annales de l’Institut Henri Poincaré C, Analyse non-linéaire, 34 (2017), pp. 1–29.
  * [20] O. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uralceva, Linear and quasilinear equations of parabolic type, Translations of mathematical monographs, American Mathematical Society, Providence, 1968.
  * [21] F. Pacard and A. Unterreiter, A variational analysis of the thermal equilibrium state of charged quantum fluids, Commun. Part. Diff. Eqs, 20 (1995), pp. 885–900.
  * [22] B. Perthame, Mathematical tools for kinetic equations, Bulletin of the American Math. Soc., 41 (2004), pp. 205–244.
  * [23] G. S. Redner, M. F. Hagan, and A. Baskaran, Structure and Dynamics of a Phase-Separating Active Colloidal Fluid, Phys. Rev. Lett., 110 (2013), p. 055701.
  * [24] P. Romanczuk, W. Ebeling, B. Lindner, and L. Schimansky-Geier, Active Brownian particles, Eur. Phys. J. Special Topics, 202 (2012), pp. 1–162.
  * [25] T. Speck, A. M. Menzel, J. Bialké, and H. Löwen, Dynamical mean-field theory and weakly non-linear analysis for the phase separation of active Brownian particles., J. Chem. Phys., 142 (2015), p. 224109.
  * [26] J. Stenhammar, R. Wittkowski, D. Marenduzzo, and M. E. Cates, Activity-Induced Phase Separation and Self-Assembly in Mixtures of Active and Passive Particles, Phys. Rev. Lett., 114 (2015), p. 018301.
  * [27] J. Yeomans, The hydrodynamics of active systems, Rivista del Nuovo Cimento, 40 (2017), pp. 1–31.