remarkRemark hypothesisHypothesis claimClaim Integro-differential model for
active particlesM. Bruna, M. Burger, A. Esposito and S. M. Schulz

# Well-posedness of an integro-differential model for active Brownian
particles††thanks:  The first author was partially funded by a Royal Society
University Research Fellowship (URF/R1/180040) and a Humboldt Research
Fellowship from the Alexander von Humboldt Foundation. The second and the
third author gratefully acknowledge support by the German Science Foundation
(DFG) through CRC TR 154 “Mathematical Modelling, Simulation and Optimization
Using the Example of Gas Networks”. AE was supported by the Advanced Grant
Nonlocal-CPD of the ERC under the EU’s Horizon 2020 research and innovation
programme (grant agreement No. 883363). The fourth author was funded by the
Royal Society Award (RGF/EA/181043).

Maria Bruna Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge CB3 0WA, UK<EMAIL_ADDRESS>Martin Burger Department Mathematik, Friedrich-Alexander Universität Erlangen-
Nürnberg, Cauerstr. 11, D 91058 Erlangen, Germany<EMAIL_ADDRESS>Antonio Esposito Mathematical Institute, University of Oxford, Oxford OX2 6GG,
UK<EMAIL_ADDRESS>Simon Schulz Department of Mathematics,
University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Dr, Madison, WI
53706, USA<EMAIL_ADDRESS>
###### Abstract

We propose a general strategy for solving nonlinear integro-differential
evolution problems with periodic boundary conditions, where no direct
maximum/minimum principle is available. This is motivated by the study of
recent macroscopic models for active Brownian particles with repulsive
interactions, consisting of advection-diffusion processes in the space of
particle position and orientation. We focus on one of such models, namely a
semilinear parabolic equation with a nonlinear active drift term, whereby the
velocity depends on the particle orientation and angle-independent overall
particle density (leading to a nonlocal term by integrating out the angular
variable). The main idea of the existence analysis is to exploit a-priori
estimates from (approximate) entropy dissipation. The global existence and
uniqueness of weak solutions is shown using a two-step Galerkin approximation
with appropriate cutoff in order to obtain nonnegativity, an upper bound on
the overall density and preserve a-priori estimates. Our analysis naturally
includes the case of finite systems, corresponding to the case of a finite
number of directions. The Duhamel principle is then used to obtain additional
regularity of the solution, namely continuity in time-space. Motivated by the
class of initial data relevant for the application, which includes perfectly
aligned particles (same orientation), we extend the well-posedness result to
very weak solutions allowing distributional initial data with low regularity.

###### keywords:

active particles, space-periodic problems, parabolic equations, Galerkin
approximation, periodic heat kernel

35K20, 35K58, 35Q70, 35Q92

##  1 Introduction

In this paper, we present a well-posedness theory (existence, uniqueness, and
regularity) for an integro-differential equation describing a system of
interacting active Brownian particles. Active Brownian particles are a model
system for self-propelled particles, and have wide-ranging applications such
as molecular motors, bacteria, animal swarms and pedestrian flows (cf. e.g.
[5, 8, 12, 13]). Denote by ${\bf X}_{i}\in\Omega\subset\mathbb{R}^{2}$ and
$\Theta_{i}\in\mathbb{T}:=\mathbb{R}/(2\pi\mathbb{Z})$ the position and
orientation of particle number $i$, respectively. The evolution of the system
with $N$ indistinguishable particles is given by the following stochastic
differential equations:

(1a) $\displaystyle\mathrm{d}{\bf X}_{i}$
$\displaystyle=\sqrt{2D_{T}}\mathrm{d}{\bf W}_{i}+v_{0}{\bf
e}(\Theta_{i})\mathrm{d}t-\sum_{j\neq i}\nabla u(({\bf X}_{i}-{\bf
X}_{j})/\varepsilon)\mathrm{d}t,$ (1b) $\displaystyle\mathrm{d}\Theta_{i}$
$\displaystyle=\sqrt{2D_{R}}\mathrm{d}W_{i},$

where $D_{T}$ and $D_{R}$ are the translational and rotational diffusion
coefficients, respectively, ${\bf W}_{i}$ and $W_{i}$ are $N$ independent
Brownian motions in $\mathbb{R}^{2}$ and $\mathbb{R}$ respectively, $v_{0}$ is
the self-propulsion speed, ${\bf e}(\theta)=(\cos\theta,\sin\theta)$, and $u$
is the pairwise interaction potential, assumed to be radially symmetric,
purely repulsive and short-ranged ($\varepsilon\ll 1$). In contrast to passive
Brownian particles (that is $D_{R}=0$ in Eq. 1b), active Brownian particles
perform a biased Brownian walk in the direction given by their orientation.

Starting from Eq. 1 and considering the associated Fokker–Planck equation for
the joint probability density in $(\Omega\times\mathbb{T})^{N}$, one can
derive the BBGKY hierarchy of coupled equations for the marginal
probabilities. Depending on the assumptions on the interaction $u$, one can
obtain different continuum models for the one-particle marginal density [4].
One of such models is studied by Speck et al. [16], where they obtain a closed
equation by assuming that the system is homogeneous and neglecting the time-
dependence of the pair correlation function. Following a rescaling of time and
space, leading to a rescaled velocity or Péclet number
$\text{Pe}=v_{0}/\sqrt{D_{R}D_{T}}\geq 0$, the resulting equation is (see [4]
for details)

(2) $\displaystyle\partial_{t}f+\text{Pe}\nabla\cdot[f(1-\rho){\bf
e}(\theta)]=D_{e}\Delta f+\partial_{\theta}^{2}f,$

where $D_{e}\in(0,1]$ is an effective translational diffusion, the unknown
$f=f(t,{\bf x},\theta)$ is the phase space density and $\rho=\rho(t,{\bf x})$
is the spatial density obtained through

$\displaystyle\rho(t,{\bf x})=\int_{0}^{2\pi}f(t,{\bf
x},\theta)~{}\mathrm{d}\theta.$

Both densities are scaled such that they have mass $\phi\in[0,1)$; this
represents the effective occupied fraction of particles in $\Omega$. By
integrating Eq. 2 with respect to $\theta$, we obtain the following equation
for $\rho$

(3) $\displaystyle\partial_{t}\rho+\text{Pe}\nabla\cdot[(1-\rho){\bf
p}]=D_{e}\Delta\rho,$

where ${\bf p}(t,{\bf x})$ is the polarisation, defined by ${\bf p}(t,{\bf
x})=\int_{0}^{2\pi}{\bf e}(\theta)f(t,{\bf x},\theta)\mathrm{d}\theta$. As we
shall see later on, the equation for $\rho$, Eq. 3, plays an important role in
the analysis of Eq. 2 in order to determine the sign of the nonlocality (in
angle) $1-\rho$, which is the reason why usual maximum/minimum principle does
not work in our case. It is also interesting to consider Eq. 2 in one spatial
dimension, $\Omega\subset\mathbb{R}$. In this case, the set of possible
orientations reduces to $\theta=0,\pi$, or right- and left-moving particles,
and the rotational diffusion reduces to discrete jumps between these two
orientations. The one-dimensional version of Eq. 2 reads

(4) $\displaystyle\begin{aligned}
\partial_{t}f_{R}+\text{Pe}\partial_{x}[f_{R}(1-\rho)]&=\partial_{xx}f_{R}+f_{L}-f_{R},\\\
\partial_{t}f_{L}-\text{Pe}\partial_{x}[f_{L}(1-\rho)]&=\partial_{xx}f_{L}+f_{R}-f_{L}.\end{aligned}$

where $f_{R}$ and $f_{L}$ are the densities of right- and left-moving
particles, respectively, and the space density is $\rho=f_{R}+f_{L}$. This
model is derived in [4] from the one-dimensional version of Eq. 1, and in [14]
as the hydrodynamic limit of an active lattice gas. In fact, the two-
dimensional model Eq. 2 can also be derived starting from an active lattice
gas model, with a partial exclusion rule (only one particle is allowed by
site, but particles can exchange sites in the diffusive step). This explains
why models Eqs. 2 and 4 present linear diffusion. If instead one considers a
complete exclusion rule (resulting in an asymmetric simple exclusion process),
nonlinearities of cross-diffusion type (involving terms $\rho\nabla
f,f\nabla\rho$) appear in the continuum model [4]. Equation 4 coincides with
the crowded version (with the additional $1-\rho$ term in the advection) of
the Goldstein–Taylor model [11, 17], and for this reason we refer to it as the
_crowded Goldstein–Taylor model_. We shall see that well-posedness for the
above system follows directly from that of Eq. 2, as consequence of our
strategy based on a two-step approximation.

To the best of our knowledge, most of the results in literature focus on time-
periodic evolution equations or initial-value problems with Neumann or zero-
flux boundary conditions, cf. e.g. [15], while periodicity in space has not
been studied in detail. Moreover, the mobility $f(1-\rho)$ in the drift term
does not yield a maximum principle due to the nonlocal dependence of $\rho$ on
$f$. In particular, it is not immediately clear whether one can use the theory
of Ladyzhenskaya and collaborators [15], or that established by Amann [1],
based on fixed point theorems. Moreover, we note that the equation does not
present a usual Wasserstein gradient flow structure, even in the absence of
angular diffusion, again because of the mobility $f(1-\rho)$ in the transport
drift term, cf. for instance, [3]. In this regard, we also point out that this
problem does not fall in the theory of generalised Wasserstein gradient flows
for equations with nonlinear mobility studied for instance in [9, 7] and the
references therein, as the mobility in Eq. 2 is nonlocal in addition to being,
effectively, nonlinear.

For these reasons, we provide a valuable strategy to tackle space-periodic
nonlinear evolution PDEs with a nonlocal in angle drift term and periodic
boundary conditions by means of a double space-discretisation purely using
periodicity. More precisely, in order to prove the existence and uniqueness of
solutions in the sense of Definition 2.2, we first consider a Galerkin
approximation with respect to the angular variable $\theta$. We emphasise
that, in the sequel, we make the choice to expand in terms of the sine and
cosine basis instead of the complex exponentials; this is so that all
coefficients are real-valued, which is required to make sense of the positive
part in the parabolic system that we study; cf. Eq. 13 in Section 3. Existence
of solutions to such a system is proven by means of a further Galerkin
approximation in the space variable, again taking into account periodicity of
the problem in the choice of the basis.

In this paper we provide the first rigorous result on a macroscopic model for
Brownian active particles in phase space, thus starting with the possibly
simplest model without cross-diffusion effects. However, it still carries two
characteristic issues of these problems, namely the nonlinear drift and the
nonlocal effects due to the overall density $\rho$. Note that here we work in
the setting of a continuous angular variable, with discrete angles or
directions as in the one-dimensional version of the model we directly see the
relation to nonlinear parabolic systems. We stress that this strategy may be
extended to the analysis of a larger class of space-periodic evolution
equations, even some cross-diffusion systems, see [4]. This will indeed be the
object of further investigation. Let us mention at this point the importance
of the space-periodic setting also for pattern formation effects. A closed
system with no-flux boundary conditions might lead to very different
stationary flux behaviour and may not lead to the phase separation phenomena
observed in many microscopic simulations and discussed in [4].

This article is organised as follows. In Section 2 we provide the general set
of assumptions and the summary of the main results. Section 3 contains the
proof of existence and uniqueness of weak solutions of Eq. 2 and Eq. 4. Here
we provide a detailed explanation of the strategy used, which can be
potentially adapted to a larger class of space-periodic nonlinear evolution
problems. More precisely, well-posedness is proven via a two-step Galerkin
discretisation. This involves solving a semilinear parabolic system outside
what is covered by standard parabolic theory, not due to particular
complications, but rather the periodic nature of the problem. In Section 4 we
obtain higher regularity of solutions by applying the Duhamel principle.
Section 5 is devoted to enlarging the class of initial data, among them those
presenting a Dirac delta in the angular variable. This is relevant to the
application of active Brownian particles, as it corresponds to all particles
being originally perfectly aligned (same orientation). Finally, the appendices
collect some technical results used in the paper.

##  2 Preliminaries and main results

In what follows, we refer to Eq. 2 and Eq. 4 as 2D and 1D problems,
respectively, in reference to the dimension of the spatial domain $\Omega$.
Note that, mathematically, the two-dimensional problem Eq. 2 lives in the
three-dimensional space $\Upsilon:=\Omega\times\mathbb{T}$. We choose the
spatial domain $\Omega$ to be the $d$-dimensional torus $\mathbb{T}^{d}$,
where $d\in\\{1,2\\}$ and $\mathbb{T}=\mathbb{R}/(2\pi\mathbb{Z})$. This
imposes periodic boundary conditions as considered in [4, 16]. The
$2\pi$-period is chosen by mathematical convenience, so that in the two-
dimensional problem $\Upsilon\equiv\mathbb{T}^{3}$. In the sequel, we say that
a function $f$ is _$\mathbb{T}^{M}$ -periodic_ (for $M\in\\{1,2,3\\}$) if, for
all $\bm{\xi}\in\mathbb{R}^{M}$ and all $j\in\\{1,\dots,M\\}$, there holds
$f(\bm{\xi})=f(\bm{\xi}+2\pi\mathbf{e}_{j})$, where
$\\{\mathbf{e}_{j}\\}_{j=1}^{M}$ are the standard unit vectors of
$\mathbb{R}^{M}$.

######  Definition 2.1 (Function spaces).

Let $M\in\\{1,2,3\\}$. We define the following Banach spaces

$\displaystyle
L^{p}_{per}(\mathbb{T}^{M}):=\\{f:\mathbb{R}^{M}\to\mathbb{R}\big{|}f\text{ is
a.e.~{}}\mathbb{T}^{M}\text{-periodic, and }f\in
L^{p}_{loc}(\mathbb{R}^{M})\\},$ $\displaystyle
H^{m}_{per}(\mathbb{T}^{M}):=\\{f:\mathbb{R}^{M}\to\mathbb{R}\big{|}f\text{ is
a.e.~{}}\mathbb{T}^{M}\text{-periodic, and }f\in
H^{m}_{loc}(\mathbb{R}^{M})\\},$

with $p\geq 1,m\in\mathbb{N}\cup\\{0\\}$ endowed with the norms
$\|\cdot\|_{L^{p}((0,2\pi)^{M})}$ and $\|\cdot\|_{H^{m}((0,2\pi)^{M})}$,
respectively. The topological dual of $H^{m}_{per}(\mathbb{T}^{M})$ is denoted
by $(H^{m}_{per})^{\prime}(\mathbb{T}^{M})$. Similarly, we define the spaces

$C^{k}_{per}(\bar{\mathbb{T}}^{M}):=\\{f:\mathbb{R}^{M}\to\mathbb{R}\big{|}f\text{
is }\mathbb{T}^{M}\text{-periodic, and }f\in C^{k}(\mathbb{R}^{M})\\},$

for $k\in\mathbb{N}\cup\\{0\\}\cup\\{\infty\\}$, equipped with their
respective norm $\|\cdot\|_{C^{k}([0,2\pi]^{M})}$.

For every $f,g:\Upsilon\to\mathbb{R}$, we use the notation
$\left\langle\cdot,\cdot\right\rangle_{\Upsilon}$ to denote both the $L^{2}$
scalar product in $\Upsilon$ and duality between $(H^{1})^{\prime}$ and
$H^{1}$ as a Gelfand triple with $L^{2}$, while for any
$a,b:\Omega\to\mathbb{R}$ all such pairings (in $L^{2}$ and $H^{1}$) are
denoted by $\left\langle a,b\right\rangle_{\Omega}$. Moreover, let us specify
that, by slight abuse of notation, any curve $f:[0,T]\to B$ will map
$t\in[0,T]\mapsto f(t,\cdot):=f(t)\in B$, for any Banach space $B$ we
consider.

######  Definition 2.2 (Weak solution to 2D model).

Let $f_{0}\in L^{2}_{per}(\Upsilon)$ be such that

(5)
$\rho_{0}(\mathbf{x}):=\int_{0}^{2\pi}f_{0}(\mathbf{x},\theta)\,\mathrm{d}\theta\in[0,1]\qquad\text{for
a.e.~{}}\mathbf{x}\in\Omega.$

A curve $f$ is a _weak solution_ to Eq. 2 if $f\in
C([0,T];L^{2}_{per}(\Upsilon))\cap L^{2}([0,T];H^{1}_{per}(\Upsilon))$,
$f^{\prime}\in L^{2}([0,T];(H^{1}_{per})^{\prime}(\Upsilon))$, and for a.e.
$t\in[0,T]$ and any $\varphi\in H^{1}_{per}(\Upsilon)$, there holds

(6)
$\begin{split}\left\langle\partial_{t}f(t),\varphi\right\rangle_{\Upsilon}\\!&=\\!\left\langle\text{Pe}(1\\!-\\!\rho(t))f(t){\bf
e}(\theta),\nabla\varphi\right\rangle_{\Upsilon}\\!-\\!\left\langle
D_{e}\nabla
f(t),\nabla\varphi\right\rangle_{\Upsilon}\\!-\\!\left\langle\partial_{\theta}f(t),\partial_{\theta}\varphi\right\rangle_{\Upsilon},\\\
f(0,{\bf x},\theta)&=f_{0}({\bf x},\theta),\end{split}$

with periodic boundary conditions on $\Upsilon$, where
$\rho(t,\mathbf{x})=\int_{0}^{2\pi}f(t,{\bf x},\theta)\,\mathrm{d}\theta$, and
the initial data is achieved in the sense $f(0)=f_{0}$ in
$L^{2}_{per}(\Upsilon)$.

Our first main result is the following.

######  Theorem 2.3 (Existence and uniqueness for the 2D model).

Let $T>0$ and $f_{0}\in L^{2}_{per}(\Upsilon)$ be nonnegative and such that
$\rho_{0}({\bf x})=\int_{0}^{2\pi}f_{0}({\bf
x},\theta)\,\mathrm{d}{\theta}\in[0,1]$ for a.e. ${\bf x}\in\Omega$. Then,
there exists a unique weak solution to Eq. 6 in the sense of Definition 2.2.

######  Remark 2.4.

In the proof of Theorem 2.3 we prove that $f\in
L^{\infty}([0,T];L^{2}_{per}(\Upsilon))\cap
L^{2}([0,T];H^{1}_{per}(\Upsilon))$. Then, $f\in C([0,T];L^{2}_{per}(\Omega))$
is deduced from [18, Ch. 3 Sec. 1.4 Lem. 1.2], since $f\in
L^{2}([0,T];H^{1}_{per}(\Upsilon))$ and $f^{\prime}\in
L^{2}([0,T];(H^{1}_{per})^{\prime}(\Upsilon))$.

An analogous theorem is obtained for the 1D model Eq. 4, see Theorem 3.7 in
Section 3.5. We then consider the regularity of weak solutions. More
precisely, we provide the following regularity result up to the initial time.

######  Theorem 2.5 (Regularity for the 2D model).

Let $f_{0}\in L^{\infty}(\Upsilon)$. Then the unique weak solution of Eq. 6
provided by Theorem 2.3 satisfies $f\in C((0,T]\times\bar{\Upsilon})$.
Furthermore, if $f_{0}\in C(\bar{\Upsilon})$, the unique weak solution to Eq.
6 belongs to $C([0,T]\times\bar{\Upsilon})$.

The previous statement also holds true for the 1D model Eq. 4, see Section
4.1.

Finally, we extend the concept of solution in order to allow for initial data
with a fixed orientation (see Definition 5.1). Note that initial data of this
form are of significantly lower regularity than considered in the original
Theorem 2.3; indeed, they are distributional in the angle variable, and hence
$\rho_{0}$ must be defined by duality. Our existence result for initial data
of this kind entails defining _very weak solutions_ , and we refer the reader
to Section 5 for further details; for clarity of presentation, we do not
include the definition of very weak solutions here. The main result proved
therein is as follows.

######  Theorem 2.6 (Existence of very weak solutions for 2D model).

Let $T>0$, $f_{0}$ be a nonnegative element of
$L^{2}_{per}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))$ and $\rho_{0}({\bf
x}):=\langle f_{0}({\bf x},\cdot),1\rangle\in[0,1]$ for a.e. ${\bf
x}\in\Omega$. Then there exists a very weak solution of Eq. 2 in the sense of
Definition 5.1, $f\in
L^{2}([0,T];H^{1}_{per}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi)))\cap
L^{\infty}([0,T];L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi)))\cap
L^{2}((0,T)\times\Upsilon)$.

Note that the result of Theorem 2.6 implies an instantaneous smoothing
phenomenon, since the initial data is distributional in the angle variable,
while the solution $f$ belongs to $L^{2}((0,T)\times\Upsilon)$.

##  3 Well-posedness

In this section we consider the analysis of the 1D and 2D models. We tackle
the 2D model first, and show in Section 3.5 that the result for the 1D model
follows from that of the 2D model.

A crucial ingredient for the well-posedness of Eq. 2 and Eq. 4 is that
$\rho\in[0,1]$ for all times. Equation Eq. 3 for the density $\rho$ does not
have an obvious maximum or minimum principle, since the drift term neither has
a sign nor is zero. One can show that any solution to Eq. 3 (provided it
exists) satisfies $\rho\leq 1$, starting with an initial datum
$\rho_{0}\in[0,1]$. However, at this early stage of the analysis, the
nonnegativity of the solution is a delicate issue. For this reason we consider
a variant of Eq. 2, where we have $(1-(\rho)_{+})_{+}$ in the transport term.
It will turn out that we are indeed solving the original equation, i.e. Eq. 2.
This final step is shown in Step 5 of the Proof of Theorem 2.3; cf. Section
3.3. The modified version of Eq. 2 we start with is

(7) $\displaystyle\partial_{t}f+\text{Pe}\nabla\cdot(f(1-(\rho)_{+})_{+}{\bf
e}(\theta))=D_{e}\Delta f+\partial_{\theta}^{2}f,$ $\displaystyle
f(0,\mathbf{x},\theta)=f_{0}({\bf x},\theta),$

in $\Upsilon=\mathbb{T}^{3}$, where $(a)_{+}=\max\\{a,0\\}$, for any
$a\in\mathbb{R}$. Later we show that we recover the solution to the original
equation Eq. 2 (see Section 3.3). We remind the reader that the nonlocality in
the drift term means that there is no inherent maximum/minimum principle for
Eq. 7. Our strategy for solving the equation is to construct an approximating
sequence $(f^{n})_{n\in\mathbb{N}}$ whose limit solves Eq. 7.

###  3.1 Approximated system in angle

We introduce the following Galerkin approximation in $\theta$. Given the
periodic boundary conditions and the term ${\bf e}(\theta)$ in Eq. 7, it is
convenient to consider the basis
$\\{1,\cos(k(\cdot)),\sin(k(\cdot))\\}_{k\in\mathbb{N}}$ of
$L^{2}_{per}([0,2\pi))$. Since $f_{0}({\bf x},\cdot)\in L^{2}_{per}([0,2\pi))$
for a.e. ${\bf x}\in\Omega$, we can write

(8) $f_{0}({\bf x},\theta)=\frac{1}{2\pi}a_{0}(0,{\bf
x})+\frac{1}{\pi}\sum_{k\geq 1}\big{(}a_{k}(0,{\bf
x})\cos(k\theta)+b_{k}(0,\mathbf{x})\sin(k\theta)\big{)},$

where $a_{k}(0,\cdot),b_{k}(0,\cdot)$ are the Fourier coefficients of
$f_{0}(\cdot,\theta)$. Note that the requirement $0\leq\rho_{0}({\bf x})\leq
1$ (cf. Definition 2.2) implies that $0\leq a_{0}(0,{\bf x})\leq 1$ for a.e.
${\bf x}\in\Omega$.

For some fixed $n\in\mathbb{N}$, we consider the finite-dimensional space

(9) $X_{n}=\text{span}\\{\cos(k(\cdot)),\sin(k(\cdot))\\}_{k=0,\dots,n}.$

We try to find a solution $f^{n}$ to the weak-formulation Eq. 6 adapted to Eq.
7 of the approximated system given by

(10)
$\left\langle\partial_{t}f^{n},\varphi\right\rangle_{\Upsilon}\\!=\\!\left\langle\text{Pe}(1\\!-\\!(\rho^{n})_{+})_{+}f^{n}(t){\bf
e}(\theta),\nabla\varphi\right\rangle_{\Upsilon}\\!-\\!\left\langle
D_{e}\nabla
f^{n},\nabla\varphi\right\rangle_{\Upsilon}\\!-\\!\left\langle\partial_{\theta}f^{n},\partial_{\theta}\varphi\right\rangle_{\Upsilon},$

where $\rho^{n}=\int f^{n}\mathrm{d}\theta$, for test functions
$\varphi=\chi(\theta)\psi({\bf x})$ with $\chi\in X_{n}$ and $\psi\in
H^{1}_{\text{per}}(\Omega)$. We can express $f^{n}$ as

(11) $f^{n}(t,{\bf
x},\theta):=\frac{1}{2\pi}a_{0}^{n}(t,\mathbf{x})+\frac{1}{\pi}\sum_{k=1}^{n}\big{(}a_{k}^{n}(t,\mathbf{x})\cos(k\theta)+b_{k}^{n}(t,\mathbf{x})\sin(k\theta)\big{)},$

where the Fourier coefficients $a_{k}^{n}(t,{\bf x}),b_{k}^{n}(t,{\bf x})$
need to be determined. We note that $\rho^{n}$ above corresponds to the
constant coefficient, that is,

(12) $\rho^{n}(t,{\bf x})=\int_{0}^{2\pi}f^{n}(t,{\bf
x},\theta)\,\mathrm{d}\theta\equiv a_{0}^{n}(t,{\bf x}).$

From Eq. 10 we obtain a $(2n+1)$-dimensional system of semilinear parabolic
equations for $a_{k}^{n}(t,{\bf x}),b_{k}^{n}(t,{\bf x})$. In particular,
testing Eq. 10 for each $\chi(\theta)$ in the basis of $X_{n}$ gives the
equation for the corresponding Fourier coefficient of $f^{n}$. More precisely,
choosing $\chi=1$ results in

(13a)
$\partial_{t}a_{0}^{n}+\text{Pe}\nabla\cdot\left[(1-(a_{0}^{n})_{+})_{+}{\bf
J}_{0}^{n}\right]=D_{e}\Delta a_{0}^{n},\quad{\bf
J}_{0}^{n}=(a_{1}^{n},b_{1}^{n}).$ Repeating for
$\chi=\cos(k\theta),\sin(k\theta)$ for $k=1,\dots,n$ leads to (13b)
$\displaystyle\partial_{t}a_{k}^{n}+\frac{\text{Pe}}{2}\nabla\cdot\left[(1-(a_{0}^{n})_{+})_{+}{\bf
J}_{k}^{n}\right]$ $\displaystyle=D_{e}\Delta a_{k}^{n}-k^{2}a_{k}^{n},$ (13c)
$\displaystyle\partial_{t}b_{k}^{n}+\frac{\text{Pe}}{2}\nabla\cdot\left[(1-(a_{0}^{n})_{+})_{+}{\bf
Q}_{k}^{n}\right]$ $\displaystyle=D_{e}\Delta b_{k}^{n}-k^{2}b_{k}^{n},$ in
$\mathbb{T}^{2}$, where the velocities ${\bf J}_{k}^{n}$ and ${\bf Q}_{k}^{n}$
are given by

for $l=2,\dots,n-1$, with periodic boundary conditions on $\Omega$ and initial
conditions $a_{k}^{n}(0,{\bf x}),b_{k}^{n}(0,{\bf x})$ given by the
corresponding Fourier coefficients of $f_{0}$ in Eq. 8.

Provided that Eq. 13 can be solved, we obtain an explicit description of each
$f^{n}$ in terms of its Galerkin coefficients $a_{k}^{n},b_{k}^{n}$. In order
to prove Theorem 2.3, we need to show the convergence (in a sense to be made
precise) of the sequence $(f^{n})_{n\in\mathbb{N}}$ towards some limit $f$,
and we must show that $f$ solves Eq. 7 in the weak sense prescribed by
Definition 2.2. As previously mentioned, we will see that we are indeed
solving Eq. 2.

###  3.2 Existence of solutions to the semilinear parabolic system with
spatially-periodic boundary conditions

In order to solve the semilinear parabolic system Eq. 13, we perform a further
Galerkin approximation in space, with the aim of rewriting it as a system of
ODEs. While the well-posedness for systems similar to Eq. 13 is considered by
Ladyzhenskaya et al. [15] and Amann [2], it is not obvious how these results
can be applied to space-periodic problems or to systems without a maximum
principle (resp. $L^{\infty}$-bounds). This motivates our study. We define

${\bf c}^{n}(t,{\bf
x})=(a_{0}^{n},a_{1}^{n},\dots,a_{n}^{n},b_{1}^{n},\dots,b_{n}^{n}).$

Since throughout this section we keep $n\in\mathbb{N}$ fixed, in order to
avoid confusion we drop the superscript $n$.

######  Theorem 3.1 (Solution to Eq. 13).

Let ${\bf c}(0,{\bf x})\in(L^{2}_{per}(\Omega))^{2n+1}$ be the initial data.
For any $T>0$ there exist curves $\bf c$ weak solutions to system Eq. 13 such
that

(13na) $\displaystyle{\bf c}$
$\displaystyle\in(L^{\infty}([0,T];L^{2}_{per}(\Omega))^{2n+1}\cap
L^{2}([0,T];H^{1}_{per}(\Omega)))^{2n+1},$ (13nb) $\displaystyle{\bf
c}^{\prime}$
$\displaystyle\in(L^{2}([0,T];(H^{1}_{per})^{\prime}(\Omega)))^{2n+1}.$

With
$C_{0,n}=\sum_{k=0}^{n}(\|a_{k}(0)\|_{L^{2}(\Omega)}^{2}+\|b_{k}(0)\|_{L^{2}(\Omega)}^{2})$,
there holds

(13oa)
$\displaystyle\sum_{k=0}^{n}\left(\|a_{k}(t)\|_{L^{2}(\Omega)}^{2}+\|b_{k}(t)\|_{L^{2}(\Omega)}^{2}\right)\leq
C_{0,n}\exp\left(2CT\frac{\text{Pe}^{2}}{D_{e}}\right),$ (13ob)
$\begin{split}\sum_{k=0}^{n}\int_{0}^{T}\\!\\!\\!\\!k^{2}\big{(}\|a_{k}(t)\|_{L^{2}(\Omega)}^{2}\\!+\\!\|b_{k}(t)\|_{L^{2}(\Omega)}^{2}\big{)}&\\!+\\!D_{e}\big{(}\|\nabla
a_{k}(t)\|_{L^{2}(\Omega)}^{2}\\!+\\!\|\nabla
b_{k}(t)\|_{L^{2}(\Omega)}^{2}\big{)}\mathrm{d}{t}\\\ &\leq
C(C_{0,n},\text{Pe},D_{e},T),\end{split}$ (13oc)
$\|a^{\prime}_{0}\|_{L^{2}([0,T];(H^{1}_{per})^{\prime}(\Omega))}\leq\bar{C}(\text{Pe},\phi,T,C_{0,n}).$

######  Proof 3.2.

Step 1 (approximation of the system): We look for an approximate solution
${\bf c}^{m}$ of Eq. 13 in $X_{m}^{2}$ Eq. 9

$X_{m}^{2}=\text{span}\\{\cos(px),\sin(px),\cos(qy),\sin(qy)\\}_{p,q=0,\dots,m}.$

From the $(2n+1)$-dimensional system of PDEs Eq. 13, this further
approximation leads to a $(2n+1)(2m+1)^{2}$-dimensional system of ordinary
differential equations (ODEs) for the Fourier coefficients of the elements of
${\bf c}^{m}$. Defining the vector of unknowns by $\Lambda^{n,m}(t)$ and its
initial data by $\Lambda^{n,m}_{0}$ (which are given explicitly in Appendix
A), we obtain an ODE of the following form:

(13p) $\left\\{\begin{aligned}
&\frac{d\Lambda^{n,m}}{dt}(t)=F(t,\Lambda^{n,m}(t))\qquad\forall t\in[0,T],\\\
&\Lambda^{n,m}(0)=\Lambda^{n,m}_{0},\end{aligned}\right.$

where $F$ is locally Lipschitz. Therefore the Cauchy–Lipschitz Theorem implies
that, for each $m\in\mathbb{N}$, there exists a solution to Eq. 13p in
$C^{1}([0,T])$.

Step 2 (uniform estimates for the approximating solution): Testing the weak
formulations for $\\{a_{k}^{n,m},b_{k}^{n,m}\\}$ to perform the classical
$L^{2}$ parabolic estimate, we obtain the following estimates for a.e.
$t\in[0,T]$ :

(13q)
$\begin{split}\frac{1}{2}\frac{d}{dt}\|a_{0}^{n,m}\|_{L^{2}(\Omega)}^{2}&=\left\langle\partial_{t}a_{0}^{n,m},a_{0}^{n,m}\right\rangle_{\Omega}\\\
&\leq-\frac{D_{e}}{2}\|\nabla
a_{0}^{n,m}\|_{L^{2}(\Omega)}^{2}+\frac{\text{Pe}^{2}}{2D_{e}}\left(\|a_{1}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{1}^{n,m}\|_{L^{2}(\Omega)}^{2}\right),\end{split}$
(13r)
$\begin{split}\frac{1}{2}&\frac{d}{dt}\left(\|a_{1}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{1}^{n,m}\|_{L^{2}(\Omega)}^{2}\right)\\\
&=-D_{e}\left(\|\nabla a_{1}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|\nabla
b_{1}^{n,m}\|_{L^{2}(\Omega)}^{2}\right)-\left(\|a_{1}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{1}^{n,m}\|_{L^{2}(\Omega)}^{2}\right)\\\
&\quad+\frac{\text{Pe}}{2}\left\langle(1\\!-\\!(a_{0}^{n,m})_{+})_{+}{\bf
J}_{1}^{n,m},\nabla
a_{1}^{n,m}\right\rangle_{\Omega}\\!+\\!\frac{\text{Pe}}{2}\left\langle(1\\!-\\!(a_{0}^{n,m})_{+})_{+}{\bf
Q}_{1}^{n,m},\nabla b_{1}^{n,m}\right\rangle_{\Omega}\\\
&\leq\\!-\frac{3D_{e}}{4}\left(\|\nabla
a_{1}^{n,m}\|_{L^{2}(\Omega)}^{2}\\!+\\!\|\nabla
b_{1}^{n,m}\|_{L^{2}(\Omega)}^{2}\right)\\!-\\!\left(\|a_{1}^{n,m}\|_{L^{2}(\Omega)}^{2}\\!+\\!\|b_{1}^{n,m}\|_{L^{2}(\Omega)}^{2}\right)\\\
&\quad+\frac{4\text{Pe}^{2}}{D_{e}}\|a_{0}^{n,m}\|_{L^{2}}^{2}+\frac{\text{Pe}^{2}}{D_{e}}\left(\|a_{2}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{2}^{n,m}\|_{L^{2}(\Omega)}^{2}\right),\end{split}$

while, for $k\in\\{2,\dots,n-1\\}$,

(13s)
$\begin{split}&\frac{1}{2}\frac{d}{dt}\left(\|a_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\right)\\\
&\leq\\!-\frac{3D_{e}}{4}\left(\|\nabla
a_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|\nabla
b_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\right)\\!-\\!k^{2}\left(\|a_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\right)\\\
&\quad+\frac{\text{Pe}^{2}}{D_{e}}\left(\|a_{k+1}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|a_{k-1}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{k+1}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{k-1}^{n,m}\|_{L^{2}(\Omega)}^{2}\right),\end{split}$

and

(13t)
$\begin{split}\frac{1}{2}\frac{d}{dt}\left(\|a_{n}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{n}^{n,m}\|_{L^{2}(\Omega)}^{2}\right)&\leq-\frac{3D_{e}}{4}\left(\|\nabla
a_{n}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|\nabla
b_{n}^{n,m}\|_{L^{2}(\Omega)}^{2}\right)\\\
&\quad-n^{2}\left(\|a_{n}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{n}^{n,m}\|_{L^{2}(\Omega)}^{2}\right)\\\
&\quad+\frac{\text{Pe}^{2}}{2D_{e}}\left(\|a_{n-1}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{n-1}^{n,m}\|_{L^{2}(\Omega)}^{2}\right).\end{split}$

By summing up Eqs. Eq. 13q–Eq. 13t , we obtain

(13u)
$\begin{split}&\frac{1}{2}\\!\frac{d}{dt}\\!\sum_{k=0}^{n}\\!\left(\\!\|a_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\\!+\\!\|b_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\\!\right)\\!+\\!\sum_{k=0}^{n}\\!k^{2}\\!\left(\\!\|a_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\\!+\\!\|b_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\\!\right)\\\
&\\!+\\!D_{e}\\!\sum_{k=0}^{n}\\!\left(\\!\|\nabla
a_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\\!+\\!\|\nabla
b_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\\!\right)\\\ &\qquad\qquad\leq
C\frac{\text{Pe}^{2}}{D_{e}}\sum_{k=0}^{n}\left(\|a_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\right),\end{split}$

for some positive constant $C$ independent of $n,m$ and Pe. Since on the left-
hand side all the quantities but the time-derivative are positive, we also
have

$\frac{d}{dt}\\!\sum_{k=0}^{n}\\!\left(\\!\|a_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\\!+\\!\|b_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\\!\right)\\!\leq\\!2C\frac{\text{Pe}^{2}}{D_{e}}\\!\sum_{k=0}^{n}\\!\left(\\!\|a_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\\!+\\!\|b_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\\!\right),$

whence, by Grönwall’s inequality, using also Eq. 13bm and the Plancherel
Theorem,

(13v)
$\sum_{k=0}^{n}\left(\|a_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\right)\leq
C_{0,n}\exp\left(2CT\frac{\text{Pe}^{2}}{D_{e}}\right),$

where,
$C_{0,n}=\sum_{k=0}^{n}\|a_{k}(0)\|_{L^{2}(\Omega)}^{2}+\|b_{k}(0)\|_{L^{2}(\Omega)}^{2}$,
which is bounded by $\|f_{0}\|_{L^{2}(\Upsilon)}^{2}$. From Eq. 13v we deduce
that $\\{a_{k}^{n,m},b_{k}^{n,m}\\}_{k=1}^{n}$ are bounded in
$L^{\infty}([0,T];L_{per}^{2}(\Omega))$ independently of $m$. As direct
consequence, by integrating Eq. 13u in time,

(13w)
$\begin{split}\sum_{k=0}^{n}k^{2}\int_{0}^{T}\big{(}&\|a_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\big{)}\mathrm{d}{t}\\\
&+D_{e}\sum_{k=0}^{n}\int_{0}^{T}\left(\|\nabla
a_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|\nabla
b_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\right)\mathrm{d}{t}\\\ &\leq
C\frac{\text{Pe}^{2}}{D_{e}}\sum_{k=0}^{n}\int_{0}^{T}\left(\|a_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}+\|b_{k}^{n,m}\|_{L^{2}(\Omega)}^{2}\right)\mathrm{d}{t}+C_{0,n}\\\
&\leq
CT\frac{\text{Pe}^{2}}{D_{e}}C_{0,n}\exp\left(2CT\frac{\text{Pe}^{2}}{D_{e}}\right)+C_{0,n}=:C(C_{0,n},\text{Pe},D_{e},T),\end{split}$

independent of $m$, and $\\{a_{k}^{n,m},b_{k}^{n,m}\\}_{k=1}^{n}$ are
uniformly bounded in $L^{\infty}([0,T];L^{2}_{per}(\Omega))\cap
L^{2}([0,T];H^{1}_{per}(\Omega))$. Next, we show
$\|{a_{k}^{m}}^{\prime}\|_{L^{2}([0,T];(H^{1}_{per})^{\prime}(\Omega))}$,
$\|{b_{k}^{m}}^{\prime}\|_{L^{2}([0,T];(H^{1}_{per})^{\prime}(\Omega))}$, for
$0\leq k\leq n$, are uniformly bounded in $m$.

Let us consider a test function $\psi\in H^{1}_{per}(\Omega)$ such that
$\|\psi\|_{H^{1}(\Omega)}\leq 1$. Since $a_{0}^{m}$ is a weak solution to Eq.
13a for $k=0$, we have for a.e. $t\in[0,T]$

(13x)
$\begin{split}|\left\langle\partial_{t}a_{0}^{m},\psi\right\rangle_{\Omega}|&\leq
D_{e}|\left\langle\nabla
a_{0}^{m},\nabla\psi\right\rangle_{\Omega}|+|\left\langle\text{Pe}(1-(a_{0}^{m})_{+})_{+}(a_{1}^{m},b_{1}^{m}),\nabla\psi\right\rangle_{\Omega}|\\\
&\leq D_{e}\|\nabla
a_{0}^{m}\|_{L^{2}(\Omega)}\|\nabla\psi\|_{L^{2}(\Omega)}+\text{Pe}\|a_{1}^{m}\|_{L^{2}(\Omega)}\|\nabla\psi\|_{L^{2}(\Omega)}\\\
&\quad+\text{Pe}\|b_{1}^{m}\|_{L^{2}(\Omega)}\|\nabla\psi\|_{L^{2}(\Omega)}\\\
&\leq\max\left\\{\text{Pe},D_{e}\right\\}\left(\|a_{1}^{m}\|_{L^{2}(\Omega)}+\|b_{1}^{m}\|_{L^{2}(\Omega)}+\|\nabla
a_{0}^{m}\|_{L^{2}(\Omega)}\right).\end{split}$

By taking the supremum over all $\psi\in H^{1}_{per}(\Omega)$ such that
$\|\psi\|_{H^{1}(\Omega)}\leq 1$, squaring and integrating in time, we obtain

(13y)
$\begin{split}\int_{0}^{T}\|\partial_{t}a_{0}^{m}&\|_{(H^{1}_{per})^{\prime}(\Omega)}^{2}\mathrm{d}{t}\leq\bar{C}(\text{Pe},\phi,T,C_{0}),\end{split}$

since $a_{1}^{m},b_{1}^{m}$ are uniformly bounded in
$L^{\infty}([0,T];L^{2}_{per}(\Omega))\cap L^{2}([0,T];H^{1}_{per}(\Omega))$,
and the boundedness of $\|\nabla
a_{0}^{m}\|_{L^{2}([0,T];L^{2}_{per}(\Omega))}$ follows from Eq. 13w. In turn,
this gives that the sequence
$\|{a_{0}^{m}}^{\prime}\|_{L^{2}([0,T];(H^{1}_{per})^{\prime}(\Omega))}$ is
uniformly bounded in $m$. With a similar computation we can prove a uniform
bound in $m$ for
$\|{a_{k}^{m}}^{\prime}\|_{L^{2}([0,T];(H^{1}_{per})^{\prime}(\Omega))}$,
$\|{b_{k}^{m}}^{\prime}\|_{L^{2}([0,T];(H^{1}_{per})^{\prime}(\Omega))}$,
where $1\leq k\leq n$.

Step 3 (convergence of the approximating solutions): For $0\leq k\leq n$, the
sequence $\\{a_{k}^{m},b_{k}^{m}\\}_{m}$ is uniformly bounded in $m$ in
$L^{\infty}([0,T];L^{2}_{per}(\Omega))\cap L^{2}([0,T];H^{1}_{per}(\Omega))$.
The Banach–Alaoglu Theorem yields that there exist subsequences
$\\{a_{k}^{m_{l}},b_{k}^{m_{l}}\\}_{l}$ and curves $a_{k},b_{k}\in
L^{\infty}([0,T];L^{2}_{per}(\Omega))\cap L^{2}([0,T];H^{1}_{per}(\Omega))$,
for $0\leq k\leq n$, such that

(13za) $\displaystyle a_{k}^{m_{l}}\overset{\ast}{\rightharpoonup}a_{k},\quad
b_{k}^{m_{l}}\overset{\ast}{\rightharpoonup}b_{k}\quad\mbox{ in
}L^{\infty}([0,T];L^{2}_{per}(\Omega))\text{ as }m_{l}\to\infty,$ (13zb)
$\displaystyle a_{k}^{m_{l}}\rightharpoonup a_{k},\quad
b_{k}^{m_{l}}\rightharpoonup b_{k}\quad\mbox{ in
}L^{2}([0,T];H^{1}_{per}(\Omega))\text{ as }m_{l}\to\infty.$

The limits coincide since $H^{1}\subset
L^{2}\equiv(L^{2})^{\prime}\subset(H^{1})^{\prime}$ and $\varphi\in
L^{2}([0,T];L^{2}_{per}(\Omega))$ is a common test function. Since
$\|{a_{0}^{m}}^{\prime}\|$, $\|{a_{k}^{m}}^{\prime}\|$,
$\|{b_{k}^{m}}^{\prime}\|$ are bounded independently of $m$, for $1\leq k\leq
n$, in $L^{2}([0,T];(H^{1}_{per})^{\prime}(\Omega))$, the Banach–Alaoglu
Theorem implies

(13aa)
$\displaystyle{a_{k}^{m_{l}}}^{\prime}\overset{\ast}{\rightharpoonup}g_{a_{k}}\quad{b_{k}^{m_{l}}}^{\prime}\overset{\ast}{\rightharpoonup}g_{b_{k}}\quad\mbox{
in }L^{2}([0,T];(H^{1}_{per})^{\prime}(\Omega)),$

for some $g_{a_{k}},g_{b_{k}}\in L^{2}([0,T];(H^{1}_{per})^{\prime}(\Omega))$,
$0\leq k\leq n$. By testing against a smooth compactly supported test function
defined on $(0,T)$ and using the weak convergence
$a_{k}^{m_{l}}\rightharpoonup a_{k}$, $b_{k}^{m_{l}}\rightharpoonup b_{k}$ in
$L^{2}([0,T];H^{1}_{per}(\Omega))$, we get $g_{a_{k}}=a^{\prime}_{k}$,
$g_{b_{k}}=b^{\prime}_{k}$. Applying the Aubin–Lions Lemma to the sequence
$\\{a_{0}^{m}\\}_{m\in\mathbb{N}}$, due to the uniform bound of
$\|a^{\prime}_{0,m}\|_{L^{2}([0,T];(H^{1}_{per})^{\prime}(\Omega))}$ from Eq.
13y, $a_{0}^{m}$ strongly converges in $L^{2}([0,T];L^{2}_{per}(\Omega))$.

The strong convergence of $a_{0}^{m}$, Eq. 13z, and Eq. 13aa allow us to pass
to the limit in the weak form of system Eq. 13, thus to obtain existence of
weak solutions. We stress that strong convergence for $a_{0}^{m}$ is needed in
order to allow convergence through the positive part in the drift term.
Indeed, the positive part function is continuous and it grows at most
linearly, thus $(1-(a_{0}^{m})_{+})_{+}\to(1-(a_{0})_{+})_{+}$ strongly in
$L^{2}([0,T];L^{2}_{per}(\Omega))$. As consequence of the lower semicontinuity
of the norm we obtain the regularity Eq. 13n and the uniform estimates Eq.
13o.

###  3.3 Consistency of the approximating scheme

We now proceed with the proof of Theorem 2.3. We split the proof into two
parts: existence and uniqueness. The proof of existence consists of five
distinct steps.

######  Proof 3.3 (Proof of Theorem 2.3 \- existence).

First of all, let us remind the reader we consider the variant Eq. 7 of the
original problem, with periodic boundary conditions on $\Upsilon$. We will see
that we recover the solution to the original problem Eq. 2 in Step 5.

Step 1 (approximating solution): As previously outlined, for each
$n\in\mathbb{N}$, we consider the approximation of Eq. 7 given by Eq. 11 where
the functions $\\{a_{k}^{n},b_{k}^{n}\\}_{k=0}^{n}$ are a solution the
semilinear parabolic equations Eq. 13. Note that Theorem 3.1 provides
existence for the coefficients $\\{a_{k}^{n},b_{k}^{n}\\}_{k=0}^{n}$
satisfying Eq. 13n and Eq. 13o. In the next step we will use that, for each
$n\in\mathbb{N}$, there holds

(13aba)
$\displaystyle\sum_{k=0}^{n}\left(\|a_{k}^{n}(t)\|_{L^{2}(\Omega)}^{2}+\|b_{k}^{n}(t)\|_{L^{2}(\Omega)}^{2}\right)\leq
C_{0}\exp\left(2CT\frac{\text{Pe}^{2}}{D_{e}}\right),$ (13abb)
$\begin{split}&\sum_{k=0}^{n}k^{2}\int_{0}^{T}\big{(}\|a_{k}^{n}(t)\|_{L^{2}(\Omega)}^{2}\\!+\\!\|b_{k}^{n}(t)\|_{L^{2}(\Omega)}^{2}\big{)}\mathrm{d}{t}\\\
&+D_{e}\sum_{k=0}^{n}\int_{0}^{T}\\!\left(\|\nabla
a_{k}^{n}(t)\|_{L^{2}(\Omega)}^{2}+\|\nabla
b_{k}^{n}(t)\|_{L^{2}(\Omega)}^{2}\right)\mathrm{d}{t}\leq
C(C_{0},\text{Pe},D_{e},T),\end{split}$ (13abc)
$\|{a_{0}^{n}}^{\prime}\|_{L^{2}([0,T];(H^{1})^{\prime}(\Omega))}\leq\bar{C}(\text{Pe},D_{e},T,C_{0}),$

being
$C_{0}=\sum_{k=0}^{\infty}(\|a_{k}(0,\cdot)\|_{L^{2}(\Omega)}^{2}+\|b_{k}(0,\cdot)\|_{L^{2}(\Omega)}^{2})=\|f_{0}\|_{L^{2}(\Omega)}^{2}$
and $C$ a constant independent of $n$ (see Eqs. Eq. 13v-Eq. 13w in the proof
of Theorem 3.1).

Step 2 (uniform estimates for the approximating solution): The uniform bound
in $L^{\infty}([0,T];L^{2}_{per}(\Upsilon))$ for the approximation $f^{n}$ of
Eq. 11 follows directly from Eq. 13ab. Indeed, or any $t\in[0,T]$, there holds

$\begin{split}\|f^{n}(t)\|_{L^{2}(\Upsilon)}^{2}\\!\leq\\!\frac{1}{\pi}\|a_{0}^{n}(t)\|_{L^{2}(\Omega)}^{2}\\!+\\!\frac{4}{\pi}\sum_{k=1}^{n}\left(\|a_{k}^{n}(t)\|_{L^{2}(\Omega)}^{2}\\!+\\!\|b_{k}^{n}(t)\|_{L^{2}(\Omega)}^{2}\right)\\!\leq\\!\frac{C_{0}}{\pi}e^{2CT\frac{\text{Pe}^{2}}{D_{e}}}.\end{split}$

Note that in the penultimate inequality we used
$|\sin(k\theta)|,|\cos(k\theta)|\leq 1$ for any $k\in\mathbb{N}$, Fubini
Theorem and the orthogonality conditions for the trigonometric functions.
Similarly, we obtain a uniform bound for $f^{n}$ in
$L^{2}([0,T];H^{1}_{per}(\Upsilon))$. Indeed,
$\int_{0}^{T}\|\nabla_{\xi}f^{n}(t)\|_{L^{2}(\Upsilon)}^{2}\,\mathrm{d}{t}\\!=\\!\int_{0}^{T}\int_{\Upsilon}|\nabla_{{\bf
x}}f^{n}(t,{\bf x},\theta)|^{2}\,\mathrm{d}{{\bf
x}}\,\mathrm{d}{\theta}\,\mathrm{d}{t}\\!+\\!\int_{0}^{T}\int_{\Upsilon}|\partial_{\theta}f^{n}(t,{\bf
x},\theta)|^{2}\,\mathrm{d}{{\bf x}}\,\mathrm{d}{\theta}\,\mathrm{d}{t}$, from
which it follows that, with $\bm{\xi}=({\bf x},\theta)$,

$\begin{split}\int_{0}^{T}\|\nabla_{\bm{\xi}}f^{n}(t)\|_{L^{2}(\Upsilon)}^{2}\,\mathrm{d}{t}&\leq\frac{1}{\pi}\int_{0}^{T}\|\nabla
a_{0}^{n}(t)\|_{L^{2}(\Omega)}^{2}\mathrm{d}{t}\\\
&\quad+\frac{4}{\pi}\sum_{k=1}^{n}\int_{0}^{T}\left(\|\nabla
a_{k}^{n}(t)\|_{L^{2}(\Omega)}^{2}+\|\nabla
b_{k}^{n}(t)\|_{L^{2}(\Omega)}^{2}\right)\mathrm{d}{t}\\\
&\quad+\frac{2}{\pi}\sum_{k=1}^{n}k^{2}\int_{0}^{T}\left(\|a_{k}^{n}(t)\|_{L^{2}(\Omega)}^{2}+\|b_{k}^{n}(t)\|_{L^{2}(\Omega)}^{2}\right)\mathrm{d}{t}\\\
&\overset{\lx@cref{creftype~refnum}{eq:nh1-bound}}{\leq}C(C_{0},\text{Pe},D_{e},T),\end{split}$

where this final constant is independent of $n$.

Finally, we prove that
$\|(f^{n})^{\prime}\|_{L^{2}([0,T];(H^{1}_{per})^{\prime}(\Upsilon))}$ is
uniformly bounded. This will be a crucial observation in order to pass to the
limit in $n$ in the equation, by applying the Aubin–Lions Lemma. Let
$\varphi\in H^{1}_{per}(\Upsilon)$ be a test function such that
$\|\varphi\|_{H^{1}(\Upsilon)}\leq 1$. Since $f^{n}$ is a weak solution to Eq.
7, we obtain for a.e. $t\in[0,T]$

(13ac)
$\begin{split}|\left\langle\partial_{t}f^{n}(t),\varphi\right\rangle_{\Upsilon}|&\leq
D_{e}|\left\langle\nabla
f^{n}(t),\nabla\varphi\right\rangle_{\Upsilon}|+|\left\langle\partial_{\theta}f^{n}(t),\partial_{\theta}\varphi\right\rangle_{\Upsilon}|\\\
&\quad+|\left\langle\text{Pe}(1-(a_{0}^{n})_{+})_{+}f^{n}(t){\bf
e}(\theta),\nabla\varphi\right\rangle_{\Upsilon}|\\\ &\leq D_{e}\|\nabla
f^{n}(t)\|_{L^{2}(\Upsilon)}\|\nabla\varphi\|_{L^{2}(\Upsilon)}+\|\partial_{\theta}f^{n}(t)\|_{L^{2}(\Upsilon)}\|\partial_{\theta}\varphi\|_{L^{2}(\Upsilon)}\\\
&\quad+\text{Pe}\|f^{n}(t)\|_{L^{2}(\Upsilon)}\|\nabla\varphi\|_{L^{2}(\Upsilon)}\\\
&\leq\max\left\\{\text{Pe},D_{e}\right\\}\left(\|f^{n}(t)\|_{L^{2}(\Upsilon)}+\|\nabla
f^{n}(t)\|_{L^{2}(\Upsilon)}+\|\partial_{\theta}f^{n}(t)\|_{L^{2}(\Upsilon)}\right).\end{split}$

Taking the supremum over all $\varphi\in H^{1}_{per}(\Upsilon)$ with
$\|\varphi\|_{H^{1}(\Upsilon)}\leq 1$,
$\|\partial_{t}f^{n}(t)\|_{(H^{1}_{per})^{\prime}(\Upsilon)}\leq
C(\text{Pe},D_{e})\left(\|f^{n}(t)\|_{L^{2}(\Upsilon)}+\|\nabla
f^{n}(t)\|_{L^{2}(\Upsilon)}+\|\partial_{\theta}f^{n}(t)\|_{L^{2}(\Upsilon)}\right)$,
whence

(13ad)
$\begin{split}\int_{0}^{T}\|\partial_{t}f^{n}(t)\|_{(H^{1}_{per})^{\prime}(\Upsilon)}^{2}\mathrm{d}{t}&\leq\tilde{C}(\text{Pe},D_{e})\int_{0}^{T}\|f^{n}(t)\|_{H^{1}(\Upsilon)}^{2}\mathrm{d}{t}\leq\bar{C}(\text{Pe},\phi,T,C_{0}),\end{split}$

due to the uniform bound on $\|f^{n}\|_{L^{2}([0,T];H^{1}_{per}(\Upsilon))}$.
Therefore, we have the desired bound for $(f^{n})^{\prime}$ in
$L^{2}([0,T];(H^{1}_{per})^{\prime}(\Upsilon))$.

Step 3 (convergence of the approximation): Since the sequence
$\\{f^{n}\\}_{n\in\mathbb{N}}$ of Eq. 11 is uniformly bounded in
$L^{\infty}([0,T];L^{2}_{per}(\Upsilon))\cap
L^{2}([0,T];H^{1}_{per}(\Upsilon))$ we infer from the Banach–Alaoglu Theorem
that there exist a subsequence $\\{f^{n_{k}}\\}_{k\in\mathbb{N}}$ and a curve
$f\in L^{\infty}([0,T];L^{2}_{per}(\Upsilon))\cap
L^{2}([0,T];H^{1}_{per}(\Upsilon))$ such that

(13ae) $\begin{split}&f^{n_{k}}\overset{\ast}{\rightharpoonup}f\quad\mbox{ in
}L^{\infty}([0,T];L^{2}_{per}(\Upsilon)),\quad f^{n_{k}}\rightharpoonup
f\quad\mbox{ in }L^{2}([0,T];H^{1}_{per}(\Upsilon)).\end{split}$

The limits coincide since $H^{1}\subset
L^{2}\equiv(L^{2})^{\prime}\subset(H^{1})^{\prime}$ and any $\varphi\in
L^{2}([0,T];L^{2}_{per}(\Upsilon))$ is a common test function. The uniform
boundedness of
$\|(f^{n})^{\prime}\|_{L^{2}([0,T];(H^{1}_{per})^{\prime}(\Upsilon))}$ and the
Banach–Alaoglu Theorem also yields that

(13af)
$\displaystyle(f^{n_{k}})^{\prime}\overset{\ast}{\rightharpoonup}f^{\prime}\quad\mbox{
in }L^{2}([0,T];(H^{1}_{per})^{\prime}(\Upsilon)),$

where we verify that this latter limit is indeed $f^{\prime}$ by testing
against a smooth compactly supported test function defined on $(0,T)$ and
using the weak convergence $f^{n_{k}}\rightharpoonup f$ in
$L^{2}([0,T];H^{1}_{per}(\Upsilon))$. Henceforth, we do not relabel further
subsequences.

Additionally, the aforementioned boundedness of
$\|(f^{n})^{\prime}\|_{L^{2}([0,T];(H^{1}_{per})^{\prime}(\Upsilon))}$ yields,
taking a further subsequence if necessary, by an application of the
Aubin–Lions Lemma that $f^{n}\to f$ strongly in
$L^{2}([0,T];L^{2}_{per}(\Upsilon))$ as $n\to\infty$. We take a further
subsequence such that, for a.e. $t\in[0,T]$, $f^{n}(t)\to f(t)$ strongly in
$L^{2}_{per}(\Upsilon)$ as $n\to\infty$. Moreover, by further applying the
theorem of Banach-Alaoglu and the Aubin–Lions Lemma to the sequence of curves
$\\{a_{0}^{n}\\}_{n\in\mathbb{N}}$, due to the uniform bounds from Eq. 13ab,
we also know that $a_{0}^{n}$ strongly converges in
$L^{2}([0,T];L^{2}_{per}(\Omega))$. In other words, up to a subsequence,
$\\{\rho^{n}\\}_{n\in\mathbb{N}}$ converges strongly in
$L^{2}([0,T];L^{2}_{per}(\Omega))$. Moreover, defining
$\rho:=\int_{0}^{2\pi}f\,\mathrm{d}\theta$, where $f$ is the aforementioned
strong limit in $L^{2}([0,T];L^{2}_{per}(\Omega))$, we have, for the
convergent subsequence in question,

$\displaystyle\|\rho-\rho^{n}\|_{L^{2}([0,T];L^{2}(\Omega))}^{2}$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\bigg{|}\int_{0}^{2\pi}\big{(}f(t,{\bf
x},\theta)-f^{n}(t,{\bf
x},\theta)\big{)}\,\mathrm{d}\theta\bigg{|}^{2}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\leq 2\pi\|f-f^{n}\|^{2}_{L^{2}([0,T];L^{2}(\Upsilon))}\to
0\qquad\text{as }n\to\infty,$

where we applied Jensen’s inequality, thus $\rho^{n}\to\rho$ strongly in
$L^{2}([0,T];L^{2}_{per}(\Omega))$.

This allows us to pass to the limit in the weak form of Eq. 7, thus to obtain
that $f$ is a weak solution to the revised Eq. 7. We stress that strong
convergence for $a_{0}^{n}$, i.e., $\\{\rho^{n}\\}_{n\in\mathbb{N}}$, is
needed in order to allow convergence through the positive part in the drift
term. Indeed, the positive part function is continuous and it grows at most
linearly, thus $(1-(\rho^{n})_{+})_{+}\to(1-(\rho)_{+})_{+}$ strongly in
$L^{2}([0,T];L^{2}_{per}(\Omega))$. More precisely, for a function $\psi\in
C^{1}([0,T])$ such that $\psi(0)=\psi(T)=0$, there holds (using, e.g., [10,
App. E Th. 8])
$\int_{0}^{T}\langle\partial_{t}f^{n}(t),\varphi\rangle_{\Upsilon}\psi(t)\,\mathrm{d}t=-\int_{0}^{T}\langle
f^{n}(t),\varphi\rangle_{\Upsilon}\psi^{\prime}(t)\,\mathrm{d}t$, and hence

$\displaystyle\int_{0}^{T}\\!\langle
f^{n}(t),\\!\varphi\rangle_{\Upsilon}\psi^{\prime}(t)\mathrm{d}{t}\\!=\\!$
$\displaystyle-\\!\int_{0}^{T}\\!\left\langle\text{Pe}(1\\!-\\!(a_{0}^{n}(t))_{+}\\!)_{+}\\!f^{n}(t){\bf
e}(\theta),\\!\nabla\varphi\right\rangle_{\Upsilon}\\!\psi(t)\mathrm{d}{t}$
$\displaystyle+\\!D_{e}\\!\int_{0}^{T}\\!\left\langle\nabla
f^{n}(t),\\!\nabla\varphi\right\rangle_{\Upsilon}\\!\psi(t)\mathrm{d}{t}\\!+\\!\int_{0}^{T}\\!\left\langle\partial_{\theta}f^{n}(t),\\!\partial_{\theta}\varphi\right\rangle_{\Upsilon}\\!\psi(t)\mathrm{d}{t}.$

Using Eq. 13ae-Eq. 13af, the aforementioned strong convergence
$(1-(\rho^{n})_{+})_{+}\to(1-(\rho)_{+})_{+}$ in
$L^{2}(0,T;L^{2}_{per}(\Omega))$, we pass to the limit $n\to+\infty$ in the
above and obtain, after another integration by parts in $t$,

$\displaystyle\int_{0}^{T}\\!\left\langle\partial_{t}f(t),\varphi\right\rangle_{\Upsilon}\psi(t)\mathrm{d}{t}$
$\displaystyle\\!=\\!\int_{0}^{T}\left\langle\text{Pe}(1-(a_{0}(t))_{+})_{+}f(t){\bf
e}(\theta),\nabla\varphi\right\rangle_{\Upsilon}\psi(t)\mathrm{d}{t}$
$\displaystyle\quad-D_{e}\int_{0}^{T}\left\langle\nabla
f(t),\nabla\varphi\right\rangle_{\Upsilon}\psi(t)\mathrm{d}{t}-\int_{0}^{T}\left\langle\partial_{\theta}f(t),\partial_{\theta}\varphi\right\rangle_{\Upsilon}\psi(t)\mathrm{d}{t}.$

Hence, since $\psi$ was arbitrary, $f\in L^{2}([0,T];H^{1}_{per}(\Upsilon))$
with $f^{\prime}\in L^{2}([0,T];(H^{1}_{per})^{\prime}(\Upsilon))$ solves Eq.
7 weakly (i.e. duality with $H^{1}_{per}(\Upsilon)$ for a.e. $t\in[0,T]$).

Step 4 (initial data): Recall that $f_{0}$ has Fourier expansion Eq. 8 and
$f^{n}(0,\cdot,\cdot)\to f_{0}$ strongly in $L^{2}_{per}(\Upsilon)$. Moreover,
by extracting a subsequence after applying the Aubin–Lions Lemma under Eq.
13af, $f^{n}(t)\to f(t)$ strongly in $L^{2}_{per}(\Upsilon)$ for a.e.
$t\in[0,T]$. By Remark 2.4, $f^{n},f\in C([0,T];L^{2}_{per}(\Upsilon))$.
Hence, $t=0$ is a Lebesgue point, and $f^{n}(0)\to f(0)$ strongly in
$L^{2}_{per}(\Upsilon)$. By uniqueness of limits, $f(0)=f_{0}$ in
$L^{2}_{per}(\Upsilon)$.

Step 5 (solution to the original equation and nonnegativity): As direct
consequence of the previous step, by integrating the equation Eq. 7 in the
angle variable, we infer that $\rho(t,{\bf x})=\int_{0}^{2\pi}f({\bf
x},t,\theta)\mathrm{d}{\theta}$ is a curve belonging to
$L^{\infty}([0,T];L^{2}_{per}(\Omega))\cap L^{2}([0,T];H^{1}_{per}(\Omega))$,
and is a weak solution of

(13ag) $\partial_{t}\rho+\text{Pe}\nabla\cdot((1-(\rho)_{+})_{+}{\bf
p})=D_{e}\Delta\rho.$

Moreover, by the weak-* lower semicontinuity of the norm, $\rho^{\prime}\in
L^{2}([0,T];(H^{1}_{per})^{\prime}(\Omega))$ as it is the weak-* limit curve
of the sequence $\\{{a_{0}^{n}}^{\prime}\\}_{n\in\mathbb{N}}$. Begin by noting
that $(1-(\rho)_{+})_{-}=-(1-\rho\mathds{1}_{\rho>0})\mathds{1}_{\rho>1}$,
which implies that, in the sense of distributions,
$\partial_{t}(1-(\rho)_{+})_{-}=\partial_{t}\rho\mathds{1}_{\rho>1}$ and
$\nabla(1-(\rho)_{+})_{-}=\nabla\rho\mathds{1}_{\rho>1}$. Testing Eq. 13ag
with $(1-(\rho)_{+})_{-}$,

$\frac{1}{2}\frac{d}{dt}\int_{\Omega}(1-(\rho)_{+})_{-}^{2}\,\mathrm{d}{\bf
x}=\text{Pe}\int_{\Omega}\nabla(1-(\rho)_{+})_{-}\cdot(1-(\rho)_{+})_{+}{\bf
p}\,\mathrm{d}{\bf
x}-D_{e}\int_{\Omega}|\nabla\rho|^{2}\mathds{1}_{\rho>1}\,\mathrm{d}{\bf x}.$

Observe that the first term on the right-hand side is null, since the supports
of the two terms in the integrand are disjoint, while the final term is
nonpositive. We thereby deduce
$\frac{1}{2}\int_{\Omega}(1-(\rho(t))_{+})_{-}^{2}\,\mathrm{d}{\bf x}\leq 0$,
since we initially have $(1-\rho_{0}({\bf x}))_{-}=0$ for a.e. $x\in\Omega$.
As a result, $\rho(t,{\bf x})\leq 1$ for a.e. $(t,{\bf
x})\in(0,T)\times\Omega$, as required.

Returning to Eq. 7 and testing with $(f)_{-}$, we get, after integrating by
parts,

$\displaystyle\frac{1}{2}\\!\frac{d}{dt}\\!\int_{\Upsilon}\\!(f)_{-}^{2}\mathrm{d}\bm{\xi}$
$\displaystyle=\\!\text{Pe}\\!\int_{\Upsilon}\\!\nabla(\\!f\\!)_{-}\\!\cdot\\!(\\!f\\!)_{-}(1\\!-\\!(\rho)_{+}\\!)_{+}{\bf
e}(\theta)\mathrm{d}\bm{\xi}\\!-\\!D_{e}\int_{\Upsilon}|\nabla(\\!f\\!)_{-}|^{2}\,\mathrm{d}\bm{\xi}\\!-\\!\int_{\Upsilon}\\!|\partial_{\theta}(\\!f\\!)_{-}|^{2}\mathrm{d}\bm{\xi}$
$\displaystyle\leq
C(\text{Pe},D_{e})\int_{\Upsilon}\frac{1}{2}(f)_{-}^{2}\,\mathrm{d}\bm{\xi}-\frac{1}{2}D_{e}\int_{\Upsilon}|\nabla(f)_{-}|^{2}\,\mathrm{d}\bm{\xi}-\int_{\Upsilon}|\partial_{\theta}(f)_{-}|^{2}\,\mathrm{d}\bm{\xi},$

where $\bm{\xi}=(\bm{{\bf x}},\theta)$. Dropping the two nonpositive terms,
Grönwall’s Lemma implies

$\int_{\Upsilon}(f(t,{\bf
x},\theta))_{-}^{2}\,\mathrm{d}\bm{\xi}\leq\exp(Ct)\int_{\Upsilon}(f_{0}({\bf
x},\theta))_{-}^{2}\,\mathrm{d}\bm{\xi}=0\qquad\text{a.e.}~{}t\in(0,T),$

since $f_{0}$ is nonnegative a.e. on $\Upsilon$. Hence, $f(t,{\bf
x},\theta)\geq 0$ for a.e. $(t,{\bf x},\theta)\in\Upsilon\times(0,T)$ and thus
the space density $\rho(t,{\bf x})\in[0,1]$ for a.e. $(t,{\bf
x})\in\Upsilon\times(0,T)$. Hence the limiting curve $f$ is a weak solution in
the sense of Definition 2.2 to the original equation Eq. 2. We emphasise that
$f\in C([0,T];L^{2}_{per}(\Upsilon))$ is deduced from $f\in
L^{2}([0,T];H^{1}_{per}(\Upsilon))$ and $f^{\prime}\in
L^{2}([0,T];(H^{1}_{per})^{\prime}(\Upsilon))$; cf. Remark 2.4.

###  3.4 Uniqueness of solutions

######  Lemma 3.4.

Given $f_{0}\in L^{2}_{per}(\Upsilon)$ nonnegative such that its corresponding
$\rho_{0}$ satisfies $\rho_{0}({\bf x})=\int_{0}^{2\pi}f_{0}({\bf
x},\theta)\,\mathrm{d}\theta\in[0,1]$ for a.e. ${\bf x}\in\Omega$, there
exists at most one periodic weak solution, in the sense of Definition 2.2, of
the problem Eq. 6.

######  Proof 3.5.

Let us consider two weak solutions to Eq. 2, $f_{1},f_{2}$, in the sense of
Definition 2.2, and set $\bar{f}:=f_{1}-f_{2}$ to be their difference. We
refer to $\rho_{1},\rho_{2}$ as the corresponding space densities, and to
$\bar{\rho}:=\rho_{1}-\rho_{2}$ as their difference. By means of a direct
computation, $\bar{f}$ is a weak solution to

$\displaystyle\partial_{t}\bar{f}+\text{Pe}\nabla\cdot((1-\rho_{1})\bar{f}{\bf
e}(\theta)-\bar{\rho}f_{2}{\bf
e}(\theta))=D_{e}\Delta\bar{f}+\partial_{\theta}^{2}\bar{f},$

and $\bar{\rho}$ solves
$\partial_{t}\bar{\rho}+\text{Pe}\nabla\cdot((1-\rho_{1})\bar{{\bf
p}}-\bar{\rho}{\bf p}_{2})=D_{e}\Delta\bar{\rho}$, where we remind the reader
$\bar{{\bf p}}(t,{\bf x})=\int_{0}^{2\pi}\bar{f}(t,{\bf
x},\theta)\mathrm{d}{\theta}$ and ${\bf p}_{2}(t,{\bf
x})=\int_{0}^{2\pi}f_{2}(t,{\bf x},\theta)\mathrm{d}{\theta}$, for any
$(t,{\bf x},\theta)\in[0,T]\times\Upsilon$. Note that, by directly integrating
the previous equation for $\bar{\rho}$, $\int_{\Omega}\bar{\rho}({\bf
x})\mathrm{d}{\bf x}=0$, and thus $\int_{\Omega}\int_{0}^{2\pi}\bar{f}(t,{\bf
x},\theta)\,\mathrm{d}\theta\,\mathrm{d}{\bf x}=0$. Motivated by this latter
“mean-zero condition”, we consider the space

$H^{1}_{z}(\Upsilon):=\left\\{f\in
H^{1}(\Upsilon):\int_{\Upsilon}f(\xi)\mathrm{d}\xi=0\right\\},$

and its dual $(H_{z}^{1})^{\prime}(\Upsilon)$, where subscript $z$ refers to
the mean-zero property in question. Let
$L:=(-\Delta)^{-1}:(H^{1}_{z})^{\prime}(\Upsilon)\to(H^{1}_{z})(\Upsilon)$ be
the inverse of the solution operator for the Poisson equation with periodic
boundary condition, that is $Lg=u$, where $\int_{\Upsilon}\nabla
u\cdot\nabla\varphi=\left\langle g,\varphi\right\rangle$ for all $\varphi\in
H^{1}_{z}(\Upsilon)$. In the following we shall use that $L$ maps continuously
$L^{2}_{z}(\Upsilon)$ into $H^{2}_{z}(\Upsilon)$. Moreover, we note the self-
adjointness property

$\left\langle\varphi,L\varphi\right\rangle=\left\langle(-\Delta)\circ(-\Delta)^{-1}\varphi,L\varphi\right\rangle=\|\nabla
L\varphi\|_{L^{2}(\Upsilon)}^{2}=\langle L\varphi,\varphi\rangle,$

for every $\varphi\in H^{1}_{z}(\Upsilon)$, where the duality product is
understood in the sense of $L^{2}(\Upsilon)$. Similarly, using the
commutativity of the partial derivatives, for any $\varphi\in
H^{1}_{z}(\Upsilon)$,

$\langle\partial_{\theta}\varphi,\partial_{\theta}L\varphi\rangle=\langle(-\Delta)\partial_{\theta}L\varphi,\partial_{\theta}L\varphi\rangle=\|\nabla\partial_{\theta}L\varphi\|^{2}_{L^{2}(\Upsilon)}.$

Let $\lambda>0$. For the next lines of computation, we use Definition 2.2,
Young’s inequality, the bound $\rho_{1}\leq 1$, and the self-adjointness
property. We drop the time dependence for ease of presentation. We obtain

(13ah)
$\begin{split}\frac{1}{2}\\!\frac{\mathrm{d}}{\mathrm{d}{t}}\\!\left(\\!\|\nabla
L\bar{f}\|_{L^{2}(\Upsilon)}^{2}\\!+\\!\lambda\|\bar{\rho}\|_{L^{2}(\Omega)}^{2}\\!\right)&\\!=\left\langle\partial_{t}\bar{f},L\bar{f}\right\rangle+\lambda\left\langle\partial_{t}\bar{\rho},\bar{\rho}\right\rangle\\\
&\\!=\\!-D_{e}\\!\|\bar{f}\|_{L^{2}(\Upsilon)}^{2}\\!-\\!\|\nabla\partial_{\theta}L\bar{f}\|_{L^{2}(\Upsilon)}^{2}\\!-\\!\lambda\\!D_{e}\\!\|\nabla\bar{\rho}\|_{L^{2}}^{2}\\\
&\quad\\!+\\!\text{Pe}\left\langle(1-\rho_{1})\bar{f}{\bf e}(\theta),\nabla
L\bar{f}\right\rangle\\!-\\!\text{Pe}\lambda\\!\left\langle\bar{\rho}{\bf
p}_{2},\\!\nabla\bar{\rho}\right\rangle\\\
&\quad+\\!\text{Pe}\lambda\left\langle(1\\!-\\!\rho_{1})\bar{{\bf
p}},\nabla\bar{\rho}\right\rangle\\!-\\!\text{Pe}\\!\left\langle\bar{\rho}f_{2}{\bf
e}(\theta),\\!\nabla L\bar{f}\right\rangle\\\
&\\!\leq\\!-D_{e}\left(1-\frac{1}{2\varepsilon}-\frac{\lambda}{\sigma}\pi\right)\|\bar{f}\|_{L^{2}(\Upsilon)}^{2}\\\
&\quad-\\!\lambda\\!\left(\\!D_{e}\\!-\\!\frac{\text{Pe}^{2}\sigma}{D_{e}}\right)\\!\|\nabla\bar{\rho}\|_{L^{2}(\Omega)}^{2}\\!+\\!\frac{\lambda
D_{e}}{2\sigma}\|\bar{\rho}\|_{L^{2}(\Omega)}^{2}\\\
&\quad+\frac{\text{Pe}^{2}\varepsilon}{2D_{e}}\|\nabla
L\bar{f}\|_{L^{2}(\Upsilon)}^{2}\\!-\\!\text{Pe}\\!\left\langle\bar{\rho}f_{2}{\bf
e}(\theta),\\!\nabla L\bar{f}\right\rangle,\end{split}$

where we also used $\|\bar{{\bf p}}\|_{L^{2}(\Omega)}^{2}\leq
2\pi\|\bar{f}\|_{L^{2}(\Upsilon)}^{2}$, and noticed that for any $(t,{\bf
x})\in\Omega\times[0,T]$, $|{\bf p}_{2}(t,{\bf
x})|\leq\int_{0}^{2\pi}f_{2}(t,{\bf
x},\theta)\mathrm{d}{\theta}=\rho_{2}(t,{\bf x})\leq 1$, which implies
$\|\bar{\rho}{\bf
p}_{2}\|_{L^{2}(\Omega)}^{2}\leq\|\bar{\rho}\|_{L^{2}(\Omega)}^{2}$. In order
to apply the Grönwall inequality and conclude the argument we must estimate
the final term in Eq. 13ah. Since the operator $L$ maps continuously
$L^{2}_{z}(\Upsilon)$ into $H^{2}_{z}(\Upsilon)$, we have $L\bar{f}\in
H^{2}_{z}(\Upsilon)$, thus $\nabla L\bar{f}\in H^{1}_{z}(\Upsilon)$. By means
of the Sobolev embedding $H^{1}(\Upsilon)\hookrightarrow L^{p_{*}}(\Upsilon)$
for $p_{*}=6$, and that $|\Upsilon|<+\infty$, we infer that $\nabla
L\bar{f}\in L^{p^{\prime}}(\Upsilon)$ for any $p^{\prime}\leq p_{*}=6$, and
$f_{2}\in L^{q}(\Upsilon)$ for any $q\leq p_{*}=6$. In particular, for
$\frac{6}{5}\leq p\leq\frac{3}{2}$ and $q=\frac{2p}{2-p}\leq 6$ , using the
Cauchy–Young inequality to get from the second line to the third, and the
Gagliardo–Nirenberg inequality to get the final line, we see that the term
$\text{Pe}\left|\left\langle\bar{\rho}f_{2}{\bf e}(\theta),\nabla
L\bar{f}\right\rangle\right|$ is bounded by

(13ai) $\begin{split}\text{Pe}\|\bar{\rho}f_{2}\|_{L^{p}(\Upsilon)}\|\nabla
L\bar{f}\|_{L^{p^{\prime}}(\Upsilon)}&\leq
C(\text{Pe},|\Upsilon|,p^{\prime},p_{*})\|\bar{f}\|_{L^{2}(\Upsilon)}\|\bar{\rho}f_{2}\|_{L^{p}(\Upsilon)}\\\
&\leq\frac{\lambda
D_{e}}{2}\|\bar{f}\|_{L^{2}(\Upsilon)}^{2}+\frac{\tilde{c}}{\lambda
D_{e}}\|f_{2}\|_{H^{1}(\Upsilon)}^{2}\|\bar{\rho}\|_{L^{2}(\Omega)}^{2},\end{split}$

where the constants $\lambda,\tilde{c}$ depend only on $\Upsilon,p,q$ via the
embeddings. Using Eq. 13ai in Eq. 13ah, we get

$\begin{split}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}{t}}\left(\|\nabla
L\bar{f}\|_{L^{2}(\Upsilon)}^{2}+\lambda\|\bar{\rho}\|_{L^{2}(\Omega)}^{2}\right)\\\
&\qquad\leq-
D_{e}\left(1-\frac{\lambda}{2}-\frac{1}{2\varepsilon}-\frac{\lambda}{\sigma}\pi\right)\|\bar{f}\|_{L^{2}(\Upsilon)}^{2}-\lambda\left(D_{e}-\frac{\text{Pe}^{2}\sigma}{D_{e}}\right)\|\nabla\bar{\rho}\|_{L^{2}(\Omega)}^{2}\\\
&\qquad\quad+\frac{\text{Pe}^{2}\varepsilon}{2D_{e}}\|\nabla
L\bar{f}\|_{L^{2}(\Upsilon)}^{2}+\frac{\lambda
D_{e}}{2\sigma}\|\bar{\rho}\|_{L^{2}(\Omega)}^{2}+\frac{\tilde{c}}{\lambda
D_{e}}\|f_{2}\|_{H^{1}(\Upsilon)}^{2}\|\bar{\rho}\|_{L^{2}(\Omega)}^{2}\\\
&\qquad\leq\frac{M(f_{2})}{\lambda^{2}\sigma}\left(\|\nabla
L\bar{f}\|_{L^{2}(\Upsilon)}^{2}+\lambda\|\bar{\rho}\|_{L^{2}(\Omega)}^{2}\right),\end{split}$

where we set
$\sigma=\frac{D_{e}^{2}}{\text{Pe}^{2}}$,$\lambda=({1+\frac{\text{Pe}^{2}}{D_{e}^{2}}\pi})^{-1}$,
$\varepsilon=\frac{1}{\lambda}$, and
$M(f_{2}):=\max\\{\frac{\text{Pe}^{2}\lambda\sigma}{2D_{e}},\frac{\lambda^{2}D_{e}}{2}+\frac{\tilde{c}\sigma}{D_{e}}\|f_{2}\|_{H^{1}(\Upsilon)}^{2}\\}$.
Finally, since $f_{2}\in L^{2}([0,T];H^{1}_{per}(\Upsilon))$, by means of
Grönwall inequality we obtain uniqueness, as for any $t\in[0,T]$

$\|\nabla
L\bar{f}(t)\|_{L^{2}(\Upsilon)}^{2}+\lambda\|\bar{\rho}(t)\|_{L^{2}(\Omega)}^{2}\leq\underbrace{\left(\|\nabla
L\bar{f}(0)\|_{L^{2}(\Upsilon)}^{2}+\lambda\|\bar{\rho}(0)\|_{L^{2}(\Omega)}^{2}\right)}_{=0}e^{2\int_{0}^{t}\frac{M(f_{2}(s))}{\lambda^{2}\sigma}\mathrm{d}{s}}.$

###  3.5 Existence and uniqueness of solutions to the 1D Model

The analysis carried out previously for Eq. 2 can be applied to obtain well-
posedness for the 1D model Eq. 4. Therefore, we will not provide proofs but
only point out the main differences.

Let $f_{L}(x,t)$ and $f_{R}(x,t)$ be the densities of the left- and right-
moving particles, respectively. The (rescaled) one dimensional version of Eq.
2 is

(13aj)
$\begin{cases}\partial_{t}f_{R}+\text{Pe}\partial_{x}(f_{R}(1-\rho))=\partial_{xx}f_{R}+f_{L}-f_{R}\\\
\partial_{t}f_{L}-\text{Pe}\partial_{x}(f_{L}(1-\rho))=\partial_{xx}f_{L}+f_{R}-f_{L}.\end{cases}$

The _space density_ and the _polarisation_ are, respectively,

$\rho:=f_{R}+f_{L},\qquad p:=f_{R}-f_{L}.$

######  Definition 3.6 (Weak solution to 1D model).

Let $f_{R,0},f_{L,0}\in L^{2}_{per}([0,2\pi])$ be nonnegative functions such
that

$\rho_{0}(x)=f_{R,0}(x)+f_{L,0}(x)\in[0,1],\qquad\mbox{for a.e }x\in[0,2\pi].$

A _weak solution_ to system Eq. 13aj is a pair of curves $(f_{R},f_{L})$ such
that $f_{R},f_{L}\in L^{2}([0,T];H^{1}_{per}([0,2\pi]))$ and
$f^{\prime}_{R},f^{\prime}_{L}\in
L^{2}([0,T];(H^{1}_{per})^{\prime}([0,2\pi]))$, and for a.e. $t\in[0,T]$ and
any $\varphi,\phi\in H^{1}_{per}([0,2\pi])$ it holds

$\begin{split}&\langle\partial_{t}f_{R}(t),\varphi\rangle_{\Omega}\\!=\\!\text{Pe}\langle
f_{R}(t)(1\\!-\\!\rho(t)),\partial_{x}\varphi\rangle_{\Omega}\\!-\\!\langle\partial_{x}f_{R}(t),\partial_{x}\varphi\rangle_{\Omega}\\!+\langle
f_{L}(t)\\!-\\!f_{R}(t),\varphi\rangle_{\Omega}\\\
&\langle\partial_{t}f_{L}(t),\phi\rangle_{\Omega}\\!=\\!-\text{Pe}\langle
f_{L}(t)(1\\!-\\!\rho(t)),\partial_{x}\phi\rangle_{\Omega}\\!-\\!\langle\partial_{x}f_{L}(t),\partial_{x}\phi\rangle_{\Omega}\\!+\langle
f_{R}(t)\\!-\\!f_{L}(t),\phi\rangle_{\Omega},\\\
&(f_{R},f_{L})(x,0)=(f_{R,0},f_{L,0}),\end{split}$

with periodic boundary conditions on $\Omega$, where
$\rho(t,x)=f_{R}(t,x)+f_{L}(t,x)$, and the initial data is achieved in the
sense $f_{R}(0)=f_{R,0}$ and $f_{L}(0)=f_{L,0}$ in $L^{2}_{per}([0,2\pi])$.

Notice that adding/subracting the equations in system Eq. 13aj yields

(13ak)
$\begin{cases}\partial_{t}\rho+\text{Pe}\partial_{x}(p(1-\rho))=\partial_{xx}\rho\\\
\partial_{t}p+\text{Pe}\partial_{x}(\rho(1-\rho))=\partial_{xx}p-2p.\end{cases}$

As already mentioned, existence and uniqueness of solutions to Eq. 13aj can be
proved with a strategy similar to that used for Eq. 2. The main difference is
the absence of the angular variable, which simplifies the discretisation of
the system; indeed the angular variable is already discretised, as there are
only two possible directions (left and right). It therefore suffices to write
a Galerkin approximation only in space, as in Section 3.2. Moreover, the
solution $(f_{R},f_{L})$ directly inherits the bound of the density $\rho$
(unlike in the higher dimensional case). This simplifies the uniqueness proof,
since we need not employ the operator $(-\Delta)^{-1}$. The result is stated
below.

######  Theorem 3.7 (Well-posedness for 1D model).

Let $(f_{R,0},f_{L,0})\in(L^{2}_{per}([0,2\pi]))^{2}$ be a pair of nonnegative
functions such that

$\rho_{0}(x)=f_{R,0}(x)+f_{L,0}(x)\in[0,1]\quad\mbox{for a.e. }x\in[0,2\pi].$

For any $T>0$, there exists a unique weak solution to Eq. 13aj in the sense of
Definition 3.6. Moreover, $f_{R}(t,x),f_{L}(t,x)\in[0,1]$ for a.e. $t\in[0,T]$
and $x\in[0,2\pi]$.

######  Remark 3.8.

As per Remark 2.4, we deduce $f_{R},f_{L}\in C([0,T];L^{2}_{per}([0,2\pi]))$.

##  4 Regularity

Having established well-posedness in Section 3, we turn to the issue of
regularity. More precisely, we provide here the proof of Theorem 2.5, which is
performed in Lemmas 4.10 and 4.12. For ease of presentation we start with the
one-dimensional model.

###  4.1 Regularity for the 1D model

Our strategy is to apply the Duhamel principle so as to treat Eq. 13aj as a
forced one-dimensional heat equation. Inspired by [6, Chapter 9], we make the
following definition.

######  Definition 4.1.

Define $\Phi:(0,\infty)\times\mathbb{R}\to\mathbb{R}$ by the explicit formula

(13al)
$\Phi(t,x):=\frac{1}{2\pi}+\frac{1}{\pi}\sum_{n=1}^{\infty}e^{-n^{2}t}\cos
nx\qquad\forall(t,x)\in(0,\infty)\times\mathbb{R}.$

Note that the above may be rewritten in terms of the complex exponential basis
as
$\Phi(t,x)=\frac{1}{2\pi}+\sum_{n\in\mathbb{Z}\setminus\\{0\\}}\frac{e^{-n^{2}t}}{2\pi}e^{inx}$
for all $(t,x)\in(0,\infty)\times\mathbb{R}$. In what follows it will be
useful to rewrite $\Phi$ in terms of the Jacobi theta-3 function (denoted by
$\theta_{3}$), and in terms of the Jacobi theta function (denoted by
$\vartheta$, also sometimes by $\vartheta_{00}$),

(13am)
$\Phi(t,x)=\frac{1}{2\pi}\theta_{3}\left(\frac{x}{2},e^{-t}\right)=\frac{1}{2\pi}\vartheta\left(\frac{x}{2\pi},\frac{it}{\pi}\right)\qquad\forall(t,x)\in(0,\infty)\times\mathbb{R}.$

######  Lemma 4.2.

The function $\Phi$ of Definition 4.1 is a smooth $2\pi$-periodic function
that satisfies $\partial_{t}\Phi-\partial_{xx}\Phi=0$ as a pointwise equality
in $(0,\infty)\times\mathbb{R}$, is nonnegative in
$(0,\infty)\times\mathbb{R}$, and $\|\Phi(t,\cdot)\|_{L^{1}([0,2\pi])}=1$ for
every $t\in(0,\infty)$. Additionally,

(13an)
$\int_{0}^{t}\|\Phi(\tau,\cdot)\|^{2}_{L^{2}([0,2\pi])}\,\mathrm{d}\tau=t+\sum_{n=1}^{\infty}\frac{1}{2n^{2}}(1-e^{-2n^{2}t})\qquad\forall
t>0,$

and
$\int_{0}^{t}\int_{0}^{2\pi}\Phi(t-s,x-y)^{2}\,\mathrm{d}y\,\mathrm{d}s=\int_{0}^{t}\|\Phi(\tau,\cdot)\|^{2}_{L^{2}([0,2\pi])}\,\mathrm{d}\tau$
for all $(t,x)\in(0,T]\times[0,2\pi]$. Finally, for any $2\pi$-periodic
$C^{2}$ function $\psi$, $\lim_{t\to
0^{+}}\int_{0}^{2\pi}\Phi(t,x)\psi(x)\,\mathrm{d}x=\psi(0)$.

The proof of Lemma 4.2 is in Appendix B. We use the function $\Phi$ as a
periodic heat kernel, in the sense made precise by the following result. The
initial data $\psi$ is chosen in $L^{1}_{loc}(\mathbb{R})$ to justify the
expansion in terms of the sine/cosine basis.

######  Lemma 4.3.

Let $\psi\in L^{1}_{per}([0,2\pi])$. Then,
$\Psi(t,x):=\int_{0}^{2\pi}\Phi(t,x-y)\psi(y)\,\mathrm{d}y$ for all
$(t,x)\in(0,\infty)\times\mathbb{R}$ is a smooth $2\pi$-periodic function
defined on the full-space $(0,\infty)\times\mathbb{R}$, which solves the
Cauchy problem

(13ao) $\left\\{\begin{aligned}
&\partial_{t}\Psi-\partial_{xx}\Psi=0\qquad&&\text{in
}(0,\infty)\times\mathbb{R},\\\ &\Psi|_{t=0}=\psi\qquad&&\text{on
}\mathbb{R},\end{aligned}\right.$

where the initial data is achieved in $L^{1}_{loc}$, i.e., $\lim_{t\to
0^{+}}\|\Psi(t,\cdot)-\psi\|_{L^{1}([0,2\pi])}=0$.

The proof of this result is in Appendix B. An alternative way of generating
periodic solutions of the heat equation is to convolve the classical one-
dimensional heat kernel with $2\pi$-periodic test functions. However, when
performing integration by parts in the Duhamel formula (cf. Eq. 13ap in the
lemma below) the boundary terms do not vanish unless both the kernel and the
“test function” are periodic. Hence, the kernel of Definition 4.1 is better
suited to our analysis.

######  Lemma 4.4 (Regularity away from initial time).

Suppose that the periodic initial data for problem Eq. 13aj is such that
$f_{R,0},f_{L,0}\in L^{\infty}(\Omega)$. Then, the unique weak solutions
$f_{R}$ and $f_{L}$ of Eq. 13aj belong to $C((0,T]\times\bar{\Omega})$.

######  Proof 4.5.

Recall from Theorem 3.7 that we have $f_{R},f_{L},\rho\in
L^{\infty}([0,T]\times\Omega)$.

The key observation is that the equations Eq. 13aj can be interpreted as heat
equations with a source on the right-hand side. Since we already derived the
fundamental solution of the periodic heat equation in Lemmas 4.2 and 4.3, and
we have already shown uniqueness in the relevant space of solutions to Eq.
13aj, we are in a position to apply the Duhamel principle and obtain implicit
integral relations for $f_{R}$ and $f_{L}$. In the first step of the proof, we
justify that these Duhamel formulas are well-defined. In a second step, we
prove the continuity of these functions away from the initial time.

Step 1 (Duhamel formula): We justify that the formula for $f_{R}$

(13ap) $\displaystyle f_{R}(t,x)\\!=$
$\displaystyle\int_{0}^{2\pi}\Phi(t,x-y)f_{R,0}(y)\,\mathrm{d}y$
$\displaystyle-$
$\displaystyle\\!\int_{0}^{t}\\!\int_{0}^{2\pi}\\!\Phi(t\\!-\\!s,\\!x\\!-\\!y)\Big{(}\text{Pe}\partial_{y}\\{f_{R}(s,y)[1-\rho(s,y)]\\}+f_{R}(s,y)\\!-\\!f_{L}(s,y)\Big{)}\mathrm{d}y\mathrm{d}s,$

is well-defined, for a.e. $(t,x)\in[0,T]\times\bar{\Omega}$. To begin with,
whenever $t>0$, the function $\Phi(t,\cdot)$ is smooth and is therefore
square-integrable on the compact interval $[0,2\pi]$. This shows that the
integrand in the first integral in Eq. 13ap is integrable. Meanwhile, the
equality Eq. 13an shows that $\Phi\in L^{2}([0,T];L^{2}([0,2\pi]))$, whence,
using the boundedness of $\rho$,

$\displaystyle\int_{0}^{t}\int_{0}^{2\pi}\Phi(t-s,x-y)\big{|}\partial_{y}\\{f_{R}(s,y)[1-\rho(s,y)]\\}\big{|}\,\mathrm{d}y\,\mathrm{d}s$
$\displaystyle\leq\\!2\\!\int_{0}^{t}\\!\bigg{[}\\!\int_{0}^{2\pi}\\!\Phi(t-s,x-y)^{2}\,\mathrm{d}y\\!\bigg{]}^{1/2}\\!\bigg{[}\\!\int_{0}^{2\pi}\\!|\partial_{y}f_{R}(s,y)|^{2}\\!+\\!f_{R}(s,y)^{2}|\partial_{y}\rho(s,y)|^{2}\,\mathrm{d}y\bigg{]}^{1/2}\,\mathrm{d}s$
$\displaystyle\leq\\!\int_{0}^{t}\\!\int_{0}^{2\pi}\\!\Phi(t\\!-\\!s,\\!x\\!-\\!y)^{2}\,\mathrm{d}y\,\mathrm{d}s\\!+\\!\|f_{R}\|^{2}_{L^{2}(0,T;H^{1}(\Omega))}\\!+\\!\int_{0}^{t}\\!\int_{0}^{2\pi}\\!f_{R}(s,y)^{2}|\partial_{y}\rho(s,y)|^{2}\,\mathrm{d}y\,\mathrm{d}s,$

where we have used the Young inequality. As before, we use a change of
variable and the constraint $x\in[0,2\pi]$ to bound the first term on the
right-hand side of the above by $2\|\Phi\|^{2}_{L^{2}(0,T;L^{2}([0,2\pi]))}$.
Meanwhile,

$\int_{0}^{t}\int_{0}^{2\pi}f_{R}(s,y)^{2}|\partial_{y}\rho(s,y)|^{2}\,\mathrm{d}y\,\mathrm{d}s\leq\|f_{R}\|^{2}_{L^{\infty}((0,T)\times\Omega)}\|\rho\|^{2}_{L^{2}(0,T;H^{1}(\Omega))},$

using that $f_{R}\in L^{\infty}((0,T)\times\Omega)$ from Theorem 3.7.
Similarly, using Lemma 4.2

$\int_{0}^{t}\\!\int_{0}^{2\pi}\\!\Phi(t\\!-\\!s,\\!x\\!-\\!y)|f_{R}(s,y)\\!-\\!f_{L}(s,y)|\,\mathrm{d}y\,\mathrm{d}s\leq
2\|\Phi\|_{L^{2}(0,T;L^{2}(\Omega))}\|f_{R}\\!-\\!f_{L}\|_{L^{2}(0,T;L^{2}(\Omega))}.$

Therefore, the integrand in the second integral of Eq. 13ap is also
integrable. It is an exercise to verify that Eq. 13ap, and the equivalent
formula for $f_{L}$ satisfy Eq. 13aj and are periodic on $\Omega$. These
formulas therefore coincide with the unique solutions of Eq. 13aj.

Step 2 (continuity away from initial time): Fix
$(t,x)\in(0,T]\times\bar{\Omega}$ and let $(t_{n},x_{n})\in(0,T]\times\Omega$
be converging to $(t,x)$. Using the integrability from Step 1, the continuity
of $\Phi$ for $t>0$, and the Dominated Convergence Theorem on Eq. 13ap and the
analogous formula for $f_{L}$, we directly obtain
$|f_{R}(t_{n},x_{n})-f_{R}(t,x)|\to 0$ and $|f_{L}(t_{n},x_{n})-f_{L}(t,x)|\to
0$ as $n\to\infty$. It follows that $f_{R},f_{L}\in
C((0,T]\times\bar{\Omega})$. In turn, we also obtain $\rho\in
C((0,T]\times\bar{\Omega})$.

######  Lemma 4.6.

Suppose that $f_{R,0},f_{L,0}\in C(\bar{\Omega})$ and are periodic. Then, the
unique weak solutions $f_{R}$ and $f_{L}$ of Eq. 13aj belong to
$C([0,T]\times\bar{\Omega})$.

######  Proof 4.7.

We only write the proof for $f_{R}$, as the proof for $f_{L}$ is identical. In
view of the uniform continuity of $f_{R}$ on the compact $\bar{\Omega}$, given
$\varepsilon>0$ there exists $\delta=\delta(\varepsilon)$ for which if
$x,y\in[0,2\pi]$ are such that $|x-y|<\delta$, then
$|f_{R,0}(x)-f_{R,0}(y)|<\varepsilon/4$. Now, given $t\in(0,\infty)$ and
$x^{*}\in(0,2\pi)$, there holds from Eq. 13ap

$\displaystyle|f_{R}(t,x)-f_{R,0}(x^{*})|\leq$
$\displaystyle\bigg{|}\int_{0}^{2\pi}\Phi(t,x-y)f_{R,0}(y)\,\mathrm{d}y-f_{R,0}(x^{*})\bigg{|}$
$\displaystyle+\text{Pe}\int_{0}^{t}\int_{0}^{2\pi}\Phi(t-s,x-y)|\partial_{y}\big{(}f_{R}(s,y)(1-\rho(s,y))\big{)}|\,\mathrm{d}y\,\mathrm{d}s$
$\displaystyle+k\int_{0}^{t}\int_{0}^{2\pi}\Phi(t-s,x-y)\big{|}f_{R}(s,y)-f_{L}(s,y)\big{|}\,\mathrm{d}y\,\mathrm{d}s$
$\displaystyle=:$ $\displaystyle I_{1}(t,x)+I_{2}(t,x)+I_{3}(t,x),$

where we used the nonnegativity of $\Phi$ and
$\int_{0}^{2\pi}\Phi(t,x)\,\mathrm{d}x=1$ for every $t>0$ from Lemma 4.2.
Using Hölder’s inequality, the $L^{2}$-boundedness of
$\partial_{x}f_{R},f_{R},f_{L}$, and Lemma 4.2, we obtain $\lim_{t\to
0^{+}}\|I_{2}(t,\cdot)\|_{L^{\infty}([0,2\pi])}=\lim_{t\to
0^{+}}\|I_{3}(t,\cdot)\|_{L^{\infty}([0,2\pi])}=0$.

It remains to control $I_{1}$. Recall that since $f_{R,0}\in C(\bar{\Omega})$
is periodic, there exists a sequence $(\psi_{m})_{m\in\mathbb{N}}$ of periodic
elements of $C^{2}(\bar{\Omega})$ such that
$\|f_{R,0}-\psi_{m}\|_{L^{\infty}(\bar{\Omega})}\to 0$ as $n\to\infty$. Then,
since $\psi_{m}\in C^{2}(\bar{\Omega})$ for each $m\in\mathbb{N}$, we know
that it has Fourier coefficients $a_{m,n},b_{m,n}$ that decay as $n^{-2}$; cf.
Eq. 13bp-Eq. 13bq. Then, using the convolution result for Fourier series,

$\displaystyle\bigg{|}\psi_{m}(x)-\int_{0}^{2\pi}\Phi(t,x-y)\psi_{m}(y)\,\mathrm{d}y\bigg{|}$
$\displaystyle=\frac{1}{\pi}\bigg{|}\sum_{n=1}^{\infty}(1-e^{-n^{2}t})\big{(}a_{m,n}\cos
nx+b_{m,n}\sin nx\big{)}\bigg{|}$ $\displaystyle\leq
C_{m}\sum_{n=1}^{\infty}\frac{1}{n^{2}}(1-e^{-n^{2}t})\to 0\qquad\text{as
}t\to 0^{+},$

where we applied the Dominated Convergence Theorem applied to the atomic
measure in the final line. Note that

$\displaystyle I_{1}(t$
$\displaystyle,x)\leq|f_{R,0}(x^{*})-f_{R,0}(x)|+|f_{R,0}(x)-\psi_{m}(x)|$
$\displaystyle+\bigg{|}\psi_{m}(x)-\int_{0}^{2\pi}\Phi(t,x-y)\psi_{m}(y)\,\mathrm{d}y\bigg{|}+\bigg{|}\int_{0}^{2\pi}\Phi(t,x-y)\big{(}\psi_{m}(y)-f_{R,0}(y)\big{)}\,\mathrm{d}y\bigg{|},$

and elementary manipulations (cf. Step 3 of Proof of Lemma 4.2) relying also
on the periodicity of $\psi_{m}$ yield
$\int_{0}^{2\pi}\Phi(t,x-y)\psi_{m}(y)\,\mathrm{d}y=\int_{0}^{2\pi}\Phi(t,x-y)\psi_{m}(x+y)\,\mathrm{d}y$.
Thus, given $\varepsilon>0$, by first picking $x$ such that $|x-x^{*}|<\delta$
and then choosing $m$ sufficiently large such that
$\|f_{R,0}-\psi_{m}\|_{L^{\infty}(\bar{\Omega})}<\varepsilon/4$, and finally
taking $t$ sufficiently close to zero for this particular $m$ (cf. convergence
to initial data of Lemma 4.2), it follows that

$I_{1}(t,x)<\frac{3}{4}\varepsilon+\|\psi_{m}-f_{R,0}\|_{L^{\infty}(\bar{\Omega})}\int_{0}^{2\pi}\Phi(t,x-y)\,\mathrm{d}y<\varepsilon,$

where we used the unit integral property of $\Phi$. Hence,
$\lim_{(t,x)\to(0^{+},x^{*})}I_{1}(t,x)=0$, and we conclude
$\lim_{(t,x)\to(0^{+},x^{*})}|f_{R,0}(x^{*})-f_{R}(t,x)|=0$, which verifies
the required continuity at the initial time. The same argument shows the
result for $f_{L}$.

######  Remark 4.8.

We showed that the weak solutions of the 1D model, which (cf. Remark 3.8) in
principle only belong to $C([0,T];L^{2}_{per}([0,2\pi]))\cap
L^{2}([0,T];H^{1}_{per}([0,2\pi])$, are in fact continuous under mild
assumptions on the initial data. Although the authors suspect that it is
possible to show higher regularity of the solutions, the method of proof
required for such a result would be rather different to the current approach.
For this reason, we leave this for further investigation.

###  4.2 Regularity for the 2D model

In accordance with the expression for the one-dimensional periodic kernel in
terms of complex exponentials (cf. Section 4.1), we find that the periodic
heat kernel in three spatial dimensions (i.e. when $\Upsilon=[0,2\pi]^{3}$) is
given by the explicit formula

(13aq) $\tilde{\Phi}(t,{\bf
x})=\frac{1}{(2\pi)^{3}}+\sum_{\mathbf{n}\in\mathbb{Z}^{3}\setminus\\{0\\}}\frac{e^{-|\mathbf{n}|^{2}t}}{(2\pi)^{3}}e^{i\mathbf{n}\cdot{\bf
x}}\qquad\forall(t,{\bf x})\in(0,\infty)\times\mathbb{R}^{3}.$

Note that the above may be rewritten in terms of the Jacobi theta-3 function
as $\tilde{\Phi}(t,{\bf
x})=(2\pi)^{-3}\theta_{3}({x_{1}}/{2},e^{-t})\theta_{3}({x_{2}}/{2},e^{-t})\theta_{3}({x_{3}}/{2},e^{-t})$
for every $(t,{\bf x})\in(0,\infty)\times\mathbb{R}^{3}$. As per the proof of
Lemma 4.2, we obtain that $\tilde{\Phi}$ is nonnegative in
$(0,\infty)\times\mathbb{R}^{3}$ with

(13ar)
$\|\tilde{\Phi}(t,\cdot)\|_{L^{1}(\Upsilon)}=1\quad\text{and}\quad\int_{0}^{t}\|\tilde{\Phi}(\tau,\cdot)\|^{2}_{L^{2}(\Upsilon)}\,\mathrm{d}\tau=t+\sum_{\mathbf{n}\in\mathbb{Z}^{3}}\frac{1}{2|\mathbf{n}|^{2}}(1-e^{-|\mathbf{n}|^{2}t})\qquad\forall
t>0,$

and that $\tilde{\Phi}$ is a smooth $\Upsilon$-periodic solution of the heat
equation in $(0,\infty)\times\mathbb{R}^{3}$. In turn, an identical proof to
that of Lemma 4.3 yields the following result.

######  Lemma 4.9.

Let $\psi\in L^{1}_{per}(\Upsilon)$. Then,
$\tilde{\Psi}(t,\bm{\xi}):=\int_{\Upsilon}\tilde{\Phi}(t,\bm{\xi}-\mathbf{z})\psi(\mathbf{z})\,\mathrm{d}\mathbf{z}$
for all $(t,\bm{\xi})\in(0,\infty)\times\mathbb{R}^{3}$ is a smooth
$\Upsilon$-periodic function defined on the full-space
$(0,\infty)\times\mathbb{R}^{3}$, and solves the Cauchy problem

(13as) $\left\\{\begin{aligned}
&\partial_{t}\tilde{\Psi}-\Delta_{\bm{\xi}}\tilde{\Psi}=0\qquad&&\text{in
}(0,\infty)\times\mathbb{R}^{3},\\\ &\Psi|_{t=0}=\psi\qquad&&\text{on
}\mathbb{R}^{3},\end{aligned}\right.$

where the initial data is achieved in $L^{1}_{loc}$, i.e., $\lim_{t\to
0^{+}}\|\tilde{\Psi}(t,\cdot)-\psi\|_{L^{1}(\Upsilon)}=0$.

In turn, we have the following regularity result.

######  Lemma 4.10.

Suppose the initial data for problem Eq. 2 is such that $f_{0}\in
L^{\infty}_{per}(\Upsilon)$. Then unique weak solution $f$ of Eq. 2 supplied
by Theorem 2.3 belongs to $C((0,T]\times\bar{\Upsilon})$.

######  Proof 4.11.

There are two steps to the proof: the first is to show the boundedness of $f$
(which requires building an appropriate test function); the second is to argue
via a Duhamel formula (cf. Proof of Lemma 4.4).

Step 1 (boundedness of $f$): Recall from Theorem 2.3 that the weak solution of
Eq. 2, $f$, a priori only belongs to the space
$L^{\infty}([0,T];L^{2}_{per}(\Upsilon))\cap
L^{2}([0,T];H^{1}_{per}(\Upsilon))$. We show that $f\in
L^{\infty}((0,T)\times\Upsilon)$.

For any $k>0$, one verifies that $(f-k)_{+}\in
L^{\infty}([0,T];L^{2}_{per}(\Omega))\cap L^{2}([0,T];H^{1}_{per}(\Upsilon))$
and $\partial_{t}(f-k)_{+}\in L^{2}([0,T];(H^{1}_{per})^{\prime}(\Upsilon))$.
Indeed, there holds $(f-k)_{+}=\mathds{1}_{\\{f>k\\}}(f-k)$, from which it
follows that, in the sense of distributions, $\nabla_{t,{\bf
x},\theta}(f-k)_{+}=\mathds{1}_{\\{f>k\\}}\nabla_{t,{\bf x},\theta}f$ a.e. in
$(0,T)\times\Upsilon$, where $\nabla_{t,{\bf x},\theta}$ is the vector of
derivatives $(\partial_{t},\nabla,\partial_{\theta})$, and we directly deduce
the boundedness of the relevant norms. Thus, $(f(t)-k)_{+}$ is an admissible
test function for the weak formulation of Definition 2.2. Note additionally
that, by [18, Ch. 2 Lem. 1.2], there holds in the sense of distributions, for
a.e. $t\in(0,T)$,
$\frac{d}{dt}\frac{1}{2}\int_{\Upsilon}(f(t)-k)_{+}^{2}\,\mathrm{d}{\bf
x}=\langle\partial_{t}f(t),(f(t)-k)_{+}\rangle_{\Upsilon}$. Using
$(f(t)-k)_{+}$ as the test function in the weak formulation of Definition 2.2,
integrating in ${\bf x}$, using the boundedness of $\rho$, Hölder’s
inequality, and the Young inequality, we get

$\displaystyle\frac{d}{dt}\left(\frac{1}{2}\|(f(t)-k)_{+}\|_{L^{2}(\Upsilon)}^{2}\right)+\int_{\Upsilon}\bigg{(}\frac{D_{e}}{2}|\nabla(f(t)-k)_{+}|^{2}+$
$\displaystyle|\partial_{\theta}(f(t)-k)_{+}|^{2}\bigg{)}\,\mathrm{d}{\bf
x}\,\mathrm{d}\theta$
$\displaystyle\leq\frac{\text{Pe}^{2}}{2D_{e}}\|(f(t)-k)_{+}\|_{L^{2}(\Upsilon)}^{2},$

for a.e. $t\in(0,T)$. It follows from the Grönwall Lemma that there holds

$\displaystyle\frac{1}{2}\|(f(t)-k)_{+}\|_{L^{2}(\Upsilon)}^{2}\leq\frac{1}{2}\|(f_{0}-k)_{+}\|_{L^{2}(\Upsilon)}^{2}\exp\left({\frac{\text{Pe}^{2}}{D_{e}}t}\right)\qquad\text{a.e.~{}}t\in(0,T).$

By selecting $k=\|f_{0}\|_{L^{\infty}(\Upsilon)}$, the right-hand side of the
above is null. It follows that $f(t)\leq\|f_{0}\|_{L^{\infty}(\Upsilon)}$ for
a.e. $t\in(0,T)$, i.e.,
$\|f\|_{L^{\infty}((0,T)\times\Upsilon)}\leq\|f_{0}\|_{L^{\infty}(\Upsilon)}$.

Step 2 (Duhamel formula for $f$): Without loss of generality for this final
step we only consider the case where we have normalised the Eq. 2 by the
positive constant $D_{e}$, so that
$\partial_{t}-D_{e}\Delta-\partial^{2}_{\theta}$ may be rewritten as
$\partial_{t}-\Delta_{\bm{\xi}}$ with $\bm{\xi}$ the usual 3-dimensional
vector.

As per the proof of Lemma 4.4, in view of the uniqueness provided by Theorem
2.3, it follows from Eq. 2 and the Duhamel principle that $f$ satisfies the
following implicit equation, where the kernel $\tilde{\Phi}$ was defined in
Eq. 13aq (cf. Lemma 4.9),

(13at)
$f(t,\bm{\xi})\\!=\\!\int_{\Upsilon}\tilde{\Phi}(t,\bm{\xi}-\mathbf{z})f_{0}(\mathbf{z})\,\mathrm{d}\mathbf{z}+\text{Pe}\int_{0}^{t}\int_{\Upsilon}\tilde{\Phi}(t-s,\bm{\xi}-\mathbf{z})\nabla_{\mathbf{z}}\cdot\big{(}f(s,\mathbf{z})(1-\rho(s,\mathbf{z}))\mathbf{e}(\theta)\big{)}\mathrm{d}\mathbf{z}\mathrm{d}s$

for a.e. $(t,\bm{\xi})\in[0,T]\times\bar{\Upsilon}$. Note that the above is
well-defined: using the boundedness of $f$ from Step 1, we proceed to show the
integrability of the integrand in Eq. 13at as per Step 1 of the proof of Lemma
4.4. We now deduce that $f\in C((0,T]\times\bar{\Omega})$ using the continuity
of $\tilde{\Phi}$ for $t>0$ and the Dominated Convergence Theorem, exactly as
we did in Step 2 of the proof of Lemma 4.4.

Finally, by following the strategy of proof of Lemma 4.6 to the letter, we
obtain the following result concerning continuity up to the initial time.

######  Lemma 4.12.

Suppose that the periodic initial data $f_{0}$ for problem Eq. 2 belongs to
$C(\bar{\Upsilon})$. Then, the unique weak solution $f$ of Eq. 2 belongs to
$C([0,T]\times\bar{\Upsilon})$.

We note, as per Remark 4.8, that it is likely possible to improve Lemma 4.12.
However, the method for proving such a result is different from the current
approach, which uses the Duhamel principle, and so we leave this for further
investigation.

##  5 Extension to very weak solutions for the 2D model

The assumption on the initial data in $L^{2}_{per}(\Upsilon)$ rules out
initial configurations with a fixed orientation, that is,
$f_{0}(x,\theta)=h^{0}(x)\delta_{\theta=\theta^{*}}$ for some
$\theta^{*}\in[0,2\pi)$ and $h^{0}\in L^{2}_{per}(\Omega)$, where the latter
is the usual Dirac delta in the angular variable only. Since such initial data
belongs to $L^{2}(\Omega;H^{s}(0,2\pi))$ for $s<-1/2$, we turn our attention
to initial data $f_{0}\in L^{2}_{per}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))$.
For the convenience of the reader, we clarify that, for any normed vector
space $V$,

(13au) $f\in L^{2}_{per}(\Omega;V)\iff\left\\{\begin{aligned}
&f:\mathbb{R}^{2}\to V,\\\ &\int_{\Omega}\|f({\bf
x},\cdot)\|_{V}^{2}\,\mathrm{d}{\bf x}<+\infty,\text{ and}\\\ &f({\bf
x}-\mathbf{w},\cdot)=f({\bf x},\cdot)\quad\text{a.e. }{\bf
x}\in\Omega,\quad\forall\mathbf{w}\in(2\pi\mathbb{Z})^{2},\end{aligned}\right.$

where the final equality on the right-hand side holds in $V$.

In view of the previous discussion, we define the following Hilbert spaces, to
give meaning to a weaker formulation of the equation (cf. Definition 5.1),

(13av) $\displaystyle X:=L^{2}([0,T];H^{1}_{per}(\Upsilon))\cap
L^{2}([0,T];L^{2}_{per}(\Omega;H^{2}_{per}(0,2\pi))),$ $\displaystyle
Y:=H^{1}_{per}(\Upsilon)\cap L^{2}_{per}(\Omega;H^{2}_{per}(0,2\pi)).$

Note that
$\|u\|_{X^{\prime}}^{2}=\int_{0}^{T}\|u(t)\|^{2}_{Y^{\prime}}\,\mathrm{d}t$,
and $\|u(t)\|_{Y^{\prime}}=\sup_{\overset{\varphi\in
C^{\infty}_{per}(\bar{\Upsilon})}{\|\varphi\|_{Y}\leq 1}}|\langle
u(t),\varphi\rangle_{\Upsilon}|$.

The initial density must now be defined in duality, i.e.,

(13aw) $\rho_{0}({\bf x}):=\langle f_{0}({\bf
x},\cdot),1\rangle_{(H^{1}_{per})^{\prime}\times H^{1}_{per}}\qquad\mbox{a.e.
}x\in\Omega,$

where $\langle\cdot,\cdot\rangle_{(H^{1}_{per})^{\prime}\times H^{1}_{per}}$
is the duality product on $(0,2\pi)$. Note that for $f_{0}({\bf
x},\theta)=h^{0}({\bf x})\delta_{\theta=\theta^{*}}$, the initial space
density is $\rho_{0}({\bf x})=h^{0}({\bf x})$.

We now extend the concept of solution established in Definition 2.2 to admit a
larger class of functions (i.e. smaller class of test functions). With slight
abuse of terminology, we name these solutions _very weak solutions_ , even
though only the regularity in the angular variable is affected.

######  Definition 5.1.

A _very weak solution_ of Eq. Eq. 2 is a curve $f$ belonging to
$L^{2}([0,T];H^{1}_{per}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi)))\cap
L^{\infty}([0,T];L^{2}_{per}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi)))\cap
L^{2}((0,T)\times\Upsilon)$, with $\partial_{t}f\in X^{\prime}$ such that, for
every test function $\varphi\in Y$, there holds for a.e. $t\in[0,T]$

(13ax)
$\begin{split}&\left\langle\partial_{t}f(t),\varphi\right\rangle_{\Upsilon}\\!=\\!\left\langle\text{Pe}(1-\rho(t))f(t){\bf
e}(\theta),\nabla\varphi\right\rangle_{\Upsilon}\\!-\\!\left\langle
D_{e}\nabla f(t),\nabla\varphi\right\rangle_{\Upsilon}+\left\langle
f(t),\partial_{\theta}^{2}\varphi\right\rangle_{\Upsilon},\\\ &f(0,{\bf
x},\theta)=f_{0}({\bf x},\theta),\qquad\text{on
}\Upsilon\times\\{0\\},\end{split}$

with periodic boundary conditions on $\Upsilon$, where
$\rho(t,\mathbf{x})=\int_{0}^{2\pi}f(t,{\bf x},\theta)\,\mathrm{d}\theta$, and
where the initial data is achieved in the sense $f(0)=f_{0}$ in $Y^{\prime}$.

Since $Y\subset L^{2}_{per}(\Omega;H^{2}_{per}(0,2\pi))$, we have the
inclusions $L^{2}_{per}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi)))\subset
L^{2}_{per}(\Omega;(H_{per}^{2})^{\prime}(0,2\pi))\subset Y^{\prime}$. It
follows that $f\in
L^{2}([0,T];L^{2}_{per}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi)))$ and
$\partial_{t}f\in X^{\prime}$ imply $f\in H^{1}([0,T];Y^{\prime})$. Hence, by
[10, Sec. 5.9.2 Th. 2], $f\in C([0,T];Y^{\prime})$. Therefore, $f(0)\in
Y^{\prime}$ is well-defined in the previous definition.

We now give the proof of Theorem 2.6; the main theorem of this section. Since
a very weak solution is an element of $L^{2}((0,T)\times\Upsilon)$, this
result implies that the solutions never concentrate in angle to a Dirac mass,
even when the initial data does. This illustrates an instantaneous smoothing
property of the equation.

######  Proof 5.2 (Proof of Theorem 2.6).

The idea of the proof is to construct a sequence of regularised initial data
that approaches $f_{0}$ in the required norm, and to show that the
corresponding regularised solutions converge to a very weak solution.

Step 1 (regularised sequence of initial data and solutions): Due to Lemma C.1
in Appendix C, there exists $(f_{0}^{m})_{m\in\mathbb{N}}$ nonnegative
elements of $C^{\infty}_{per}(\bar{\Upsilon})$ such that

(13ay) $\|f_{0}-f_{0}^{m}\|_{L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))}\to
0\qquad\text{as }m\to\infty,$

with $\rho_{0}^{m}({\bf x}):=\int_{0}^{2\pi}f_{0}^{m}({\bf
x},\theta)\,\mathrm{d}\theta\in[0,1]$ for a.e. ${\bf x}\in\Upsilon$. Note that
Eq. 13ay of course implies that there exists a positive constant $M$
independent of $m$ such that

(13az)
$\sup_{m}\|f_{0}^{m}\|_{L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))}\leq
M<+\infty.$

Since each $f_{0}^{m}$ is an admissible initial datum for Theorem 2.3, we
deduce that there exists a unique $f_{m}\in
L^{\infty}([0,T];L^{2}_{per}(\Upsilon))\cap
L^{2}([0,T];H^{1}_{per}(\Upsilon))$ with $\partial_{t}f_{m}\in
L^{2}([0,T];(H^{1}_{per})^{\prime}(\Upsilon))$ solving Eq. 2 in the weak sense
given by by Definition 2.2. Moreover, we have the uniform bound
$0\leq\rho_{m}(t,{\bf x})\leq 1$ for a.e. $(t,x)\in(0,T)\times\Omega$. Note
however that we do not have a uniform estimate on
$\|f_{m}\|_{L^{2}((0,T);H^{1}(\Upsilon))}$ or even on
$\|f_{m}\|_{L^{2}((0,T)\times\Upsilon)}$, since these latter estimates depend
on $\|f_{0}^{m}\|_{L^{2}(\Upsilon)}$, which explodes in the limit as
$m\to\infty$. We therefore require a different estimate on
$(f_{m})_{m\in\mathbb{N}}$.

Step 2 (an auxiliary Dirichlet problem): In contrast to the inverse Laplacian
introduced in Section 3.4, we introduce
$L_{\theta}:=(-\partial^{2}_{\theta})^{-1}:(H^{1}_{per})^{\prime}(0,2\pi)\to
H^{1}_{per}(0,2\pi)$, which is understood to be the unique solution of the
Dirichlet problem:

(13ba) $\left\\{\begin{aligned}
&(-\partial_{\theta}^{2})L_{\theta}f=f\qquad\text{in }(0,2\pi),\\\
&L_{\theta}f(0)=L_{\theta}f(2\pi)=0.\end{aligned}\right.$

Let us note that for any function $f\in(H^{1}_{per})^{\prime}(0,2\pi)$, there
holds

$\|f\|_{(H^{1}_{per})^{\prime}(0,2\pi)}=\sup_{{\overset{\varphi\in
H^{1}_{per}(0,2\pi)}{\|\varphi\|_{H^{1}(0,2\pi)}\leq 1}}}\langle
f,\varphi\rangle_{(H^{1}_{per})^{\prime}\times
H^{1}_{per}}\leq\|\partial_{\theta}L_{\theta}f\|_{L^{2}_{per}(0,2\pi)}.$

Indeed, using the property $(-\partial_{\theta}^{2})L_{\theta}f=f$, we have,
where the duality product is between $H^{1}_{per}$ and its dual,

$\begin{split}|\langle
f,\varphi\rangle|&=|\langle(-\partial_{\theta}^{2})\circ(-\partial_{\theta}^{2})^{-1}f,\varphi\rangle|=|\langle\partial_{\theta}L_{\theta}f,\partial_{\theta}\varphi\rangle_{L^{2}\times
L^{2}}|\leq\|\partial_{\theta}L_{\theta}f\|_{L^{2}}\|\partial_{\theta}\varphi\|_{L^{2}}.\end{split}$

Note that the integration by parts in the above involve no additional boundary
term due to the periodicity of $f$ and $L_{\theta}f$. By choosing the specific
test function $\varphi=L_{\theta}f\in H^{1}_{per}(0,2\pi)$ in the previous
estimate, we obtain the maximum value in the duality product defining the
$(H^{1}_{per})^{\prime}$ norm, hence

(13bb)
$\|f\|_{(H^{1}_{per})^{\prime}(0,2\pi)}=\|\partial_{\theta}L_{\theta}f\|_{L^{2}(0,2\pi)}.$

Since $L_{\theta}f\in H^{1}_{0}(0,2\pi)$, the Poincaré inequality in one
dimension yields

(13bc) $\|L_{\theta}f\|_{L^{2}(0,2\pi)}^{2}\leq
C_{P}\|\partial_{\theta}L_{\theta}f\|^{2}_{L^{2}(0,2\pi)}=C_{P}\|f\|^{2}_{(H^{1}_{per})^{\prime}(0,2\pi)}\leq
C_{P}\|f\|^{2}_{L^{2}_{per}(0,2\pi)},$

where the constant $C_{P}$ depends only on the domain $(0,2\pi)$, and Eq. 13bb
was used to obtain the equality. In turn, given a function $f$ such that
$\nabla f\in L^{2}_{per}(0,2\pi)\subset(H^{1}_{per})^{\prime}(0,2\pi)$, there
holds the estimate

(13bd) $\|L_{\theta}\nabla f\|_{L^{2}(0,2\pi)}^{2}\leq C_{P}\|\nabla
f\|^{2}_{(H^{1}_{per})^{\prime}(0,2\pi)}\leq C_{P}\|\nabla
f\|^{2}_{L^{2}(0,2\pi)},$

and, directly from Eq. 13bb,

(13be) $\|\partial_{\theta}L_{\theta}\nabla f\|_{L^{2}(0,2\pi)}^{2}=\|\nabla
f\|^{2}_{(H^{1}_{per})^{\prime}(0,2\pi)}\leq\|\nabla f\|^{2}_{L^{2}(0,2\pi)}.$

Step 3 (energy estimate on first derivative independent of $m$): We now apply
the operator $L_{\theta}$ to the sequence $(f_{m})_{m\in\mathbb{N}}$. Note
that, for a.e. $t\in(0,T)$,

$\|L_{\theta}f_{m}(t)\|_{H^{1}(\Upsilon)}^{2}=\|L_{\theta}f_{m}(t)\|_{L^{2}(\Upsilon)}^{2}+\|\nabla
L_{\theta}f_{m}(t)\|^{2}_{L^{2}(\Upsilon)}+\|\partial_{\theta}L_{\theta}f_{m}(t)\|^{2}_{L^{2}(\Upsilon)}.$

It then follows from Eq. 13bc and Eq. 13bd that

$\displaystyle\|L_{\theta}f_{m}(t)\|_{H^{1}(\Upsilon)}^{2}\leq$
$\displaystyle\int_{\Omega}\|L_{\theta}f_{m}(t,{\bf
x},\cdot)\|_{L^{2}(0,2\pi)}^{2}\,\mathrm{d}{\bf x}+\int_{\Omega}\|\nabla
L_{\theta}f_{m}(t,{\bf x},\cdot)\|^{2}_{L^{2}(0,2\pi)}\,\mathrm{d}{\bf x}$
$\displaystyle+\int_{\Omega}\|\partial_{\theta}L_{\theta}f_{m}(t,{\bf
x},\cdot)\|^{2}_{L^{2}(0,2\pi)}\,\mathrm{d}{\bf x}$ $\displaystyle\leq$
$\displaystyle 2C_{P}\int_{\Omega}\big{(}\|f_{m}(t,{\bf
x},\cdot)\|^{2}_{L^{2}(0,2\pi)}+\|\nabla f_{m}(t,{\bf
x},\cdot)\|^{2}_{L^{2}(0,2\pi)}\big{)}\,\mathrm{d}{\bf x}.$

Thus, $\|L_{\theta}f_{m}(t)\|_{H^{1}(\Upsilon)}^{2}\leq
2C_{P}\|f_{m}\|_{H^{1}(\Upsilon)}^{2}<+\infty$. Therefore,
$L_{\theta}f_{m}(t)$ is an admissible test function for the weak formulation
of Definition 2.2. This latter observation enables us to test the equation
with $L_{\theta}f_{m}$ and to obtain an estimate independent of $m$. We get,
for a.e. $(t,{\bf x})\in(0,T)\times\Omega$,

$\begin{split}\frac{1}{2}\frac{d}{dt}\|f_{m}(t,{\bf
x},\cdot)\|_{(H^{1}_{per})^{\prime}(0,2\pi)}^{2}=&\frac{1}{2}\frac{d}{dt}\|\partial_{\theta}L_{\theta}f_{m}(t,{\bf
x},\cdot)\|_{L^{2}(0,2\pi)}^{2}\\\ =&\langle\partial_{t}f_{m}(t,{\bf
x},\cdot),L_{\theta}f_{m}(t,{\bf x},\cdot)\rangle_{(H^{1})^{\prime}\times
H^{1}},\end{split}$

where we used that $\partial_{t}f\in
L^{2}([0,T];(H^{1}_{per})^{\prime}(\Upsilon))$ and $L_{\theta}f_{m}\in
L^{2}([0,T];H^{1}_{per}(\Upsilon))$. Hence, by integrating the previous line
with respect to the space variable and using the weak formulation, also noting
that $\langle\nabla f_{m},\nabla
L_{\theta}f_{m}\rangle_{\Upsilon}=\|\nabla\partial_{\theta}L_{\theta}f_{m}\|_{L^{2}(\Upsilon)}^{2}$
and
$\langle\partial_{\theta}f_{m},\partial_{\theta}L_{\theta}f_{m}\rangle_{\Upsilon}=\|f_{m}\|_{L^{2}(\Upsilon)}^{2}$,

(13bf)
$\begin{split}\frac{1}{2}\frac{d}{dt}\|f_{m}(t)\|_{L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))}^{2}=\text{Pe}\langle(1\\!&-\\!\rho_{m})f_{m}{\bf
e}(\theta),\\!\nabla L_{\theta}f_{m}\rangle_{\Upsilon}\\!\\\
&-\\!\|\nabla\partial_{\theta}L_{\theta}f_{m}\|_{L^{2}(\Upsilon)}^{2}-\\!\|f_{m}\|_{L^{2}(\Upsilon)}^{2},\end{split}$

where we used the equality $(-\partial^{2}_{\theta})L_{\theta}f_{m}=f_{m}$,
integrated by parts in $\theta$ the last two duality products, and used the
Dirichlet problem Eq. 13ba. Next, observe that $f_{m}{\bf
e}(\theta)=(-\partial_{\theta}^{2})L_{\theta}f_{m}{\bf
e}(\theta)=-\partial_{\theta}(\partial_{\theta}L_{\theta}f_{m}{\bf
e}(\theta))-\partial_{\theta}L_{\theta}f_{m}{\bf e}^{\perp}(\theta)$, where
${\bf e}^{\perp}(\theta)=(-\sin\theta,\cos\theta)$. Using the fact that
$\rho_{m}$ is independent of the angle variable, it follows from an
integration by parts that

$\begin{split}\langle(1-\rho_{m})f_{m}{\bf e}(\theta),\nabla
L_{\theta}f_{m}\rangle_{\Upsilon}=\langle(1-\rho_{m})&\partial_{\theta}L_{\theta}f_{m}{\bf
e}(\theta),\nabla\partial_{\theta}L_{\theta}f_{m}\rangle_{\Upsilon}\\\
&-\langle(1-\rho_{m})\partial_{\theta}L_{\theta}f_{m}{\bf
e}^{\perp}(\theta),\nabla L_{\theta}f_{m}\rangle_{\Upsilon},\end{split}$

whence the Hölder inequality and the bound for $\rho_{m}$ give
$|\langle(1-\rho_{m})f_{m}{\bf e}(\theta),\nabla
L_{\theta}f_{m}\rangle_{\Upsilon}|\leq\|\partial_{\theta}L_{\theta}f_{m}\|_{L^{2}(\Upsilon)}\|\nabla\partial_{\theta}L_{\theta}f_{m}\|_{L^{2}(\Upsilon)}+\|\partial_{\theta}L_{\theta}f_{m}\|_{L^{2}(\Upsilon)}\|\nabla
L_{\theta}f_{m}\|_{L^{2}(\Upsilon)}$. Thus, by applying the Poincaré
inequality Eq. 13bc to bound $\nabla L_{\theta}f_{m}$ by
$\nabla\partial_{\theta}L_{\theta}f_{m}$ and Young’s inequality,

$\displaystyle|\langle(1\\!-\\!\rho_{m})f_{m}{\bf e}(\theta),\\!\nabla
L_{\theta}f_{m}\rangle_{\Upsilon}|\leq\text{Pe}(1\\!+\\!C_{P})\|\partial_{\theta}L_{\theta}f_{m}\|_{L^{2}(\Upsilon)}^{2}\\!+\\!\frac{1}{2\text{Pe}}\|\partial_{\theta}\nabla
L_{\theta}f_{m}\|_{L^{2}(\Upsilon)}^{2}.$

Combining the above with Eq. 13bf, using Eq. 13be to rewrite
$\|\nabla\partial_{\theta}L_{\theta}f_{m}\|_{L^{2}(\Upsilon)}^{2}$ and Eq.
13bb to rewrite $\|\partial_{\theta}L_{\theta}f_{m}\|^{2}_{L^{2}(\Upsilon)}$,
and integrating in time, we get, for a.e. $t\in(0,T)$,

$\displaystyle\|f_{m}(t)\|_{L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))}^{2}+\int_{0}^{t}\|\nabla
f_{m}(\tau)\|_{L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))}^{2}\,\mathrm{d}\tau+2\int_{0}^{t}\|f_{m}(\tau)\|_{L^{2}(\Upsilon)}^{2}\,\mathrm{d}\tau$
$\displaystyle\quad\leq\|f_{0}^{m}\|_{L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))}^{2}+\text{Pe}^{2}(1+C_{P})\int_{0}^{t}\|f_{m}(\tau)\|_{L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))}^{2}\,\mathrm{d}\tau$
.

Grönwall’s Lemma yields
$\|f_{m}(t)\|_{L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))}^{2}\leq
M^{2}\exp(\text{Pe}^{2}(1+C_{P})T)$ a.e. $t\in(0,T)$, and we recall from Eq.
13az that the right-hand side is independent of $m$. Hence, we obtain, for
some positive constant $M_{1}$ independent of $m$,

(13bg)
$\displaystyle\operatorname{\mathrm{esssup}}_{t\in[0,T]}\\!\|f_{m}(t)\|_{L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))}^{2}$
$\displaystyle+\\!\int_{0}^{T}\\!\|\nabla
f_{m}(t)\|_{L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))}^{2}\,\mathrm{d}t$
$\displaystyle+\\!\int_{0}^{T}\\!\|f_{m}(t)\|_{L^{2}(\Upsilon)}^{2}\,\mathrm{d}t\leq
M_{1}.$

Thus,
$\|f_{m}\|_{L^{\infty}([0,T];L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi)))}$,$\|f_{m}\|_{L^{2}([0,T];H^{1}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi)))}$,
$\|f_{m}\|_{L^{2}((0,T)\times\Upsilon)}$ are bounded independently of $m$.

Step 4 (corresponding estimate on the time derivatives): We now return to Eq.
2 in order to obtain a similar uniform estimate on the sequence of time
derivatives $(\partial_{t}f_{m})_{m\in\mathbb{N}}$. Fix $\varphi$ to be smooth
and periodic on $\bar{\Upsilon}$. Such a function is an admissible test
function for the weak formulation of the equation, and we get

$\displaystyle|\langle\partial_{t}f_{m}(t),\varphi\rangle_{\Upsilon}|\leq$
$\displaystyle|\text{Pe}\langle(1-\rho_{m})f_{m}{\bf
e}(\theta),\nabla\varphi\rangle_{\Upsilon}|+|\langle\nabla
f_{m},\nabla\varphi\rangle_{\Upsilon}|+|\langle\partial_{\theta}f_{m},\partial_{\theta}\varphi\rangle_{\Upsilon}|$
$\displaystyle\leq$
$\displaystyle\text{Pe}\|f_{m}\|_{L^{2}(\Upsilon)}\|\nabla\varphi\|_{L^{2}(\Upsilon)}+\|f_{m}\|_{L^{2}(\Upsilon)}\|\partial_{\theta}^{2}\varphi\|_{L^{2}(\Upsilon)}$
$\displaystyle+\|\nabla
f_{m}\|_{L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))}\|\nabla\varphi\|_{L^{2}(\Omega;H^{1}(0,2\pi))}$
$\displaystyle\leq$
$\displaystyle\big{(}(1+\text{Pe})\|f_{m}(t)\|_{L^{2}(\Upsilon)}+\|\nabla
f_{m}(t)\|_{L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))}\big{)}\big{(}\|\varphi\|_{H^{1}(\Upsilon)}$
$\displaystyle+\|\varphi\|_{L^{2}(\Omega;H^{2}(0,2\pi))}\big{)},$

for a.e. $t\in(0,T)$. We therefore obtain, using Eq. 13av and Eq. 13av,
$\|\partial_{t}f_{m}(t)\|_{Y^{\prime}}\leq((1+\text{Pe})\|f_{m}(t)\|_{L^{2}(\Upsilon)}+\|\nabla
f_{m}(t)\|_{L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))})$, whence, using Eq.
13bg to bound the right-hand side of the above, we get, with $M_{2}$ a
positive constant independent of $m$,

(13bh) $\|\partial_{t}f_{m}\|_{X^{\prime}}\leq M_{2}.$

Step 5 (estimates on $\rho$ independent of $m$): We infer uniform estimates on
the sequence $(\rho_{m})_{m\in\mathbb{N}}$ from those on
$(f_{m})_{m\in\mathbb{N}}$. Firstly, $\rho_{m}(t,{\bf
x})=\int_{0}^{2\pi}f_{m}(t,{\bf x},\theta)\,\mathrm{d}\theta\qquad$ for a.e.
$(t,{\bf x})\in(0,T)\times\Omega$, from which it is easy to see, since the
constant function $1/2\pi$ belongs to $H^{1}_{per}(0,2\pi)$ and has norm equal
to $1$ therein,

(13bi) $|\rho_{m}(t,{\bf x})|\leq
2\pi\sup_{\|\varphi\|_{H^{1}_{per}(0,2\pi)}\leq
1}\bigg{|}\int_{0}^{2\pi}f_{m}(t,{\bf
x},\theta)\varphi(\theta)\,\mathrm{d}\theta\bigg{|}=2\pi\|f_{m}(t,{\bf
x},\cdot)\|_{(H^{1}_{per})^{\prime}(0,2\pi)},$

for a.e. $(t,{\bf x})\in(0,T)\times\Omega$. The same estimate on the
derivative yields $|\nabla\rho_{m}(t,{\bf x})|\leq 2\pi\|\nabla f_{m}(t,{\bf
x},\cdot)\|_{(H^{1}_{per})^{\prime}(0,2\pi)}$ a.e. in $(0,T)\times\Omega$. It
follows directly from the comment under the uniform estimate Eq. 13bg that
there holds

(13bj)
$\|\rho_{m}\|_{L^{\infty}([0,T];L^{2}_{per}(\Omega))}+\|\rho_{m}\|_{L^{2}([0,T];H^{1}_{per}(\Omega))}\leq
M_{3},$

for some $M_{3}$ independent of $m$. Finally, for the time derivatives, by
returning to the weak formulation of Eq. 3 and arguing as we did to obtain Eq.
13bh, we obtain

$|\langle\partial_{t}\rho_{m}(t),\varphi\rangle_{\Omega}|\leq\text{Pe}|\Omega|^{1/2}\|\nabla\varphi\|_{L^{2}(\Omega)}+\|\nabla\rho_{m}(t)\|_{L^{2}(\Omega)}\|\nabla\varphi\|_{L^{2}(\Omega)},$

whence
$\|\partial_{t}\rho_{m}(t)\|_{(H^{1}_{per})^{\prime}(\Omega)}\leq\text{Pe}|\Omega|^{1/2}+\|\nabla\rho_{m}(t)\|_{L^{2}(\Omega)}$
for a.e. $t\in(0,T)$. As a result, we deduce from Eq. 13bj the uniform
estimate

(13bk)
$\|\partial_{t}\rho_{m}\|_{L^{2}([0,T];(H^{1}_{per})^{\prime}(\Omega))}\leq
M_{4},$

for some positive constant $M_{4}$ also independent of $m$.

Step 6 (compactness and passage to the limit in $m$): Due to the uniform
estimate $\sup_{m}\|f_{m}\|_{L^{2}((0,T)\times\Upsilon)}<+\infty$ deduced
under Eq. 13bg, we have from the Banach–Alaoglu Theorem that there exists
$f\in L^{2}((0,T)\times\Upsilon)$ such that

$f_{m}\rightharpoonup f\qquad\text{weakly in }L^{2}((0,T)\times\Upsilon),$

up to a subsequence, which, with slight abuse of notation, we still label as
$(f_{m})_{m\in\mathbb{N}}$. In turn, there holds $\rho_{m}\rightharpoonup\rho$
weakly in $L^{2}((0,T)\times\Omega)$, where $\rho(t,{\bf
x})=\int_{0}^{2\pi}f(t,{\bf x},\theta)\,\mathrm{d}\theta$ for a.e. $(t,{\bf
x})\in(0,T)\times\Omega$. Meanwhile, given the estimates Eq. 13bj and Eq.
13bk, the Aubin–Lions Lemma implies the strong convergence

$\rho_{m}\to\rho\qquad\text{strongly in }L^{2}((0,T)\times\Omega).$

The term $(1-\rho_{m})f_{m}$ is therefore the product of a strongly converging
sequence and a weakly converging sequence in $L^{2}((0,T)\times\Omega)$. We
are therefore able to pass to the limit weakly in $L^{2}((0,T)\times\Omega)$
in the nonlinear drift term.

Additionally, taking further subsequences if necessary, the remaining uniform
estimates from Eq. 13bg and Eq. 13bh, as well as the Banach–Alaoglu Theorem,
imply the convergences
$\partial_{t}f_{m}\overset{*}{\rightharpoonup}\partial_{t}f$ in $X^{\prime}$,
$\nabla f_{m}\overset{*}{\rightharpoonup}\nabla f$ in
$L^{2}([0,T];L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi)))$, which proves the
convergence of the relevant duality products to the limit equation. For the
last duality product, we use an integration by parts to write
$\langle\partial_{\theta}f_{m},\partial_{\theta}\varphi\rangle_{\Upsilon}=-\langle
f_{m},\partial^{2}_{\theta}\varphi\rangle_{\Upsilon}\to\langle
f,\partial^{2}_{\theta}\varphi\rangle_{\Upsilon}$ as $m\to\infty$, where we
used the weak convergence $f_{m}\rightharpoonup f$ in
$L^{2}((0,T)\times\Upsilon)$. Since the test function $\varphi$ is chosen to
belong to the space $X$, we obtain the limit equation as $m\to\infty$, as
required.

Step 7 (initial data): Recalling also the discussion under Definition 5.1, the
estimates Eq. 13bg and Eq. 13bh imply that $f_{m},f\in
H^{1}([0,T];Y^{\prime})$. By [10, Sec. 5.9.2 Th. 2], $f_{m},f\in
C([0,T];Y^{\prime})$. Recall from Eq. 13ay
$\|f_{0}-f_{0}^{m}\|_{L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))}\to 0$,
from which $\|f_{0}-f_{0}^{m}\|_{Y^{\prime}}\to 0$; since
$L^{2}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))\hookrightarrow Y^{\prime}$
continuously. From Eq. 13bg, $(f_{m})_{m\in\mathbb{N}}$ is bounded in
${L^{2}([0,T];L^{2}_{per}(\Upsilon))}$. Using the Rellich–Kondrachov
Compactness Embedding and properties of the adjoint map,
$L^{2}_{per}(\Upsilon)$ embeds compactly into $Y^{\prime}$; since
$Y\hookrightarrow H^{1}_{per}(\Upsilon)$ continuously, and
$H^{1}_{per}(\Upsilon)$ is compactly embedded in $L^{2}_{per}(\Upsilon)$.
Meanwhile, the sequence $(\partial_{t}f_{m})_{m\in\mathbb{N}}$ is uniformly
bounded in $X^{\prime}=L^{2}([0,T];Y^{\prime})$. Thus the Aubin–Lions Lemma
implies $f_{m}\to f$ strongly in $X^{\prime}$, and hence, up to taking a
further subsequence, $f_{m}(t)\to f(t)$ strongly in $Y^{\prime}$ a.e.
$t\in[0,T]$. Uniqueness of limits in $Y^{\prime}$ at the Lebesgue point $t=0$
implies $f(0)=f_{0}$ in $Y^{\prime}$.

##  Appendix A The system of ODEs for the Galerkin approximation in space

Throughout this section, $m\in\mathbb{N}$ is fixed. For $k\in\\{0,\dots,n\\}$,
we define

(13bl) $\displaystyle a_{k}^{n,m}(t,x,y):=$
$\displaystyle\sum_{q=0}^{m}\sum_{p=0}^{m}\alpha_{1,p,q}^{k,m}(t)\cos px\cos
qy+\sum_{q=1}^{m}\sum_{p=0}^{m}\alpha_{2,p,q}^{k,m}(t)\cos px\sin qy$
$\displaystyle+\sum_{q=0}^{m}\sum_{p=1}^{m}\alpha_{3,p,q}^{k,m}(t)\sin px\cos
qy+\sum_{q=1}^{m}\sum_{p=1}^{m}\alpha_{4,p,q}^{k,m}(t)\sin px\sin qy,$
$\displaystyle b_{k}^{n,m}(t,x,y):=$
$\displaystyle\sum_{q=0}^{m}\sum_{p=0}^{m}\beta_{1,p,q}^{k,m}(t)\cos px\cos
qy+\sum_{q=1}^{m}\sum_{p=0}^{m}\beta_{2,p,q}^{k,m}(t)\cos px\sin qy$
$\displaystyle+\sum_{q=0}^{m}\sum_{p=1}^{m}\beta_{3,p,q}^{k,m}(t)\sin px\cos
qy+\sum_{q=1}^{m}\sum_{p=1}^{m}\beta_{4,p,q}^{k,m}(t)\sin px\sin qy,$

where $\\{\alpha_{j,p,q}^{k,m},\beta_{j,p,q}^{k,m}\\}$ are chosen to be the
solutions of a specific ODE system.

Recall that the products of sines and cosines form an orthogonal basis of
$L^{2}_{per}(\Omega)$, which motivates the approximations Eq. 13bl. Our aim is
to recover the coefficients $\\{a_{k},b_{k}\\}_{k=0}^{n}$ as the limits as
$m\to\infty$ of the coefficients defined in Eq. 13bl. To do so, we impose that
$\\{a_{k}^{n,m},b_{k}^{n,m}\\}$ solve Eq. 13 weakly (i.e. in duality with
$H^{1}_{per}(\Omega)$) and with initial data
$\\{a_{k}^{n,m}(0),b_{k}^{n,m}(0)\\}$ computed by testing
$\\{a_{k}^{0},b_{k}^{0}\\}$ with products of sines and cosines up to
frequencies $m$ inclusive and adding these contributions, i.e.,

(13bm) $\displaystyle a^{0}_{k,m}({\bf x})=$
$\displaystyle\sum_{q=0}^{m}\sum_{p=0}^{m}C_{1,p,q}\cos px\cos
qy+\sum_{q=1}^{m}\sum_{p=0}^{m}C_{2,p,q}\cos px\sin qy$
$\displaystyle+\sum_{q=0}^{m}\sum_{p=1}^{m}C_{3,p,q}\sin px\cos
qy+\sum_{q=1}^{m}\sum_{p=1}^{m}C_{4,p,q}\sin px\sin qy,$

where $C_{1,p,q}=\int_{\Omega}\cos px\cos qya^{0}_{k}({\bf x})\,\mathrm{d}{\bf
x}$, $C_{2,p,q}=\int_{\Omega}\cos px\sin qya^{0}_{k}({\bf x})\,\mathrm{d}{\bf
x}$, $C_{3,p,q}=\int_{\Omega}\sin px\cos qya^{0}_{k}({\bf x})\,\mathrm{d}{\bf
x}$, and $C_{4,p,q}=\int_{\Omega}\sin px\sin qya^{0}_{k}({\bf
x})\,\mathrm{d}{\bf x}$ (up to multiples of $2\pi$). The expression for
$b^{0}_{k,m}$ is similar. By construction,
$(a^{0}_{k,m},b^{0}_{k,m})\to(a^{0}_{k}b^{0}_{k})$ strongly in $L^{2}(\Omega)$
as $m\to\infty$.

By substituting appropriate test functions into the aforementioned weak
formulations for $a_{k}^{n,m}$ and $b_{k}^{n,m}$, we obtain a system of ODEs
with $\alpha^{k,m}_{j,p,q},\beta^{k,m}_{j,p,q}$ as the unkowns.

Testing against $\varphi=1$ in the equation for $a_{0}^{m}$ in Eq. 13 we find
$4\pi^{2}\frac{d\alpha^{0,m}_{1,0,0}}{dt}(t)=0$, whence
$\alpha^{0,m}_{1,0,0}(t)=\alpha^{0,m}_{1,0,0}(0)$ for all $t\in[0,T]$.
Moreover, testing against $\varphi=1$ in the equations for $a_{1}^{m}$ and
$b_{1}^{m}$ in Eq. 13, we obtain
$4\pi^{2}\frac{d\alpha^{k,m}_{1,0,0}}{dt}(t)=-4\pi^{2}k^{2}\alpha^{k,m}_{1,0,0}(t)$
,
$4\pi^{2}\frac{d\beta^{k,m}_{1,0,0}}{dt}(t)=-4\pi^{2}k^{2}\beta^{k,m}_{1,0,0}(t)$
for $k\in\\{1,\dots,n\\}$. All coefficients
$\\{\alpha^{k,m}_{1,0,0},\beta^{k,m}_{1,0,0}\\}_{k=0}^{n}$ are therefore
determined.

Then, testing the equations for $a_{k}^{n,m},b_{k}^{n,m}$ in Eq. 13 with
$\varphi=\cos px$ for $p\in\\{1,\dots,m\\}$, we get

$2\pi^{2}\frac{d\alpha^{0,m}_{1,p,0}}{dt}(t)-\text{Pe}\big{\langle}(1-(a_{0}^{m}(t))_{+})_{+}a_{1}^{m}(t),(-p\sin
px)\big{\rangle}=-D_{e}p^{2}2\pi^{2}\alpha^{0,m}_{1,p,0}(t),$ $\displaystyle
2\pi^{2}\frac{d\alpha^{1,m}_{1,p,0}}{dt}(t)-\frac{\text{Pe}}{2}\big{\langle}(1-(a_{0}^{m}(t))_{+})_{+}(2a_{0}^{m}(t)+$
$\displaystyle a_{2,m}(t)),(-p\sin px)\big{\rangle}=$ $\displaystyle-
D_{e}p^{2}2\pi^{2}\alpha^{1,m}_{1,p,0}(t)-2\pi^{2}\alpha^{1,m}_{1,p,0}(t),$
$\displaystyle
2\pi^{2}\frac{d\beta^{1,m}_{1,p,0}}{dt}(t)-\frac{\text{Pe}}{2}\big{\langle}(1-(a_{0}^{m}(t))_{+})_{+}b_{2,m}(t),(-$
$\displaystyle p\sin
px)\big{\rangle}=-D_{e}p^{2}2\pi^{2}\beta^{1,m}_{1,p,0}(t)$
$\displaystyle-2\pi^{2}\beta^{1,m}_{1,p,0}(t),$

while, for $1<k<n$,

$\displaystyle
2\pi^{2}\frac{d\alpha^{k,m}_{1,p,0}}{dt}(t)-\frac{\text{Pe}}{2}\big{\langle}(1-(a_{0}^{m}(t))_{+})_{+}(a_{k-1,m}(t)+$
$\displaystyle a_{k+1,m}(t)),(-p\sin px)\big{\rangle}=$ $\displaystyle-
D_{e}p^{2}2\pi^{2}\alpha^{k,m}_{1,p,0}(t)$
$\displaystyle-2\pi^{2}k^{2}\alpha^{k,m}_{1,p,0}(t),$ $\displaystyle
2\pi^{2}\frac{d\beta^{k,m}_{1,p,0}}{dt}(t)-\frac{\text{Pe}}{2}\big{\langle}(1-(a_{0}^{m}(t))_{+})_{+}(b_{k-1,m}(t)+$
$\displaystyle b_{k+1,m}(t)),(-p\sin px)\big{\rangle}=$ $\displaystyle-
D_{e}p^{2}2\pi^{2}\beta^{k,m}_{1,p,0}(t)$
$\displaystyle-2\pi^{2}k^{2}\beta^{k,m}_{1,p,0}(t),$

and

$\displaystyle
2\pi^{2}\frac{d\alpha^{n,m}_{1,p,0}}{dt}(t)-\frac{\text{Pe}}{2}\big{\langle}(1-(a_{0}^{m}(t))_{+})_{+}$
$\displaystyle a_{n-1,m}(t),(-p\sin
px)\big{\rangle}=-D_{e}p^{2}2\pi^{2}\alpha^{n,m}_{1,p,0}(t)$
$\displaystyle-2\pi^{2}n^{2}\alpha^{n,m}_{1,p,0}(t),$ $\displaystyle
2\pi^{2}\frac{d\beta^{n,m}_{1,p,0}}{dt}(t)-\frac{\text{Pe}}{2}\big{\langle}(1-(a_{0}^{m}(t))_{+})_{+}$
$\displaystyle b_{n-1,m}(t),(-p\sin
px)\big{\rangle}=-D_{e}p^{2}2\pi^{2}\beta^{n,m}_{1,p,0}(t)$
$\displaystyle-2\pi^{2}n^{2}\beta^{n,m}_{1,p,0}(t).$

The ODEs for the coefficients
$\\{\alpha_{3,p,0}^{k,m},\beta_{3,p,0}^{k,m}\\}_{k=0}^{n}$ are obtained
similarly by testing the equations with $\varphi=\sin px$ for
$p\in\\{1,\dots,m\\}$ and have a similar structure to those for
$\\{\alpha_{1,p,0}^{k,m},\alpha_{1,p,0}^{k,m}\\}_{k=0}^{n}$. The same is true
for the ODEs for the coefficients
$\\{\alpha_{1,0,q}^{k,m},\beta_{1,0,q}^{k,m}\\}_{k=0}^{n}$ (obtained by
testing Eq. 13 with $\varphi=\cos qy$ for $q\in\\{1,\dots,m\\}$) and those for
the coefficients $\\{\alpha_{2,0,q}^{k,m},\beta_{2,0,q}^{k,m}\\}_{k=0}^{n}$
(obtained by testing Eq. 13 $\varphi=\sin qy$ for $q\in\\{1,\dots,m\\}$).

Similarly, the ODEs for the coefficients
$\\{\alpha_{1,p,q}^{k,m},\beta_{1,p,q}^{k,m}\\}_{k=0}^{n}$ are obtained by
testing Eq. 13 with $\varphi=\cos px\cos qy$ for $p,q\in\\{1,\dots,m\\}$. In
detail, we obtain

$\displaystyle\frac{d\alpha^{0,m}_{1,p,q}}{dt}\\!=\\!\frac{\text{Pe}}{\pi^{2}}\big{\langle}(1\\!-\\!(a_{0}^{m}(t))_{+})_{+}$
$\displaystyle(a_{1}^{m}(t),b_{1}^{m}(t)),\\!(\\!-p\sin px\cos qy,\\!-q\sin
qy\cos px)\big{\rangle}_{\Omega}$ $\displaystyle-
D_{e}(p^{2}+q^{2})\alpha^{0,m}_{1,p,q}(t),$
$\displaystyle\pi^{2}\frac{d\alpha^{1,m}_{1,p,q}}{dt}(t)+D_{e}(p^{2}+q^{2})\pi^{2}\alpha^{1,m}_{1,p,q}(t)+\pi^{2}\alpha^{1,m}_{1,p,q}(t)=$
$\displaystyle\frac{\text{Pe}}{2}\big{\langle}(1\\!-\\!(a_{0}^{m}(t))_{+})_{+}(2a_{0}^{m}(t)\\!+\\!a_{2,m}(t),b_{2,m}(t)),(\\!-p\sin
px\cos qy,\\!-q\sin qy\cos px)\big{\rangle}_{\Omega},$
$\displaystyle\pi^{2}\frac{d\beta^{1,m}_{1,p,q}}{dt}(t)+D_{e}(p^{2}+q^{2})\pi^{2}\beta^{1,m}_{1,p,q}(t)+\pi^{2}\beta^{1,m}_{1,p,q}(t)=$
$\displaystyle\frac{\text{Pe}}{2}\big{\langle}(1\\!-\\!(a_{0}^{m}(t))_{+})_{+}(b_{2,m}(t),2a_{0}^{m}(t)\\!-\\!a_{2,m}(t)),(\\!-p\sin
px\cos qy,\\!-q\sin qy\cos px)\big{\rangle}_{\Omega},$

while, for $1<k<n$,

$\displaystyle\frac{2\pi^{2}}{\text{Pe}}\frac{d\alpha^{k,m}_{1,p,q}}{dt}(t)\\!+\\!\frac{2D_{e}}{\text{Pe}}(p^{2}\\!+\\!q^{2})\pi^{2}\alpha^{k,m}_{1,p,q}(t)+\frac{2\pi^{2}}{\text{Pe}}k^{2}\alpha^{k,m}_{1,p,q}(t)=$
$\displaystyle\big{\langle}\\!(\\!1\\!-\\!(a_{0}^{m})_{+})_{+}(a_{k-1,m}\\!+\\!a_{k+1,m},b_{k+1,m}\\!-\\!b_{k-1,m}),\\!(\\!-p\sin
px\cos qy,\\!-q\sin qy\cos px)\big{\rangle}_{\Omega},$
$\displaystyle\frac{2\pi^{2}}{\text{Pe}}\frac{d\beta^{k,m}_{1,p,q}}{dt}(t)\\!+\\!\frac{2D_{e}}{\text{Pe}}(p^{2}\\!+\\!q^{2})\pi^{2}\beta^{k,m}_{1,p,q}(t)+\frac{2\pi^{2}}{\text{Pe}}k^{2}\beta^{k,m}_{1,p,q}(t)=$
$\displaystyle\big{\langle}\\!(\\!1\\!-\\!(a_{0}^{m})_{+})_{+}(b_{k-1,m}\\!+\\!b_{k+1,m},a_{k-1,m}\\!-\\!a_{k+1,m}),\\!(\\!-p\sin
px\cos qy,\\!-q\sin qy\cos px)\big{\rangle}_{\Omega},$

and

$\displaystyle\pi^{2}\frac{d\alpha^{n,m}_{1,p,q}}{dt}(t)+D_{e}(p^{2}+q^{2})\pi^{2}\alpha^{n,m}_{1,p,q}(t)+\pi^{2}n^{2}\alpha^{n,m}_{1,p,q}(t)=$
$\displaystyle\frac{\text{Pe}}{2}\big{\langle}(1-(a_{0}^{m}(t))_{+})_{+}(a_{n-1,m}(t),-b_{n-1,m}(t)),(-p\sin
px\cos qy,-q\sin qy\cos px)\big{\rangle}_{\Omega},$
$\displaystyle\pi^{2}\frac{d\beta^{n,m}_{1,p,q}}{dt}(t)+D_{e}(p^{2}+q^{2})\pi^{2}\beta^{n,m}_{1,p,q}(t)+\pi^{2}n^{2}\beta^{n,m}_{1,p,q}(t)=$
$\displaystyle\frac{\text{Pe}}{2}\big{\langle}(1-(a_{0}^{m}(t))_{+})_{+}(b_{n-1,m}(t),a_{n-1,m}(t)),(-p\sin
px\cos qy,-q\sin qy\cos px)\big{\rangle}_{\Omega}.$

The ODEs for the coefficients
$\\{\alpha_{3,p,q}^{k,m},\beta_{3,p,q}^{k,m}\\}_{k=0}^{n}$ are obtained
similarly by testing the equations with $\varphi=\sin px\cos qy$ for
$p,q\in\\{1,\dots,m\\}$, and have a similar structure to those of
$\\{\alpha^{n,m}_{1,p,q},\beta^{n,m}_{1,p,q}\\}_{k=0}^{n}$. The same is true
for the ODEs for the coefficients
$\\{\alpha_{2,p,q}^{k,m},\beta_{2,p,q}^{k,m}\\}_{k=0}^{n}$ (obtained by
testing the equations with $\varphi=\cos px\sin qy$ for
$p,q\in\\{1,\dots,m\\}$) and those for the coefficients
$\\{\alpha_{4,p,q}^{k,m},\beta_{4,p,q}^{k,m}\\}_{k=0}^{n}$ (obtained by
testing the equations with $\varphi=\sin px\sin qy$ for
$p,q\in\\{1,\dots,m\\}$).

As was already mentioned in Step 1 of Section 3.2, the previous system may be
rewritten in the form of Eq. 13p, i.e.
$\Lambda_{m}^{\prime}(t)=F(t,\Lambda_{m}(t))$ for $t\in[0,T]$, with initial
condition $\Lambda_{m}(0)=\Lambda_{m,0}$, where
$\Lambda_{m}=\\{\alpha^{k,m}_{j,p,q}\\}_{k,j,p,q}$ is the tensor of
coefficients, with initial value $\Lambda_{m,0}$, and $F$ is locally Lipschitz
since the function $x\mapsto x_{+}^{2}$ is locally Lipschitz. The initial data
is given by the explicit formulas

(13bn) $\displaystyle
4\pi^{2}\alpha^{0,m}_{1,0,0}(0)=\int_{\Omega}a^{0}_{0,m}\,\mathrm{d}x\,\mathrm{d}y,$
$\displaystyle
4\pi^{2}\alpha_{1,0,0}^{k,m}(0)=\int_{\Omega}a^{0}_{k,m}(x,y)\,\mathrm{d}x\mathrm{d}y,$

where we recall the expression for $a^{0}_{k,m}$ in Eq. 13bm, and, for
$p,q\in\\{1,\dots,m\\}$,

(13bo)
$\displaystyle\alpha_{1,p,0}^{k,m}(0)\\!=\\!\int_{\Omega}\frac{a^{0}_{k,m}(x,y)}{2\pi^{2}}\cos
px\,\mathrm{d}x\,\mathrm{d}y,$
$\displaystyle\alpha_{3,p,0}^{k,m}(0)\\!=\\!\int_{\Omega}\frac{a^{0}_{k,m}(x,y)}{2\pi^{2}}\sin
px\,\mathrm{d}x\,\mathrm{d}y,$
$\displaystyle\alpha_{1,0,q}^{k,m}(0)\\!=\\!\int_{\Omega}\frac{a^{0}_{k,m}(x,y)}{2\pi^{2}}\cos
qy\,\mathrm{d}x\,\mathrm{d}y,$
$\displaystyle\alpha_{2,0,q}^{k,m}(0)\\!=\\!\int_{\Omega}\frac{a^{0}_{k,m}(x,y)}{2\pi^{2}}\sin
qy\,\mathrm{d}x\,\mathrm{d}y,$
$\displaystyle\alpha_{1,p,q}^{k,m}(0)\\!=\\!\int_{\Omega}\\!\frac{a^{0}_{k,m}(x,y)}{\pi^{2}}\\!\cos
px\\!\cos qy\mathrm{d}x\mathrm{d}y,\\!$
$\displaystyle\alpha_{3,p,q}^{k,m}(0)\\!=\\!\int_{\Omega}\\!\frac{a^{0}_{k,m}(x,y)}{\pi^{2}}\\!\sin
px\\!\cos qy\mathrm{d}x\mathrm{d}y,$
$\displaystyle\alpha_{2,p,q}^{k,m}(0)\\!=\\!\int_{\Omega}\\!\frac{a^{0}_{k,m}(x,y)}{\pi^{2}}\\!\cos
px\\!\sin qy\mathrm{d}x\mathrm{d}y,\\!$
$\displaystyle\alpha_{4,p,q}^{k,m}(0)\\!=\\!\int_{\Omega}\\!\frac{a^{0}_{k,m}(x,y)}{\pi^{2}}\\!\sin
px\\!\sin qy\mathrm{d}x\mathrm{d}y,$

and the formulas for the coefficients $\\{\beta_{i,p,q}^{k,m}\\}_{i,p,q,k,m}$
are identical, with the exception that $b^{0}_{k,m}$ replaces $a^{0}_{k,m}$ on
the right-hand sides of the previous equations. Note that the above are well-
defined real numbers due to the integrability of the initial data
$a^{0}_{k,m},b^{0}_{k,m}$ which is inherited from the integrability of
$f_{0}\in L^{2}_{per}(\Upsilon)$.

##  Appendix B Properties of the periodic heat kernel

Below we provide proofs of Lemma 4.2 and Lemma 4.3, respectively.

######  Proof B.1 (Proof of Lemma 4.2).

Step 1 (smoothness away from initial time): Begin by noting that, by a
standard argument using the Weierstraß M-test, the series representation Eq.
13al for $\Phi$ is well-defined, and moreover can repeatedly be differentiated
term-by-term for positive times. As such, $\Phi\in
C^{\infty}((0,\infty)\times\mathbb{R})$, and
$\partial_{t}\Phi(t,x)=-\frac{1}{\pi}\sum_{n=1}^{\infty}n^{2}e^{-n^{2}t}\cos
nx=\partial_{xx}\Phi(t,x)$ for all $(t,x)\in(0,\infty)\times\mathbb{R}$.

Step 2 (nonnegativity and $L^{1}$-norm): Recall (cf. [19, Section 21.3]) the
infinite product representation for the Jacobi theta-3 function. For every
$(t,x)\in(0,\infty)\times\mathbb{R}$,

$\Phi(t,x)=\frac{1}{2\pi}\theta_{3}\left(\frac{x}{2},e^{-t}\right)=\frac{1}{2\pi}\prod_{n=1}^{\infty}(1-e^{-2nt})(1+2e^{-t(2n-1)}\cos
x+e^{-t(4n-2)}).$

By the Young inequality, $1+e^{-t(4n-2)}\geq 2e^{-t(2n-1)}$, and hence every
term in the above product is nonnegative. As such, $\Phi$ itself is
nonnegative in $(0,\infty)\times\mathbb{R}$. Meanwhile, integrating the series
term by term, we obtain $\int_{0}^{2\pi}\Phi(t,x)\,\mathrm{d}x=1$ for every
$t\in(0,\infty)$. It therefore follows that
$\|\Phi(t,\cdot)\|_{L^{1}([0,2\pi])}=\int_{0}^{2\pi}\Phi(t,x)\,\mathrm{d}x=1$
for all $t\in(0,\infty)$. Also, the Plancherel Theorem yields
$\|\Phi(t,\cdot)\|^{2}_{L^{2}([0,2\pi])}=1+\sum_{n=1}^{\infty}e^{-2n^{2}t}$.
Integrating in time (using also $\sum n^{-2}<+\infty$) yields Eq. 13an.

Step 3 (constant $L^{2}$-norm): Furthermore, given any
$(t,x)\in(0,T]\times[0,2\pi]$, successive changes of variables yield

$\displaystyle\int_{0}^{t}\int_{0}^{2\pi}\Phi(t-s,x-y)^{2}\,\mathrm{d}y\,\mathrm{d}s$
$\displaystyle=\int_{0}^{t}\int_{0}^{2\pi}\Phi(\tau,x-y)^{2}\,\mathrm{d}y\,\mathrm{d}\tau$
$\displaystyle=\\!\int_{0}^{t}\int_{0}^{x}\\!\Phi(\tau,w)^{2}\,\mathrm{d}w\,\mathrm{d}\tau\\!+\\!\int_{0}^{t}\int_{x-2\pi}^{0}\\!\Phi(\tau,w\\!+\\!2\pi)^{2}\,\mathrm{d}w\,\mathrm{d}\tau$
$\displaystyle=\int_{0}^{t}\int_{0}^{2\pi}\Phi(\tau,w)^{2}\,\mathrm{d}w\,\mathrm{d}\tau,$

as required, where we also used the periodicity of $\Phi$ to obtain the second
equality.

Step 4 (convergence to initial data): It remains to verify the initial data.
Fix any $2\pi$-periodic $C^{2}$ function $\psi$. Then, the elementary theory
of Fourier series shows that $\psi$ admits the Fourier representation
$\psi(x)=\frac{1}{2\pi}a_{0}+\frac{1}{\pi}\sum_{n=1}^{\infty}a_{n}\cos
nx+\frac{1}{\pi}\sum_{n=1}^{\infty}b_{n}\sin nx$ a.e. $x\in\mathbb{R}$, where,
for $n\in\mathbb{N}$,

(13bp) $\displaystyle a_{0}=\int_{0}^{2\pi}\psi(x)\,\mathrm{d}x,$
$\displaystyle a_{n}=\int_{0}^{2\pi}\psi(x)\cos nx\,\mathrm{d}x,\quad
b_{n}=\int_{0}^{2\pi}\psi(x)\sin nx\,\mathrm{d}x.$

Note that, due to the $C^{2}$-regularity of $\psi$ and its periodicity, two
consecutive integration by parts yields
$a_{n}=-\frac{1}{n^{2}}\int_{0}^{2\pi}\psi^{\prime\prime}(x)\cos
nx\,\mathrm{d}x$ and
$b_{n}=-\frac{1}{n^{2}}\int_{0}^{2\pi}\psi^{\prime\prime}(x)\sin
nx\,\mathrm{d}x$ so that there exists a positive constant $C_{\psi}$ depending
on $\psi$ alone such that

(13bq) $|a_{n}|+|b_{n}|\leq C_{\psi}n^{-2}\qquad\forall n\in\mathbb{N}.$

Hence,
$|\psi(0)-\int_{0}^{2\pi}\Phi(t,x)\psi(x)\,\mathrm{d}x|=\frac{1}{\pi}|\sum_{n=1}^{\infty}(1-e^{-n^{2}t})a_{n}|\leq\frac{C_{\psi}}{\pi}\sum_{n=1}^{\infty}n^{-2}(1-e^{-n^{2}t})$,
and, by the Dominated Convergence Theorem applied to the atomic measure (using
that $\sum n^{-2}<+\infty$), the right-hand side vanishes in the limit as
$t\to 0^{+}$.

######  Proof B.2 (Proof of Lemma 4.3).

Recall from Lemma 4.2 that $\Phi\in C^{\infty}((0,\infty)\times\mathbb{R})$.
Since $\psi\in L^{1}([0,2\pi])$, an application of the Dominated Convergence
Theorem shows that $\Psi\in C^{\infty}((0,\infty)\times\mathbb{R})$, and that
for any $m,n\in\mathbb{N}\cup\\{0\\}$, there holds
$\partial_{t}^{(m)}\partial_{x}^{(n)}\Psi(t,x)=\int_{0}^{2\pi}\partial_{t}^{(m)}\partial_{x}^{(n)}\Phi(t,x-y)\psi(y)\,\mathrm{d}y$
for all $(t,x)\in(0,\infty)\times\mathbb{R}$. In turn, we directly obtain that
$\partial_{t}\Psi-\partial_{xx}\Psi=0$ pointwise in
$(0,\infty)\times\mathbb{R}$.

It remains to verify the initial condition. Suppose for the time being that
$\psi$ is smooth enough, say $\psi\in C^{2}_{per}([0,2\pi])$. From the Fourier
series decomposition Eq. 13bp for $\psi$ we compute, using the Fubini–Tonelli
Theorem (or directly using the convolution result for Fourier series), the
Fourier coefficients of $\Psi$:

(13br)
$\displaystyle\tilde{a}_{0}(t)=a_{0},\quad\tilde{a}_{n}(t)=e^{-n^{2}t}a_{n},\quad\tilde{b}_{n}(t)=e^{-n^{2}t}b_{n}\qquad\text{for
}n\in\mathbb{N}.$

Using the Plancherel Theorem for Fourier series of square integrable functions
we get (up to a multiplicative constant)

(13bs)
$\|\Psi(t,\cdot)-\psi\|^{2}_{L^{2}([0,2\pi])}=\sum_{n=1}^{\infty}(1-e^{-n^{2}t})^{2}(a_{n}^{2}+b_{n}^{2})\to
0\qquad\text{as }t\to 0^{+},$

where we used the decay estimate Eq. 13bq and the Dominated Convergence
Theorem applied to the atomic measure.

When $\psi$ is only $L^{1}_{per}([0,2\pi])$, we consider
$\\{\psi_{m}\\}_{m\in\mathbb{N}}$ a sequence in $C^{2}_{per}([0,2\pi])$
approximating $\psi$ in $L^{1}([0,2\pi])$. Then, given any $\varepsilon>0$,
there exists $N(\varepsilon)\in\mathbb{N}$ such that
$\|\psi-\psi_{m}\|_{L^{1}([0,2\pi])}<\varepsilon/3$ $\forall m\geq
N(\varepsilon)$. We then have

$\displaystyle\|\Psi(t,\cdot)-\psi\|_{L^{1}([0,2\pi])}\leq\\!\|\Psi(t,\\!\cdot)-\Psi_{m}(t,\cdot)\|_{L^{1}([0,2\pi])}$
$\displaystyle+\|\Psi_{m}(t,\cdot)-\psi_{m}\|_{L^{1}([0,2\pi])}$
$\displaystyle+\|\psi_{m}-\psi\|_{L^{1}([0,2\pi])},$

where $\Psi_{m}=\int_{0}^{2\pi}\Phi(t,x-y)\psi_{m}(y)\,\mathrm{d}y$. Note that
the second term in the previous expression vanishes in the limit as $t\to
0^{+}$ by the argument in the previous paragraph. Next, using the
Fubini–Tonelli Theorem to exchange the order of the integrals,
$\|\Psi(t,\cdot)-\Psi_{m}(t,\cdot)\|_{L^{1}([0,2\pi])}\leq\|\Phi(t,\cdot)\|_{L^{1}([0,2\pi])}\|\psi-\psi_{m}\|_{L^{1}([0,2\pi])}=\|\psi-\psi_{m}\|_{L^{1}([0,2\pi])},$
where we used from Lemma 4.2 that $\|\Phi(t,\cdot)\|_{L^{1}([0,2\pi])}=1$ for
every $t\in(0,\infty)$. It follows that
$\|\Psi(t,\cdot)-\psi\|_{L^{1}([0,2\pi])}<2\varepsilon/3+\|\Psi_{m}-\psi_{m}\|_{L^{1}([0,2\pi])}$
for all $m\geq N(\varepsilon)$. Thus, having fixed $m$ large enough, by now
letting $t\to 0^{+}$ and using the result from the previous paragraph, we get
$\|\Psi(t,\cdot)-\psi\|_{L^{1}([0,2\pi])}$ vanishes in the limit as $t\to
0^{+}$.

##  Appendix C Approximation Lemma

######  Lemma C.1.

Let $f\in L^{2}_{per}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))$ be nonnegative
and such that $\rho({\bf x}):=\langle f({\bf
x},\cdot),1\rangle_{(H^{1}_{per})^{\prime}\times H^{1}_{per}}\in[0,1]$ for
a.e. ${\bf x}\in\Omega$. There exists a sequence
$(f_{\varepsilon})_{\varepsilon>0}$ of nonnegative elements of
$C^{\infty}_{per}(\bar{\Upsilon})$ such that

(13bt)
$\|f-f_{\varepsilon}\|_{L^{2}_{per}(\Omega;(H^{1}_{per})^{\prime}(0,2\pi))}\to
0\qquad\text{as }\varepsilon\to 0,$

with $\rho_{\varepsilon}({\bf x})=\int_{0}^{2\pi}f_{\varepsilon}({\bf
x},\theta)\,\mathrm{d}\theta\in[0,1]$ for a.e. ${\bf x}\in\Omega$.

######  Proof C.2.

Step 1 (defining the approximation): Let $\beta\in
C^{\infty}_{c}(\mathbb{R}^{2})$ and $\gamma\in C^{\infty}_{c}(\mathbb{R})$ be
nonnegative smooth bump functions, chosen such that $\beta$ is compactly
supported inside $\Omega$ and $\gamma$ is compactly supported inside the
interval $[0,2\pi)$, and without loss of generality

(13bu)
$\bigg{(}\int_{\Omega}\beta(\mathbf{y})\,\mathrm{d}\mathbf{y}\bigg{)}\bigg{(}\int_{0}^{2\pi}\gamma(\theta)\,\mathrm{d}\theta\bigg{)}=1.$

Moreover, we impose that $\beta$ is radially symmetric about the point
$(\pi,\pi)$ and that $\gamma$ is symmetric about the point $\pi$. Consider the
associated sequences $(\beta_{\varepsilon})_{\varepsilon>0}$ and
$(\gamma_{\varepsilon})_{\varepsilon>0}$

$\beta_{\varepsilon}({\bf x}):=\varepsilon^{-2}\beta({\bf
x}/\varepsilon),\qquad\gamma_{\varepsilon}(\theta):=\varepsilon^{-\alpha}\gamma(\theta/\varepsilon^{\alpha}),$

for $\alpha>2$. These functions are smooth and nonnegative, and
$\beta_{\varepsilon}$ is compactly supported in the square $\varepsilon\Omega$
while $\gamma_{\varepsilon}$ is compactly supported in
$[0,2\pi\varepsilon^{\alpha}]$. We define $\tilde{\gamma}_{\varepsilon}$ to be
the $2\pi$-periodic extension of $\gamma_{\varepsilon}$ from
$[0,2\pi\varepsilon^{\alpha}]$ to all of $\mathbb{R}$, so that
$\tilde{\gamma}_{\varepsilon}\in H^{1}_{per}(0,2\pi)$. Similarly, we define
$\tilde{\beta}_{\varepsilon}$ to be the $\Omega$-periodic extension of
$\beta_{\varepsilon}$ to all of $\mathbb{R}^{2}$.

In turn, we define the sequence of approximations
$(f_{\varepsilon})_{\varepsilon>0}$ by the explicit formula

$f_{\varepsilon}({\bf
x},\theta):=\int_{\Omega}\tilde{\beta}_{\varepsilon}({\bf
x}-\mathbf{y})\langle
f(\mathbf{y},\cdot),\tilde{\gamma}_{\varepsilon}(\theta-\cdot)\rangle\,\mathrm{d}\mathbf{y}\qquad\text{a.e.~{}}({\bf
x},\theta)\in\Upsilon,$

where, throughout the rest of this proof, the duality product is understood to
be in the angle variable only, and denotes the duality product of
$H^{1}_{per}(0,2\pi)$.

Observe firstly that, due to the assumption that $f$ is nonnegative, we
immediately have $f_{\varepsilon}$ nonnegative a.e. in $\Upsilon$. Moreover,
one may readily check (for instance by means of difference quotients) that
$f_{\varepsilon}$ is smooth. Similarly, it is straightforward to verify that
$f_{\varepsilon}$ is itself $\Upsilon$-periodic. Thus, $f_{\varepsilon}\in
C^{\infty}_{per}(\bar{\Upsilon})$ for every $\varepsilon>0$.

Step 2 (boundedness the approximate density): Note that, in view of the
aforementioned nonnegativity of $f_{\varepsilon}$, there holds the inequality
$0\leq\int_{0}^{2\pi}f_{\varepsilon}({\bf
x},\theta)\,\mathrm{d}\theta=\rho_{\varepsilon}({\bf x})$. Additionally, we
have $\rho_{\varepsilon}({\bf
x})=\int_{0}^{2\pi}(\int_{\Omega}\tilde{\beta}_{\varepsilon}({\bf
x}-\mathbf{y})\langle
f(\mathbf{y},\cdot),\tilde{\gamma}_{\varepsilon}(\theta-\cdot)\rangle\,\mathrm{d}\mathbf{y})\,\mathrm{d}\theta$,
and a standard argument involving Riemann sums and the Arzelà–Ascoli Theorem
(using also the Fubini–Tonelli Theorem and the Dominated Convergence Theorem)
leads to being able to commute the duality bracket with the integral in the
angle variable, i.e.,

(13bv) $\rho_{\varepsilon}({\bf
x})=\int_{\Omega}\tilde{\beta}_{\varepsilon}({\bf
x}-\mathbf{y})\big{\langle}f(\mathbf{y},\cdot),\int_{0}^{2\pi}\tilde{\gamma}_{\varepsilon}(\theta-\cdot)\,\mathrm{d}\theta\big{\rangle}\,\mathrm{d}\mathbf{y}.$

Observe now that the integral inside the duality bracket is a constant.
Indeed, since $\tilde{\gamma}_{\varepsilon}$ is smooth, and therefore bounded
over any compact interval, the Dominated Convergence Theorem yields
$\frac{d}{ds}\int_{0}^{2\pi}\tilde{\gamma}_{\varepsilon}(\theta-s)\,\mathrm{d}\theta=-\int_{0}^{2\pi}\tilde{\gamma}_{\varepsilon}^{\prime}(\theta-s)\,\mathrm{d}\theta=\tilde{\gamma}_{\varepsilon}(-s)-\tilde{\gamma}_{\varepsilon}(2\pi-s)=0$,
where the final equality follows from the periodicity of
$\tilde{\gamma}_{\varepsilon}$; recall that it is the periodic extension to
all of $\mathbb{R}$ of the original mollifier $\gamma_{\varepsilon}$. It
follows that

(13bw)
$\displaystyle\int_{0}^{2\pi}\tilde{\gamma}_{\varepsilon}(\theta-s)\,\mathrm{d}\theta=\int_{0}^{2\pi}\tilde{\gamma}_{\varepsilon}(\theta)\,\mathrm{d}\theta=\int_{0}^{2\pi}\gamma_{\varepsilon}(\theta)\,\mathrm{d}\theta=\int_{0}^{2\pi}\gamma(\theta)\,\mathrm{d}\theta\qquad\forall
s\in\mathbb{R},$

where we recall in passing that $\gamma_{\varepsilon}$ is supported in
$[0,2\pi\varepsilon^{\alpha}]$, while $\gamma$ is supported in $[0,2\pi]$.
Hence, we get $\rho_{\varepsilon}({\bf
x})=(\int_{0}^{2\pi}\gamma(\theta)\,\mathrm{d}\theta)(\int_{\Omega}\tilde{\beta}_{\varepsilon}({\bf
x}-\mathbf{y})\langle f(\mathbf{y},\cdot),1\rangle\,\mathrm{d}\mathbf{y})$.

Also, applying the Dominated Convergence Theorem and the Fubini–Tonelli
Theorem gives $\partial_{x_{1}}\int_{\Omega}\tilde{\beta}_{\varepsilon}({\bf
x}-\mathbf{y})\,\mathrm{d}\mathbf{y}=\int_{0}^{2\pi}\int_{0}^{2\pi}\partial_{1}\tilde{\beta}_{\varepsilon}(x_{1}-y_{1},x_{2}-y_{2})\,\mathrm{d}y_{1}\,\mathrm{d}y_{2}=-\int_{0}^{2\pi}(\tilde{\beta}_{\varepsilon}(x_{1}-2\pi,x_{2}-y_{2})-\tilde{\beta}_{\varepsilon}(x_{1},x_{2}-y_{2}))\,\mathrm{d}y_{2}=0$,
where the final equality follows from the periodicity of
$\tilde{\beta}_{\varepsilon}$; recall that this latter function is the
$\Omega$-periodic extension of $\beta_{\varepsilon}$. An identical argument
shows that $\partial_{x_{2}}\int_{\Omega}\tilde{\beta}_{\varepsilon}({\bf
x}-\mathbf{y})\,\mathrm{d}\mathbf{y}=0$. Arguing as per Eq. 13bw and making
use of the symmetry property of $\beta$ about the point $(\pi,\pi)$, we deduce

(13bx) $\int_{\Omega}\tilde{\beta}_{\varepsilon}({\bf
x}-\mathbf{y})\,\mathrm{d}\mathbf{y}=\int_{\Omega}\beta(\mathbf{y})\,\mathrm{d}\mathbf{y}\qquad\forall{\bf
x}\in\mathbb{R}^{2}.$

We emphasise that the above property is not true for the function