# Nonlocal-to-local convergence of the Cahn-Hilliard equation with degenerate
mobility and the Flory-Huggins potential.

Charles Elbar Sorbonne Université, CNRS, Université de Paris, Inria,
Laboratoire Jacques-Louis Lions, F-75005 Paris, France email:
<EMAIL_ADDRESS>Jakub Skrzeczkowski Mathematical
Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, United
Kingdom; Faculty of Mathematics, Informatics and Mechanics, University of
Warsaw, Stefana Banacha 2, 02-097 Warsaw, Polandemail:
<EMAIL_ADDRESS>
###### Abstract

The Cahn-Hilliard equation is a fundamental model for phase separation
phenomena. Its rigorous derivation from the nonlocal aggregation equation,
motivated by the desire to link interacting particle systems and continuous
descriptions, has received much attention in recent years. In the recent
article, we showed how to treat the case of degenerate mobility for the first
time. Here, we discuss how to adapt the exploited tools to the case of the
mobility $m(u)=u\,(1-u)$ as in the original works of Giacomin-Lebowitz and
Elliott-Garcke. The main additional information is the boundedness of $u$,
implied by the form of mobility, which allows handling the nonlinear terms. We
also discuss the case of (mildly) singular kernels and a model of cell-cell
adhesion with the same mobility.

2010 Mathematics Subject Classification. 35B40, 35D30, 35K25, 35K55.
Keywords and phrases. degenerate Cahn-Hilliard equation; nonlocal Cahn-
Hilliard equation; aggregation-diffusion; singular limit; cell-cell adhesion.

##  1 Introduction

The question of nonlocal-to-local convergence for the Cahn-Hilliard equation
is of fundamental nature, linking the models derived from particle processes
or interacting particle systems [40] and from physical principles [7, 8]. This
question was recently addressed by several authors [20, 18, 50, 19] but only
for the case of constant mobility. Our recent article [25] was the first one
to establish rigorously the convergence for the degenerate mobility $m(u)=u$
and smooth potential as a consequence of nonlocal compactness results
developed by Bourgain-Brezis-Mironcescu [3] and Ponce [55]. In the current
paper we conclude our analysis by showing that the reasoning from [25] can be
extended to the mobility $m(u)=u\,(1-u)$ and the logarithmic Flory-Huggins
potential as in the original works of Giacomin-Lebowitz [40] and Elliot-Garcke
[27]. We also consider singular nonlocal interaction kernels in the spirit of
[20]. In the last section, using similar techniques, we also discuss the
nonlocal-to-local limit of a model of cell-cell adhesion with the same
mobility introduced in [13].

We consider the nonlocal Cahn-Hilliard equation with degenerate mobility

$\begin{split}\partial_{t}u=\operatorname{div}(m(u)\nabla\mu)\quad\text{in}\quad&(0,+\infty)\times\mathbb{T}^{d},\\\
\mu=B[u]+F^{\prime}(u)\quad\text{in}\quad&(0,+\infty)\times\mathbb{T}^{d},\end{split}$
(1.1)

equipped with an initial datum $u^{0}\geq 0$. Here, the mobility is
$m(u)=u(1-u)$, $\mathbb{T}^{d}$ is the $d$-dimensional flat torus, which is
particularly useful when treating nonlocal operators, $B$ is the nonlocal
operator $B=B_{\varepsilon}$ defined with

$B_{\varepsilon}[u](x)=(J_{\varepsilon}\ast 1)u(x)-J_{\varepsilon}\ast
u(x)=\int_{\mathbb{T}^{d}}J_{\varepsilon}(x-y)(u(x)-u(y))\mathop{}\\!\mathrm{d}y.$
(1.2)

The convolution kernel $J_{\varepsilon}$ is of the form

$J_{\varepsilon}(x)=\frac{\omega_{\varepsilon}(x)}{\varepsilon^{2-\alpha}|x|^{\alpha}}$
(1.3)

for $0\leq\alpha<d-1$, where $\\{\omega_{\varepsilon}\\}_{\varepsilon}$ is the
family of standard mollifiers:

$\omega_{\varepsilon}(x)=\frac{1}{\varepsilon^{d}}\omega\left(\frac{x}{\varepsilon}\right)$
(1.4)

with $\omega$ being a nonnegative smooth radially symmetric kernel which is
compactly supported in the unit ball of $\mathbb{R}^{d}$ satisfying

$\int_{\mathbb{T}^{d}}\frac{\omega(y)y_{i}y_{j}}{|y|^{\alpha}}\mathop{}\\!\mathrm{d}y=\frac{W}{d}\delta_{i,j},$
(1.5)

for some $W>0$.

######  Remark 1.1.

The range $\alpha\in[0,d-1)$ ensures that the kernel is in
$W^{1,1}_{loc}(\mathbb{R}^{d})$ and therefore we can apply the existing theory
of the existence of weak solutions for the nonlocal equation. This type of
kernel has for instance been considered in [19].

The potential $F$ is the usual double-well logarithmic potential

$F(s)=s\log(s)+(1-s)\log(1-s)-\theta\left(s-\frac{1}{2}\right)^{2}+k,$ (1.6)

where $\theta>1$ and $k$ is a constant such that $F(s)\geq 0$ for all
$s\in(0,1)$. Note in particular the identities

$F^{\prime}(s)=\log\left(\frac{s}{1-s}\right)+\theta(1-2s),\quad
F^{\prime\prime}(s)=\frac{1}{s(1-s)}-2\theta,\quad
m(s)F^{\prime\prime}(s)=1-2\theta s(1-s).$ (1.7)

We impose the constraint on the parameter $\theta$

$2\,\theta<\inf_{\varepsilon<\varepsilon_{0}}(J_{\varepsilon}\ast 1)$ (1.8)

for some $\varepsilon_{0}$. This is always satisfied for $\varepsilon_{0}$
small enough since

$J_{\varepsilon}\ast
1=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{d}}\frac{\omega(y)}{|y|^{\alpha}}\mathop{}\\!\mathrm{d}y.$

As we intend to send $\varepsilon\to 0$, we can assume that this condition is
always satisfied. The condition (1.8) ensures that the coefficient
$(J_{\varepsilon}\ast 1)+F^{\prime\prime}(s)$ (standing next to $\nabla
u_{\varepsilon}$ in (1.1)) is strictly positive and the existence as well as
uniqueness of the weak solution follows.

Our target is to prove that as $\varepsilon\to 0$, the solutions of

$\displaystyle\partial_{t}u_{\varepsilon}=\operatorname{div}(m(u_{\varepsilon})\nabla\mu_{\varepsilon})\quad\text{in}\quad$
$\displaystyle(0,+\infty)\times\mathbb{T}^{d},$ (1.9)
$\displaystyle\mu_{\varepsilon}=B_{\varepsilon}[u_{\varepsilon}]+F^{\prime}(u_{\varepsilon})\quad\text{in}\quad$
$\displaystyle(0,+\infty)\times\mathbb{T}^{d},$ (1.10)

tend to the weak solution of the degenerate Cahn-Hilliard equation

$\displaystyle\partial_{t}u=\operatorname{div}(m(u)\nabla\mu)\quad\text{in}\quad$
$\displaystyle(0,+\infty)\times\mathbb{T}^{d},$ (1.11)
$\displaystyle\mu=-\frac{W}{2}\Delta u+F^{\prime}(u)\quad\text{in}\quad$
$\displaystyle(0,+\infty)\times\mathbb{T}^{d}.$ (1.12)

Our main result reads as follows.

######  Theorem 1.2 (Convergence of nonlocal to local Cahn-Hilliard equation
on the torus).

Let $u^{0}\geq 0$ be an initial datum with finite energy and entropy
$\sup_{\varepsilon}E_{\varepsilon}(u^{0}),\sup_{\varepsilon}\Phi_{\varepsilon}(u^{0})<\infty$
defined in (3.1) and (3.2). Let $\\{u_{\varepsilon}\\}_{\varepsilon}$, with
$\varepsilon<\min(\varepsilon_{A},\varepsilon_{B})$ defined in Lemmas C.1-C.3,
be a sequence of solutions of the nonlocal Cahn-Hilliard equation (1.9)-(1.10)
as in Proposition 3.1. Then, along a subsequence not relabeled,

$u_{\varepsilon}\to u\mbox{ in
}L^{2}(0,T;H^{1}(\mathbb{T}^{d}))\cap{L^{p}((0,T)\times\mathbb{T}^{d})},\,\forall
p\in[1,\infty),$

where $u$ is a weak solution of the degenerate Cahn-Hilliard equation
(1.11)-(1.12) as defined in Definition 4.1.

######  Remark 1.3.

If $u_{0}\in H^{1}(\mathbb{T}^{d})$, then
$\sup_{\varepsilon}E_{\varepsilon}(u^{0}),\sup_{\varepsilon}\Phi_{\varepsilon}(u^{0})<\infty$,
see [3, Theorem 1].

The main contribution of Theorem 1.2 is the possibility to consider more
general, nonlinear mobilities $m(u)$. In [25], where we considered the
particular mobility $m(u)=u$, we exploited the formula

$\nabla m(u)\,\Delta u=\nabla u\,\Delta u=\operatorname{div}(\nabla
u\otimes\nabla u)-\frac{1}{2}\nabla|\nabla u|^{2}$

in a crucial way. This is no longer available when $m(u)=u\,(1-u)$. However,
the more general term can be still handled because the singular Flory-Huggings
potential ensures that the sequence $\\{u_{\varepsilon}\\}$ is uniformly
bounded in $L^{\infty}((0,T)\times\mathbb{T}^{d})$.

We also would like to point out that while the setting above is formulated for
the standard choice $m(u)=u\,(1-u)$, the same is also true for more general
case $m(u)=u^{k}\,(1-u)^{l}$ where $k,l\geq 1$. In particular, the proof of
Theorem 1.2 and of the crucial Lemma 4.5, is formulated in so general way that
it uses only the fact that $m$ is a $C^{1}$ function.

The structure of the paper is as follows. In Section 2 we review the existing
literature on the problem while in Section 3 we recall the basic properties of
the system (1.9)-(1.10) (well-posedness, energy, and entropy estimates).
Section 4 is dedicated to the proof of the main results. In Section 5 we
present numerical simulations which show certain qualitative differences
between the local and nonlocal models. The last section, Section 6, is devoted
to a model of cell-cell adhesion phenomena from [13], which also includes the
mobility $m(u)=u\,(1-u)$. For this model we also obtain a nonlocal-to-local
convergence result.

##  2 Motivations and litterature review

The non-local Cahn-Hilliard equation was initially derived by Giacomin and
Lebowitz [40, 41] through a microscopic approach. Their model is based on a
$d$-dimensional lattice gas that evolves through Kawasaki exchange dynamics,
which is a Poisson process that exchanges nearest neighbors. In the
hydrodynamic limit, the average of the occupation numbers over a small
macroscopic volume element tends towards a solution of a non-local Cahn-
Hilliard equation. This equation is an approximation of the local Cahn-
Hilliard equation, as demonstrated in Theorem 1.2. The literature concerning
the nonlocal Cahn-Hilliard equation is quite well-developed and we refer to
[37, 39, 2, 42, 56, 15, 38, 54, 33, 34] and [31, 35, 47, 36, 48, 29, 30, 32]
and the references therein for the cases of non-degenerate and degenerate
mobilities respectively. For the local case, we refer for instance to [52, 27,
53, 57].

The convergence of the nonlocal Cahn-Hilliard equation to its local
counterpart is a relatively new topic of research. The first result in this
area was achieved in 2019 by Melchiona, Ranetbauer, Scarpa, and Trussardi on
the torus with constant mobilities [50]. The study was subsequently extended
to different types of kernels, potentials, and Neumann boundary conditions in
[18, 20, 19, 51]. Furthermore, due to the higher regularity of solutions for
the constant mobility, very recently Abels and Hurm obtained an explicit rate
of convergence in this case [1]. The proof does not work in the case of
degenerate mobility.

Concerning the degenerate mobility, in [25], we investigated the case $m(u)=u$
with a smooth potential, motivated by biological applications [24, 23]. This
result has been extended to the case of systems in [10] which appear in
modeling of cell-cell adhesion [28, 13]. The result presented in this paper
(Theorem 1.2) completes the nonlocal-to-local convergence program by examining
the mobility $m(u)=u(1-u)$ and logarithmic potential, which is a typical
framework in the analysis of the Cahn-Hilliard equation.

Finally, we wish to put our research in a broader context. First, we remark
that (1.9)–(1.10) can be interpreted as a gradient flow in the Wasserstein
space with nonlinear mobility. Such problems have been extensively studied
recently [12, 49, 46, 45, 4, 44, 21, 14]. Second, the question of passing to
the limit from a nonlocal equation to a local one attracts researchers
studying different equations. This includes porous media equation [11, 5, 43,
22, 9], hyperbolic conservation laws [16] and cross-diffusion systems [26,
17].

##  3 Existence of solutions

In [41], Giacomin and Lebowitz proved the existence and uniqueness of a weak
solution to (1.9)-(1.10) on the torus. In [31] Frigeri, Gal, and Grasselli
obtain the existence and uniqueness in the case of a bounded domain with
Neumann boundary conditions. The following two propositions can be adapted
from their work.

######  Proposition 3.1 (Existence of weak solutions).

[41, Theorem 4.1] or [31, Theorem 2.3]. Let $u^{0}$ be a measurable function
such that $F(u_{0})\in L^{1}(\mathbb{T}^{d})$ and $\phi(u_{0})\in
L^{1}(\mathbb{T}^{d})$ where $\phi\in C^{2}(0,1)$ is defined as
$m^{\prime\prime}(s)\phi(s)=1$, $\phi(0)=\phi^{\prime}(0)=0$. Then there
exists a unique weak solution $u$ of (1.9)-(1.10) such that $u\in
L^{2}(0,T;H^{1}(\mathbb{T}^{d}))$, $0\leq u\leq 1$ a.e. in
$(0,T)\times\mathbb{T}^{d}$ and for all $\varphi\in
C_{c}^{\infty}([0,T)\times\mathbb{T}^{d})$ we have

$-\int_{0}^{T}\int_{\mathbb{T}^{d}}u\,\partial_{t}\varphi\mathop{}\\!\mathrm{d}t-\int_{\mathbb{T}^{d}}u^{0}(x)\,\varphi(0,x)\mathop{}\\!\mathrm{d}x+\int_{0}^{T}\int_{\mathbb{T}^{d}}m(u)F^{\prime\prime}(u)\nabla
u\nabla\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t+\\\
+\int_{0}^{T}\int_{\mathbb{T}^{d}}m(u)\nabla((J_{\varepsilon}\ast
1)u-J_{\varepsilon}\ast
u)\nabla\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=0.$

######  Remark 3.2.

The weak solutions defined in Proposition 3.1 can be shown to have higher
regularity (under some assumptions on the initial condition). In particular,
it is possible to prove the existence/uniqueness of a strong solution
satisfying the strict separation property [31].

…
Let us sketch quickly the proof of the bound $0\leq u\leq 1$ which is the most
important for the current work. Let $0\leq u^{0}\leq 1$ and let
$\mbox{sign}_{+}(u-1)=\mathds{1}_{u>1}$. Multiplying (1.9)-(1.10) by
$\mbox{sign}_{+}(u-1)$ and using the notation
$|u-1|_{+}=(u-1)\mathds{1}_{u>1}$ we get

$\partial_{t}|u_{\varepsilon}-1|_{+}=-\operatorname{div}(u_{\varepsilon}\,|u_{\varepsilon}-1|_{+}\,\nabla\mu_{\varepsilon})$

so that
$\partial_{t}\int_{\mathbb{T}^{d}}|u_{\varepsilon}-1|_{+}\mathop{}\\!\mathrm{d}x=0$.
The bound follows from the one on the initial condition.

It can be further proved that the solutions in Proposition 3.1 satisfy the
following energy/entropy estimates (at least as inequalities) uniformly in
$\varepsilon$:

######  Proposition 3.3 (Energy/ entropy estimates).

Let $u_{\varepsilon}$ be a sequence of weak solutions as defined in
Proposition 3.1 with same initial condition $u^{0}$ (one can also consider a
sequence of initial conditions which converges strongly). Then, defining the
energy and entropy:

$\displaystyle
E_{\varepsilon}[u]:=\int_{\mathbb{T}^{d}}F(u)\mathop{}\\!\mathrm{d}x+\frac{1}{4}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{\omega_{\varepsilon}(y)}{\varepsilon^{2-\alpha}|y|^{\alpha}}|u(x)-u(x-y)|^{2}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y,$
(3.1)
$\displaystyle\Phi[u]:=\int_{\mathbb{T}^{d}}\phi(u)\mathop{}\\!\mathrm{d}x,\quad\phi^{\prime\prime}(u)=\frac{1}{m(u)},\,\phi(0)=\phi^{\prime}(0)=0,$
(3.2)

we have

$\displaystyle\int_{\mathbb{T}^{d}}u_{\varepsilon}(t,\cdot)\mathop{}\\!\mathrm{d}x=\int_{\mathbb{T}^{d}}u^{0}\mathop{}\\!\mathrm{d}x,$
(3.3)
$\displaystyle\frac{d}{dt}E[u_{\varepsilon}]+\int_{\mathbb{T}^{d}}m(u_{\varepsilon})\,|\nabla\mu_{\varepsilon}|^{2}\leq
0,$ (3.4)
$\displaystyle\frac{d}{dt}\Phi[u_{\varepsilon}]+\frac{1}{2}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{\omega_{\varepsilon}(y)}{\varepsilon^{2-\alpha}|y|^{\alpha}}\,|\nabla
u_{\varepsilon}(x)-\nabla
u_{\varepsilon}(x-y)|^{2}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y+\int_{\mathbb{T}^{d}}F^{\prime\prime}(u_{\varepsilon})|\nabla
u_{\varepsilon}|^{2}\mathop{}\\!\mathrm{d}x\leq 0,$ (3.5)

where $\mu_{\varepsilon}$ is the chemical potential defined in (1.10).

Roughly speaking, to see these estimates, one multiplies Equation (1.9) by
$\mu$ to obtain the energy and by $\phi^{\prime}(u)$ to obtain the entropy.

Since $F$ is not convex (i.e. $F^{\prime\prime}$ can be negative), we need to
justify that the entropy estimates (3.5) indeed provide a priori estimates.
More precisely, we need to prove that the negative part of the integral
$\int_{\mathbb{T}^{d}}F^{\prime\prime}(u_{\varepsilon})|\nabla
u_{\varepsilon}|^{2}\mathop{}\\!\mathrm{d}x$ is controlled where the negative
part of $F^{\prime\prime}$ equals $-2\,\theta$. By (3.4), we know that the
sequence $\\{u_{\varepsilon}\\}$ is bounded in
$L^{\infty}(0,T;L^{2}(\mathbb{T}^{d}))$. Hence, the sufficient control comes
from Lemma C.3 with $\gamma=\frac{1}{8\theta}$

$2\,\theta\int_{\mathbb{T}^{d}}|\nabla
u_{\varepsilon}|^{2}\leq\frac{1}{4}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{|\nabla
u_{\varepsilon}(x)-\nabla
u_{\varepsilon}(x-y)|^{2}}{\varepsilon^{2-\alpha}|y|^{\alpha}}\omega_{\varepsilon}(|y|)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y+C(\theta)\|u_{\varepsilon}\|^{2}_{L^{2}(\mathbb{T}^{d})}.$
(3.6)

With the existence of a unique weak solution with energy/entropy bounds for
all $\varepsilon$ small enough, we are now in a position to pass to the local
limit $\varepsilon\to 0$.

##  4 Convergence $\varepsilon\to 0$

We first define the weak solutions to the local Cahn-Hilliard equation.

######  Definition 4.1.

We say that $u$ is a weak solution of (1.11)-(1.12) if

$u\in L^{2}(0,T;H^{2}(\mathbb{T}^{d}))\cap
L^{\infty}(0,T;H^{1}(\mathbb{T}^{d})),$ $|u|\leq 1\,\text{a.e. in
$(0,T)\times\mathbb{T}^{d}$},$

and if for all $\varphi\in C_{c}^{\infty}([0,T)\times\mathbb{T}^{d})$ we have

$-\int_{0}^{T}\int_{\mathbb{T}^{d}}u\,\partial_{t}\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t-\int_{\mathbb{T}^{d}}u^{0}(x)\,\varphi(0,x)\mathop{}\\!\mathrm{d}x=-\frac{W}{2}\int_{0}^{T}\int_{\mathbb{T}^{d}}\Delta
u\,\nabla
m(u)\cdot\nabla\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t\\\
-\frac{W}{2}\int_{0}^{T}\int_{\mathbb{T}^{d}}m(u)\,\Delta
u\,\Delta\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t-\int_{0}^{T}\int_{\mathbb{T}^{d}}m(u)\,F^{\prime\prime}(u)\,\nabla
u\cdot\nabla\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t.$

###  4.1 Uniform estimates and compactness

In order to prove Theorem 1.2 we first collect uniform estimates in
$\varepsilon$:

######  Lemma 4.2 (Uniform estimates).

The following sequences are bounded uniformly in $\varepsilon$:

  1. (A)

$|u_{\varepsilon}|\leq 1$ a.e. in $(0,T)\times\mathbb{T}^{d}$,

  2. (B)

$\\{\nabla u_{\varepsilon}\\}_{\varepsilon}$ in
$L^{2}((0,T)\times\mathbb{T}^{d})$,

  3. (C)

$\\{\sqrt{m(u_{\varepsilon})}\,\nabla\mu_{\varepsilon}\\}_{\varepsilon}$ in
$L^{2}((0,T)\times\mathbb{T}^{d})$,

  4. (D)

$\\{\partial_{t}u_{\varepsilon}\\}_{\varepsilon}$ in
$L^{2}(0,T;H^{-1}(\mathbb{T}^{d}))$,

  5. (E)

$\\{\partial_{t}\nabla u_{\varepsilon}\\}_{\varepsilon}$ in
$L^{2}(0,T;H^{-2}(\mathbb{T}^{d}))$,

  6. (F)

$\left\\{\frac{\sqrt{\omega(y)}}{|y|^{\frac{\alpha}{2}}}\,\frac{u_{\varepsilon}(x)-u_{\varepsilon}(x-\varepsilon\,y)}{\varepsilon}\right\\}_{\varepsilon}$
in $L^{\infty}(0,T;L^{2}(\mathbb{T}^{d}\times\mathbb{T}^{d}))$,

  7. (G)

$\left\\{\frac{\sqrt{\omega(y)}}{|y|^{\frac{\alpha}{2}}}\,\frac{\nabla
u_{\varepsilon}(x)-\nabla
u_{\varepsilon}(x-\varepsilon\,y)}{\varepsilon}\right\\}_{\varepsilon}$ in
$L^{2}((0,T)\times\mathbb{T}^{d}\times\mathbb{T}^{d})$.

###### Proof.

The estimates are a consequence of Propositions 3.1, 3.3 and Lemma C.1. To be
more precise, (A) is a consequence of Proposition 3.1 while (C), (F), (G)
follows from the energy/entropy identities (3.4)–(3.5) and a change of
variables $y\to\varepsilon y$ (note that the negative part in the entropy
identity (3.5) is controlled as explained in (3.5)). Then, (B) can be easily
deduced by the Poincaré’s inequality in Lemma C.1 because the average of the
gradient on the torus is 0. Next, (D) follow from the PDE (1.9)–(1.10) by
writing

$\partial_{t}u_{\varepsilon}=\operatorname{div}(\sqrt{m(u_{\varepsilon})}\,\sqrt{m(u_{\varepsilon})}\,\nabla
u_{\varepsilon})$

and using estimates (A), (C). Finally, (E) is a direct consequence of (D).
∎

######  Lemma 4.3 (compactness).

There exists $u\in L^{\infty}((0,T)\times\mathbb{T}^{d})\cap
L^{2}(0,T;H^{2}(\mathbb{T}^{d}))$, $0\leq u\leq 1$, such that, up to a
subsequence not relabeled, $u_{\varepsilon}\to u\mbox{ a.e., strongly in
}L^{2}(0,T;H^{1}(\mathbb{T}^{d}))$ and
$L^{p}((0,T)\times\mathbb{T}^{d})\,\mbox{ for all }p\in[1,\infty).$

###### Proof.

First, by a classical Aubin-Lions lemma and estimates (B), (D) we obtain
strong convergence in $L^{2}((0,T)\times\mathbb{T}^{d})$ (and a.e.). Then,
arguing by interpolation and (A), we deduce convergence in all
$L^{p}((0,T)\times\mathbb{T}^{d})$. For the strong convergence in
$L^{2}(0,T;H^{1}(\mathbb{T}^{d}))$, we first obtain compactness in space of
the gradient with (G) and Theorem B.1. Compactness in time follows from (E).
An Aubin-Lions argument yields the result (see [25, Lemma D.1] for the precise
statement). ∎

###  4.2 Nonlocal gradient $S_{\varepsilon}$ and its convergence properties

The limit equation that we obtain is a fourth-order equation. The number of
derivatives can be reduced by the integration by parts in a weak formulation.
Therefore, to obtain this weak formulation we need to mimic the same
integration by parts at the nonlocal level (i.e. with $B_{\varepsilon}[u]$
instead of $-\Delta u$). This can be done if we introduce a nonlocal
approximation of $\nabla$. Thus we define the operator (see e.g. [25])

$S_{\varepsilon}[\varphi](x,y):=\frac{\sqrt{\omega_{\varepsilon}(y)}}{\sqrt{2}\,\varepsilon^{1-\frac{\alpha}{2}}|y|^{\frac{\alpha}{2}}}(\varphi(x-y)-\varphi(x)),$
(4.1)

which has the following algebraic properties:

######  Lemma 4.4.

The operator $S_{\varepsilon}$ satisfies:

  1. (S1)

$S_{\varepsilon}$ is a linear operator that commutes with derivatives with
respect to $x$,

  2. (S2)

for all functions $f,g:\mathbb{T}^{d}\to\mathbb{R}$ we have the Leibniz rule
up to an error:

$\displaystyle S_{\varepsilon}[fg](x,y)-S_{\varepsilon}[f](x,y)g(x)$
$\displaystyle-S_{\varepsilon}[g](x,y)f(x)=$
$\displaystyle=\frac{\sqrt{\omega_{\varepsilon}(y)}}{\sqrt{2}\,\varepsilon^{1-\frac{\alpha}{2}}|y|^{\frac{\alpha}{2}}}(f(x-y)-f(x))\,(g(x-y)-g(x)).$

  3. (S3)

for all $u,\varphi\in L^{2}(\mathbb{T}^{d})$

$\langle
B_{\varepsilon}[u](\cdot),\varphi(\cdot)\rangle_{L^{2}(\mathbb{T}^{d})}=\langle
S_{\varepsilon}[u](\cdot,\cdot),S_{\varepsilon}[\varphi](\cdot,\cdot)\rangle_{L^{2}(\mathbb{T}^{d}\times\mathbb{T}^{d})}.$

For the simple proof, we refer to [25, Lemma 3.4].

Due to the property (S3), when proving the convergence $\varepsilon\to 0$, we
will need to deal with the operator $S_{\varepsilon}$. Its asymptotic
properties are summarized in the following result. We remark that it is a
substantial upgrade of [25, Lemma 3.4 (S4)] in two directions: allowing to
consider singular potentials and including the case of general mobility
$m(u)$. To simplify the notation, we skip
$\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t$ in the
integrals in the statement of Lemma 4.5 and its proof.

######  Lemma 4.5.

Let $f\in C^{1}(\mathbb{R},\mathbb{R})$. Suppose that $\\{u_{\varepsilon}\\}$
converges strongly in $L^{2}(0,T;H^{1}(\mathbb{T}^{d}))$ to some $u\in
L^{2}(0,T;H^{2}(\mathbb{T}^{d}))$. Moreover, assume that $0\leq
u_{\varepsilon}\leq 1$ and that the sequence

$\left\\{\frac{\sqrt{\omega(y)}}{|y|^{\frac{\alpha}{2}}}\,\frac{\nabla
u_{\varepsilon}(x)-\nabla
u_{\varepsilon}(x-\varepsilon\,y)}{\varepsilon}\right\\}\mbox{ is bounded in
}L^{2}((0,T)\times\mathbb{T}^{d}\times\mathbb{T}^{d}).$ (4.2)

Let
$W:=\int_{\mathbb{T}^{d}}\frac{\omega(y)}{|y|^{\alpha-2}}\mathop{}\\!\mathrm{d}y$.
Then, when $\varepsilon\to 0$, for all smooth
$\varphi(t,x):[0,T]\times\mathbb{T}^{d}\to\mathbb{R}$:

$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[f(u_{\varepsilon})]\,S_{\varepsilon}[\nabla
u_{\varepsilon}]\cdot\nabla\varphi(t,x)\to\frac{W}{2}\,\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla
f(u(t,x))\cdot D^{2}u(t,x)\nabla\varphi(t,x),$ (4.3)
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[f(u_{\varepsilon})]\,S_{\varepsilon}[u_{\varepsilon}]\,\varphi(t,x)\to\frac{W}{2}\,\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla
f(u(t,x))\cdot\nabla u(t,x)\,\varphi(t,x),$ (4.4)
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[u_{\varepsilon}]\nabla
f(u_{\varepsilon})\cdot
S_{\varepsilon}[\nabla\varphi(t,x)]\to\frac{W}{2}\,\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla
f(u(t,x))\cdot D^{2}\varphi(t,x)\nabla u(t,x),$ (4.5)
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[u_{\varepsilon}]f(u_{\varepsilon})\,S_{\varepsilon}[\varphi(t,x)]\to\frac{W}{2}\,\int_{0}^{T}\int_{\mathbb{T}^{d}}f(u(t,x))\,\nabla
u(t,x)\cdot\nabla\varphi(t,x).$ (4.6)

###### Proof.

We write

$\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[f(u_{\varepsilon})]S_{\varepsilon}[\nabla
u_{\varepsilon}]\cdot\nabla\varphi(x)=\\\
=\frac{1}{2}\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{\omega(y)}{|y|^{\alpha}}\frac{f(u_{\varepsilon}(x))-f(u_{\varepsilon}(x-\varepsilon
y))}{\varepsilon}\frac{\nabla u_{\varepsilon}(x)-\nabla
u_{\varepsilon}(x-\varepsilon y)}{\varepsilon}\cdot\nabla\varphi(x).$

From Lemma A.1 we know that

$\frac{v_{\varepsilon}(x)-v_{\varepsilon}(x-\varepsilon
y)}{\varepsilon\,|y|}\to\nabla v\cdot\frac{y}{|y|}\mbox{ strongly in
}L^{\infty}_{y}(\mathbb{T}^{d};L^{2}_{(t,x)}((0,T)\times\mathbb{T}^{d}))$
(4.7)

for any sequence $\\{v_{\varepsilon}\\}$ compact in
$L^{2}(0,T;H^{1}(\mathbb{T}^{d}))$ and converging to some $v$, in particular
for $v_{\varepsilon}=f(u_{\varepsilon})$. Moreover, by (4.2), we know that
$\frac{\sqrt{\omega(y)}}{|y|^{\frac{\alpha}{2}}}\frac{\nabla
u_{\varepsilon}(x)-\nabla u_{\varepsilon}(x-\varepsilon y)}{\varepsilon}$ is
uniformly bounded in $L^{2}((0,T)\times\mathbb{T}^{d}\times\mathbb{T}^{d})$ so
that, up to a subsequence, it has a weak limit. To identify the limit, we
compute for an arbitrary smooth vector field $\phi=\phi(t,x,y)$

$\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{\sqrt{\omega(y)}}{|y|^{\frac{\alpha}{2}}}\frac{\nabla
u_{\varepsilon}(x)-\nabla u_{\varepsilon}(x-\varepsilon
y)}{\varepsilon}\cdot\phi=-\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{\sqrt{\omega(y)}}{|y|^{\frac{\alpha}{2}}}\frac{u_{\varepsilon}(x)-u_{\varepsilon}(x-\varepsilon
y)}{\varepsilon}\operatorname{div}_{x}\phi.$

Using (4.7) (with $v_{\varepsilon}=u_{\varepsilon}$) and
$\frac{\sqrt{\omega(y)}}{|y|^{\frac{\alpha}{2}-1}}\in
L^{2}(\mathbb{T}^{d})\subset L^{1}(\mathbb{T}^{d})$ (because we assume
$\alpha<d-1$ but even $\alpha<d+2$ would be sufficient here) we can pass to
the limit and obtain

$\lim_{\varepsilon\to
0}\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{\sqrt{\omega(y)}}{|y|^{\frac{\alpha}{2}}}\frac{u_{\varepsilon}(x)-u_{\varepsilon}(x-\varepsilon
y)}{\varepsilon}\operatorname{div}_{x}\phi=\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{\sqrt{\omega(y)}}{|y|^{\frac{\alpha}{2}}}\nabla
u(x)\cdot y\operatorname{div}_{x}\phi$

Moreover, using $H^{2}$ regularity of $u$, we can integrate by parts backwards
to conclude

$\frac{\sqrt{\omega(y)}}{|y|^{\frac{\alpha}{2}}}\frac{\nabla
u_{\varepsilon}(x)-\nabla u_{\varepsilon}(x-\varepsilon
y)}{\varepsilon}\rightharpoonup\frac{\sqrt{\omega(y)}}{|y|^{\frac{\alpha}{2}}}\,D^{2}u(x)y\mbox{
in }L^{2}((0,T)\times\mathbb{T}^{d}\times\mathbb{T}^{d}).$ (4.8)

Now, it remains to use again that
$\frac{\sqrt{\omega(y)}}{{|y|^{\frac{\alpha}{2}-1}}}\in L^{2}(\mathbb{T}^{d})$
and apply (4.7)–(4.8).

Concerning (4.4), we compute

$\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[u_{\varepsilon}]S_{\varepsilon}[f(u_{\varepsilon})]\,\varphi(t,x)=\\\
=\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{\omega(y)}{2\,|y|^{\alpha-2}}\frac{u_{\varepsilon}(x)-u_{\varepsilon}(x-\varepsilon
y)}{\varepsilon|y|}\frac{f(u_{\varepsilon}(x))-f(u_{\varepsilon}(x-\varepsilon
y))}{\varepsilon|y|}\,\varphi(t,x)$

Clearly $\frac{\omega(y)}{|y|^{\alpha-2}}\in L^{1}(\mathbb{T}^{d})$. Since $f$
is $C^{1}$ and $0\leq u_{\varepsilon}\leq 1$, we can apply (4.7) both to
$u_{\varepsilon}$ and to $f(u_{\varepsilon})$ so that we obtain

$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[u_{\varepsilon}]S_{\varepsilon}[f(u_{\varepsilon})]\,\varphi(t,x)\to\frac{W}{2}\,\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla
u(x)\cdot\nabla f(u(x))\,\varphi(t,x).$

Next, we demonstrate (4.5). We write the integral as

$\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{\omega(y)}{2|y|^{\alpha-2}}\frac{u_{\varepsilon}(x)-u_{\varepsilon}(x-\varepsilon
y)}{\varepsilon\,|y|}\,\nabla
f(u_{\varepsilon}(x))\cdot\frac{\nabla\varphi(t,x)-\nabla\varphi(t,x-\varepsilon
y)}{\varepsilon\,|y|}.$

Observe that $\nabla f(u_{\varepsilon})$ is weakly compact in
$L^{2}((0,T)\times\mathbb{T}^{d})$,
$\frac{u_{\varepsilon}(x)-u_{\varepsilon}(x-\varepsilon y)}{\varepsilon\,|y|}$
is compact due to (4.7) while $\frac{\varphi(t,x)-\varphi(t,x-\varepsilon
y)}{\varepsilon\,|y|}$ is compact in
$L^{\infty}((0,T)\times\mathbb{T}^{d}\times\mathbb{T}^{d})$. Hence, using
integrability of $\frac{\omega(y)}{2|y|^{\alpha-2}}$ as above, we easily pass
to the limit $\varepsilon\to 0$.

Finally, the proof of (4.6) is similar to that of (4.5): indeed, we have the
same compactness estimates for the sequence $\\{f(u_{\varepsilon})\\}$ as for
$\\{\nabla f(u_{\varepsilon})\\}$. ∎

###### Proof of Theorem 1.2.

We only have to explain how to pass to the limit in the term

$\int_{0}^{T}\int_{\mathbb{T}^{d}}\operatorname{div}(m(u_{\varepsilon})\nabla\mu_{\varepsilon})\,\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t$

where $\varphi\in C^{3}([0,T]\times\mathbb{T}^{d})$. Integrating by parts, we
obtain

$\begin{split}\int_{0}^{T}\int_{\mathbb{T}^{d}}\operatorname{div}&(m(u_{\varepsilon})\nabla\mu_{\varepsilon})\,\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=-\int_{0}^{T}\int_{\mathbb{T}^{d}}m(u_{\varepsilon})\nabla\mu_{\varepsilon}\cdot\nabla\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=\\\
=&-\int_{0}^{T}\int_{\mathbb{T}^{d}}m(u_{\varepsilon})\nabla(B_{\varepsilon}[u_{\varepsilon}]+F^{\prime}(u_{\varepsilon}))\cdot\nabla\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t\\\
=&\int_{0}^{T}\int_{\mathbb{T}^{d}}B_{\varepsilon}[u_{\varepsilon}]\,\nabla
m(u_{\varepsilon})\cdot\nabla\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t+\int_{0}^{T}\int_{\mathbb{T}^{d}}B_{\varepsilon}[u_{\varepsilon}]\,m(u_{\varepsilon})\Delta\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t\\\
&-\int_{0}^{T}\int_{\mathbb{T}^{d}}m(u_{\varepsilon})F^{\prime\prime}(u_{\varepsilon})\nabla
u_{\varepsilon}\cdot\nabla\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=:I_{1}+I_{2}+I_{3}.\end{split}$
(4.9)

We choose a subsequence as in Lemma 4.3. Using (1.7) we can pass to the limit
in $I_{3}$. It remains to prove convergence of $I_{1}$ and $I_{2}$.

Step 1: Convergence of term $I_{1}$. Using (S3) in Lemma 4.4 we write

$\displaystyle I_{1}=$
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[u_{\varepsilon}]\,S_{\varepsilon}[\nabla
m(u_{\varepsilon})\cdot\nabla\varphi]\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}t$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[u_{\varepsilon}]\,S_{\varepsilon}[\nabla
m(u_{\varepsilon})]\cdot\nabla\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}t+\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[u_{\varepsilon}]\,\nabla
m(u_{\varepsilon})\cdot
S_{\varepsilon}[\nabla\varphi]\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}t$
$\displaystyle+R_{\varepsilon}^{(1)}=:J_{1}^{(1)}+J_{2}^{(1)}+R_{\varepsilon}^{(1)},\phantom{int_{0}^{T}\int_{\mathbb{T}^{d}}}$

where $R_{\varepsilon}^{(1)}$ is an error defined as

$R_{\varepsilon}^{(1)}=\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[u_{\varepsilon}]\left(S_{\varepsilon}[\nabla
m(u_{\varepsilon})\cdot\nabla\varphi]-S_{\varepsilon}[\nabla
m(u_{\varepsilon})]\cdot\nabla\varphi-\nabla m(u_{\varepsilon})\cdot
S_{\varepsilon}[\nabla\varphi]\right)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}t.$

For the term $J_{1}^{(1)}$ we integrate by parts

$J_{1}^{(1)}=-\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[u_{\varepsilon}]\,S_{\varepsilon}[m(u_{\varepsilon})]\,\Delta\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}t-\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[m(u_{\varepsilon})]\,S_{\varepsilon}[\nabla
u_{\varepsilon}]\cdot\nabla\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}t$

For the first term we use (4.4) while for the second term we apply (4.3) to
get

$J_{1}^{(1)}\to-\frac{W}{2}\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla
m(u)\cdot\nabla
u\,\Delta\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t-\frac{W}{2}\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla
m(u)\cdot D^{2}u\nabla\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t.$
(4.10)

For the term $J_{2}^{(1)}$ we simply apply (4.5) and obtain

$J_{1}^{(2)}\to\frac{W}{2}\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla m(u)\cdot
D^{2}\varphi\nabla u\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t.$ (4.11)

Finally, concerning the term $R_{\varepsilon}^{(1)}$, we will prove that it
converges to 0. We first use (S2) to write

$R_{\varepsilon}^{(1)}=\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{\omega_{\varepsilon}(y)}{2|y|^{\alpha}\,\varepsilon^{2-\alpha}}(u_{\varepsilon}(x)-u_{\varepsilon}(x-y))(\nabla
m(u_{\varepsilon})(x)-\nabla
m(u_{\varepsilon})(x-y))\cdot(\nabla\varphi(x)-\nabla\varphi(x-y)).$

Integrating by parts we get that $R_{\varepsilon}^{(1)}$ equals to:

$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{\omega_{\varepsilon}(y)}{2|y|^{\alpha}\,\varepsilon^{2-\alpha}}(m(u_{\varepsilon})(x)-m(u_{\varepsilon})(x-y))(\nabla
u_{\varepsilon}(x)-\nabla
u_{\varepsilon}(x-y))\cdot(\nabla\varphi(x)-\nabla\varphi(x-y))$
$\displaystyle+\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{\omega_{\varepsilon}(y)}{2|y|^{\alpha}\,\varepsilon^{2-\alpha}}(m(u_{\varepsilon})(x)-m(u_{\varepsilon})(x-y))(u_{\varepsilon}(x)-u_{\varepsilon}(x-y))\,(\Delta\varphi(x)-\Delta\varphi(x-y)).$

We consider the first term which is harder and then we briefly comment how to
handle the second one. We change variables to have

$\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{\omega(y)}{2|y|^{\alpha}\,\varepsilon^{2}}(m(u_{\varepsilon})(x)-m(u_{\varepsilon})(x-\varepsilon
y))(\nabla u_{\varepsilon}(x)-\nabla u_{\varepsilon}(x-\varepsilon
y))\cdot(\nabla\varphi(x)-\nabla\varphi(x-\varepsilon y))$

Note that since $\\{u_{\varepsilon}\\}$ is bounded,
$|m(u_{\varepsilon})(x)-m(u_{\varepsilon})(x-y)|\leq
C\,|u_{\varepsilon}(x)-u_{\varepsilon}(x-y)|$ so that due to (F) in Lemma 4.2,
the sequence

$\left\\{\frac{\sqrt{\omega(y)}}{|y|^{\frac{\alpha}{2}}}\frac{|m(u_{\varepsilon})(x)-m(u_{\varepsilon})(x-\varepsilon\,y)|}{\varepsilon}\right\\}\mbox{
is bounded in }L^{\infty}(0,T;L^{2}(\mathbb{T}^{d}\times\mathbb{T}^{d})).$
(4.12)

Moreover,
$|\nabla\varphi(x)-\nabla\varphi(x-\varepsilon\,y)|,\,|\Delta\varphi(x)-\Delta\varphi(x-\varepsilon\,y)|\leq
C\,\varepsilon\,|y|$. Hence, using (4.12) and (G) in Lemma 4.2, the term of
interest is bounded by $C\,\varepsilon$ so it converges to 0 when
$\varepsilon\to 0$. Finally, for the second term in the definition of
$R_{\varepsilon}^{(1)}$, we argue in the same way using (4.12) and (F) (Lemma
4.2).

From (4.10) and (4.11), using the identity

$\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla m(u)\cdot\left[D^{2}\varphi\nabla
u-D^{2}u\nabla\varphi\right]\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla
m(u)\cdot\left[\nabla u\,\Delta\varphi-\nabla\varphi\,\Delta
u\right]\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t,$

easily verifiable by integration by parts, we conclude that the limit can be
simplified to

$I_{1}\to-\frac{W}{2}\,\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla
m(u)\cdot\nabla\varphi\,\Delta u.$

Step 2: Convergence of term $I_{2}$. We write

$\displaystyle I_{2}$
$\displaystyle=\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[u_{\varepsilon}]\,S_{\varepsilon}[m(u_{\varepsilon})\Delta\varphi]\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[u_{\varepsilon}]S_{\varepsilon}[m(u_{\varepsilon})]\Delta\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}t+\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[u_{\varepsilon}]\,m(u_{\varepsilon})S_{\varepsilon}[\Delta\varphi]\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}t+R_{\varepsilon}^{(2)}$
$\displaystyle=J_{1}^{(2)}+J_{2}^{(2)}+R_{\varepsilon}^{(2)},$

where $R_{\varepsilon}^{(2)}$ equals

$R_{\varepsilon}^{(2)}=\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}S_{\varepsilon}[u_{\varepsilon}]\left(S_{\varepsilon}[m(u_{\varepsilon})\,\Delta\varphi]-S_{\varepsilon}[m(u_{\varepsilon})]\,\Delta\varphi-m(u_{\varepsilon})\,S_{\varepsilon}[\Delta\varphi]\right)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}t.$

Applying (4.4) and (4.6) we obtain that the first two terms converge to

$J_{1}^{(2)}+J_{2}^{(2)}\to\frac{W}{2}\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla
m(u)\cdot\nabla
u\,\Delta\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t+\frac{W}{2}\int_{0}^{T}\int_{\mathbb{T}^{d}}m(u)\nabla
u\cdot\nabla\Delta\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=\\\
=-\frac{W}{2}\int_{0}^{T}\int_{\mathbb{T}^{d}}m(u)\,\Delta
u\,\Delta\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t$

so it remains to prove that $R_{\varepsilon}^{(2)}\to 0$. In fact, using (S2)
in Lemma 4.4 we realize it equals

$\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{\omega_{\varepsilon}(y)}{2|y|^{\alpha}\,\varepsilon^{2-\alpha}}(m(u_{\varepsilon})(x)-m(u_{\varepsilon})(x-y))(u_{\varepsilon}(x)-u_{\varepsilon}(x-y))\,(\Delta\varphi(x)-\Delta\varphi(x-y))$

which was already studied in the case of $R_{\varepsilon}^{(1)}$. We obtain
that

$I_{2}\to-\frac{W}{2}\int_{0}^{T}\int_{\mathbb{T}^{d}}m(u)\,\Delta
u\,\Delta\varphi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t$

and the proof is concluded. ∎

##  5 Numerical simulations

To illustrate the behavior of the solutions as $\varepsilon\to 0$, we propose
numerical simulations. To keep things simple, we assume that the mobility is
constant, $m(u)=1$, and we use a smooth approximation of the double-well
potential $F(u)=u^{2}(1-u)^{2}$ in Equations (1.9)-(1.10) and (1.11)-(1.12).
We refer to [6], where authors investigate how the non-smooth nature of the
double-well potential (such as the double-obstacle potential) generates sharp
interfaces.

Our simulations are made in both one and two dimensions. The kernel reads

$J_{\varepsilon}(x)=\frac{1}{\varepsilon^{d+2}}\exp\left(-\frac{|x|^{2}}{\varepsilon^{2}}\right).$

We consider three specific scenarios: $\varepsilon=1$, $\varepsilon=0.7$, and
$\varepsilon=0.1$. In the one-dimensional case, we superimpose the curve of
the nonlocal solution (represented in red) onto the curve of the local
solution (depicted in blue). We observe their evolution over time at $t=1$,
$t=10$, and $t=100$.

In the two-dimensional case, the first row showcases the evolution of the
nonlocal Cahn-Hilliard equation at $t=10$, $t=100$, and $t=1000$. The last row
represents the evolution of the local equation.

The numerical implementation was made using Python, and numpy.fft, numpy.ifft
to efficiently compute the Fourier and inverse Fourier transforms. The Fourier
transform is particularly useful for terms like $\Delta^{2}u$ and the
convolution $\Delta(u-J\ast u)$.

Focusing on the numerical results, the observations made in both one and two
dimensions are the same. As $\varepsilon$ is sent to zero, the solutions of
the non-local equation converge to those of the local equation. When
$\varepsilon$ is set to 0.1, the difference becomes invisible to the naked
eye.

We opted for a random initial condition with low regularity. Within the
simulations for both one-dimensional and two-dimensional scenarios with
$\varepsilon=1$, we observe two interesting aspects. First, the regularity of
solutions to the nonlocal equation is lower than that of the local one and the
initial condition requires a longer time to be smoothed. From a theoretical
point of view, this can be explained by the $H^{1}$ regularity (a porous
medium character) for the nonlocal equation compared to the $H^{2}$ regularity
for the local equation. The second point of interest lies in the formation of
an interface in a long time which is of different nature for both equations.
For, $\varepsilon=1$, at the final time step, the interface is sharp, unlike
the diffuse interface observed for the local equation.

$\varepsilon=1$, $t=1$

$\varepsilon=1$, $t=10$

$\varepsilon=1$, $t=100$

$\varepsilon=0.7$, $t=1$

$\varepsilon=0.7$, $t=10$

$\varepsilon=0.7$, $t=100$

$\varepsilon=0.1$, $t=1$

$\varepsilon=0.1$, $t=10$

$\varepsilon=0.1$, $t=100$

$\varepsilon=1$, $t=10$

$\varepsilon=1$, $t=100$

$\varepsilon=1$, $t=1000$

local, $t=10$

local, $t=10$

local, $t=1000$

##  6 The nonlocal-to-local limit for the cell-cell adhesion model

Here, we explain, using similar ideas, how to treat a recently introduced
model in the theory of cell-cell adhesion from [13, eq. (15)] which also
involves the mobility $m(u)=u\,(1-u)$. More precisely, we consider the PDE
posed on $[0,T]\times\mathbb{T}^{d}$

$\partial_{t}u=\operatorname{div}(u\nabla
u)-a\,\operatorname{div}(u\,(1-u)\,K[u]),$ (6.1)

where the parameter $a$ measures the relative cell-cell adhesion strength
while $K$ is the nonlocal kernel $K$ defined as

$K[u]=\int_{0}^{1}\int_{S^{d-1}}u(x+r\,\eta)\,V(r\,\eta)\,r^{d-1}\,\eta\mathop{}\\!\mathrm{d}S(\eta)\,\mathop{}\\!\mathrm{d}r.$

Here, $V$ is a smooth, radially symmetric and compactly supported kernel with
$\operatorname*{\operatorname{supp}}V\subset\mathbb{T}^{d}$ and $S^{d-1}$ is a
unit sphere in $\mathbb{R}^{d}$. It is useful to observe that, with the
assumptions on $V$, $K$ can be written as a convolution operator

$K[u](x)=\int_{\mathbb{T}^{d}}u(x-y)\,V(y)\,\frac{y}{|y|}\mathop{}\\!\mathrm{d}y=u\ast\left(V(\cdot)\,\left(\frac{\cdot}{|\cdot|}\right)\right)$

We now rescale the kernel by taking
$V(x):=\frac{1}{\varepsilon}\,\omega_{\varepsilon}(x)$ in (6.1) where
$\omega_{\varepsilon}(x)=\frac{1}{\varepsilon^{d}}\omega\left(\frac{x}{\varepsilon}\right)$
and $\omega:\mathbb{R}^{d}\to\mathbb{R}$ is a smooth, radially symmetric and
compactly supported kernel. The scaling factor $\frac{1}{\varepsilon}$ is
chosen to obtain a meaningful PDE in the limit $\varepsilon\to 0$ as explained
below. Hence, we consider solutions to the PDE

$\partial_{t}u_{\varepsilon}=\operatorname{div}(u_{\varepsilon}\nabla
u_{\varepsilon})-a\,\operatorname{div}(u_{\varepsilon}\,(1-u_{\varepsilon})\,K_{\varepsilon}[u_{\varepsilon}])$
(6.2)

with

$K_{\varepsilon}[u]=\frac{1}{\varepsilon}\int_{\mathbb{T}^{d}}u(x-y)\,\omega_{\varepsilon}(y)\,\frac{y}{|y|}\mathop{}\\!\mathrm{d}y$
(6.3)

and we want to understand the limit $\varepsilon\to 0$. To guess what is the
expected PDE, let $\psi\in C_{c}^{\infty}((0,T)\times\mathbb{T}^{d})$. Then,
by Taylor’s expansion,

$\begin{split}K_{\varepsilon}[\psi]&=\frac{1}{\varepsilon}\int_{\mathbb{T}^{d}}(\psi(x-y)-\psi(x))\,\omega_{\varepsilon}(y)\,\frac{y}{|y|}\mathop{}\\!\mathrm{d}y=\\\
&=\nabla\psi(x)\,\frac{1}{\varepsilon}\int_{\mathbb{T}^{d}}\omega_{\varepsilon}(y)\,|y|\mathop{}\\!\mathrm{d}y+R_{\varepsilon}=\nabla\psi(x)\,\left(\int_{\mathbb{T}^{d}}\omega(y)\,|y|\mathop{}\\!\mathrm{d}y\right)+R_{\varepsilon},\end{split}$
(6.4)

where $R_{\varepsilon}\leq C(\omega)\,\|D^{2}\psi\|_{\infty}\,\varepsilon\to
0$ when $\varepsilon\to 0$. Hence, we expect that the limiting PDE reads

$\partial_{t}u=\operatorname{div}(u\nabla
u)-C_{\omega}\,a\,\operatorname{div}(u\,(1-u)\,\nabla u),$ (6.5)

where
$C_{\omega}:=\int_{\mathbb{T}^{d}}\omega(y)\,|y|\mathop{}\\!\mathrm{d}y$.

To have sufficient uniform a priori estimates with respect to
$\varepsilon\in(0,1)$, we have to assume that the parameter $a$ satisfies

$|a|\,\sqrt{C}<1,$ (6.6)

where $C$ is a constant in the inequality

$\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{|f(x)-f(y)|^{2}}{\varepsilon^{2}}\omega_{\varepsilon}(x-y)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\leq
C\,\|f\|_{H^{1}(\mathbb{T}^{d})}^{2},$ (6.7)

which holds true by Lemma C.2. The inequality (6.7) implies also the uniform
bound on $K_{\varepsilon}[u]$: using that
$\int_{\mathbb{T}^{d}}\omega_{\varepsilon}(y)\frac{y}{|y|}\mathop{}\\!\mathrm{d}y=0$
we have

$K_{\varepsilon}[u]=\frac{1}{\varepsilon}\int_{\mathbb{T}^{d}}u(x-y)\,\omega_{\varepsilon}(y)\,\frac{y}{|y|}\mathop{}\\!\mathrm{d}y=\frac{1}{\varepsilon}\int_{\mathbb{T}^{d}}(u(x-y)-u(x))\,\omega_{\varepsilon}(y)\,\frac{y}{|y|}\mathop{}\\!\mathrm{d}y$

so that by Jensen’s inequality

$\|K_{\varepsilon}[u]\|_{L^{2}(\mathbb{T}^{d})}^{2}\leq\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{(u(x-y)-u(x))^{2}}{\varepsilon^{2}}\,\omega_{\varepsilon}(y)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}x\leq
C\,\|\nabla u\|_{L^{2}(\mathbb{T}^{d})}^{2}.$ (6.8)

For the nonlocal-to-local limit, the main idea is again the same as in the
case of the Cahn-Hilliard system: the mobility $m(u)=u\,(1-u)$ ensures that
the solution remains bounded so that we can easily handle the nonlinear terms.
We also want to comment that in [13] Authors derive also systems of PDEs of
the form (6.2) but these are not treatable mathematically because, in full
generality, there is currently no mathematical theory of such systems. Here we
focus on a single equation.

###  6.1 Existence of weak solutions to (6.2)

######  Definition 6.1.

We say that a function $u_{\varepsilon}:[0,T]\times\mathbb{T}^{d}\to[0,1]$ is
a weak solution to (6.2) with initial condition $u^{0}$ if $\nabla
u_{\varepsilon}\in L^{2}((0,T)\times\mathbb{T}^{d})$ and for all $\psi\in
C_{c}^{\infty}([0,T)\times\mathbb{T}^{d})$ we have

$\int_{\mathbb{T}^{d}}u^{0}\,\psi(0,x)\mathop{}\\!\mathrm{d}x+\int_{0}^{T}\int_{\mathbb{T}^{d}}u_{\varepsilon}\,\partial_{t}\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=\int_{0}^{T}\int_{\mathbb{T}^{d}}(u_{\varepsilon}\,\nabla
u_{\varepsilon}-a\,(1-u_{\varepsilon})\,u_{\varepsilon}\,K_{\varepsilon}[u_{\varepsilon}])\cdot\nabla\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t.$

######  Lemma 6.2.

Let $0\leq u^{0}\leq 1$ and let the parameter $a$ satisfy (6.6). Then, there
exists a weak solution to (6.2) with the initial condition $u^{0}$. Moreover,
the following sequences are bounded uniformly with respect to
$\varepsilon\in(0,1)$:

  1. (E1)

$\\{u_{\varepsilon}\\}$ in $L^{\infty}((0,T)\times\mathbb{T}^{d})$ with $0\leq
u_{\varepsilon}\leq 1$,

  2. (E2)

$\\{\nabla u_{\varepsilon}\\}$ in $L^{2}((0,T)\times\mathbb{T}^{d})$,

  3. (E3)

$\\{\partial_{t}u_{\varepsilon}\\}$ in $L^{2}(0,T;H^{-1}(\mathbb{T}^{d}))$.

###### Proof.

Step 1: An auxiliary problem. For fixed $\varepsilon\in(0,1)$ we consider an
auxiliary problem

$\partial_{t}u_{\delta}=\operatorname{div}(u_{\delta}\nabla
u_{\delta})-a\,\operatorname{div}((1-u_{\delta})\,(u_{\delta}\,K_{\varepsilon}[u_{\delta}])\ast\varphi_{\delta})$
(6.9)

where $\\{\varphi_{\delta}\\}_{\delta\in(0,1)}$ is a standard mollifying
sequence. A solution to (6.9) with regularity $u_{\delta}\in
L^{2}(0,T;H^{1}(\mathbb{T}^{d}))$ can be obtained by standard methods, for
example by the Schauder’s fixed point theorem. We want to pass with $\delta\to
0$ in (6.9).

Step 2: Uniform bounds with respect to $\delta$ and $\varepsilon$. We claim
that the following sequences are uniformly bounded with respect to
$\delta,\varepsilon\in(0,1)$:

  1. (F1)

$\\{u_{\delta}\\}$ in $L^{\infty}((0,T)\times\mathbb{T}^{d})$ with $0\leq
u_{\delta}\leq 1$,

  2. (F2)

$\\{\nabla u_{\delta}\\}$ in $L^{2}((0,T)\times\mathbb{T}^{d})$,

  3. (F3)

$\\{\partial_{t}u_{\delta}\\}$ in $L^{2}(0,T;H^{-1}(\mathbb{T}^{d}))$.

Let $p$ be a function such that $p(s)=0$ for $s<1$ and $p^{\prime}(s)\geq 0$
for $s\geq 1$. Let $P(s)=\int_{1}^{s}p(r)\mathop{}\\!\mathrm{d}r$. Integrating
in space we deduce

$\partial_{t}\int_{\mathbb{T}^{d}}P(u_{\delta})\mathop{}\\!\mathrm{d}x=-\int_{\mathbb{T}^{d}}u_{\delta}\,p^{\prime}(u_{\delta})\,|\nabla
u_{\delta}|^{2}\mathop{}\\!\mathrm{d}x-a\,\int_{\mathbb{T}^{d}}p(u_{\delta})\,\operatorname{div}((1-u_{\delta})\,(u_{\delta}\,K_{\varepsilon}[u_{\delta}])\ast\varphi_{\delta})\mathop{}\\!\mathrm{d}x.$

Since $\int_{\mathbb{T}^{d}}u_{\delta}\,p^{\prime}(u_{\delta})\,|\nabla
u_{\delta}|^{2}\mathop{}\\!\mathrm{d}x\geq 0$ and $u_{\delta}\in
L^{2}(0,T;H^{1}(\mathbb{T}^{d}))$, we can approximate function
$\mbox{sign}_{+}(u-1)=\mathds{1}_{u>1}$ with $p$ and deduce

$\partial_{t}\int_{\mathbb{T}^{d}}|u_{\delta}-1|_{+}\mathop{}\\!\mathrm{d}x\leq-a\,\int_{\mathbb{T}^{d}}\mbox{sign}_{+}(u_{\delta}-1)\,\operatorname{div}((1-u_{\delta})\,(u_{\delta}\,K_{\varepsilon}[u_{\delta}])\ast\varphi_{\delta})\mathop{}\\!\mathrm{d}x,$

where $|u-1|_{+}=(u-1)\,\mathds{1}_{u>1}$. The integrand on the (RHS) can be
written again in the divergence form as
$-\operatorname{div}((1-u_{\delta})_{+}\,(u_{\delta}\,K_{\varepsilon}[u_{\delta}])\ast\varphi_{\delta})$
so the respective integral vanish and we conclude that $u_{\delta}\leq 1$. In
a similar way we prove that $u_{\delta}\geq 0$.

Next, we multiply by $\log u_{\delta}$ and integrate by parts to get

$\begin{split}\partial_{t}\int_{\mathbb{T}^{d}}u_{\delta}\log
u_{\delta}\mathop{}\\!\mathrm{d}x+&\int_{\mathbb{T}^{d}}|\nabla
u_{\delta}|^{2}\mathop{}\\!\mathrm{d}x\leq|a|\,\int_{\mathbb{T}^{d}}|\nabla
u_{\delta}|\,|u_{\delta}\,K_{\varepsilon}[u_{\delta}]|\ast\varphi_{\delta}\mathop{}\\!\mathrm{d}x\leq\\\
\phantom{\int_{\mathbb{T}^{d}}}&\leq|a|\,\|\nabla
u_{\delta}\|_{L^{2}(\mathbb{T}^{d})}\,\|u_{\delta}\,K_{\varepsilon}[u_{\delta}]\|_{L^{2}(\mathbb{T}^{d})}\leq|a|\,\|\nabla
u_{\delta}\|_{L^{2}(\mathbb{T}^{d})}\,\|K_{\varepsilon}[u_{\delta}]\|_{L^{2}(\mathbb{T}^{d})}.\end{split}$
(6.10)

Using (6.8), we obtain

$\partial_{t}\int_{\mathbb{T}^{d}}u_{\delta}\log
u_{\delta}\mathop{}\\!\mathrm{d}x+\int_{\mathbb{T}^{d}}|\nabla
u_{\delta}|^{2}\mathop{}\\!\mathrm{d}x\leq|a|\,\int_{\mathbb{T}^{d}}|\nabla
u_{\delta}|\,|u_{\delta}\,K_{\varepsilon}[u_{\delta}]|\ast\varphi_{\delta}\mathop{}\\!\mathrm{d}x\leq|a|\,\sqrt{C}\,\|\nabla
u_{\delta}\|_{L^{2}(\mathbb{T}^{d})}^{2}.$

Hence, since $|a|\,\sqrt{C}<1$, we obtain (F2).

Finally, multiplying (6.9) by $\psi\in
C^{\infty}_{c}((0,T)\times\mathbb{T}^{d})$ and using that $0\leq
u_{\delta}\leq 1$ as well as (6.8) we have

$\displaystyle\left|\int_{0}^{T}\int_{\mathbb{T}^{d}}u_{\delta}\,\partial_{t}\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t\right|$
$\displaystyle\leq\left|\int_{0}^{T}\int_{\mathbb{T}^{d}}u_{\delta}\,\nabla
u_{\delta}\,\nabla\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t\right|+\left|\int_{0}^{T}\int_{\mathbb{T}^{d}}(1-u_{\delta})\,(u_{\delta}\,K_{\varepsilon}[u_{\delta}])\ast\varphi_{\delta}\,\nabla\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t\right|$
$\displaystyle\leq\left(\|\nabla
u_{\delta}\|_{L^{2}((0,T)\times\mathbb{T}^{d})}+\|K_{\varepsilon}[u_{\delta}]\|_{L^{2}((0,T)\times\mathbb{T}^{d})}\right)\,\|\nabla\psi\|_{L^{2}((0,T)\times\mathbb{T}^{d})}\phantom{\int_{\mathbb{T}^{d}}}$
$\displaystyle\leq(1+\sqrt{C})\,\|\nabla
u_{\delta}\|_{L^{2}((0,T)\times\mathbb{T}^{d})}\,\|\nabla\psi\|_{L^{2}((0,T)\times\mathbb{T}^{d})},\phantom{\int_{\mathbb{T}^{d}}}$

so that (F3) follows by (F2).

Step 3: The limit $\delta\to 0$ and the estimates (E1)–(E3). By the Aubin-
Lions lemma, the Banach-Alaoglu theorem and interpolation in $L^{p}$ spaces,
there exists a subsequence such that

$\begin{split}&u_{\delta}\to u\mbox{ strongly in
}L^{p}((0,T)\times\mathbb{T}^{d})\mbox{ for }p\in[1,\infty),\\\ &\nabla
u_{\delta}\rightharpoonup u\mbox{ weakly in
}L^{2}((0,T)\times\mathbb{T}^{d}),\\\
&\partial_{t}u_{\delta}\rightharpoonup\partial_{t}u\mbox{ weakly in
}L^{2}(0,T;H^{-1}(\mathbb{T}^{d})).\end{split}$ (6.11)

This is sufficient to pass to the limit $\delta\to 0$ in the expression

$\int_{\mathbb{T}^{d}}u^{0}\,\psi(0,x)\mathop{}\\!\mathrm{d}x+\int_{0}^{T}\int_{\mathbb{T}^{d}}u_{\delta}\,\partial_{t}\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=\int_{0}^{T}\int_{\mathbb{T}^{d}}(u_{\delta}\,\nabla
u_{\delta}-a\,(1-u_{\delta})\,(u_{\delta}\,K_{\varepsilon}[u_{\delta}])\ast\varphi_{\delta})\cdot\nabla\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t$

(keep in mind that $\varepsilon$ is fixed!) for $\psi\in
C^{\infty}_{c}([0,T);\mathbb{T}^{d})$ so that $u$ satisfies Definition 6.1.
Finally, since bounds (F1)–(F3) are preserved along the limit (6.11),
(E1)–(E3) follow.

∎

###  6.2 Passing to the limit $\varepsilon\to 0$ in (6.2).

######  Theorem 6.3.

Let $0\leq u^{0}\leq 1$ and let the parameter $a$ satisfy (6.6). Let
$\\{u_{\varepsilon}\\}_{\varepsilon\in(0,1)}$ be a sequence of solutions to
(6.2) constructed in Lemma 6.2 satisfying uniform bounds (E1)–(E3). Then,
there exists a subsequence (not relabelled) such that

$\begin{split}&u_{\varepsilon}\to u\mbox{ strongly in
}L^{p}((0,T)\times\mathbb{T}^{d})\mbox{ for }p\in[1,\infty),\\\ &\nabla
u_{\varepsilon}\rightharpoonup u\mbox{ weakly in
}L^{2}((0,T)\times\mathbb{T}^{d}),\end{split}$ (6.12)

where $u$ satisfies

$\int_{\mathbb{T}^{d}}u^{0}\,\psi(0,x)\mathop{}\\!\mathrm{d}x+\int_{0}^{T}\int_{\mathbb{T}^{d}}u\,\partial_{t}\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=\int_{0}^{T}\int_{\mathbb{T}^{d}}(u\,\nabla
u-C_{\omega}\,a\,(1-u)\,u\,\nabla
u)\cdot\nabla\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t,$

for all $\psi\in C^{\infty}_{c}([0,T);\mathbb{T}^{d})$, i.e. $u$ is a
distributional solution of (6.5).

###### Proof.

As in the proof of Lemma 6.2, by the Aubin-Lions lemma, the Banach-Alaoglu
theorem, interpolation in $L^{p}$ spaces and the bounds (E1)–(E3), there
exists a subsequence as in (6.12).

We want to pass to the limit $\varepsilon\to 0$ in Definition 6.1. The only
difficulty is to justify the limit in the term

$\int_{0}^{T}\int_{\mathbb{T}^{d}}(1-u_{\varepsilon})\,u_{\varepsilon}\,K_{\varepsilon}[u_{\varepsilon}]\cdot\nabla\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t$
(6.13)

due to the singularity $\frac{1}{\varepsilon}$ in the nonlocal operator
$K_{\varepsilon}[u_{\varepsilon}]$. By the bound (6.8), there exists $\xi$
such that

$K_{\varepsilon}[u_{\varepsilon}]\rightharpoonup\xi\mbox{ weakly in
}L^{2}((0,T)\times\mathbb{T}^{d}).$

To identify $\xi$, let $\psi\in C_{c}^{\infty}((0,T)\times\mathbb{T}^{d})$.
Since $K_{\varepsilon}$ is a convolution operator with an antisymmetric kernel
(see 6.3)

$\int_{0}^{T}\int_{\mathbb{T}^{d}}K_{\varepsilon}[u_{\varepsilon}]\,\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=-\int_{0}^{T}\int_{\mathbb{T}^{d}}u_{\varepsilon}\,K_{\varepsilon}[\psi]\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t.$
(6.14)

By (6.4), $K_{\varepsilon}[\psi]\to C_{\omega}\,\nabla\psi$ so that passing to
the limit in (6.14) we obtain

$\int_{0}^{T}\int_{\mathbb{T}^{d}}\xi\,\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=-C_{\omega}\,\int_{0}^{T}\int_{\mathbb{T}^{d}}u\,\nabla\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t$

which implies $\xi=C_{\omega}\,\nabla u$. Furthermore, by (6.12),

$u_{\varepsilon}\,(1-u_{\varepsilon})\to u\,(1-u)\mbox{ strongly in
}L^{2}((0,T)\times\mathbb{T}^{d})$

so that we can pass to the limit in (6.13)

$\lim_{\varepsilon\to
0}\int_{0}^{T}\int_{\mathbb{T}^{d}}(1-u_{\varepsilon})\,u_{\varepsilon}\,K_{\varepsilon}[u_{\varepsilon}]\cdot\nabla\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=C_{\omega}\int_{0}^{T}\int_{\mathbb{T}^{d}}(1-u)\,u\,\nabla
u\cdot\nabla\psi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t$

and the proof is concluded. ∎

## Acknowledgements

Jakub Skrzeczkowski was supported by the Advanced Grant Nonlocal-CPD (Nonlocal
PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and
Synchronization) of the European Research Council Executive Agency (ERC) under
the European Union’s Horizon 2020 research and Innovation programme (grant
agreement No. 883363).

##  Appendix A Results from classical analysis

######  Lemma A.1.

Let $\\{u_{\varepsilon}\\}_{\varepsilon}$ be a sequence strongly compact in
$L^{2}(0,T;H^{1}(\mathbb{T}^{d}))$. Then,

$\frac{u_{\varepsilon}(t,x-\varepsilon
y)-u_{\varepsilon}(t,x)}{\varepsilon|y|}\to-\nabla
u(t,x)\cdot\frac{y}{|y|}\mbox{ strongly in
}L^{\infty}_{y}(\mathbb{T}^{d};L^{2}_{(t,x)}((0,T)\times\mathbb{T}^{d})).$

###### Proof.

We write

$\displaystyle\frac{u_{\varepsilon}(t,x-\varepsilon
y)-u_{\varepsilon}(t,x)}{\varepsilon|y|}$ $\displaystyle=-\int_{0}^{1}\nabla
u_{\varepsilon}(t,x-\varepsilon sy)\cdot\frac{y}{|y|}\mathop{}\\!\mathrm{d}s$
$\displaystyle=-\nabla u_{\varepsilon}(t,x)\frac{y}{|y|}-\int_{0}^{1}[\nabla
u_{\varepsilon}(t,x-\varepsilon sy)-\nabla
u_{\varepsilon}(t,x)]\cdot\frac{y}{|y|}\mathop{}\\!\mathrm{d}s.$

Therefore

$\displaystyle\left\|\frac{u_{\varepsilon}(t,x-\varepsilon
y)-u_{\varepsilon}(t,x)}{\varepsilon|y|}+\nabla
u(t,x)\frac{y}{|y|}\right\|_{L^{2}_{t,x}}$ $\displaystyle\leq\left\|(\nabla
u_{\varepsilon}-\nabla u)\cdot\frac{y}{|y|}\right\|_{L^{2}_{t,x}}$
$\displaystyle+\int_{0}^{1}\int_{(0,T)\times\mathbb{T}^{d}}|\nabla
u_{\varepsilon}(t,x-\varepsilon sy)-\nabla
u_{\varepsilon}(t,x)|^{2}\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}s$

With the assumptions of the lemma, we deduce that the first term converges to
0 uniformly with respect to $y$. Concerning the second term, we use the
Fréchet-Kolmogorov theorem to deduce that the translation converge to 0
uniformly: there exists a modulus of continuity
$\gamma:[0,\infty)\to[0,\infty)$ such that $\lim_{r\to 0}\gamma(r)=0$ and

$\int_{(0,T)\times\mathbb{T}^{d}}|\nabla u_{\varepsilon}(t,x-\varepsilon
sy)-\nabla
u_{\varepsilon}(t,x)|^{2}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t\leq\gamma(\varepsilon\,s\,|y|)\leq\gamma(\varepsilon).$

This proves the result. ∎

##  Appendix B Bourgain-Brézis-Mironescu and Ponce compactness result

We consider a sequence of radial functions
$\\{\rho_{\varepsilon}\\}_{\varepsilon}$ such that $\rho_{\varepsilon}\geq 0$,
$\int_{\mathbb{R}^{d}}\rho_{\varepsilon}\mathop{}\\!\mathrm{d}x=1$ and

$\lim_{\varepsilon\to
0}\int_{|x|>\delta}\rho_{\varepsilon}(x)\mathop{}\\!\mathrm{d}x=0\mbox{ for
all }\delta>0.$

For the formulation of the compactness result, we use another sequence
$\\{\varphi_{\delta}\\}_{\delta\in(0,1)}\subset
C_{c}^{\infty}(\mathbb{R}^{d})$ of standard mollifiers with mass 1 such that
$\varphi_{\delta}(x)=\frac{1}{\delta^{d}}\varphi(\frac{x}{\delta})$ with
$\varphi$ of mass 1 and compactly supported.

######  Theorem B.1.

Let $d\geq 2$. Let $\\{f_{\varepsilon}\\}$ be a sequence bounded in
$L^{2}((0,T)\times\mathbb{T}^{d})$. Suppose that there exists a sequence
$\\{\rho_{\varepsilon}\\}$ as above such that

$\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{|f_{\varepsilon}(t,x)-f_{\varepsilon}(t,y)|^{2}}{|x-y|^{2}}\rho_{\varepsilon}(|x-y|)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}t\leq
C$ (B.1)

for some constant $C$. Then, $\\{f_{\varepsilon}\\}$ is compact in space in
$L^{2}((0,T)\times\mathbb{T}^{d})$, i.e.

$\lim_{\delta\to 0}\limsup_{\varepsilon\to
0}\int_{0}^{T}\int_{\mathbb{T}^{d}}|f_{\varepsilon}\ast\varphi_{\delta}(t,x)-f_{\varepsilon}(t,x)|^{2}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=0.$
(B.2)

######  Remark B.2.

Let $\omega:\mathbb{R}^{d}\to\mathbb{R}$ be a function as in (1.4)-(1.5).
Consider
$\omega_{\varepsilon}=\frac{1}{\varepsilon^{d}}\omega\left(\frac{x}{\varepsilon}\right)$.
Suppose that

$\int_{0}^{T}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{|f_{\varepsilon}(x)-f_{\varepsilon}(y)|^{2}}{\varepsilon^{2-\alpha}|y|^{\alpha}}\omega_{\varepsilon}(|x-y|)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}t\leq\widetilde{C}.$

Then, (B.1) is satisfied. Indeed, we consider

$\rho_{\varepsilon}(x)=\frac{\omega_{\varepsilon}(|x|)\,|y|^{2-\alpha}}{\varepsilon^{2-\alpha}\,\int_{\mathbb{R}^{d}}\omega(y)|y|^{2-\alpha}\mathop{}\\!\mathrm{d}y}$
(B.3)

so that (B.1) holds true with
$\frac{\widetilde{C}}{\int_{\mathbb{R}^{d}}\omega(y)|y|^{2-\alpha}\mathop{}\\!\mathrm{d}y}$.

##  Appendix C Nonlocal Poincaré inequalities

Let $\omega:\mathbb{R}^{d}\to\mathbb{R}$ be a smooth function, supported in
the unit ball such that
$\int_{\mathbb{R}^{d}}\omega(x)\mathop{}\\!\mathrm{d}x=1$. Consider
$\omega_{\varepsilon}=\frac{1}{\varepsilon^{d}}\omega\left(\frac{x}{\varepsilon}\right)$.

######  Lemma C.1.

For all $\alpha\in[0,2]$, there exists $C_{p}$ and $\varepsilon_{0}^{A}$ such
that

$\int_{\mathbb{T}^{d}}|f-(f)_{\mathbb{T}^{d}}|^{2}\leq\frac{1}{4C_{p}}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{|f(t,x)-f(t,y)|^{2}}{\varepsilon^{2-\alpha}|x-y|^{\alpha}}\omega_{\varepsilon}(|x-y|)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y$

for every $f\in L^{2}(\mathbb{T}^{d})$ and
$\varepsilon\leq\varepsilon_{0}^{A}$.

For the proof, we refer to Ponce [55, Theorem 1.1]. We also have an opposite
inequality from [3, Theorem 1]:

######  Lemma C.2.

For all $\alpha\in[0,2]$, there exists a constant $C=C(\mathbb{T}^{d},\alpha)$
such that for all $f\in H^{1}(\mathbb{T}^{d})$

$\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{|f(x)-f(y)|^{2}}{\varepsilon^{2-\alpha}|x-y|^{\alpha}}\omega_{\varepsilon}(x-y)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\leq
C\,\|f\|_{H^{1}(\mathbb{T}^{d})}^{2}.$

Finally, we formulate a variant of Lemma C.1 which does not require an average
on the left-hand side. The proof can be adapted from [25, Lemma C.3].

######  Lemma C.3.

For each $\gamma\in(0,1)$ there exists ${\varepsilon}_{0}^{B}$ and constant
$C(\gamma)$ such that for all $\varepsilon\in(0,{\varepsilon}_{0}^{B})$ and
all $f\in H^{1}(\mathbb{T}^{d})$ we have

$\|f\|^{2}_{H^{1}(\mathbb{T}^{d})}\leq\gamma\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}\frac{|\nabla
f(x)-\nabla
f(y)|^{2}}{\varepsilon^{2-\alpha}|x-y|^{\alpha}}\omega_{\varepsilon}(|x-y|)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y+C(\gamma)\|f\|^{2}_{L^{2}(\mathbb{T}^{d})}.$

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