for a suitably small $\xi>0$. The result can be obtained by replacing
$W\ast(\Phi_{\bar{\rho}}^{\prime}{\bar{\rho}}^{\delta})=\int_{\mathbb{T}^{d}}W(x-y)\Phi_{\bar{\rho}}^{\prime}(y){\bar{\rho}}^{\delta}(y)dy$
with
$\int_{\mathbb{T}^{d}}W(y-x)\Phi_{\bar{\rho}}^{\prime}(y){\bar{\rho}}^{\delta}(y)dy$
in II and (3.25), and then estimating the resulting difference as

$\displaystyle\left|\Phi^{\delta}(x)-\int_{\mathbb{T}^{d}}W(y-x)\Phi_{\bar{\rho}}^{\prime}(y){\bar{\rho}}^{\delta}(y)dy\right|$
$\displaystyle\quad\leqslant
2\|\Phi^{\prime\prime}\|_{\infty}\|W\|_{L^{2}(\mathbb{T}^{d})}^{2}\int_{\mathbb{T}^{d}}({\bar{\rho}}^{\delta})^{2}dx+\|\Phi^{\prime}\|_{\infty}\int_{\mathbb{T}^{d}}\left|{\bar{\rho}}^{\delta}(y)(W(x-y)-W(y-x))\right|dy,$

where the first term is as in the proof above and the last term pops up due to
the lack of symmetry of $W$. By a similar argument to the last paragraphs of
the proof of Proposition 4.4 for the four population model (4.1), we get the
additional condition (3.26), cf. (4.14).

##  4 Extension of the main results to a model for grid cells

The stochastic neural field PDE system (1.1) was first proposed in [11] in the
following more general form, with four populations of neurons with an
orientation preference:

$\displaystyle\tau_{c}\frac{\partial\rho^{\beta}}{\partial
t}=-\frac{\partial}{\partial
s}\Bigg{(}\Big{[}\Phi^{\beta}(x,t)-s\Big{]}\rho^{\beta}\Bigg{)}+\sigma\frac{\partial^{2}\rho^{\beta}}{\partial
s^{2}},$ (4.1)

where $\Phi^{\beta}(x,t)$ is given by

$\displaystyle\Phi^{\beta}(x,t)=\Phi\left(\frac{1}{4}\sum_{\beta^{\prime}=1}^{4}\int_{\mathbb{T}^{d}}W^{\beta^{\prime}}(x-y)\int_{0}^{\infty}s\rho^{\beta^{\prime}}(y,s,t)\,\textrm{d}s\textrm{d}y+B^{\beta}(t)\right),$
(4.2)

where the orientation preference in physical space is represented by the index
$\beta=1,2,3,4$ (north, west, south, east), and with corresponding no-flux
boundary and mass normalisation conditions. In the case $d=2$, the system
(4.1) models a network of noisy grid cells [11]. The orientation preference is
modelled by $\beta-$dependent shifts $r^{\beta}$ and connectivity
$W^{\beta}(x-y)=W(x-y-r^{\beta})$. The movement of an animal can be connected
to this PDE system (4.1) through the orientation and time dependent input
$B^{\beta}(t)$. The system was rigorously derived from a stochastic particle
system in [8] and its noise-driven bifurcations from the homogeneous
stationary state were rigorously characterised in [13].

In the same fashion as in [13], the main analytical results obtained on the
solutions of (1.1)-(1.2) can be extended to the more general system (4.1). By
checking the key points of the proof and without writing all the technical
details, we first extend the local and global in time existence results of
Theorem 1.5 to the four population system to a very general setting, before
the asymptotic convergence results of Theorem 1.6 are extended to (4.1).

###  4.1 Global-in-time solutions for Lipschitz modulation functions

The proof of local and global existence of solutions for the four population
model (4.1)-(4.2) being a direct extension of our work in Section 2, we will
only go through the main steps in order to check that no additional difficulty
endangers the method.

We can apply the same change of variables as before to each population

$y=e^{t}s\qquad\tau=\dfrac{1}{2}(e^{2t}-1),\qquad
q^{\beta}(x,y,\tau)=\alpha(\tau)\rho^{\beta}\Big{(}x,y\alpha(\tau),-\log(\alpha(\tau))\Big{)},$

and then we can apply separately to each population the change of variable

$\displaystyle\gamma^{\beta}(x,\tau)=-\int_{0}^{\tau}\Phi^{\beta}\left(\alpha(\eta)\sum_{\beta=1}^{4}W\ast\bar{q}^{\beta}(x,\eta)+\lambda^{\beta}(\eta)\right)\alpha(\eta)d\eta,$
(4.3) $\displaystyle
u^{\beta}(x,z,\tau)=q^{\beta}(x,y,\tau),\qquad\bar{u}^{\beta}=\int_{\gamma^{\beta}(x,\tau)}^{+\infty}zu^{\beta}(z,\tau)dz,\quad
v^{\beta}(x,\tau)=u^{\beta}(x,\gamma(x,\tau),\tau).$ (4.4)

Then, we obtain a similar free boundary Stefan problem with non-local Robin
boundary condition: for $\beta\in\\{1,2,3,4\\}$,

$\left\\{\begin{array}[]{rcl}\displaystyle\dfrac{\partial
u^{\beta}}{\partial\tau}(x,z,\tau)&=&\dfrac{\partial^{2}u^{\beta}}{\partial
z^{2}}(x,z,\tau),\qquad\qquad\qquad\
z\in(\gamma^{\beta}(x,\tau),+\infty),\\\\[5.69054pt]
\displaystyle\gamma^{\beta}(x,\tau)&=&-\displaystyle\int_{0}^{\tau}\Psi^{\beta}(x,\eta)d\eta,\\\
\dfrac{\partial u^{\beta}}{\partial
y}(x,\gamma(x,\tau),\tau)&=&\Psi^{\beta}(x,\tau)u^{\beta}(x,\gamma(x,\tau),\tau),\\\
\Psi^{\beta}(x,\tau)&=&\displaystyle\Phi\left(\alpha(\tau)\sum_{\beta=1}^{4}W^{\beta}\ast[\bar{u}^{\beta}(x,\tau)-\gamma^{\beta}(x,\tau)]+\lambda^{\beta}(\tau)\right)\alpha(\tau),\\\
\displaystyle u^{\beta}(x,z,0)&=&u^{0,\beta}(x,z),\qquad\qquad\qquad\qquad
z\in\,(0,+\infty).\end{array}\right.$ (4.5)

The exact same method that we used in Section 2 for the one population model
allows us to derive Duhamel formulae for $u^{\beta},v^{\beta},\gamma^{\beta}$.

For the solution:

$u^{\beta}(x,z,\tau)=\int_{0}^{+\infty}G(z,\tau,\xi,0)u^{0,\beta}(x,\xi)d\xi+\int_{0}^{\tau}\dfrac{\partial
G}{\partial\xi}(z,\tau,\gamma^{\beta}(x,\eta),\eta)\,v^{\beta}(x,\eta)d\eta.$
(4.6)

For the boundary value:

$\displaystyle v^{\beta}(x,\tau)=$
$\displaystyle\,2\int_{0}^{+\infty}G(\gamma^{\beta}(x,\tau),\tau,\xi,0)u^{0,\beta}(x,\xi)d\xi$
$\displaystyle\,+2\int_{0}^{\tau}\dfrac{\partial
G}{\partial\xi}(\gamma^{\beta}(x,\tau),\tau,\gamma^{\beta}(x,\eta),\eta)\,v^{\beta}(x,\eta)d\eta.$
(4.7)

For the average:

$\displaystyle\bar{u}^{\beta}(x,\tau)=$
$\displaystyle\,\int_{0}^{+\infty}\int_{\gamma^{\beta}(x,\tau)}^{+\infty}zG(z,\tau,\xi,0)dzu^{0,\beta}(x,\xi)d\xi$
$\displaystyle\,+\int_{0}^{\tau}\int_{\gamma(x,\tau)}^{+\infty}z\dfrac{\partial
G}{\partial\xi}(z,\tau,\gamma^{\beta}(x,\eta),\eta)dz\,v^{\beta}(x,\eta)d\eta.$
(4.8)

For the motion of the boundary:

$\gamma^{\beta}(x,\tau)=-\int_{0}^{\tau}\Phi\left(\alpha(\eta)\sum_{\beta=1}^{4}W^{\beta}\ast[\bar{u}^{\beta}(x,\eta)-\gamma^{\beta}(x,\eta)]+\lambda^{\beta}(\eta)\right)\alpha(\eta)d\eta.$
(4.9)

Hence, if we denote in capital letters the vectors containing entries for all
$\beta$, we can write the same kind of closed system as before

$\left\\{\begin{array}[]{rcl}V(x,\tau)&=&F_{v}[V,\Gamma,\bar{U}](x,\tau)\\\
\Gamma(x,\tau)&=&F_{\gamma}[V,\Gamma,\bar{U}](x,\tau)\\\
\bar{U}(x,\tau)&=&F_{\bar{u}}[V,\Gamma,\bar{U}](x,\tau).\end{array}\right.$
(4.10)

We can then consider the space

$\mathcal{C}_{\tau_{0},m}=\\{\textbf{w}\in
C^{0}(\mathbb{T}^{d}\times[0,\tau_{0}])^{8}\ |\
\left\lVert\textbf{w}\right\rVert_{\infty}<m\\},$

where we denote

$\left\lVert\textbf{w}\right\rVert_{\infty}=\max_{i\in\\{1,\dots,8\\}}\
\left\lVert w_{i}\right\rVert_{L^{\infty}(\mathbb{T}^{d}\times[0,\tau_{0}])}.$

On this space, we consider the functional

$\begin{array}[]{rccl}\mathcal{T}:&\mathcal{C}_{\tau_{0},m}&\to&C^{0}(\mathbb{T}^{d}\times[0,\tau_{0}])^{8}\\\
&(V,\bar{U})&\mapsto&(F_{v}[V,\Gamma,\bar{U}],F_{\bar{u}}[V,\Gamma,\bar{U}]),\end{array}$
(4.11)

where $\Gamma=(\gamma^{1},\gamma^{2},\gamma^{3},\gamma^{4})$ is the unique
continuous solution of the coupled initial value problems

$\left\\{\begin{array}[]{rcl}\dfrac{\partial\gamma^{\beta}}{\partial\tau}(x,\tau)&=&\displaystyle-\Phi\left(\alpha(\tau)\sum_{\beta=0}^{4}W^{\beta}\ast[\bar{u}^{\beta}(x,\tau)-\gamma^{\beta}(x,\tau)]+\lambda^{\beta}(\tau)\right)\alpha(\tau),\\\
\gamma^{\beta}(x,0)&=&\gamma^{0,\beta}(x).\end{array}\right.$ (4.12)

Like previously in Lemma 2.1, we can prove that if $\Phi$ is locally
Lipschitz, $W^{\beta}\in L^{1}(\mathbb{T}^{d})$ and $m>0$ is fixed, then there
exists $\tau_{0}\in(0,1]\,$ small enough depending only on
$\Phi,\lambda^{\beta}$ and $m$ such that for all $\bar{u}\in
C^{0}(\mathbb{T}^{d}\times[0,\tau_{0}])$ and $\gamma^{0}\in
C^{0}(\mathbb{T}^{d})$ satisfying

$\max_{\beta\in\\{1,2,3,4\\}}\left\lVert\bar{u}^{\beta}\right\rVert_{L^{\infty}(\mathbb{T}^{d}\times[0,\tau_{0}])}\leqslant
m,\qquad\mathrm{and}\qquad\max_{\beta\in\\{1,2,3,4\\}}\left\lVert\gamma^{0,\beta}\right\rVert_{L^{\infty}(\mathbb{T}^{d})}\leqslant\frac{m}{2},$

there exists a unique solution $\Gamma\in
C^{0}(\mathbb{T}^{d}\times[0,\tau_{0}])^{4}$ to (4.12). For all
$x\in\mathbb{T}^{d}$, $\tau\mapsto\Gamma(x,\tau)$ is a $C^{1}$ function and,
denoting $\Psi=(\Psi^{1},\Psi^{2},\Psi^{3},\Psi^{4})$,

$\left\lVert\Psi\right\rVert_{L^{\infty}(\mathbb{T}^{d}\times[0,\tau_{0}])^{4}}\leqslant
C_{\Psi},\qquad\mathrm{and}\qquad\left\lVert\Gamma\right\rVert_{L^{\infty}(\mathbb{T}^{d}\times[0,\tau_{0}])^{4}}\leqslant
m.$

Now we can assume without loss of generality that $\Gamma^{0}\equiv(0,0,0,0)$
like we did above for the one population case, and choose

$m=\max_{\beta\in\\{1,2,3,4\\}}\left(\max\left(\ 2\left\lVert
u^{0,\beta}\right\rVert_{L^{\infty}(\mathbb{T}^{d}\times\mathbb{R}_{+})}\ ,\
\left\lVert
zu^{0,\beta}\right\rVert_{L^{\infty}(\mathbb{T}^{d},L^{1}(\mathbb{R}_{+}))}\
\right)+1\right).$

Then, we can follow, for each component $\beta\in\\{1,2,3,4\\}$, the proof of
Proposition 2.3. The interaction appearing only in the terms $\Psi^{\beta}$
which are uniformly controlled, no additional difficulty can arise and the
exact same techniques lead to a local existence result for the closed system
(4.10).

Then, for global existence, as we argued before, it is enough to obtain
uniform bounds on $\Gamma$ or $\bar{U}$ for any compact interval of time. We
can follow the proof of Theorem 2.9: step 1 can be done with the sole
modification of bounding the sum in oder to obtain

$\left\lVert\Gamma\right\rVert_{L^{\infty}(\mathbb{T}^{d}\times[0,\tau^{*}))^{4}}\leqslant
C^{*};$

then, for step 2, we can apply the same test function $h$ in original
variables and obtain

$\forall
x\in\mathbb{T}^{d},\quad\int_{0}^{+\infty}h(s)\rho^{\beta}(x,s,t)ds=\int_{0}^{+\infty}h(s)\rho^{0,\beta}(x,s)ds\\\
+\frac{1}{\tau_{c}}\int_{0}^{t}\int_{0}^{+\infty}\left[(\Phi_{\bar{\rho}}^{\beta}(x,t^{\prime})-s)\dfrac{dh}{ds}(s)+\sigma\dfrac{d^{2}h}{ds^{2}}(s)\right]\rho(x,s,t^{\prime})dsdt^{\prime},$
(4.13)

and like before get to

$\displaystyle\bar{\rho}^{\beta}(x,t)\leqslant\dfrac{1}{L^{d}}$
$\displaystyle+\int_{0}^{+\infty}h(s)\rho^{0,\beta}(x,s)ds$
$\displaystyle+\dfrac{T^{*}}{\tau_{c}L^{d}}\left(C_{1}^{*}\max_{\beta\in\\{1,2,3,4\\}}\left(\left\lVert
W\right\rVert_{L^{1}}\left\lVert\bar{\rho}^{\beta}\right\rVert_{L^{\infty}(\mathbb{T}^{d}\times[0,t))}+\left\lVert
B\right\rVert_{L^{\infty}([0,T^{*}))}\right)+C_{2}^{*}\right);$

taking the supremum on $\beta$ in the left-hand side and $T^{*}$ small enough
yields a locally uniform in time bound on
$(\bar{\rho}^{1},\bar{\rho}^{2},\bar{\rho}^{3},\bar{\rho}^{4})$. Hence, we can
claim the following result.

######  Theorem 4.1 (Solutions for the four population model).

Assume $\Phi$ is locally Lipschitz, $B^{\beta}\in C^{0}(\mathbb{R}_{+})$,
$\beta\in\\{1,2,3,4\\}$, and $W\in L^{1}(\mathbb{T}^{d})$. Then, for any
compatible initial data $\rho^{0,\beta}$, $\beta\in\\{1,2,3,4\\}$, equation
(4.1) admits a unique maximal solution in the sense of Definition 1.4 for each
component. Moreover, if $\Phi$ is globally Lipschitz,

  * •

the solution is global-in-time;

  * •

if $\Phi$ non-negative, then

$\forall\beta\in\\{1,2,3,4\\},\ \forall t\in\mathbb{R}_{+},\ \forall
x\in\mathbb{T}^{d},\quad\rho^{\beta}(x,0,t)\leqslant\dfrac{1}{L^{d}}\sqrt{\dfrac{2}{\pi\sigma}}\dfrac{1}{\sqrt{1-e^{-\frac{2t}{\tau_{c}}}}};$

  * •

if $\Phi$ is non-positive, then for all $\beta\in\\{1,2,3,4\\}$,

$\forall t\in\mathbb{R}_{+},\ \forall
x\in\mathbb{T}^{d},\quad\bar{\rho}^{\beta}(x,t)\leqslant\dfrac{1}{L^{d}}(1+\sigma-\sigma
e^{-\frac{2t}{\tau_{c}}})+e^{-\frac{2t}{\tau_{c}}}\sup_{x\in\mathbb{T}^{d}}\int_{0}^{+\infty}s^{2}\rho^{0,\beta}(x,s)ds.$

######  Remark 4.2.

An alternative route for the extension of existence to the four population
system would be, denoting
$\Lambda=(\Lambda^{1},\Lambda^{2},\Lambda^{3},\Lambda^{4})$ a vector of four
continuous functions of time, to introduce the following functional: consider
for each $\beta$ the solution of the one population problem (2.3) with
exterior input $\Lambda^{\beta}$. Given the four solutions of these one
population problems, we define $\mathcal{F}(\Lambda)$ as the vector whose
entries $\beta\in\\{1,2,3,4\\}$ are

$\mathcal{F}(\Lambda)_{\beta}=\sum_{\beta^{\prime}=1,\
\beta^{\prime}\neq\beta}^{4}W^{\beta}\ast[\bar{u}^{\beta}(x,\tau)-\gamma^{\beta}(x,\tau)]+\lambda^{\beta}(\tau).$

From the regularities proved in the one population existence proof, it is
possible to prove that this functional is a contraction on a suitable Banach
space. Applications of the Banach fixed point theorem then yields a unique
fixed point $(\Lambda^{1},\Lambda^{2},\Lambda^{3},\Lambda^{4})$. Using these
four functions as exterior input in the one-population existence theorem and
assembling these carefully crafted pieces, we recover the unique solution of
the four population model.

Yet another way to frame the extension is to introduce, instead of the vector
$\bar{U}=(\bar{u}^{1},\bar{u}^{2},\bar{u}^{3},\bar{u}^{4})$, the scalar
variable

$\bar{U}=\sum_{\beta^{\prime}=1}^{4}W^{\beta^{\prime}}\ast\bar{u}^{\beta^{\prime}},$

and then to sum the Duhamel formulae for $\bar{u}^{\beta}$ in order to include
this quantity in the fixed point functional (4.11), thus symplifying the space
upon which it is defined. This last extension framework was used in [13] to
extend bifurcation results from the one population system to the model for
grid cells.

###  4.2 Long-time behaviour for constant and identical input functions

Note that due to the arguments of [13, Sec. 4], the stationary states for the
four different orientation preferences $\beta$ are identical. Checking the
proof of Proposition 3.1, the result can be straightforwardly extended to
(4.1):

######  Corollary 4.3.

Let $B^{\beta}$ be constant and identical for $\beta=1,2,3,4$. Let $\Phi$ be
Lipschitz, $W\in L^{2}(\mathbb{T}^{d})$, and let $\rho_{\infty}$ be a
stationary state of (4.1), equal for $\beta=1,2,3,4$, given a $\sigma$. Assume
that $\sigma$ satisfies

$\displaystyle\|\Phi^{\prime}\|_{\infty}\|W\|_{L^{2}({\mathbb{T}^{d}})}\left(\sup_{x\in{\mathbb{T}^{d}}}M_{\infty}^{\nicefrac{\mathrm{1}}{\mathrm{2}}}(x)\right)<2\sigma\tilde{\gamma}(\rho_{\infty})^{\nicefrac{\mathrm{1}}{\mathrm{2}}},$

where
$\tilde{\gamma}(\rho_{\infty})=\inf_{x\in\mathbb{T}^{d}}\gamma(\rho_{\infty}^{\beta}(x))$,
and $\gamma(\rho_{\infty}(x))$ is the Poincaré constant of (3.5) for
$\rho_{\infty}(x)$, and $M_{\infty}$ is defined in (3.4). Then, the sum of the
$L^{1}$ norms of the relative entropies,

$\displaystyle\sum_{\beta}\int_{\mathbb{T}^{d}}\int_{0}^{+\infty}\left(\dfrac{\rho^{\beta}(x,s,t)-\rho_{\infty}(x,s)}{\rho_{\infty}(x,s)}\right)^{2}\rho_{\infty}(x,s)dsdx,$

decays exponentially fast whenever the initial data $\rho_{0}^{\beta}$ are
close enough to the stationary state $\rho_{\infty}$ in relative entropy.

Extending the result of Proposition 3.7 to the system (4.1) requires a bit
more work as the stability condition in this case will depend on the size of
the shifts $r^{\beta}$ as well.

######  Proposition 4.4.

Let $B^{\beta}$ be constant and identical for $\beta=1,2,3,4$, $\Phi\in
C^{2}(\mathbb{R})\cap\mathcal{W}^{2,\infty}(\mathbb{R})$, and $W\in
L^{2}(\mathbb{T}^{d})\cap\mathcal{W}^{1,\infty}(\mathbb{T}^{d})$. Assume that
$\rho_{\infty}$ is a spatially homogeneous stationary state of (4.1), equal
for $\beta=1,2,3,4$, given a $\sigma$. Furthermore, assume that $W$ is
componentwise symmetric, and that $\sigma$ satisfies

$\displaystyle\sum_{\beta}\int_{\mathbb{T}^{d}}(1-\alpha)(g^{\beta})^{2}(x)-\frac{\Phi^{\delta}_{g}(x)}{\sigma}g^{\beta}(x)\,dx>0,\quad
g^{\beta}\in L^{2}(\mathbb{T}^{d}),$

where

$\Phi^{\delta}_{g}(x)=\Phi\left(\frac{M_{\infty}}{4}\sum_{\beta}W\ast
g^{\beta}(x-r^{\beta})+W_{0}{\bar{\rho}}_{\infty}+B\right)-\Phi_{0},$

and $M_{\infty}$ is defined in (3.4). Then, if the shifts $r^{\beta}$ satisfy

$\displaystyle\max_{\beta,j}|r^{\beta}+r^{j}|\leqslant\frac{\alpha^{\nicefrac{\mathrm{3}}{\mathrm{2}}}(2-4\xi)\gamma(\rho_{\infty})^{\nicefrac{\mathrm{1}}{\mathrm{2}}}\sigma}{\|\nabla
W\|_{\infty}\|\Phi^{\prime}\|_{\infty}M_{\infty}}$ (4.14)

for a suitably small $\xi>0$, the sum of the $L^{1}$ norms of the relative
entropies,

$\displaystyle\sum_{\beta}\int_{\mathbb{T}^{d}}\int_{0}^{+\infty}\left(\dfrac{\rho^{\beta}(x,s,t)-\rho_{\infty}(s)}{\rho_{\infty}(s)}\right)^{2}\rho_{\infty}(s)dsdx,$

decays exponentially fast whenever the initial data $\rho_{0}^{\beta}$ are
close enough to the stationary state $\rho_{\infty}$ in relative entropy.

###### Proof.

We follow the lines of the proof of Proposition 3.7, but this time denote

$\Phi^{\delta}=\Phi\left(\frac{1}{4}\sum_{\beta}W^{\beta}\ast{\bar{\rho}}^{\beta}+B\right)-\Phi_{0},\quad\bar{\rho}^{\delta}_{\beta}=\bar{\rho}^{\beta}-\bar{\rho}_{\infty},\quad\Phi^{\prime}_{{\bar{\rho}}}=\Phi^{\prime}\left(\frac{1}{4}\sum_{\beta}W^{\beta}\ast{\bar{\rho}}_{\beta}+B\right),$

and

$\displaystyle\mathcal{E}=\sum_{\beta}\int_{0}^{+\infty}\frac{1}{2}(h^{\beta}-1)^{2}\rho_{\infty}ds,\quad\mathrm{and}\quad\mathcal{H}=\sum_{\beta}\Phi^{\delta}\bar{\rho}^{\delta}_{\beta}.$

We have

$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{d}}2\mathcal{E}-\frac{1}{\sigma}\mathcal{H}\,dx=\,$
$\displaystyle
2\int_{\mathbb{T}^{d}}\Phi^{\delta}\sum_{\beta}\int_{0}^{+\infty}\partial_{s}h^{\beta}(h^{\beta}-1)\rho_{\infty}dsdx-2\sigma\sum_{\beta}\int_{\mathbb{T}^{d}}\int|\partial_{s}h^{\beta}|^{2}\rho_{\infty}dsdx$
$\displaystyle+3\int_{\mathbb{T}^{d}}\Phi^{\delta}\sum_{\beta}\int_{0}^{+\infty}\partial_{s}h^{\beta}\rho_{\infty}dsdx-\frac{4}{\sigma}\int_{\mathbb{T}^{d}}(\Phi^{\delta})^{2}dx$
$\displaystyle-\frac{1}{\sigma}\int_{\mathbb{T}^{d}}\left(\sum_{\beta}{\bar{\rho}}^{\delta}_{\beta}\right)\,\Phi^{\prime}_{\bar{\rho}}\frac{1}{4}\sum_{\beta}W^{\beta}\ast\left(\Phi^{\delta}-\sigma\int_{0}^{+\infty}\partial_{s}h^{\beta}\rho_{\infty}ds\right)dx$
$\displaystyle=$
$\displaystyle-\frac{2}{\sigma}\sum_{\beta}\int_{\mathbb{T}^{d}}\int_{0}^{+\infty}\left(\Phi^{\delta}-\sigma\partial_{s}h^{\beta}\right)^{2}\rho_{\infty}dsdx$
$\displaystyle+\frac{1}{\sigma}\sum_{\beta}\int_{\mathbb{T}^{d}}\Phi^{\delta}\left(\Phi^{\delta}-\sigma\int_{0}^{+\infty}\partial_{s}h^{\beta}\rho_{\infty}ds\right)dx$
$\displaystyle+2\int_{\mathbb{T}^{d}}\Phi^{\delta}\sum_{\beta}\int_{0}^{+\infty}\partial_{s}h^{\beta}(h^{\beta}-1)\rho_{\infty}dsdx$
$\displaystyle-\frac{1}{\sigma}\int_{\mathbb{T}^{d}}\left(\sum_{\beta}{\bar{\rho}}^{\delta}_{\beta}\right)\,\Phi^{\prime}_{\bar{\rho}}\frac{1}{4}\sum_{\beta}W^{\beta}\ast\left(\Phi^{\delta}-\sigma\int_{0}^{+\infty}\partial_{s}h^{\beta}\rho_{\infty}ds\right)dx\,.$

Using the symmetry of $W$,

$\displaystyle-\frac{1}{\sigma}\int_{\mathbb{T}^{d}}\left(\sum_{\beta}{\bar{\rho}}^{\delta}_{\beta}\right)\,\Phi^{\prime}_{\bar{\rho}}\frac{1}{4}\sum_{\beta}W^{\beta}\ast\left(\Phi^{\delta}-\sigma\int_{0}^{+\infty}\partial_{s}h^{\beta}\rho_{\infty}ds\right)dx$
$\displaystyle=$
$\displaystyle-\frac{1}{4\sigma}\sum_{j}\sum_{\beta}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}{\bar{\rho}}^{\delta}_{\beta}(x)\,\Phi^{\prime}_{\bar{\rho}}(x)\,W(x-y-r^{j})\left(\Phi^{\delta}(y)-\sigma\int_{0}^{+\infty}\partial_{s}h^{j}(y)\rho_{\infty}ds\right)dydx$
$\displaystyle=$
$\displaystyle-\frac{1}{4\sigma}\sum_{j}\sum_{\beta}\int_{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}{\bar{\rho}}^{\delta}_{\beta}(x)\,\Phi^{\prime}_{\bar{\rho}}(x)\,W(y+r^{j}-x)\left(\Phi^{\delta}(y)-\sigma\int_{0}^{+\infty}\partial_{s}h^{j}(y)\rho_{\infty}ds\right)dydx$
$\displaystyle=$
$\displaystyle-\frac{1}{4\sigma}\sum_{j}\sum_{\beta}\int_{\mathbb{T}^{d}}W\ast\left({\bar{\rho}}^{\delta}_{\beta}\,\Phi^{\prime}_{\bar{\rho}}\right)(y+r^{j})\,\left(\Phi^{\delta}(y)-\sigma\int_{0}^{+\infty}\partial_{s}h^{j}(y)\rho_{\infty}ds\right)dy$
$\displaystyle=$
$\displaystyle-\frac{1}{\sigma}\sum_{j}\int_{\mathbb{T}^{d}}\left(\frac{1}{4}\sum_{\beta}W\ast\big{(}\Phi_{\bar{\rho}}^{\prime}\,{\bar{\rho}}^{\delta}_{\beta}\big{)}(x)\right)\int_{0}^{+\infty}\left(\Phi^{\delta}-\sigma\partial_{s}h^{j}\right)(x-r^{j})\rho_{\infty}dsdx$

where the last equality is just a change of variable $x=y+r^{j}$. Now, taking
into account

$\displaystyle\int_{\mathbb{T}^{d}}\Phi^{\delta}\left(\Phi^{\delta}-\sigma\int_{0}^{+\infty}\partial_{s}h^{j}\rho_{\infty}ds\right)dx=\int_{\mathbb{T}^{d}}\Phi^{\delta}(x-r^{j})\int_{0}^{+\infty}\left(\Phi^{\delta}-\sigma\partial_{s}h^{j}\right)(x-r^{j})\rho_{\infty}dsdx$

and

$\int_{\mathbb{T}^{d}}\int_{0}^{+\infty}(\Phi^{\delta})^{2}\sum_{\beta}(h^{\beta}-1)\rho_{\infty}dsdx=0\,,$

we can rewrite the right hand side as

$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{d}}2\mathcal{E}-\frac{1}{\sigma}\mathcal{H}\,dx=\,$
$\displaystyle-\frac{2}{\sigma}\sum_{\beta}\int_{\mathbb{T}^{d}}\int_{0}^{+\infty}\left(\Phi^{\delta}-\sigma\partial_{s}h^{\beta}\right)^{2}\rho_{\infty}dsdx$
$\displaystyle+\frac{1}{\sigma}\sum_{j}\int_{\mathbb{T}^{d}}\left(\Phi^{\delta}(x-r^{j})-\frac{1}{4}\sum_{\beta}W\ast\big{(}\Phi_{\bar{\rho}}^{\prime}\,{\bar{\rho}}^{\delta}_{\beta}\big{)}(x)\right)$
$\displaystyle\qquad\qquad\qquad\times\int_{0}^{+\infty}\left(\Phi^{\delta}-\sigma\partial_{s}h^{j}\right)(x-r^{j})\rho_{\infty}dsdx$
$\displaystyle-\frac{2}{\sigma}\int_{\mathbb{T}^{d}}\int_{0}^{+\infty}\Phi^{\delta}\sum_{\beta}(h^{\beta}-1)\left(\Phi^{\delta}-\sigma\partial_{s}h^{\beta}\right)\rho_{\infty}dsdx$
$\displaystyle=I+II+III,$

where

$\displaystyle II$
$\displaystyle\leqslant\frac{1}{\sigma}\sum_{j}\int_{\mathbb{T}^{d}}\bigg{|}\Phi^{\delta}(x-r^{j})-\frac{1}{4}\sum_{\beta}W\ast\big{(}\Phi_{\bar{\rho}}^{\prime}{\bar{\rho}}^{\delta}_{\beta}\big{)}(x)\bigg{|}\|\Phi^{\delta}-\sigma\partial_{s}h^{j}\|_{L^{2}_{\rho_{\infty}(x)}}(x-r^{j})dx,$
$\displaystyle III$
$\displaystyle\leqslant\frac{2}{\sigma}\int_{\mathbb{T}^{d}}|\Phi^{\delta}|\sum_{\beta}\|h^{\beta}-1\|_{L^{2}_{\rho_{\infty}(x)}}\|\Phi^{\delta}-\sigma\partial_{s}h^{\beta}\|_{L^{2}_{\rho_{\infty}(x)}}dx.$

Now let

$\displaystyle\varphi^{\prime}(x)=\frac{\Phi^{\delta}(x)}{\frac{1}{4}\sum_{\beta}W^{\beta}\ast{\bar{\rho}}^{\delta}_{\beta}(x)}=\Phi^{\prime}\left(\frac{1}{4}\sum_{\beta}\left[\eta(x)(W^{\beta}\ast{\bar{\rho}}^{\beta})+(1-\eta(x))(W^{\beta}\ast{\bar{\rho}}_{\infty})\right]+B\right),$

for some $0\leqslant\eta(x)\leqslant 1$ in virtue of the mean-value theorem.
Then, through the following estimate, the shifts come into play:

$\displaystyle\bigg{|}$
$\displaystyle\Phi^{\delta}(x-r^{j})-\frac{1}{4}\sum_{\beta}W\ast\big{(}\Phi_{\bar{\rho}}^{\prime}\,{\bar{\rho}}^{\delta}_{\beta}\big{)}(x)\bigg{|}$
$\displaystyle\qquad=\frac{1}{4}\left|\sum_{\beta}\int_{\mathbb{T}^{d}}{\bar{\rho}}^{\delta}_{\beta}(y)\left(\varphi^{\prime}(x-r^{j})W(x-y-r^{\beta}-r^{j})-\Phi_{\bar{\rho}}^{\prime}(y)W(x-y)\right)dy\right|$
$\displaystyle\qquad\leqslant\frac{1}{4}\sum_{\beta}\int_{\mathbb{T}^{d}}|{\bar{\rho}}^{\delta}_{\beta}(y)|\left|\varphi^{\prime}(x-r^{j})-\Phi_{\bar{\rho}}^{\prime}(y)\right||W(x-y)|dy$
$\displaystyle\qquad\quad+\frac{1}{4}\sum_{\beta}\int_{\mathbb{T}^{d}}|{\bar{\rho}}^{\delta}_{\beta}(y)||\varphi^{\prime}(x-r^{j})|\left|W(x-y-r^{\beta}-r^{j})-W(x-y)\right|dy$
$\displaystyle\qquad=A+B.$

By using the definition of $\varphi^{\prime}(x)$ and applying the mean value
theorem to $\Phi^{\prime}$, we get

$\displaystyle\left|\varphi^{\prime}(x-r^{j})-\Phi_{\bar{\rho}}^{\prime}(y)\right|\leqslant\,$
$\displaystyle\frac{\|\Phi^{\prime\prime}\|_{\infty}}{4}\sum_{i}\left|\eta(x-r_{j})(W^{i}\ast{\bar{\rho}}^{i})(x-r_{j})\right.$
$\displaystyle\qquad\qquad\,\,\,\,\,\,\left.+(1-\eta(x-r_{j}))(W^{i}\ast{\bar{\rho}}_{\infty})-W^{i}\ast{\bar{\rho}}^{i}(y)\right|$
$\displaystyle=\,$
$\displaystyle\frac{\|\Phi^{\prime\prime}\|_{\infty}}{4}\sum_{i}\eta(x-r_{j})\left|(W^{i}\ast{\bar{\rho}}_{i}^{\delta})(x-r_{j})-W^{i}\ast{\bar{\rho}}^{\delta}_{i}(y)\right|$
$\displaystyle\leqslant\,$
$\displaystyle\frac{\|\Phi^{\prime\prime}\|_{\infty}}{2}\|W\|_{L^{2}(\mathbb{T}^{d})}\sum_{i}\|{\bar{\rho}}_{i}^{\delta}\|_{L^{2}(\mathbb{T}^{d})},$

where we used the Cauchy–Schwarz inequality in the convolutions. A further
application of the Cauchy–Schwarz inequality yields

$A\leqslant\frac{\|\Phi^{\prime\prime}\|_{\infty}}{8}\|W\|_{L^{2}(\mathbb{T}^{d})}^{2}\sum_{\beta,i}\|{\bar{\rho}}_{\beta}^{\delta}\|_{L^{2}(\mathbb{T}^{d})}\|{\bar{\rho}}_{i}^{\delta}\|_{L^{2}(\mathbb{T}^{d})}.$

We proceed to estimate $B$ as

$\displaystyle B$ $\displaystyle\leqslant\frac{1}{4}\|\nabla
W\|_{\infty}\|\Phi^{\prime}\|_{\infty}\sum_{\beta}|r^{\beta}+r^{j}|\int_{\mathbb{T}^{d}}|{\bar{\rho}}_{\beta}^{\delta}|dx\leqslant\frac{1}{4}\|\nabla
W\|_{\infty}\|\Phi^{\prime}\|_{\infty}\sum_{\beta}|r^{\beta}+r^{j}|\,\|{\bar{\rho}}_{\beta}^{\delta}\|_{L^{2}(\mathbb{T}^{d})}.$

Collecting the estimates above we finally conclude that

$\displaystyle\bigg{|}\Phi^{\delta}(x-r^{j})-\frac{1}{4}\sum_{\beta}W\ast\big{(}\Phi_{\bar{\rho}}^{\prime}\,{\bar{\rho}}^{\delta}_{\beta}\big{)}(x)\bigg{|}\leqslant
C_{1}\sum_{\beta}\|{\bar{\rho}}_{\beta}^{\delta}\|_{L^{2}(\mathbb{T}^{d})}^{2}+C_{2}\sum_{\beta}|r^{\beta}+r^{j}|\,\|{\bar{\rho}}_{\beta}^{\delta}\|_{L^{2}(\mathbb{T}^{d})},$

with
$C_{1}=\frac{1}{2}\|\Phi^{\prime\prime}\|_{\infty}\|W\|_{L^{2}(\mathbb{T}^{d})}^{2}$
and $C_{2}=\frac{1}{4}\|\nabla W\|_{\infty}\|\Phi^{\prime}\|_{\infty}$. Thus,

$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{d}}2\mathcal{E}-\frac{1}{\sigma}\mathcal{H}\,dx$
$\displaystyle\leqslant-\frac{2}{\sigma}\sum_{\beta}\|\Phi^{\delta}-\sigma
h_{s}^{\beta}\|_{L^{2}_{\rho_{\infty}}}^{2}$
$\displaystyle+\frac{1}{\sigma}\sum_{j}\int_{\mathbb{T}^{d}}\bigg{|}\Phi^{\delta}(x-r^{j})-\frac{1}{4}\sum_{\beta}W\ast\big{(}\Phi_{\bar{\rho}}^{\prime}{\bar{\rho}}^{\delta}_{\beta}\big{)}(x)\bigg{|}\|\Phi^{\delta}-\sigma\partial_{s}h^{j}\|_{L^{2}_{\rho_{\infty}(x)}}(x-r^{j})dx$
$\displaystyle+\frac{2}{\sigma}\int_{\mathbb{T}^{d}}|\Phi^{\delta}|\sum_{\beta}\|h^{\beta}-1\|_{L^{2}_{\rho_{\infty}(x)}}\|\Phi^{\delta}-\sigma\partial_{s}h^{\beta}\|_{L^{2}_{\rho_{\infty}(x)}}dx$
$\displaystyle\leqslant$
$\displaystyle\,-\frac{2}{\sigma}\sum_{j}\|\Phi^{\delta}-\sigma
h_{s}^{j}\|_{L^{2}_{\rho_{\infty}}}^{2}+\frac{1}{\sigma}\sum_{j}\|\Phi^{\delta}-\sigma
h_{s}^{j}\|_{L^{2}_{\rho_{\infty}}}\bigg{(}C_{1}\sum_{\beta}\|{\bar{\rho}}_{\beta}^{\delta}\|_{L^{2}(\mathbb{T}^{d})}^{2}$
$\displaystyle\qquad\qquad+C_{2}\sum_{\beta}|r^{\beta}+r^{j}|\|{\bar{\rho}}_{\beta}^{\delta}\|_{L^{2}(\mathbb{T}^{d})}+2\left(\int_{\mathbb{T}^{d}}|\Phi^{\delta}|^{2}\|h^{j}-1\|_{L^{2}_{\rho_{\infty}(x)}}^{2}dx\right)^{\nicefrac{\mathrm{1}}{\mathrm{2}}}\bigg{)}$
$\displaystyle\leqslant$
$\displaystyle\,-\frac{1}{\sigma}\sum_{j}\|\Phi^{\delta}-\sigma
h_{s}^{j}\|_{L^{2}_{\rho_{\infty}}}\sum_{\beta}\bigg{(}\frac{1}{2}\|\Phi^{\delta}-\sigma
h_{s}^{\beta}\|_{L^{2}_{\rho_{\infty}}}-C_{1}\|{\bar{\rho}}_{\beta}^{\delta}\|_{L^{2}(\mathbb{T}^{d})}^{2}$
$\displaystyle\qquad\qquad-
C_{2}|r^{\beta}+r^{j}|\|{\bar{\rho}}_{\beta}^{\delta}\|_{L^{2}(\mathbb{T}^{d})}-2\left(\int_{\mathbb{T}^{d}}|\Phi^{\delta}|^{2}\|h^{j}-1\|_{L^{2}_{\rho_{\infty}(x)}}^{2}dx\right)^{\nicefrac{\mathrm{1}}{\mathrm{2}}}\bigg{)}.$

Note that the third term in the parenthesis might decay at the same order as
the first term. Thus, to ensure that the right hand side of the inequality can
be made strictly smaller than zero for all $t\geqslant 0$ by enforcing a
smallness assumption on the pertubation away from the stationary state
$\rho_{\infty}$ as was done in the proof of Proposition 3.7, we need to ensure
that

$\displaystyle\sum_{\beta}\left(\frac{1}{2}\|\Phi^{\delta}-\sigma
h_{s}^{\beta}\|_{L^{2}_{\rho_{\infty}}}-C_{2}|r^{\beta}+r^{j}|\|{\bar{\rho}}_{\beta}^{\delta}\|_{L^{2}(\mathbb{T}^{d})}\right)\geqslant\xi\sum_{\beta}\|\Phi^{\delta}-\sigma
h_{s}^{\beta}\|_{L^{2}_{\rho_{\infty}}},$

for some constant $\xi>0$, which holds under assumption (4.14). This can be
checked by applying the Poincaré inequality (3.5) to the first term and then
estimates corresponding to (3.24). The remaining steps now follow by taking
extra care of the different orientation preferences, doing a change of
variables as in the proof of Proposition 3.7 and following the remaining
arguments therein. ∎

### Acknowledgements

JAC and PR were 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). JAC was also supported by the grants EP/T022132/1 and
EP/V051121/1 of the Engineering and Physical Sciences Research Council (EPSRC,
UK). Parts of this research was conducted while SS was an academic visitor at
the Mathematical Institute, University of Oxford, and the author would like to
thank the institution for its warm hospitality. The authors would like to
thank the Isaac Newton Institute for Mathematical Sciences, University of
Cambridge, for support and hospitality during the programme “Frontiers in
kinetic theory” where parts of the work on this paper was undertaken,
supported by EPSRC grant no EP/R014604/1.

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