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# Well-posedness of aggregation-diffusion systems with irregular kernels

José A. Carrillo  ,  Yurij Salmaniw  and  Jakub Skrzeczkowski  Mathematical
Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, United
Kingdom<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We consider aggregation-diffusion equations with merely bounded nonlocal
interaction potential $K$. We are interested in establishing their well-
posedness theory when the nonlocal interaction potential $K$ is neither
differentiable nor positive (semi-)definite, thus preventing application of
classical arguments. We prove the existence of weak solutions in two cases: if
the mass of the initial data is sufficiently small, or if the interaction
potential is symmetric and of bounded variation without any smallness
assumption. The latter allows one to exploit the dissipation of the free
energy in an optimal way, which is an entirely new approach. Remarkably, in
both cases, under the additional condition that $\nabla K\ast K$ is in
$L^{2}$, we can prove that the solution is smooth and unique. When $K$ is a
characteristic function of a ball, we construct the classical unique solution.
Under additional structural conditions we extend these results to the
$n$-species system.

###### Key words and phrases:

aggregation-diffusion, irregular kernel, nonlocal PDE, entropy methods, cross-
diffusion, multi-species models

###### 1991 Mathematics Subject Classification:

35K40, 35K55, 35A01, 35A02, 35B65, 35Q92

##  1\. Introduction

Aggregation-diffusion equations and systems have been used extensively as a
tool to model mean-field approximations of interacting agents or particles in
a wide array of scientific disciplines, including fluid dynamics [56],
environmental science [6], chemical engineering [9], physics [5], phase
separation in materials science [42, 37], cell-cell adhesion in biology [23,
24, 16, 15, 35], crowd dynamics [54, 51], biological aggregation [60] and
ecology [18]. Their ubiquity can be explained by their direct connection with
discrete dynamics described by stochastic differential equations [22, 46, 53,
55, 43], random walk processes [58, 57] and singular limits connecting them
with higher-order PDEs [32, 33, 21].

In the present work, we consider the general scalar aggregation-diffusion
equation

$\displaystyle\begin{cases}\frac{\partial u}{\partial
t}=\nabla\cdot\left(D\nabla u+u\nabla(K*u)\right),\\\
u(0,x)=u_{0}(x),\end{cases}$ (1.1)

and more generally the $n$-species counterpart

$\displaystyle\begin{cases}\frac{\partial u_{i}}{\partial
t}=\nabla\cdot\left(D_{i}\nabla
u_{i}+u_{i}\sum_{j=1}^{n}\nabla(K_{ij}*u_{j})\right),\\\
u_{i}(0,x)=u_{i0}(x),\end{cases}$ (1.2)

where $*$ denotes a spatial convolution

$\displaystyle
K*v(t,x):=\int_{\mathbb{R}^{d}}K(x-y)\,v(t,y)\mathop{}\\!\mathrm{d}y$ (1.3)

for some appropriate kernel(s) $K(\cdot)$ and spatial dimension $d\geq 1$. The
coefficients $D_{i}>0$ with $i=1,\ldots,n$ describe the rate at which the
$i^{th}$ population disperses via linear diffusion while the interaction term
describes the attraction/repulsion of population $u_{i}$ to/from population
$u_{j}$ depending on the characteristics of the kernel $K_{ij}$.

The primary purpose of this work is to establish a robust well-posedness
theory (global existence & uniqueness) for irregular interaction potentials,
with the motivating prototypical form being the top-hat kernel in one
dimension:

$K(x)=\begin{cases}-\frac{\alpha}{2R},\hskip 14.22636pt-R\leq x\leq R,\cr
0,\hskip 28.45274pt\text{otherwise},\end{cases}$ (1.4)

where $\alpha\in\mathbb{R}$ is the strength of attraction ($\alpha>0$) or
repulsion ($\alpha<0$).

Despite the lack of well-posedness theory for (1.1) with the top-hat kernel,
(1.1) has been used extensively as a minimal example of nonlocal animal
interactions that can be explored analytically from the perspective of pattern
formation via linear stability analysis, see e.g., [34, 58, 57, 59, 39]. In
some cases, we obtain the top-hat kernel through derivation [57]; in other
cases, we use the top-hat kernel to explore the transition between platykurtic
(no tails) to leptokurtic (fat tailed) kernels [34]. As demonstrated in [59],
several animal movement models which incorporate spatial memory (e.g., of
marks left on a landscape [47, 52], of locations of territorial interactions
[58], or of memory of previously visited locations [48]) can be reduced to
system (1.2) through an appropriate quasi-steady state approximation. System
(1.2) can then be viewed as a minimal model describing interacting populations
with perception and implicit spatial memory.

In this work we provide a well-posedness theory for a wide class of kernels,
including (1.4). Our work solves an open problem posed in [61], and answers in
the affirmative a conjecture made in [59]. Importantly, we note that (1.4)
falls outside the scope of any recent works on the topic: [44] deals with
positive definite kernels (note that (1.4) is not positive definite as its
Fourier transform is $\frac{\sin(Rx)}{Rx}$) while [38] deals with kernels
having at least one derivative in the weak sense.

From the regularity theory point of view, (1.1) includes two terms of opposite
effect: a regularizing linear diffusion and irregular advection term which
does not fall into the classical Cauchy-Lipschitz theory nor the theory of
renormalized solutions [2, 30]. The question of which of these effects wins is
delicate and we refer to [50] for examples of vector fields $V\in
C(0,T;W^{1,p}(\mathbb{T}^{d}))$ such that uniqueness of solutions with
periodic boundary conditions to

$\frac{\partial u}{\partial t}=\nabla\cdot(u\,V)+\Delta u$

fails in the class $u\in C(0,T;L^{q}(\mathbb{T}^{d}))$ for certain exponents
$p,q$, where $\mathbb{T}^{d}$ stands for the $d$-dimensional torus. Our work
answers the question when the regularizing effect of linear diffusion is so
strong that it can smooth out the advection driven by the velocity $\nabla
K\ast u$. In particular, our assumption that $K\in
L^{\infty}(\mathbb{R}^{d})\cap\operatorname{BV}(\mathbb{R}^{d})$ (see Theorem
2.2) should not be confused with the theory of renormalized solutions for
$\operatorname{BV}$ vector fields of Ambrosio [2] which requires the gradient
$\nabla K$ to be in $\operatorname{BV}(\mathbb{R}^{d})$.

We highlight some relevant efforts related to the current objectives. If $K\in
W^{2,\infty}(\mathbb{T}^{d})$ and $n=1$, the well-posedness of (1.1) is fairly
classical and we refer to [22, 25] and references therein. Concerning the case
of the $n$-species systems, the authors in [38] construct the unique mild
solution under the assumption that $K_{ij}=\alpha_{ij}K$,
$\alpha_{ij}\in\mathbb{R}$, $i,j=1,\ldots,n$, with $K$ twice differentiable
and $\nabla K\in L^{\infty}(\mathbb{T}^{d})$. They use a semigroup theory
approach, applying a contraction mapping argument with Duhamel’s formula. The
obtained solution exists globally in time only for $d=1$. The regularity
assumption on the kernel $K$ has been substantially relaxed in [44], where
weak solutions are obtained for the $n$-species case when $D_{i}=\sigma>0$,
$i=1,\ldots,n$, and the kernels $K_{ij}\in L^{s}(\mathbb{T}^{d})$ for some
$s\geq 1$ (depending on the dimension $d$) are positive-definite and satisfy
the so-called detailed balance condition (see Hypothesis (H5)). Their approach
uses entropy estimates, where the positive-definiteness is required to control
the dissipation of the entropy. Finally, (1.1) can be considered with singular
kernels in the whole space. For instance, the particular important case with
$K$ being the Newtonian potential leads to the celebrated and well-studied
Patlak-Keller-Segel system [13, 7, 45, 12, 8, 10, 31, 14, 11, 19, 40, 36, 17,
27, 28].

The remainder of the paper is organized as follows. In Section 2, we present
the main assumptions imposed on the kernel $K$ and the main results. Then, in
Section 3, we focus on the scalar equation. We briefly introduce some
preliminary notations and notions of a weak, strong, and classical solution
for problems (1.1)-(1.2). We obtain apriori estimates for the scalar equation,
improve these estimates with no further restriction, and then prove Theorems
2.1-2.2. Under additional conditions, we obtain further improved estimates
which allow us to prove Theorems 2.3-2.4. In Section 4, we generalize the
approach to the general $n$-species system case: we first state the main
theorems, and then follow a similar organisation to Section 3. In Section 5,
we discuss some relevant applications of our results to biological systems.
Numerical simulations are provided for selected cases using a positivity-
preserving finite-volume method [20, 4].

##  2\. Main Results

###  2.1. Hypotheses

In the present work, we consider the well-posedness of system (1.2) on the
whole space for any dimension $d\geq 1$ for merely bounded kernels, noting
that these results hold true for the $d$-dimensional torus with no essential
change to the proofs.

Given a function $g(\cdot)$, we denote by $\tilde{g}(\cdot):=g(-\cdot)$ its
reflection. Our hypotheses on the kernels $K_{ij}$ are as follows.

###### (H1)

$K_{ij}\in L^{1}({\mathbb{R}^{d}})\cap L^{\infty}({\mathbb{R}^{d}})$ for all
$i,j=1,\ldots,n$;

###### (H2)

there exists $C_{K_{ij}}>0$ such that for all $\phi(\cdot)$ smooth there holds
$\left\|\nabla K_{ij}*\phi\right\|_{L^{1}({\mathbb{R}^{d}})}\leq
C_{K_{ij}}\left\|\phi\right\|_{L^{1}({\mathbb{R}^{d}})}$ or equivalently, as
proven below, $K_{ij}\in\operatorname{BV}({\mathbb{R}^{d}})$ with the total
variation $\|\nabla K_{ij}\|_{\text{TV}}<\infty$ for each $i,j=1,\ldots,n$;

###### (H3)

$K_{ij}(\cdot)$ is symmetric about the origin for each $i,j=1,\ldots,n$;

###### (H4)

$\nabla(\tilde{K}_{ij}*K_{ij})\in L^{2}({\mathbb{R}^{d}})$ for all
$i,j=1,\ldots,n$;

###### (H5)

$K_{ij}$ are in detailed balance, that is, there exists constants $\pi_{i}>0$,
$i=1,\ldots,n$, such that $\pi_{i}K_{ij}(\cdot)=\pi_{j}K_{ji}(\cdot)$ for all
$i,j=1,\ldots,n$;

###### (H6)

$K_{ij}$ is compactly supported for each $i,j=1,\ldots,n$.

We briefly comment on assumptions that are not standard.
…
Hypothesis (H2). We recall from [3] that
$K\in\operatorname{BV}(\mathbb{R}^{d})$ means that its distributional gradient
$\nabla K$ is a bounded measure, i.e.

$\|\nabla K\|_{TV}=\sup_{\varphi\in
C^{1}(\mathbb{R}^{d}),\,\|\varphi\|_{\infty}\leq
1}\int_{\mathbb{R}^{d}}K\,\operatorname{div}\varphi\mathop{}\\!\mathrm{d}x<\infty.$
(2.1)

For example, when $K=\mathds{1}_{U}$ and $U\subset\mathbb{R}^{d}$ is a bounded
set with smooth boundary, the condition is satisfied because

$\|\nabla K\|_{TV}:=\sup_{\varphi\in
C^{1}(\mathbb{R}^{d}),\,\|\varphi\|_{\infty}\leq 1}\int_{\partial
U}\langle\varphi,{\bf{n}}\rangle\mathop{}\\!\mathrm{d}S\leq|\partial
U|<\infty.$

Let us check the equivalence between the statements in hypothesis (H2).
Indeed, let $\psi\in L^{\infty}(\mathbb{R}^{d})$ with
$\|\psi\|_{L^{\infty}(\mathbb{R}^{d})}\leq 1$ and assume (2.1) holds. Then,

$\int_{\mathbb{R}^{d}}\nabla
K*\phi\,\psi\mathop{}\\!\mathrm{d}x=-\int_{\mathbb{R}^{d}}K\,\operatorname{div}\phi*\psi\mathop{}\\!\mathrm{d}x\leq\|\nabla
K\|_{\text{TV}}\,\|\phi*\psi\|_{L^{\infty}(\mathbb{R}^{d})}\leq\|\nabla
K\|_{\text{TV}}\,\|\phi\|_{L^{1}(\mathbb{R}^{d})}.$

Taking supremum over all $\psi\in L^{\infty}(\mathbb{R}^{d})$ with
$\|\psi\|_{L^{\infty}(\mathbb{R}^{d})}\leq 1$ we get

$\left\|\nabla K*\phi\right\|_{L^{1}({\mathbb{R}^{d}})}\leq\|\nabla
K\|_{TV}\,\left\|\phi\right\|_{L^{1}({\mathbb{R}^{d}})}.$ (2.2)

Conversely, assume this condition holds. Let
$\\{\omega_{\varepsilon}\\}_{\varepsilon>0}$ be a standard mollifier. We have

$\int_{\mathbb{R}^{d}}K\ast\omega_{\varepsilon}\operatorname{div}\varphi\mathop{}\\!\mathrm{d}x=\int_{\mathbb{R}^{d}}\nabla
K\ast\omega_{\varepsilon}\,\varphi\mathop{}\\!\mathrm{d}x\leq
C_{K}\,\|\varphi\|_{L^{\infty}(\mathbb{R}^{d})}$

so passing to the limit $\varepsilon\to 0$ and taking supremum over all
$\varphi\in C^{1}(\mathbb{R}^{d})$ with
$\|\varphi\|_{L^{\infty}(\mathbb{R}^{d})}\leq 1$ we deduce (2.1).

Hypothesis (H4) can be better viewed in the Fourier space. Indeed, with
$K=K_{ij}$ and $\widehat{K}$ its Fourier transform, it is sufficient by
Plancherel’s theorem to check that there holds

$\int_{|\xi|>1}|\xi|^{2}\,|\widehat{K}(\xi)|^{4}\mathop{}\\!\mathrm{d}\xi<\infty,$
(2.3)

where we ignore the set $\\{|\xi|\leq 1\\}$ as the Fourier transform of an
integrable function is bounded. Condition (2.3) is satisfied for
$K(x)=\mathds{1}_{Q}$ where $Q\subset\mathbb{R}^{d}$ is a bounded cube since
one can easily compute $|\widehat{K}(\xi)|\approx{C}\,{|\xi|^{-d}}$ for large
$\xi$. Condition (2.3) is also satisfied by $K(x)=\mathds{1}_{B_{r}}$ where
$B_{r}\subset\mathbb{R}^{d}$ is a ball of radius $r$ since
$\widehat{K}(\xi)=r^{d/2}\,|\xi|^{-d/2}\,J_{d/2}(r\,|\xi|)$ where $J_{d/2}$ is
a Bessel function of the first kind (see, e.g., [63, Lemma 12.2]) and it is
known that for large arguments $\xi$ we have $|J_{d/2}(\xi)|\approx
C\,|\xi|^{-1/2}$ (see, e.g., [63, Prop. 5.6]).

We also comment that hypothesis (H4) is implied by (H1) and (H2). Indeed, as
above, we consider function $K\ast K\ast\omega_{\varepsilon}$ which is clearly
differentiable. Let $\psi$ be such that $\|\psi\|_{L^{2}(\mathbb{R}^{d})}\leq
1$. Then,

$\displaystyle\left|\int_{\mathbb{R}^{d}}\nabla K\ast
K\ast\omega_{\varepsilon}\,\psi\mathop{}\\!\mathrm{d}x\right|$
$\displaystyle=\left|\int_{\mathbb{R}^{d}}K\,\nabla
K\ast\omega_{\varepsilon}\ast\psi\mathop{}\\!\mathrm{d}x\right|\leq\|K\|_{L^{2}(\mathbb{R}^{d})}\,\|\nabla
K\ast\omega_{\varepsilon}\ast\psi\|_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle\leq\|K\|_{L^{2}(\mathbb{R}^{d})}\,\|\nabla
K\ast\omega_{\varepsilon}\|_{L^{1}(\mathbb{R}^{d})}\leq\|K\|_{L^{2}(\mathbb{R}^{d})}\,\|\nabla
K\|_{\text{TV}}$

by (2.2). It follows that $\|\nabla K\ast
K\ast\omega_{\varepsilon}\|_{L^{2}(\mathbb{R}^{d})}\leq\|K\|_{L^{2}(\mathbb{R}^{d})}\,\|\nabla
K\|_{\text{TV}}$ so that passing to the limit $\varepsilon\to 0$ we deduce
that $\nabla K\ast K$ exists as a function in $L^{2}(\mathbb{R}^{d})$.

Hypothesis (H5): in the scalar equation, this condition is superfluous. For
$n\geq 2$ interacting species, it is a moderate generalization of a simpler
symmetry condition $K_{ij}=K_{ji}$ which ensures that the system has a
gradient flow structure [44]. A detailed balance condition has also been used
to study local cross-diffusion systems, see e.g., [26, 29]. We explore this
condition in more detail in Section 5.

We can roughly divide our results into the following:

  * i)

existence of a global weak solution for small initial mass under hypothesis
(H1) only, and for arbitrary initial mass under hypotheses (H1)-(H3);

  * ii)

global existence of a unique, classical solution under the additional
hypothesis (H4) and (H6);

  * iii)

generalization of these results to treat the $n$-species system.

Throughout the manuscript we always assume that the initial data satisfies the
mild condition

$0\lneqq u_{i0}\in L^{1}(\mathbb{R}^{d})\cap L^{\infty}({\mathbb{R}^{d}}),$

and has finite entropy and finite second moment (defined in (2.5) and (3.4),
respectively) for each $i=1,\ldots,n$. Formally, for a smooth, nonnegative
solution $\mathbf{u}=(u_{1},\ldots,u_{2})$, this problem enjoys a conservation
of mass $m_{i}$ for each component $u_{i}$:

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\left\|u_{i}(t,\cdot)\right\|_{L^{1}({\mathbb{R}^{d}})}=0\Rightarrow\left\|u_{i}(t,\cdot)\right\|_{L^{1}({\mathbb{R}^{d}})}=\left\|u_{i0}\right\|_{L^{1}({\mathbb{R}^{d}})}=:m_{i}>0,$

for all $t>0$. In the scalar case $n=1$, we simply write $m_{1}=m$.

###  2.2. Statement of Main Results

Our main results for the scalar equation are as follows.

######  Theorem 2.1 (Existence of global weak solution for scalar equation
with small mass).

Assume (H1) holds and that

$\displaystyle m\left\|K\right\|_{L^{\infty}(\mathbb{R}^{d})}<D,$ (2.4)

where $m:=\left\|u_{0}\right\|_{L^{1}(\mathbb{R}^{d})}$. Then, there exists a
global weak solution $u\geq 0$ in $Q_{T}$ solving problem (1.1) in the sense
of Definition 3.1. Moreover, $u\in L^{\infty}(0,T;L^{p}({\mathbb{R}^{d}}))$
for any $1\leq p<\infty$, and $\nabla u\in
L^{2}(0,T;L^{2}({\mathbb{R}^{d}}))$.

######  Theorem 2.2 (Existence of global weak solution for scalar equation
with arbitrary mass).

Assume (H1)-(H3) hold. Then, there exists a global weak solution $u\geq 0$ in
$Q_{T}$ solving problem (1.1) in the sense of Definition 3.1. Moreover, $u\in
L^{\infty}(0,T;L^{p}({\mathbb{R}^{d}}))$ for any $1\leq p<\infty$, and $\nabla
u\in L^{2}(0,T;L^{2}({\mathbb{R}^{d}}))$.

######  Theorem 2.3 (Existence of unique strong solution).

Assume that the initial data satisfies $\nabla u_{0}\in L^{2}(\mathbb{R}^{d})$
as well as $K\ast u_{0}\in W^{2,p}(\mathbb{R}^{d})$ for some $p>2$. Under the
assumptions of Theorem 2.2 or under the assumptions of Theorem 2.1 together
with (H4), the obtained global weak solution is the unique, global strong
solution solving problem (1.1) in the sense of Definition 3.2

######  Theorem 2.4 (Existence of unique classical solution).

Assume in addition to the hypotheses of Theorem 2.3 that (H6) hold and that
the initial data satisfies $u_{0}\in C^{3}(\mathbb{R}^{d})$ as well as $\nabla
u_{0}\in L^{p}(\mathbb{R}^{d})$ for $p>d+2$. Then, the obtained unique global
strong solution is the unique, global classical solution solving problem (1.1)
in the sense of Definition 3.3. Moreover, the solution is strictly positive
over $\mathbb{R}^{d}$ for all $t>0$.

Let us briefly explain the proofs of Theorems 2.1–2.3. The existence results
are obtained via compactness, departing from regularized equations. We now
discuss the main apriori estimates leading to the proof of Theorem 2.1. We
first analyse the evolution of the entropy functional

$\displaystyle H[u]:=\int_{\mathbb{R}^{d}}u\log u\,{\rm d}x.$ (2.5)

More precisely, differentiating $H[u(t)]$ along solutions of (1.1), one can
show that

$\displaystyle\frac{{\rm d}}{{\rm
d}t}H[u(t)]+D\,\int_{\mathbb{R}^{d}}\frac{|\nabla
u|^{2}}{u}\mathop{}\\!\mathrm{d}x\leq
m\,\|K\|_{L^{\infty}(\mathbb{R}^{d})}\,\int_{\mathbb{R}^{d}}\frac{|\nabla
u|^{2}}{u}\mathop{}\\!\mathrm{d}x.$

Classical arguments using the evolution of the second moment allow us to
estimate the entropy from below and control the entropy in bounded time
intervals using the small mass condition.

Theorem 2.2 is a more subtle consequence of the gradient flow structure and so
we can only prove it for symmetric kernels (see (H3)). More precisely, (1.1)
is the Wasserstein gradient flow of the free energy functional, i.e. the
weighted sum of the entropy and the interaction energy $\mathcal{K}[u]$ of the
system, given by

$\displaystyle\mathcal{F}[u]:=D\int_{\mathbb{R}^{d}}u\log
u\,\mathop{}\\!\mathrm{d}x+\frac{1}{2}\int_{\mathbb{R}^{d}}u(K*u)\,\mathop{}\\!\mathrm{d}x=:DH[u]+\mathcal{K}[u].$
(2.6)

The dissipation of $\mathcal{F}[u(t)]$ is given by

$\frac{{\rm d}}{{\rm
d}t}\mathcal{F}[u(t)]=-\|f(t)\|_{L^{2}(\mathbb{R}^{d})}^{2}\qquad\text{and}\qquad
f:=\sqrt{u}\,\nabla(D\log u+K*u).$

Note that this allows one to estimate $\nabla u$ in
$L^{2}(0,T;L^{1}(\mathbb{R}^{d}))$ since $D\nabla u=f\,\sqrt{u}-u\,\nabla
K\ast u$. Indeed, the first term can be bounded using the dissipation of the
free energy and the conservation of mass. The second term $u\nabla K\ast u$
can be split into small values of $u$, controlled by the bounded variation of
$K$ in (H2), and large values of $u$, controlled by the uniform equi-
integrability estimate from the entropy. We conclude that $\nabla\sqrt{u}\in
L^{2}(Q_{T})$, which is precisely the information obtained in the small-mass
case.

Improved regularity is first based on the novel condition $\nabla K\ast K\in
L^{2}(\mathbb{R}^{d})$, cf. (H4). This allows one to obtain better estimates
on the vector field $\nabla K\ast u$ by testing (1.1) with $K\ast K\ast\Delta
u$. The main point here is that while $K$ itself may not be differentiable,
$K\ast K$ enjoys regularization by convolution and may be differentiable. Once
we obtain bounds on $\nabla K\ast u$ in
$L^{\infty}(0,T;L^{2}(\mathbb{R}^{d}))$ and $\Delta K\ast u$ in
$L^{2}(Q_{T})$, we can show $u$ is bounded (see Lemma 3.7) and uniqueness can
be established. The uniqueness result is obtained by usual energy methods with
a modification that we study the evolution of the mixed local-nonlocal energy

$\mathcal{E}(u):=\int_{\mathbb{R}^{d}}u^{2}\mathop{}\\!\mathrm{d}x+C\,\int_{\mathbb{R}^{d}}|K\ast
u|^{2}\mathop{}\\!\mathrm{d}x$

for some constant $C>0$. Combining two quantities allows us to prove that if
$u$ and $v$ are regular solutions to (1.1) then
$\partial_{t}\mathcal{E}(u-v)\leq C(t)\,\mathcal{E}(u-v)$ which implies
uniqueness. Under some additional technical requirements on the initial data,
we can improve the integrability of $\nabla u$ and ultimately apply Schauder
estimates to obtain the classical differentiability of the solution.

Finally, the methods discussed above are flexible enough to be extended to the
case of the $n$-species systems, see Theorems 4.1-4.3 in Section 4.

##  3\. The Scalar Equation

In this section, we will prove well-posedness in the scalar case, Theorems
2.1-2.3. This will simplify the exposition of the general $n$-species case. We
start by introducing some notations needed and the notion of weak solution we
will work with directly in the general $n$-species case.

###  3.1. Preliminaries

We consider our problem over ${\mathbb{R}^{d}}$ for $d\geq 1$, and denote by
$Q_{T}:={\mathbb{R}^{d}}\times(0,T)$ for some $T>0$ fixed. For $1\leq
p\leq\infty$, $L^{p}({\mathbb{R}^{d}})$ denotes the usual Lebesgue space over
the spatial domain ${\mathbb{R}^{d}}$, with
$\left\|u\right\|_{L^{\infty}({\mathbb{R}^{d}})}$ the essential supremum of
$u$ over ${\mathbb{R}^{d}}$. For $1\leq p,q<\infty$,
$L^{q}(0,T;L^{p}({\mathbb{R}^{d}}))$ then denotes the typical Lebesgue space
over $Q_{T}$ with norm

$\displaystyle\left\|v\right\|_{L^{q}(0,T;L^{p}({\mathbb{R}^{d}}))}:=\left(\int_{0}^{T}\left\|v(t,\cdot)\right\|_{L^{p}({\mathbb{R}^{d}})}^{q}{\rm
d}t\right)^{1/q},$ (3.1)

and when $1\leq p\leq q=\infty$

$\displaystyle\left\|v\right\|_{L^{\infty}(0,T;L^{p}({\mathbb{R}^{d}}))}:=\operatorname*{ess\,sup}_{t\in(0,T)}\left\|v(t,\cdot)\right\|_{L^{p}({\mathbb{R}^{d}})}.$
(3.2)

In cases where $p=q$, for notational brevity we simply write
$L^{p}(Q_{T}):=L^{p}(0,T;L^{p}({\mathbb{R}^{d}}))$. We denote by
$C^{1+\sigma/2,2+\sigma}_{loc}([0,T]\times\mathbb{R}^{d})$ the class a
functions which are (locally) twice differentiable in space and once
differentiable in time (in the classical sense), with its second order spatial
derivatives and first order time derivative being Hölder continuous for some
$\sigma\in(0,1)$. We understand a weak solution to problems (1.1)-(1.2) in the
following sense.

######  Definition 3.1 (weak solution).

We call $\mathbf{u}:=(u_{1},u_{2},\ldots,u_{n})$ a weak solution to problem
(1.2) corresponding to the initial data
$\mathbf{u_{0}}:=(u_{10},\ldots,u_{n0})$ if

${u_{i}}\in L^{\infty}(0,T;L^{2}(\mathbb{R}^{d})),\quad\nabla u_{i}\in
L^{2}(0,T;L^{4/3}(\mathbb{R}^{d})),\quad\sqrt{u_{i}}(K_{ij}*\nabla u_{j})\in
L^{2}(0,T;L^{2}({\mathbb{R}^{d}})),$ $\tfrac{\partial u_{i}}{\partial t}\in
L^{2}(0,T;W^{-1,q}(\mathbb{R}^{d}))\mbox{ with }q=4/3,$

for all $i,j=1,\ldots,n$, and for all test functions $\phi_{i}\in
L^{2}(0,T;W^{1,q^{\prime}}({\mathbb{R}^{d}}))$, $i=1,\ldots,n$, there holds

$\displaystyle\int_{0}^{T}\left<(u_{i})_{t},\phi_{i}\right>\mathop{}\\!\mathrm{d}t+D_{i}\iint_{Q_{T}}\nabla
u_{i}\cdot\nabla\phi_{i}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=-\iint_{Q_{T}}u_{i}\sum_{j=1}^{n}(K_{ij}*\nabla
u_{j})\cdot\nabla\phi_{i}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t,$
(3.3)

were $\left<\cdot,\cdot\right>$ denotes the dual pairing between
$W^{-1,q}({\mathbb{R}^{d}})$ and $W^{1,q^{\prime}}({\mathbb{R}^{d}})$ with
$1/q+1/q^{\prime}=1$, and the initial data is satisfied in the sense of
$W^{-1,q}(\mathbb{R}^{d})$. We call $\mathbf{u}$ a global weak solution if
(3.3) holds for all $T>0$.

It will be shown in the proof of Theorems 2.1-2.2 that the integrals in (3.3)
are well-defined under regularity assumptions in the definition.

In fact, for many kernels we obtain a strong solution to problems (1.1)-(1.2)
in the following sense.

######  Definition 3.2 (strong solution).

We call $\mathbf{u}:=(u_{1},u_{2},\ldots,u_{n})$ a strong solution to problem
(1.1) corresponding to the initial data
$\mathbf{u_{0}}:=(u_{10},\ldots,u_{n0})$ if it is a bounded weak solution with

$\nabla u\in L^{\infty}(0,T;L^{2}(\mathbb{R}^{d}));\quad\Delta u\in
L^{2}(Q_{T}),$

and the initial data is satisfied in the sense of $L^{2}(\mathbb{R}^{d})$. We
call the strong solution global if it is a strong solution for all $T>0$.

Finally, we refer to a classical solution to problems (1.1)-(1.2) in the
following sense.

######  Definition 3.3 (classical solution).

We call $\mathbf{u}:=(u_{1},u_{2},\ldots,u_{n})$ a classical solution to
problem (1.1) corresponding to the initial data
$\mathbf{u_{0}}:=(u_{10},\ldots,u_{n0})$ if

$u_{i}\in C^{1+\sigma/2,2+\sigma}_{loc}([0,T]\times\mathbb{R}^{d}),$

for some $\sigma\in(0,1)$. We call the classical solution global if it is a
classical solution for all $T>0$.

We now establish some apriori estimates for small mass and arbitrary mass
regimes for problem (1.1).

###  3.2. Apriori estimates

We first establish some apriori estimates under the assumption that we have a
classical solution $u(t,x)$ solving problem (1.1). In addition to the entropy
$H[u]$, the interaction energy $\mathcal{K}[u]$, and the free energy
functional $\mathcal{F}[u]$ defined in (2.5)-(2.6), we also denote the second
moment by:

$I[u(t)]=\int_{\mathbb{R}^{d}}u\left|x\right|^{2}\mathop{}\\!\mathrm{d}x\,.$
(3.4)

With these estimates, subsequent improvements to these preliminary estimates
will follow. For the regime of small mass, we have the following.

######  Lemma 3.4 (Apriori estimates under (H1) and small-mass condition
(2.4)).

Fix $T>0$. Assume (H1) and suppose (2.4) holds. Then for any smooth, positive
solution $u$ to problem (1.1) there holds

$\displaystyle\sup_{t\in(0,T)}I[u(t)]=\sup_{t\in(0,T)}\int_{\mathbb{R}^{d}}u\left|x\right|^{2}\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq C;$ (3.5)
$\displaystyle\left\|\nabla\sqrt{u}\right\|_{L^{2}(Q_{T})}^{2}$
$\displaystyle\leq C;$ (3.6) $\displaystyle\left\|\sqrt{u}\,\nabla
K*u\right\|_{L^{2}(Q_{T})}^{2}$ $\displaystyle\leq C,$ (3.7)
$\displaystyle\left\|\nabla u\right\|_{L^{2}(0,T;L^{1}(\mathbb{R}^{d}))}$
$\displaystyle\leq C;$ (3.8)

where
$C=C(D,T,c_{1,1}^{-1},\left\|K\right\|_{L^{\infty}(\mathbb{R}^{d})},d,m,I[u_{0}],H[u_{0}])$,
$m=\left\|u_{0}\right\|_{L^{1}(\mathbb{R}^{d})}$ and
$c_{1,1}=D-m\left\|K\right\|_{L^{\infty}(\mathbb{R}^{d})}$.

######  Lemma 3.5 (Apriori estimates under (H1)-(H3)).

Fix $T>0$. Assume (H1)-(H3). Then for any smooth, positive solution $u$ to
problem (1.1) there holds

$\displaystyle\sup_{t\in(0,T)}\int_{\mathbb{R}^{d}}u\left|x\right|^{2}\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq C;$ (3.9)
$\displaystyle\sup_{t\in(0,T)}\int_{\mathbb{R}^{d}}u\left|\log
u\right|\mathop{}\\!\mathrm{d}x+\tfrac{1}{2}\left\|f\right\|_{L^{2}(Q_{T})}^{2}$
$\displaystyle\leq C;$ (3.10) $\displaystyle\left\|\nabla
u\right\|_{L^{2}(0,T;L^{1}(\mathbb{R}^{d}))}$ $\displaystyle\leq C;$ (3.11)
$\displaystyle\left\|\sqrt{u}\,\nabla K*u\right\|_{L^{2}(Q_{T})}$
$\displaystyle\leq C;$ (3.12)
$\displaystyle\left\|\nabla\sqrt{u}\right\|_{L^{2}(Q_{T})}$ $\displaystyle\leq
C;$ (3.13)

where
$C=C(D,T,\left\|K\right\|_{L^{\infty}(\mathbb{R}^{d})},d,m,I[u_{0}],H[u_{0}])$
and $f:=\sqrt{u}\,\nabla(D\log u+K*u)$ is the dissipation.

We begin with the proof of Lemma 3.2.

###### Proof of Lemma 3.4.

Consider the entropy functional $H[u(t)]=\int_{\mathbb{R}^{d}}u\log u\,{\rm
d}x$. Taking the derivative with respect to time, using the conservation of
mass, and integrating by parts we have

$\displaystyle\frac{{\rm d}}{{\rm d}t}H[u(t)]$
$\displaystyle=-\int_{\mathbb{R}^{d}}\frac{\nabla u}{u}\left(D\,\nabla
u+u\nabla(K*u)\right)\mathop{}\\!\mathrm{d}x$
$\displaystyle=-D\int_{\mathbb{R}^{d}}\frac{\left|\nabla
u\right|^{2}}{u}\mathop{}\\!\mathrm{d}x-\int_{\mathbb{R}^{d}}\nabla
u\,\nabla(K*u)\mathop{}\\!\mathrm{d}x.$ (3.14)

We first control the second term of (3.2) using Hölder’s inequality and
Young’s convolution inequality:

$\displaystyle\left|\int_{\mathbb{R}^{d}}\nabla u\nabla(K*u){\rm d}x\right|$
$\displaystyle\leq\left\|K*\nabla
u(t,\cdot)\right\|_{L^{\infty}({\mathbb{R}^{d}})}\left\|\nabla
u(t,\cdot)\right\|_{L^{1}({\mathbb{R}^{d}})}$
$\displaystyle\leq\left\|K\right\|_{L^{\infty}({\mathbb{R}^{d}})}\left\|\nabla
u(t,\cdot)\right\|_{L^{1}({\mathbb{R}^{d}})}^{2}.$ (3.15)

Furthermore, we can estimate $\nabla u$ via Hölder’s inequality and the
conservation of mass

$\displaystyle\left\|\nabla u(t,\cdot)\right\|_{L^{1}({\mathbb{R}^{d}})}^{2}$
$\displaystyle=\left(\int_{\mathbb{R}^{d}}\sqrt{u}\,\frac{\left|\nabla
u\right|}{\sqrt{u}}\mathop{}\\!\mathrm{d}x\right)^{2}\leq\left\|u(t,\cdot)\right\|_{L^{1}({\mathbb{R}^{d}})}\int_{\mathbb{R}^{d}}\frac{\left|\nabla
u\right|^{2}}{u}\mathop{}\\!\mathrm{d}x=m\int_{\mathbb{R}^{d}}\frac{\left|\nabla
u\right|^{2}}{u}\mathop{}\\!\mathrm{d}x.$ (3.16)

Combining estimates (3.2)-(3.16), (3.2) becomes

$\displaystyle\frac{{\rm d}}{{\rm
d}t}H[u(t)]+4c_{1,1}\left\|\nabla\sqrt{u}(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}\leq
0.$ (3.17)

Integrating both sides from $0$ to $t$ yields

$\displaystyle
H[u(t)]+4c_{1,1}\left\|\nabla\sqrt{u}\right\|_{L^{2}(Q_{t})}^{2}\leq
H[u_{0}].$ (3.18)

We also notice that by Hölder’s inequality, Young’s convolution inequality and
(3.16) that

$\displaystyle\left\|\sqrt{u}\,\nabla(K*u)(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}=\int_{\mathbb{R}^{d}}u\left|K*\nabla
u\right|^{2}\mathop{}\\!\mathrm{d}x$ $\displaystyle\leq m\left\|K*\nabla
u\right\|_{L^{\infty}({\mathbb{R}^{d}})}^{2}$ $\displaystyle\leq
m\left\|K\right\|_{L^{\infty}({\mathbb{R}^{d}})}^{2}\left\|\nabla
u(t,\cdot)\right\|_{L^{1}({\mathbb{R}^{d}})}^{2}$ $\displaystyle\leq
4m^{2}\left\|K\right\|_{L^{\infty}({\mathbb{R}^{d}})}^{2}\left\|\nabla\sqrt{u}(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}$
(3.19)

We seek to combine these estimates to obtain an estimate on
$\left\|\nabla\sqrt{u}\right\|_{L^{2}(Q_{T})}^{2}$, but we must first control
the negative part of $u\log u$. This can be achieved using the second moment
$I[u(t)]$ by considering the set $\underline{\Omega}:=\\{(t,x):u\leq 1\\}$ and
splitting it for two subsets $\\{u\leq e^{-\left|x\right|^{2}/2}\\}$,
$\\{e^{-\left|x\right|^{2}/2}\leq u\leq 1\\}$:

$\displaystyle-\int_{\mathbb{R}^{d}}\chi_{\underline{\Omega}}u\log
u\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\sup_{\xi\in(0,1)}\left|\sqrt{\xi}\log\xi\right|\int_{\mathbb{R}^{d}}e^{-\left|x\right|^{2}/2}\mathop{}\\!\mathrm{d}x-\int_{\\{e^{-\left|x\right|^{2}/2}\leq
u\leq 1\\}}u\log u\mathop{}\\!\mathrm{d}x$ $\displaystyle\leq
C_{1}+\int_{\mathbb{R}^{d}}u\left|x\right|^{2}\mathop{}\\!\mathrm{d}x=C_{1}+I[u(t)],$
(3.20)

where $C_{1}>0$ depends only on the dimension $d$. Hence, we compute the time
derivative of $I[u(t)]$, integrate by parts, apply Young’s inequality with
$\varepsilon$ and use estimate (3.2) to find

$\displaystyle\frac{{\rm d}}{{\rm d}t}I[u(t)]$
$\displaystyle=-2\int_{\mathbb{R}^{d}}x\cdot(D\nabla u+u\,K*\nabla
u)\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\varepsilon^{-1}I[u(t)]+\varepsilon(8D^{2}\left\|\nabla\sqrt{u}(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}+2\left\|\sqrt{u}\,\nabla
K*u\right\|_{L^{2}(\mathbb{R}^{d})}^{2})$
$\displaystyle\leq\varepsilon^{-1}I[u(t)]+8\,\varepsilon(D^{2}+m^{2}\left\|K\right\|_{L^{\infty}(\mathbb{R}^{d})}^{2})\left\|\nabla\sqrt{u}(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2},$
(3.21)

where $\varepsilon>0$ is to be chosen. Integrating this result from $0$ to $t$
yields

$\displaystyle I[u(t)]\leq\varepsilon^{-1}\int_{0}^{t}I[u(s)]{\rm
d}s+8\,\varepsilon(D^{2}+m^{2}\left\|K\right\|_{L^{\infty}(\mathbb{R}^{d})}^{2})\left\|\nabla\sqrt{u}\right\|_{L^{2}(Q_{T})}^{2}+I[u_{0}].$
(3.22)

Writing

$\int_{\mathbb{R}^{d}}u\left|\log
u\right|\mathop{}\\!\mathrm{d}x=\int_{\mathbb{R}^{d}}u\log
u\mathop{}\\!\mathrm{d}x-2\int_{\mathbb{R}^{d}}\chi_{\underline{\Omega}}u\log
u\mathop{}\\!\mathrm{d}x,$ (3.23)

we may combine estimates (3.18), (3.2), (3.22) and (3.23) to obtain

$\displaystyle\int_{\mathbb{R}^{d}}u\left|\log
u\right|\mathop{}\\!\mathrm{d}x$
$\displaystyle\,+2c_{1,1}\left\|\nabla\sqrt{u}\right\|_{L^{2}(Q_{t})}^{2}+I[u(t)]$
$\displaystyle=$
$\displaystyle\,H[u(t)]-2\int_{\mathbb{R}^{d}}\chi_{\underline{\Omega}}u\log
u\mathop{}\\!\mathrm{d}x+2c_{1,1}\left\|\nabla\sqrt{u}\right\|_{L^{2}(Q_{t})}^{2}+I[u(t)]$
$\displaystyle\leq$
$\displaystyle\,H[u_{0}]-2c_{1,1}\left\|\nabla\sqrt{u}\right\|_{L^{2}(Q_{t})}^{2}+2C_{1}+3I[u(t)]$
$\displaystyle\leq$
$\displaystyle\,H[u_{0}]+2C_{1}+3\varepsilon^{-1}\int_{0}^{t}I[u(s)]{\rm d}s$
$\displaystyle\,+(24\varepsilon(D^{2}+4m^{2}\left\|K\right\|_{L^{\infty}(\mathbb{R}^{d})}^{2})-2c_{1,1})\left\|\nabla\sqrt{u}\right\|_{L^{2}(Q_{t})}^{2}+3I[u_{0}].$
(3.24)

Choosing $\varepsilon$ sufficiently small, the last of (3.2) is nonpositive,
and so we conclude that

$\displaystyle\int_{\mathbb{R}^{d}}u\left|\log
u\right|\mathop{}\\!\mathrm{d}x+2c_{1,1}\left\|\nabla\sqrt{u}\right\|_{L^{2}(Q_{t})}^{2}+I[u(t)]$
$\displaystyle\leq H[u_{0}]+2C_{1}+3\varepsilon^{-1}\int_{0}^{t}I[u(s)]{\rm
d}s+3I[u_{0}],$ (3.25)

whence Grönwall’s lemma yields estimate (3.5). With the bound on $I[u(t)]$, we
immediately obtain estimate (3.6), from which estimate (3.7) follows from
estimate (3.2). Finally, (3.8) follows by writing $\nabla
u=2\,(\nabla\sqrt{u})\,\sqrt{u}$ and using the conservation of mass. ∎

Next we prove Lemma 3.5.

###### Proof of Lemma 3.5.

We begin with estimates on
$\left\|u(t,\cdot)\right\|_{L^{1}(\mathbb{R}^{d})}$, the second moment
$I[u(t)]$, and $\int_{\mathbb{R}^{d}}u\left|\log u\right|{\rm d}x$ in
$L^{\infty}(0,T)$. From the conservation of mass, we already have that
$\left\|u\right\|_{L^{\infty}(0,T;L^{1}({\mathbb{R}^{d}}))}=m$. We then write
(3.40) as a gradient flow:

$u_{t}-\nabla\cdot(u\nabla(D\log u+K*u))=u_{t}-\nabla\cdot(\sqrt{u}f)=0,$

where $f$ is the dissipation defined in the statement of the Lemma. By direct
computation and hypothesis (H3) there holds

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\mathcal{F}[u(t)]+\int_{\mathbb{R}^{d}}\left|f\right|^{2}\mathop{}\\!\mathrm{d}x=0.$
(3.26)

Integrating both sides from $0\to t$, applying Young’s convolution inequality
and using the conservation of mass yields

$\displaystyle DH[u(t)]+\left\|f\right\|_{L^{2}(Q_{t})}^{2}$
$\displaystyle\leq
DH[u_{0}]+m^{2}\left\|K\right\|_{L^{\infty}({\mathbb{R}^{d}})}.$ (3.27)

As in (3.2), we may control the negative part of $u\log u$ in terms of the
second moment $I[u(t)]$ which can be estimated by using the dissipation $f$ as
follows

$\displaystyle\frac{{\rm d}}{{\rm d}t}I[u(t)]$
$\displaystyle=-2\int_{\mathbb{R}^{d}}\sqrt{u}f\cdot
x\mathop{}\\!\mathrm{d}x\leq\varepsilon^{-1}\int_{\mathbb{R}^{d}}u\left|x\right|^{2}\mathop{}\\!\mathrm{d}x+\varepsilon\int_{\mathbb{R}^{d}}\left|f\right|^{2}\mathop{}\\!\mathrm{d}x.$

After integration from $0$ to $t$ we obtain

$\displaystyle I[u(t)]$
$\displaystyle\leq\varepsilon^{-1}\int_{0}^{t}I[u(s)]{\rm
d}s+\varepsilon\left\|f\right\|_{L^{2}(Q_{t})}^{2}+I[u_{0}],$ (3.28)

where we fix $\varepsilon=\tfrac{1}{2(2D+1)}$. Using (3.2) and (3.23) as in
the proof of Lemma 3.4, estimates (3.27)-(3.28) paired with Grönwall’s lemma
gives estimate (3.9), from which estimate (3.10) follows.

Now we obtain an estimate on $\nabla u$ in
$L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))$. To this end, we write $D\nabla
u=\sqrt{u}f-u\nabla K*u$ so that we may deduce from Hölder’s inequality and
the conservation of mass that

$\displaystyle D\left\|\nabla u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}$
$\displaystyle\leq\left\|\sqrt{u}f\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}+\left\|u\nabla
K*u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}$
$\displaystyle\leq\sqrt{m}\left\|f\right\|_{L^{2}(Q_{T})}+\left\|u\nabla
K*u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}.$ (3.29)

We consider separately two sets: $\underline{{\Omega}}:=\\{(t,x):u\leq\ell\\}$
and $\overline{{\Omega}}:=\\{(t,x):u>\ell\\}$ for some $\ell>1$ to be chosen
sufficiently large. We then estimate as follows:

$\displaystyle\left\|u\nabla
K*u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}=\left\|\chi_{\underline{\Omega}}u\nabla
K*u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}+\left\|\chi_{\overline{\Omega}}u\nabla
K*u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}$
$\displaystyle\leq\ell\left\|\nabla
K*u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}+\left\|\frac{u\left|\log
u\right|}{\log\ell}\nabla K*u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}$
$\displaystyle\leq
C_{K}\ell\left\|u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}+\frac{1}{\log\ell}\left\|u\left|\log
u\right|\right\|_{L^{\infty}(0,T;L^{1}({\mathbb{R}^{d}}))}\left\|\nabla
K*u\right\|_{L^{2}(0,T;L^{\infty}({\mathbb{R}^{d}}))}$ $\displaystyle\leq
C_{K}\ell\left\|u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}+\frac{1}{\log\ell}\left\|u\left|\log
u\right|\right\|_{L^{\infty}(0,T;L^{1}({\mathbb{R}^{d}}))}\left\|K\right\|_{L^{\infty}({\mathbb{R}^{d}})}\left\|\nabla
u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}$ $\displaystyle\leq C_{K}\ell
m\sqrt{T}+\frac{C_{2}}{\log\ell}\left\|\nabla
u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}.$ (3.30)

The first inequality uses the properties of the sets $\underline{\Omega}$,
$\overline{\Omega}$; the second inequality uses hypothesis (H2) and Hölder’s
inequality; the third inequality uses Young’s convolution inequality; the
fourth inequality uses hypothesis (H1) and the previously obtained bounds on
$u$ and $u\left|\log u\right|$ in $L^{\infty}(0,T;L^{1}({\mathbb{R}^{d}}))$.
Combining (3.2)-(3.2) we have shown that

$\displaystyle D\left\|\nabla
u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}\leq\sqrt{m}\left\|f\right\|_{L^{2}(Q_{T})}+C_{K}\ell
m\sqrt{T}+\frac{C_{2}}{\log\ell}\left\|\nabla
u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))},$ (3.31)

whence choosing $\ell$ sufficiently large yields a uniform estimate on $\nabla
u$ in $L^{2}(0,T;L^{1}(\mathbb{R}^{d}))$. Consequently, Hölder’s inequality,
Young’s convolution inequality and hypothesis (H1) yields

$\displaystyle\left\|\sqrt{u}\nabla K*u\right\|_{L^{2}(Q_{T})}$
$\displaystyle\leq\left\|\sqrt{u}\right\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}))}^{1/2}\left\|K\right\|_{L^{\infty}({\mathbb{R}^{d}})}\left\|\nabla
u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}$
$\displaystyle=\sqrt{m}\left\|K\right\|_{L^{\infty}({\mathbb{R}^{d}})}\left\|\nabla
u\right\|_{L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))}$ (3.32)

and so (3.2) paired with the uniform estimate obtained from (3.31) yields
estimate (3.12).

Finally, as $2D\nabla\sqrt{u}=f-\sqrt{u}\nabla K*u$, estimate (3.2) and the
bound on the dissipation $f$ in $L^{2}(Q_{T})$ shows that $\nabla\sqrt{u}$ is
bounded in $L^{2}(Q_{T})$ and estimate (3.13) follows. ∎

###  3.3. Improved estimates

In this section we improve the estimates obtained in Lemmas 3.4-3.5. As the
the following result uses only the estimates on $\nabla\sqrt{u}$ and
$\sqrt{u}\nabla K*u$, the same argument applies assuming the conditions of
either Lemma 3.4 or Lemma 3.5 are met.

First, we obtain estimates to $u\in L^{\infty}(0,T;L^{p}({\mathbb{R}^{d}})$
for any $2\leq p<\infty$ and on $\nabla u$ in $L^{2}(Q_{T})$ with no
additional assumptions required.

######  Lemma 3.6 (Improved estimates with no further assumptions).

Assume the conditions of Lemma 3.4 (Lemma 3.5) are satisfied. Then, for any
$p\in[2,\infty)$ the following estimates hold

$\displaystyle\sup_{t\in(0,T)}\left\|u(t,\cdot)\right\|_{L^{p}({\mathbb{R}^{d}})}$
$\displaystyle\leq
e^{C\left\|K\right\|_{L^{\infty}(\mathbb{R}^{d})}^{2}p/D}\left\|u_{0}\right\|_{L^{p}({\mathbb{R}^{d}})};$
(3.33) $\displaystyle\left\|\nabla u\right\|_{L^{2}(Q_{T})}$
$\displaystyle\leq
D^{-1}\tilde{C}\left\|u_{0}\right\|_{L^{2}({\mathbb{R}^{d}})},$ (3.34)

where $\tilde{C}=\tilde{C}(C)$ and $C$ is given in Lemma 3.4 (Lemma 3.5).

###### Proof.

We consider the quantity
$\tfrac{1}{p}\left\|u(t,\cdot)\right\|_{L^{p}({\mathbb{R}^{d}})}^{p}$ for
$p\in[2,\infty)$. Taking the derivative with respect to time and integrating
by parts yields

$\displaystyle\frac{1}{p}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}u^{p}\mathop{}\\!\mathrm{d}x$
$\displaystyle=-D(p-1)\int_{\mathbb{R}^{d}}\left|\nabla
u\right|^{2}u^{p-2}\mathop{}\\!\mathrm{d}x-(p-1)\int_{\mathbb{R}^{d}}u^{p-1}\nabla
u\cdot K*\nabla u\mathop{}\\!\mathrm{d}x$
$\displaystyle=-\tfrac{4D(p-1)}{p^{2}}\int_{\mathbb{R}^{d}}\left|\nabla
u^{p/2}\right|^{2}{\rm d}x-(p-1)\int_{\mathbb{R}^{d}}u^{p-1}\nabla u\cdot
K*\nabla u\mathop{}\\!\mathrm{d}x.$ (3.35)

We use the first term on the right hand side of (3.3) to control the second
term using Cauchy’s inequality with $\varepsilon$ and Hölder’s inequality:

$\displaystyle(p-1)\int_{\mathbb{R}^{d}}u^{p-1}\nabla uK*\nabla
u\mathop{}\\!\mathrm{d}x$
$\displaystyle=\tfrac{2(p-1)}{p}\int_{\mathbb{R}^{d}}\nabla
u^{p/2}u^{p/2}K*\nabla u\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\tfrac{\varepsilon(p-1)}{p}\int_{\mathbb{R}^{d}}\left|\nabla
u^{p/2}\right|^{2}\mathop{}\\!\mathrm{d}x+\tfrac{(p-1)}{\varepsilon
p}\int_{\mathbb{R}^{d}}u^{p}\left|K*\nabla
u\right|^{2}\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\tfrac{\varepsilon(p-1)}{p}\int_{\mathbb{R}^{d}}\left|\nabla
u^{p/2}\right|^{2}\mathop{}\\!\mathrm{d}x$
$\displaystyle\quad+\tfrac{(p-1)}{\varepsilon p}\left\|(K*\nabla
u)(t,\cdot)\right\|_{L^{\infty}({\mathbb{R}^{d}})}^{2}\int_{\mathbb{R}^{d}}u^{p}\mathop{}\\!\mathrm{d}x.$
(3.36)

Choosing $\varepsilon=2D/p$ we combine estimate (3.3) with (3.3) to find

$\displaystyle\frac{1}{p}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}u^{p}\mathop{}\\!\mathrm{d}x+\tfrac{2D(p-1)}{p^{2}}\int_{\mathbb{R}^{d}}\left|\nabla
u^{p/2}\right|^{2}\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\tfrac{(p-1)}{2D}\left\|(K*\nabla
u)(t,\cdot)\right\|_{L^{\infty}({\mathbb{R}^{d}})}^{2}\int_{\mathbb{R}^{d}}u^{p}\mathop{}\\!\mathrm{d}x.$
(3.37)

Grönwall’s lemma implies that

$\displaystyle\left\|u(t,\cdot)\right\|_{L^{p}({\mathbb{R}^{d}})}$
$\displaystyle\leq
e^{C_{3}(T)p/D}\left\|u_{0}\right\|_{L^{p}({\mathbb{R}^{d}})},$ (3.38)

where $C_{3}(T)=\left\|K*\nabla
u\right\|_{L^{2}(0,T;L^{\infty}(\mathbb{R}^{d}))}^{2}\leq
C\left\|K\right\|_{L^{\infty}(\mathbb{R}^{d})}^{2}$ by estimate (3.8) (or
(3.11)). Taking the supremum over $t\in(0,T)$ yields estimate (3.33). Note
carefully that this estimate does not allow one to take $p\to\infty$ as the
coefficient on the right hand side depends critically on $p$.

Returning to estimate (3.37), we fix $p=2$ and integrate both sides from $0$
to $T$ to obtain

$\displaystyle\int_{\mathbb{R}^{d}}u^{2}\mathop{}\\!\mathrm{d}x+D\iint_{Q_{T}}\left|\nabla
u\right|^{2}\mathop{}\\!\mathrm{d}x{\rm d}t$ $\displaystyle\leq
D^{-1}C\left\|K\right\|_{L^{\infty}(\mathbb{R}^{d})}^{2}\sup_{t\in(0,T)}\left\|u(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}+\left\|u_{0}\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}.$
(3.39)

Using estimate (3.38) with $p=2$ and rearranging the result yields estimate
(3.34). ∎

###  3.4. Existence of weak solutions (proof of Theorems 2.1-2.2)

We are now prepared to prove Theorems 2.1-2.2.

###### Proof of Theorems 2.1-2.2.

We present the details for Theorem 2.1 only; the conclusion of Theorem 2.2
follows in an identical fashion by using the estimates obtained in Lemma 3.5,
which are the same as those obtained in Lemma 3.4 for the small mass regime.

To start, we consider approximate solutions for any $\varepsilon>0$ given by

$\displaystyle\frac{\partial u^{\varepsilon}}{\partial t}=D\Delta
u^{\varepsilon}+\nabla\cdot(u^{\varepsilon}\nabla
K*{\omega}^{\varepsilon}*u^{\varepsilon}),$ (3.40)

satisfying the initial data $u^{\varepsilon}(0,x)=u_{0}(x)$ for each
$\varepsilon>0$, where ${\omega}^{\varepsilon}$ denotes the standard
mollifier. We seek to pass the limit as $\varepsilon\to 0^{+}$. First note
that a unique, positive classical solution $u^{\varepsilon}(t,x)$ solving
(3.40) with $u(\cdot,x)=u_{0}(x)$ exists for each $\varepsilon>0$ by the
methods used in the proof of, e.g., [22, Theorem 2.2]. Moreover, by Young’s
convolution inequality we have that

$\left\|K*{\omega}^{\varepsilon}\right\|_{L^{\infty}({\mathbb{R}^{d}})}\leq\left\|K\right\|_{L^{\infty}({\mathbb{R}^{d}})}\left\|{\omega}^{\varepsilon}\right\|_{L^{1}({\mathbb{R}^{d}})}=\left\|K\right\|_{L^{\infty}({\mathbb{R}^{d}})},$

and so if the small mass condition (2.4) is satisfied, it is also satisfied
for the regularized problem for all $\varepsilon\geq 0$. Finally, for each
$t>0$, the solution $u^{\varepsilon}$ is strictly positive by the strong
maximum principle as in (3.80).

By Lemma 3.4, we have the following uniform bounds on the approximate
solutions $u^{\varepsilon}$:

  1. (A)

$\\{\nabla\sqrt{u^{\varepsilon}}\\}_{\varepsilon>0}$ is bounded in
$L^{2}(Q_{T})$;

  2. (B)

$\\{\sqrt{u^{\varepsilon}}\nabla
K*{\omega}^{\varepsilon}*u^{\varepsilon}\\}_{\varepsilon>0}$ is bounded in
$L^{2}(Q_{T})$;

  3. (C)

$\\{\sup_{t\in(0,T)}\int_{\mathbb{R}^{d}}u^{\varepsilon}\left|x\right|^{2}\mathop{}\\!\mathrm{d}x\\}_{\varepsilon>0}$
is bounded.

(A) implies that $2\nabla
u^{\varepsilon}=\sqrt{u^{\varepsilon}}\nabla\sqrt{u^{\varepsilon}}$ belongs to
$L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))$ uniformly in $\varepsilon$; by Lemma 3.6,
$\nabla u^{\varepsilon}$ belongs to $L^{2}(Q_{T})$ uniformly in $\varepsilon$,
and hence by interpolation belongs to $L^{2}(0,T;L^{r}(\mathbb{R}^{d}))$ for
any $r\in(1,2)$. Moreover, $u^{\varepsilon}$ belongs to
$L^{\infty}(0,T;L^{p}(\mathbb{R}^{d}))$ uniformly in $\varepsilon$ for any
$p\in(1,\infty)$, also by Lemma 3.6 and interpolation. Therefore, we are able
to conclude that $\\{\frac{\partial u^{\varepsilon}}{\partial
t}\\}_{\varepsilon>0}$ is bounded in $L^{2}(0,T;W^{-1,q}(\mathbb{R}^{d}))$ for
any $q\in(1,2)$. Indeed, given any $\phi\in
L^{2}(0,T;W^{1,q^{\prime}}(\mathbb{R}^{d}))$ Hölder’s inequality gives

$\displaystyle\left|\int_{0}^{T}\int_{\mathbb{R}^{d}}\frac{\partial
u^{\varepsilon}}{\partial
t}\phi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t\right|$
$\displaystyle\leq D\left\|\nabla
u^{\varepsilon}\right\|_{L^{2}(0,T;L^{r}(\mathbb{R}^{d}))}\left\|\nabla\phi\right\|_{L^{2}(0,T;L^{q^{\prime}}(\mathbb{R}^{d}))}$
$\displaystyle+\left\|\sqrt{u^{\varepsilon}}\nabla
K*\omega^{\varepsilon}*u^{\varepsilon}\right\|_{L^{2}(Q_{T})}\left\|u^{\varepsilon}\right\|_{L^{\infty}(0,T;L^{p/2}(\mathbb{R}^{d}))}^{1/2}\left\|\nabla\phi\right\|_{L^{2}(0,T;L^{q^{\prime}}(\mathbb{R}^{d}))}$
$\displaystyle\leq
C_{5}\left\|\phi\right\|_{L^{2}(0,T;W^{1,q^{\prime}}(\mathbb{R}^{d}))},$
(3.41)

where $q^{\prime}$, $p$ and $r$ are determined by $1/2+1/p+1/q^{\prime}=1$,
$1/r+1/q^{\prime}=1$, so that $q^{\prime}=2+2/(p-1)$, $r=2p/(p+1)$ for
$p\in(1,\infty)$ , and $C_{5}$ does not depend on $\varepsilon$. As in
Definition 3.1, we choose $p=4$, $q^{\prime}=4$ and $r=\frac{4}{3}$.

Consequently, (A) paired with the conservation of mass implies that
$u^{\varepsilon}\in L^{2}(0,T;W^{1,1}(\mathbb{R}^{d}))$) uniformly in
$\varepsilon$. We then conclude by the Aubin-Lions lemma with

$W:=\left\\{v\in L^{2}(0,T;W^{1,1}_{loc}(\mathbb{R}^{d})):\tfrac{\partial
v}{\partial t}\in L^{2}(0,T;W^{-1,q}_{loc}(\mathbb{R}^{d}))\right\\}$

that there exists a subsequence (still denoted by $\varepsilon$) such that

$\displaystyle u^{\varepsilon}\to u\quad\text{ strongly in }\quad
L^{1}_{loc}(Q_{T})$ (3.42)

as $\varepsilon\to 0^{+}$. This implies that, perhaps for another subsequence,

$\displaystyle\sqrt{u^{\varepsilon}}\to\sqrt{u}\quad\text{ strongly in }\quad
L^{2}_{loc}(Q_{T})$ (3.43)

as $\varepsilon\to 0^{+}$ by the Vitali convergence theorem. By (C) it is easy
to see that given any $\delta>0$, for $R$ chosen sufficiently large there
holds $\left\|u^{\varepsilon}\right\|_{L^{1}(0,T;L^{1}(\mathbb{R}^{d}\setminus
B_{R}))}<\delta$, independent of $\varepsilon$, and so
$\\{u^{\varepsilon}\\}_{\varepsilon>0}$ has compact closure in $L^{1}(Q_{T})$.
In particular, the strong convergence obtained in (3.42)-(3.43) holds
globally.

Next, (B) implies that there exists $\xi\in L^{2}(Q_{T})$ such that

$\displaystyle\sqrt{u^{\varepsilon}}\nabla
K*{\omega}^{\varepsilon}*u^{\varepsilon}\to\xi\quad\text{ weakly in }\quad
L^{2}(Q_{T})$ (3.44)

as $\varepsilon\to 0^{+}$. We claim that $\xi=\sqrt{u}\nabla K*u$. To identify
the limit, we may identify it in $L^{1}(0,T;L^{2}({\mathbb{R}^{d}}))$. Notice
first that $\nabla K*{\omega}^{\varepsilon}*u^{\varepsilon}$ is bounded in
$L^{2}(0,T;L^{\infty}({\mathbb{R}^{d}}))$ since $\nabla u^{\varepsilon}\in
L^{2}(0,T;L^{1}(\mathbb{R}^{d}))$, and so has a weak∗ limit in the same space.
Hence, there holds

$\nabla K*{\omega}^{\varepsilon}*u^{\varepsilon}\to\nabla K*u\quad\text{ weak*
in }\quad L^{2}(0,T;L^{\infty}({\mathbb{R}^{d}}))$

as $\varepsilon\to 0^{+}$. Furthermore, the limit can be identified as $\nabla
K*u=K*\nabla u$ because of the $L^{2}(Q_{T})$ estimate on $\nabla
u^{\varepsilon}$. Then, as $\sqrt{u^{\varepsilon}}$ converges strongly in
$L^{2}(Q_{T})$, given any $\phi\in L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}))$
there holds

$\displaystyle\left|\int_{0}^{T}\int_{\mathbb{R}^{d}}\left(\sqrt{u^{\varepsilon}}\nabla
K*{\omega}^{\varepsilon}*u^{\varepsilon}-\sqrt{u}K*\nabla
u\right)\phi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t\right|\leq$
$\displaystyle\qquad\qquad\leq\left|\int_{0}^{T}\int_{\mathbb{R}^{d}}\left([\sqrt{u^{\varepsilon}}-\sqrt{u}\,]\nabla
K*{\omega}^{\varepsilon}*u^{\varepsilon}]\right)\phi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t\right|$
$\displaystyle\qquad\qquad\quad+\left|\int_{0}^{T}\int_{\mathbb{R}^{d}}\sqrt{u}\phi\left(\nabla
K*{\omega}^{\varepsilon}*u^{\varepsilon}-K*\nabla
u\right)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t\right|$
$\displaystyle\qquad\qquad\leq\left\|\sqrt{u^{\varepsilon}}-\sqrt{u}\right\|_{L^{2}(0,T;L^{2}(\mathbb{R}^{d}))}\left\|\nabla
K*{\omega}^{\varepsilon}*u^{\varepsilon}\right\|_{L^{2}(0,T;L^{\infty}({\mathbb{R}^{d}}))}\left\|\phi\right\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}))}$
$\displaystyle\qquad\qquad\quad+\left|\int_{0}^{T}\int_{\mathbb{R}^{d}}\sqrt{u}\phi(\nabla
K*{\omega}^{\varepsilon}*u^{\varepsilon}-K*\nabla
u)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t\right|\to 0$ (3.45)

as $\varepsilon\to 0^{+}$. Notice that the second term converges to zero as
$\varepsilon\to 0^{+}$ since $\sqrt{u}\phi\in
L^{2}(0,T;L^{1}({\mathbb{R}^{d}}))$, the predual of
$L^{2}(0,T;L^{\infty}({\mathbb{R}^{d}}))$. Hence, (3.43)-(3.44) hold and
$\xi=\sqrt{u}\nabla(K*u)$ as asserted.

These convergence results allow us to conclude that in the limit as
$\varepsilon\to 0^{+}$ there holds

$\int_{0}^{T}\left<\tfrac{\partial u}{\partial
t},\phi\right>\mathop{}\\!\mathrm{d}t+D\int_{0}^{T}\int_{\mathbb{R}^{d}}\nabla
u\cdot\nabla\phi\,\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t=-\int_{0}^{T}\int_{\mathbb{R}}^{d}u\nabla(K*u)\cdot\nabla\phi\,\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t,$

for any $\phi$ smooth and compactly supported. By (A)-(B) and a density
argument, this weak formulation holds for any $\phi\in
L^{2}(0,T;W^{1,q^{\prime}}(\mathbb{R}^{d}))$ with $q^{\prime}>2$. Finally,
since $\\{u^{\varepsilon}\\}_{\varepsilon>0}$ is bounded in
$H^{1}(0,T;W^{-1,q}(\mathbb{R}^{d}))$, we have that in fact
$u^{\varepsilon}\in C(0,T;W^{-1,q}(\mathbb{R}^{d}))$ uniformly in
$\varepsilon$ and so $u(t,\cdot)\to u_{0}(\cdot)$ strongly in
$W^{-1,q}(\mathbb{R}^{d})$ as $t\to 0^{+}$, and indeed the initial data is
satisfied in the sense of $W^{-1,q}(\mathbb{R}^{d})$. As $T>0$ was arbitrary,
the solution obtained is in fact global.

To conclude, we recall that by Lemma 3.6 we have that $u\in
L^{\infty}(0,T;L^{p}({\mathbb{R}^{d}}))$ for all $1\leq p<\infty$ and $\nabla
u\in L^{2}(Q_{T})$. ∎

###  3.5. Higher order estimates

Under the additional hypothesis (H4) and further conditions on the initial
data we can obtain better estimates on $\nabla u$, $\Delta u$, and a uniform
estimate on $u$ over $Q_{T}$.

######  Lemma 3.7 (Higher order estimates).

Assume that $\nabla u_{0}\in L^{2}(\mathbb{R}^{d})$ and $K*u_{0}\in
W^{2,p}(\mathbb{R}^{d})$ for some $p>2$. Suppose hypothesis (H4) holds in
addition to the conditions of Lemma 3.4 (resp. Lemma 3.5). Then, there exists
a constant $\tilde{C}$ so that

$\displaystyle\left\|\nabla
u\right\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}))}$
$\displaystyle\leq\tilde{C};$ (3.46) $\displaystyle\left\|\Delta
u\right\|_{L^{2}(Q_{T})}$ $\displaystyle\leq\tilde{C};$ (3.47)
$\displaystyle\left\|u\right\|_{L^{\infty}(Q_{T})}$
$\displaystyle\leq\tilde{C},$ (3.48)

where
$\tilde{C}=\tilde{C}(C,\left\|\nabla(\tilde{K}*K)\right\|_{L^{2}(\mathbb{R}^{d})})$,
and $C$ is given in Lemma 3.4 (resp. Lemma 3.5).

###### Proof.

Before estimating $u$ and its derivatives, we first obtain improved estimates
on $K*\nabla u$ and $K*\Delta u$. To this end, recall that
$\tilde{K}(\cdot):=K(-\cdot)$ and multiply (1.1) by $\tilde{K}*K*\Delta u$,
integrate over ${\mathbb{R}^{d}}$, and integrate by parts to find

$\displaystyle\frac{1}{2}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}\left|\nabla(K*u)\right|^{2}\mathop{}\\!\mathrm{d}x$
$\displaystyle=-D\int_{\mathbb{R}^{d}}\left|\Delta(K*u)\right|^{2}\mathop{}\\!\mathrm{d}x+\int_{\mathbb{R}^{d}}\tilde{K}*K*\Delta
u\nabla\cdot(u\nabla(K*u))$
$\displaystyle=-D\int_{\mathbb{R}^{d}}\left|\Delta(K*u)\right|^{2}\mathop{}\\!\mathrm{d}x+\int_{\mathbb{R}^{d}}\Delta(K*u)K*(\nabla
u\cdot\nabla(K*u))\mathop{}\\!\mathrm{d}x$
$\displaystyle\quad+\int_{\mathbb{R}^{d}}u\,\Delta(K*u)(\tilde{K}*\Delta(K*u))\mathop{}\\!\mathrm{d}x.$
(3.49)

We first estimate the second term on the right hand side of (3.5) via Cauchy’s
inequality with $\varepsilon=D/2$, followed by Young’s convolution inequality
and Hölder’s inequality:

$\displaystyle\int_{\mathbb{R}^{d}}\Delta(K*u)$ $\displaystyle\,K*(\nabla
u\cdot K*\nabla u)\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\tfrac{D}{4}\left\|\Delta(K*u)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}+D^{-1}\left\|K*(\nabla
u\cdot\nabla(K*u))\right\|_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle\leq\tfrac{D}{4}\left\|\Delta(K*u)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}+D^{-1}\left\|K\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}\left\|\nabla
u\cdot\nabla(K*u)\right\|_{L^{1}({\mathbb{R}^{d}})}^{2}$
$\displaystyle\leq\tfrac{D}{4}\left\|\Delta(K*u)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}+D^{-1}\left\|K\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}\left\|\nabla
u(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}\left\|\nabla(K*u)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}$
$\displaystyle=:\tfrac{D}{4}\left\|\Delta(K*u)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}+D^{-1}a^{2}(t)\left\|\nabla(K*u)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2},$
(3.50)

where $a(t):=\left\|K\right\|_{L^{2}({\mathbb{R}^{d}})}\left\|\nabla
u(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}$. Notice that $a^{2}(t)\in
L^{1}(0,T)$ for any fixed $T>0$ by Lemma 3.6.

Next, we estimate the third term appearing in (3.5) via two applications of
Hölder’s inequality followed by Cauchy’s inequality with $\varepsilon=D/2$:

$\displaystyle\int_{\mathbb{R}^{d}}u\,\Delta(K*u)(\tilde{K}*\Delta(K*u))\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\left\|u(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}\left\|\Delta(K*u)(\tilde{K}*\Delta(K*u))\right\|_{L^{2}({\mathbb{R}^{d}})}$
$\displaystyle\leq\left\|u(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}\left\|\tilde{K}*(\Delta(K*u))\right\|_{L^{\infty}({\mathbb{R}^{d}})}\left\|\Delta(K*u)\right\|_{L^{2}({\mathbb{R}^{d}})}$
$\displaystyle\leq\frac{D}{4}\left\|\Delta(K*u)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}$
$\displaystyle+D^{-1}\left\|u(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}\left\|\tilde{K}*(\Delta(K*u))\right\|_{L^{\infty}({\mathbb{R}^{d}})}^{2},$
(3.51)

We must ensure that the second product belongs to $L^{1}(0,T)$. By Lemma 3.6,
we know that $u\in L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}))$. For the remaining
term, we use the fact that

$\displaystyle\tilde{K}*\Delta(K*u)=\nabla(\tilde{K}*K)*\nabla u.$

Hence, we may estimate via Young’s convolution inequality

$\displaystyle\left\|\tilde{K}*(\Delta(K*u))\right\|_{L^{\infty}({\mathbb{R}^{d}})}^{2}$
$\displaystyle\leq\left\|\nabla(\tilde{K}*K)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}\left\|\nabla
u(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2},$ (3.52)

from which we conclude that

$\displaystyle\int_{\mathbb{R}^{d}}u\,\Delta(K*u)(\tilde{K}*\Delta(K*u))\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\frac{D}{4}\left\|\Delta(K*u)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}$
$\displaystyle+D^{-1}b^{2}(t)$ (3.53)

where

$\displaystyle
b(t):=\left\|\nabla(\tilde{K}*K)\right\|_{L^{2}({\mathbb{R}^{d}})}\sup_{t\in(0,T)}\left\|u(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}\left\|\nabla
u(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})},$

and $b^{2}(t)\in L^{1}(0,T)$ by hypothesis (H4) and by estimate (3.34) of
Lemma 3.6.

Combining estimates (3.5) and (3.5) with (3.5), we finally obtain

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}\left|\nabla(K*u)\right|^{2}\mathop{}\\!\mathrm{d}x+D\int_{\mathbb{R}^{d}}\left|\Delta(K*u)\right|^{2}\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq
D^{-1}\left(a^{2}(t)\left\|\nabla(K*u))\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}+b^{2}(t)\right).$
(3.54)

By Gronwall’s lemma we conclude that

$\displaystyle\sup_{t\in(0,T)}\left\|\nabla(K*u)(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}$
$\displaystyle\leq
D^{-1}A(T)\left(\left\|\nabla(K*u_{0})\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}+\int_{0}^{T}b^{2}(t){\rm
d}t\right),$ (3.55)

where $A(t):=e^{\int_{0}^{t}a^{2}(s){\rm d}s}$. Hence, $\nabla(K*u)\in
L^{\infty}(0,T;L^{2}(\Omega))$.

Returning to estimate (3.54), we extract the second term on the left hand side
and use estimate (3.55) to conclude that in fact

$\displaystyle\left\|\Delta(K*u)\right\|_{L^{2}(Q_{T})}$ $\displaystyle\leq
2D^{-1}\left(\sup_{t\in(0,T)}\left\|\nabla(K*u)(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}\left\|a(\cdot)\right\|_{L^{2}(0,T)}+\left\|b(\cdot)\right\|_{L^{2}(0,T)}\right),$
(3.56)

and so $\Delta(K*u)\in L^{2}(Q_{T})$.

We now improve the estimate on $\Delta(K*u)$ from $L^{2}(Q_{T})$ to
$L^{2}(0,T;L^{p}({\mathbb{R}^{d}}))$ for some $p>2$. To this end, notice first
that if $u$ solves the heat equation

$\frac{\partial u}{\partial t}-D\Delta u=g\in
L^{2}(0,T;L^{1}({\mathbb{R}^{d}})),$

for some function $g(t,x)$, then by the linearity of the convolution we have
also that $K*u$ solves

$\frac{\partial}{\partial t}(K*u)-D\Delta(K*u)=K*g.$

In our case, using the improved estimates on $K*\nabla u$, $K*\Delta u$, we
have

$g=\nabla u\,K*\nabla u+u\,K*\Delta u\,\in
L^{2}(0,T;L^{1}({\mathbb{R}^{d}})),$

so that $K*g\in L^{2}(0,T;L^{q}({\mathbb{R}^{d}}))$ for any $q\in[1,\infty]$
since $K\in L^{1}({\mathbb{R}^{d}})\cap L^{\infty}({\mathbb{R}^{d}})$. Hence,
by the maximal regularity of the heat equation (see, e.g., [41, Theorem 3.1]),
we find that $\Delta(K*u)\in L^{2}(0,T;L^{p}({\mathbb{R}^{d}}))$ where $p$ is
an exponent such that $K*u_{0}\in W^{2,p}(\mathbb{R}^{d})$.

We are now ready to obtain estimates (3.46)-(3.47). Multiplying (1.1) by
$\Delta u$ and integrating over ${\mathbb{R}^{d}}$ yields

$\displaystyle\frac{{\rm d}}{{\rm d}t}\int_{\mathbb{R}^{d}}\left|\nabla
u\right|^{2}\mathop{}\\!\mathrm{d}x+D\int_{\mathbb{R}^{d}}\left|\Delta
u\right|^{2}\mathop{}\\!\mathrm{d}x\leq\int_{\mathbb{R}^{d}}\left|u\,\Delta
u(K*\Delta u)\right|\mathop{}\\!\mathrm{d}x+\int_{\mathbb{R}^{d}}\left|\nabla
u\,\Delta u(K*\nabla u)\right|\mathop{}\\!\mathrm{d}x.$ (3.57)

We estimate the right hand side via Cauchy’s inequality. For the first term we
have

$\displaystyle\int_{\mathbb{R}^{d}}\left|u\,\Delta u(K*\Delta
u)\right|\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\frac{D}{4}\int_{\mathbb{R}^{d}}\left|\Delta
u\right|^{2}\mathop{}\\!\mathrm{d}x+D^{-1}\int_{\mathbb{R}^{d}}\left|u\right|^{2}\left|K*\Delta
u\right|^{2}\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\frac{D}{4}\int_{\mathbb{R}^{d}}\left|\Delta
u\right|^{2}\mathop{}\\!\mathrm{d}x+D^{-1}\left\|u(t,\cdot)\right\|_{L^{q}({\mathbb{R}^{d}})}^{2}\left\|K*\Delta
u(t,\cdot)\right\|_{L^{p}({\mathbb{R}^{d}})}^{2}$
$\displaystyle\leq\frac{D}{4}\int_{\mathbb{R}^{d}}\left|\Delta
u\right|^{2}\mathop{}\\!\mathrm{d}x+D^{-1}\left\|u\right\|_{L^{\infty}(0,T;L^{q}({\mathbb{R}^{d}}))}^{2}\left\|K*\Delta
u(t,\cdot)\right\|_{L^{p}({\mathbb{R}^{d}})}^{2}$
$\displaystyle\leq\frac{D}{4}\int_{\mathbb{R}^{d}}\left|\Delta
u\right|^{2}\mathop{}\\!\mathrm{d}x+D^{-1}e^{8C(T)/D}\left\|u_{0}\right\|_{L^{q}({\mathbb{R}^{d}})}^{2}\left\|K*\Delta
u(t,\cdot)\right\|_{L^{p}({\mathbb{R}^{d}})}^{2},$ (3.58)

where we have used Hölder’s inequality and that $u\in
L^{\infty}(0,T;L^{q}({\mathbb{R}^{d}}))$ for any exponent
$q:=2p/(p-2)\in(1,\infty)$. From our improved estimates on $\Delta(K*u)$ we
have already shown that $\left\|K*\Delta
u(t,\cdot)\right\|_{L^{p}({\mathbb{R}^{d}})}^{2}\in L^{1}(0,T)$ for some
$p>2$.

For the second term, we again use Cauchy’s inequality

$\displaystyle\int_{\mathbb{R}^{d}}\left|\nabla u\Delta u(K*\nabla
u)\right|\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\frac{D}{4}\int_{\mathbb{R}^{d}}\left|\Delta
u\right|^{2}\mathop{}\\!\mathrm{d}x+D^{-1}\int_{\mathbb{R}^{d}}\left|K*\nabla
u\right|^{2}\left|\nabla u\right|^{2}\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\frac{D}{4}\int_{\mathbb{R}^{d}}\left|\Delta
u\right|^{2}\mathop{}\\!\mathrm{d}x+D^{-1}\left\|K*\nabla
u(t,\cdot)\right\|_{L^{\infty}({\mathbb{R}^{d}})}^{2}\int_{\mathbb{R}^{d}}\left|\nabla
u\right|^{2}{\rm d}x,$

where $\left\|K*\nabla u(t,\cdot)\right\|_{L^{\infty}({\mathbb{R}^{d}})}\in
L^{1}(0,T)$ since $\nabla u\in L^{2}(Q_{T})$ and $K\in
L^{1}(\mathbb{R}^{d})\cap L^{\infty}({\mathbb{R}^{d}})$. Hence, (3.57) becomes

$\displaystyle\frac{{\rm d}}{{\rm d}t}\int_{\mathbb{R}^{d}}\left|\nabla
u\right|^{2}\mathop{}\\!\mathrm{d}x+\frac{D}{2}\int_{\mathbb{R}^{d}}\left|\Delta
u\right|^{2}\mathop{}\\!\mathrm{d}x\leq
C_{1}(t)+C_{2}(t)\int_{\mathbb{R}^{d}}\left|\nabla
u\right|^{2}\mathop{}\\!\mathrm{d}x,$ (3.59)

for functions $C_{1}(t)$, $C_{2}(t)\in L^{1}(0,T)$. Grönwall’s lemma implies
$\Delta u\in L^{2}(0,T;L^{2}({\mathbb{R}^{d}}))$ and $\nabla u\in
L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}))$ from which estimates (3.46)-(3.47)
follow.

We now obtain a uniform bound on the solution $u(t,x)$ via an Alikakos type
argument [1], which arises as a consequence of $\nabla u\in
L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}))\Rightarrow K*\nabla u\in
L^{\infty}(Q_{T})$. Set $C_{0}:=\left\|K*\nabla
u\right\|_{L^{\infty}(Q_{T})}$. To begin, we return to the right hand side of
estimate (3.3) and apply Cauchy’s inequality

$\displaystyle-(p-1)\int_{\mathbb{R}^{d}}u^{p-1}\nabla uK*\nabla
u\mathop{}\\!\mathrm{d}x$
$\displaystyle=-\tfrac{2(p-1)}{p}\int_{\mathbb{R}^{d}}u^{p/2}\nabla(u^{p/2})K*\nabla
u\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\tfrac{2(p-1)}{p}C_{0}\int_{\mathbb{R}^{d}}u^{p/2}\left|\nabla
u^{p/2}\right|\mathop{}\\!\mathrm{d}x$ $\displaystyle\leq
C_{0}\int_{\mathbb{R}^{d}}\left(\varepsilon\left|\nabla
u^{p/2}\right|^{2}+\frac{1}{\varepsilon}u^{p}\right){\rm d}x,$

where $\varepsilon>0$ is to be chosen. We then recall the Nash’s inequality:
there exists $C_{N}>0$ such that for $g\in L^{1}(\mathbb{R}^{d})\cap
W^{1,2}(\mathbb{R}^{d})$ there holds

$\|g\|^{2}_{L^{2}(\mathbb{R}^{d})}\leq
C_{N}\,\|g\|^{4/(d+2)}_{L^{1}(\mathbb{R}^{d})}\,\|\nabla
g\|^{2d/(d+2)}_{L^{2}(\mathbb{R}^{d})}\leq\delta\,C_{N}^{(d+2)/d}\,\|\nabla
g\|^{2}_{L^{2}(\mathbb{R}^{d})}+\delta^{-d/2}\|g\|_{L^{1}(\mathbb{R}^{d})}^{2},$

where the second step is a consequence of the Young’s inequality with
conjugate exponents $\frac{d+2}{2}$ and $\frac{d+2}{d}$, and $\delta>0$ is to
be chosen. We apply it with $g=u^{p/2}$ and we combine it with estimate (3.3)
to obtain

$\displaystyle\frac{1}{p}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}u^{p}{\rm d}x\leq$
$\displaystyle-\tfrac{4D(p-1)}{p^{2}}\|\nabla
u^{p/2}\|^{2}_{L^{2}(\mathbb{R}^{d})}+\varepsilon\,C_{0}\|\nabla
u^{p/2}\|^{2}_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle+\frac{1}{\varepsilon}C_{0}\left(\delta\,C_{N}^{(d+2)/d}\,\|\nabla
u^{p/2}\|^{2}_{L^{2}(\mathbb{R}^{d})}+\delta^{-d/2}\|u^{p/2}\|_{L^{1}(\mathbb{R}^{d})}^{2}\right).$
(3.60)

We now set $\delta\,C_{N}^{(d+2)/d}=\varepsilon^{2}$ and choose
$\varepsilon=\tfrac{2D(p-1)}{C_{0}p^{2}}$ so that

$2C_{0}\varepsilon=\frac{4D(p-1)}{p^{2}}\Rightarrow\varepsilon=\frac{2D(p-1)}{C_{0}p^{2}}.$

It follows that the gradient term vanishes and (3.5) becomes

$\displaystyle\frac{1}{p}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}u^{p}\mathop{}\\!\mathrm{d}x$ $\displaystyle\leq
C_{6}\,\varepsilon^{-d-1}\left\|u^{p/2}(t,\cdot)\right\|_{L^{1}({\mathbb{R}^{d}})}^{2}\leq
C_{7}\,p^{d+1}\left\|u^{p/2}(t,\cdot)\right\|_{L^{1}({\mathbb{R}^{d}})}^{2},$
(3.61)

where $C_{6}$, $C_{7}$ depend on $C_{N}$, $C_{0}$, $D$ but do not depend on
$p$. Integrating both sides from $0\to t$ yields

$\displaystyle\left\|u(t,\cdot)\right\|_{L^{p}({\mathbb{R}^{d}})}^{p}\leq\left\|u_{0}\right\|_{L^{p}({\mathbb{R}^{d}})}^{p}+C_{7}\,p^{d+2}T\left\|u\right\|_{L^{\infty}(0,T;L^{p/2}({\mathbb{R}^{d}})}^{p}.$
(3.62)

Without loss of generality, for $T$ fixed we may assume that

$\left\|u_{0}\right\|_{L^{p}({\mathbb{R}^{d}})}^{p}\leq
C_{7}\,p^{d+2}\,T\left\|u\right\|_{L^{\infty}(0,T;L^{p/2}({\mathbb{R}^{d}}))}^{p},$

for all $p\geq p^{*}$ for some $p^{*}\geq 1$. Otherwise, there exists a
sequence $\\{p_{k}\\}_{k\geq 0}$ such that $p_{k}\to+\infty$ as $k\to+\infty$
so that

$\left\|u_{0}\right\|_{L^{p_{k}}({\mathbb{R}^{d}})}^{p_{k}}\geq
C_{7}\,{p_{k}}^{d+2}\,T\left\|u\right\|_{L^{\infty}(0,T;L^{p_{k}/2}({\mathbb{R}^{d}}))}^{p_{k}}$

from which the boundedness of $u$ follows. Set $C_{8}=(2C_{7}T)^{1/(d+2)}$ and
$M(p):=\left\|u\right\|_{L^{\infty}(0,T;L^{p}({\mathbb{R}^{d}}))}.$ We then
have that

$M(p)\leq(C_{8}\,p)^{(d+2)/p}M(p/2),\quad\forall p\geq p^{*}.$

Take $i\geq 1$ so that $2^{i}\geq p^{*}$. For any fixed $k$ we proceed
iteratively to obtain

$\displaystyle M(2^{k})\,$
$\displaystyle\leq\,(C_{8}2^{k})^{(d+2)/{2^{k}}}M(2^{k-1})\,\leq\,(C_{8}2^{k})^{(d+2)/{2^{k}}}(C_{8}2^{(k-1)})^{(d+2)/{2^{k-1}}}M(2^{k-2})$
$\displaystyle\ldots\leq M(2^{i-1})\prod_{j=i}^{k}(C_{8}2^{j})^{(d+2)/2^{j}}.$
(3.63)

It follows that $M(2^{k})$ is bounded independent of $k$ so long as

$\lim_{k\to\infty}\prod_{j=i}^{k}(C_{8}2^{j})^{(d+2)/2^{j}}<\infty\iff\sum_{j=i}^{\infty}\frac{(d+2)\log(C_{8}2^{j})}{2^{j}}<\infty,$

which readily follows from the fact that $2^{j}\gg(d+2)\,j$. Hence, $u$ is
uniformly bounded in $Q_{T}$ for any $T>0$ fixed and estimate (3.48) follows.
∎

######  Lemma 3.8 (Improved integrability of gradient).

Suppose $\nabla u_{0}\in L^{p}(\mathbb{R}^{d})$. Under the conditions of Lemma
3.7, for each $p\geq 2$ there exists a constant $\tilde{C}_{0}$ such that
there holds

$\displaystyle\left\|\nabla
u\right\|_{L^{\infty}(0,T;L^{p}(\mathbb{R}^{d}))}\leq\tilde{C}_{0},$ (3.64)

where $\tilde{C}_{0}=\tilde{C}_{0}(\tilde{C},p,\left\|\nabla
u_{0}\right\|_{L^{p}(\mathbb{R}^{d})})$, and $\tilde{C}$ is given in Lemma
3.7.

###### Proof.

Under the conditions of Lemma 3.7, we have that

$u\in L^{\infty}(Q_{T});\quad\nabla u\in
L^{\infty}(0,T;L^{2}(\Omega));\quad\Delta u\in L^{2}(Q_{T}).$

Assume that $u$ is a smooth solution to problem (1.1). For $p\geq 2$, we
multiply the equation for $u$ by

$-\mbox{div}\left(\left|\nabla u\right|^{p-2}\nabla u\right)=-(p-1)\,\Delta
u\,|\nabla u|^{p-2}$

and integrate over $\mathbb{R}^{d}$ to obtain

$\displaystyle\int_{\mathbb{R}^{d}}\left|\nabla u\right|^{p-2}\nabla u\,\nabla
u_{t}\mathop{}\\!\mathrm{d}x=-(p-1)\int_{\mathbb{R}^{d}}\left|\nabla
u\right|^{p-2}\,\Delta u\,(D\Delta u+\nabla uK*\nabla u+uK*\Delta
u)\mathop{}\\!\mathrm{d}x.$ (3.65)

Clearly, the left hand side of (3.65) equals
$\frac{1}{p}\int_{\mathbb{R}^{d}}\left|\nabla
u\right|^{p}\mathop{}\\!\mathrm{d}x$ and so rearrangement of (3.65) yields

$\displaystyle\frac{1}{p\,(p-1)}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}\left|\nabla u\right|^{p}{\rm d}x\leq$
$\displaystyle-D\int_{\mathbb{R}^{d}}\left|\nabla u\right|^{p-2}\left|\Delta
u\right|^{2}{\rm d}x+\int_{\mathbb{R}^{d}}\left|\nabla
u\right|^{p-1}\left|\Delta u\right|\left|K*\nabla u\right|{\rm d}x$
$\displaystyle+\int_{\mathbb{R}^{d}}\left|\nabla u\right|^{p-2}u\left|\Delta
u\right|\left|K*\Delta u\right|$ $\displaystyle=:$
$\displaystyle-D\int_{\mathbb{R}^{d}}\left|\nabla u\right|^{p-2}\left|\Delta
u\right|^{2}{\rm d}x+I_{1}+I_{2}.$ (3.66)

We estimate $I_{1}$, $I_{2}$ separately. For $I_{1}$, Cauchy’s inequality
yields

$\displaystyle I_{1}\leq$
$\displaystyle\,\frac{D}{2}\int_{\mathbb{R}^{d}}\left|\nabla
u\right|^{p-2}\left|\Delta u\right|^{2}{\rm
d}x+\frac{1}{2D}\int_{\mathbb{R}^{d}}\left|\nabla u\right|^{p}\left|K*\nabla
u\right|^{2}{\rm d}x$ $\displaystyle\leq$
$\displaystyle\frac{D}{2}\int_{\mathbb{R}^{d}}\left|\nabla
u\right|^{p-2}\left|\Delta u\right|^{2}{\rm
d}x+C_{1}\int_{\mathbb{R}^{d}}\left|\nabla u\right|^{p}{\rm d}x,$ (3.67)

where

$C_{1}:=\frac{\left\|K\right\|_{L^{2}(\mathbb{R^{d}})}^{2}\left\|\nabla
u(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}}{2D}<\infty,$

since $\nabla u\in L^{\infty}(0,T;L^{2}(\mathbb{R}^{d}))$. For $I_{2}$, we
first estimate via Cauchy’s inequality as

$\displaystyle I_{2}\leq\frac{D}{2}\int_{\mathbb{R}^{d}}\left|\nabla
u\right|^{p-2}\left|\Delta u\right|^{2}{\rm
d}x+C_{2}\int_{\mathbb{R}^{d}}\left|\nabla u\right|^{p-2}{\rm d}x,$ (3.68)

where

$C_{2}(t):=\frac{\left\|u\right\|_{L^{\infty}(Q_{T})}^{2}\left\|K\right\|_{L^{2}(\mathbb{R}^{d})}^{2}\left\|\Delta
u(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}}{2D}\in L^{1}(0,T),$

since $u\in L^{\infty}(Q_{T})$ and $\Delta u\in L^{2}(Q_{T})$.

To estimate $\int\left|\nabla u\right|^{p-2}{\rm d}x$, we choose $p>4$ (the
case $p=4$ is direct) and apply Young’s inequality for products with
$p^{\prime}=(p-2)/(p-4)$, $q^{\prime}=(p-2)/2$:

$\displaystyle\int_{\mathbb{R}^{d}}\left|\nabla u\right|^{p-2}{\rm d}x=$
$\displaystyle\int_{\mathbb{R}^{d}}\left|\nabla
u\right|^{p(p-4)/(p-2)}\left|\nabla u\right|^{4/(p-2)}{\rm d}x$
$\displaystyle\leq$ $\displaystyle\int_{\mathbb{R}^{d}}(\left|\nabla
u\right|^{p}+\left|\nabla u\right|^{2}){\rm d}x.$ (3.69)

Combining estimates (3.5)-(3.5) with estimate (3.65) leaves

$\displaystyle\frac{1}{p\,(p-1)}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}\left|\nabla u\right|^{p}{\rm d}x\leq$
$\displaystyle\left(C_{1}+C_{2}(t)\right)\int_{\mathbb{R}^{d}}\left|\nabla
u\right|^{p}{\rm d}x+C_{2}(t)\left\|\nabla
u(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}.$ (3.70)

Since $C_{2}(t)\in L^{1}(0,T)$, and since

$\nabla u\in L^{\infty}(0,T;L^{2}(\mathbb{R}^{d}))\Rightarrow
C_{2}(t)\left\|\nabla u(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}\in
L^{1}(0,T),$

we obtain the estimate (3.64) via Grönwall’s lemma. The case when $p\in(2,4)$
is obtained via interpolation between the cases $p=2$ and $p=4$. ∎

###  3.6. Uniqueness

The higher regularity results from the previous subsection allow us to show
the following uniqueness result.

######  Theorem 3.9.

Suppose hypothesis (H4) holds. Then any nonnegative solution $u$ solving
problem (1.1) belonging to the class

$\displaystyle u$ $\displaystyle\in L^{\infty}(Q_{T})\cap
L^{\infty}(0,T;L^{1}({\mathbb{R}^{d}}));$ $\displaystyle\nabla u$
$\displaystyle\in L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}));$
$\displaystyle\Delta u$ $\displaystyle\in L^{2}(Q_{T})$

is unique.

###### Proof.

Assume there are two nonnegative solutions $u$ and $v$. We write

$\displaystyle(u-v)_{t}-D\Delta(u-v)=\nabla\cdot((u-v)\nabla
K*u)+\nabla\cdot(v\nabla K*(u-v)).$ (3.71)

Set $w=u-v$. Testing (3.71) against $w$, integrating over ${\mathbb{R}^{d}}$
and integration by parts gives

$\displaystyle\frac{1}{2}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}w^{2}\mathop{}\\!\mathrm{d}x+D\int_{\mathbb{R}^{d}}\left|\nabla
w\right|^{2}\mathop{}\\!\mathrm{d}x$
$\displaystyle=-\int_{\mathbb{R}^{d}}w\nabla(K*u)\nabla
w\mathop{}\\!\mathrm{d}x$
$\displaystyle\quad-\int_{\mathbb{R}^{d}}v\nabla(K*w)\nabla
w\mathop{}\\!\mathrm{d}x$
$\displaystyle=-\frac{1}{2}\int_{\mathbb{R}^{d}}\nabla
w^{2}\nabla(K*u)\mathop{}\\!\mathrm{d}x$
$\displaystyle-\int_{\mathbb{R}^{d}}v\nabla(K*w)\nabla
w\mathop{}\\!\mathrm{d}x.$ (3.72)

We estimate each term separately. For the first term, we integrate by parts
and use that $\Delta u\in L^{2}(0,T;L^{2}({\mathbb{R}^{d}}))\Rightarrow
K*\Delta u\in L^{2}(0,T;L^{\infty}({\mathbb{R}^{d}}))$ to obtain

$\displaystyle-\frac{1}{2}\int_{\mathbb{R}^{d}}\nabla
w^{2}\nabla(K*u)\mathop{}\\!\mathrm{d}x=\frac{1}{2}\int_{\mathbb{R}^{d}}w^{2}K*\Delta
u\mathop{}\\!\mathrm{d}x\leq\frac{1}{2}\left\|K*\Delta
u(t,\cdot)\right\|_{L^{\infty}({\mathbb{R}^{d}})}\int_{\mathbb{R}^{d}}w^{2}\mathop{}\\!\mathrm{d}x.$

For the second term, we apply Cauchy’s inequality with
$\varepsilon=D\left\|v\right\|_{L^{\infty}(Q_{T})}^{-1}$ to obtain

$\displaystyle-\int_{\mathbb{R}^{d}}v\nabla(K*w)\nabla
w\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\frac{1}{2}\int_{\mathbb{R}^{d}}\left(D\left|\nabla
w\right|^{2}+\frac{\left\|v\right\|_{L^{\infty}(Q_{T})}^{2}}{D}\left|K*\nabla
w\right|^{2}\right)\mathop{}\\!\mathrm{d}x.$

Combining these estimates, (3.6) becomes

$\displaystyle\frac{1}{2}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}w^{2}\mathop{}\\!\mathrm{d}x+\frac{D}{2}\int_{\mathbb{R}^{d}}\left|\nabla
w\right|^{2}\mathop{}\\!\mathrm{d}x$ $\displaystyle\leq
C_{1}(t)\int_{\mathbb{R}^{d}}w^{2}\mathop{}\\!\mathrm{d}x+C_{2}\int_{\mathbb{R}^{d}}\left|K*\nabla
w\right|^{2}\mathop{}\\!\mathrm{d}x,$ (3.73)

where $C_{1}(t)=\left\|K*\Delta
u(t,\cdot)\right\|_{L^{\infty}({\mathbb{R}^{d}})}\in L^{1}(0,T)$ and
$C_{2}=D^{-1}\left\|v\right\|_{L^{\infty}(Q_{T})}^{2}$.

Next, we test (3.71) against $\tilde{K}*K*w$, integrate over
${\mathbb{R}^{d}}$ and integrate by parts to find

$\displaystyle\frac{1}{2}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}\left|K*w\right|^{2}\mathop{}\\!\mathrm{d}x+D\int_{\mathbb{R}^{d}}\left|\nabla
K*w\right|^{2}\mathop{}\\!\mathrm{d}x$
$\displaystyle=-\int_{\mathbb{R}^{d}}w\nabla(K*u)\nabla(\tilde{K}*K*w)\mathop{}\\!\mathrm{d}x$
$\displaystyle\quad-\int_{\mathbb{R}^{d}}v\nabla(K*w)\nabla(\tilde{K}*K*w)\mathop{}\\!\mathrm{d}x.$
(3.74)

Since $\nabla u\in L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}))$, there holds
$\nabla(K*u)\in L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}))$ so long as $K\in
L^{1}({\mathbb{R}^{d}})$. Moreover,

$\left\|\nabla(\tilde{K}*K*w)(t,\cdot)\right\|_{L^{\infty}({\mathbb{R}^{d}})}\leq\left\|\nabla(\tilde{K}*K)\right\|_{L^{2}({\mathbb{R}^{d}})}\left\|w(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}=C_{3}\left\|w(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})},$

where
$C_{3}=\left\|\nabla(\tilde{K}*K)\right\|_{L^{2}({\mathbb{R}^{d}})}<\infty$
under hypothesis (H4). Thus, the first term of (3.6) can be estimated as

$\displaystyle\left|\int_{\mathbb{R}^{d}}w\nabla(K*u)\nabla(\tilde{K}*K*w){\rm
d}x\right|$
$\displaystyle\quad\leq\left\|w(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}\left\|\nabla(K*u)(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}\left\|\nabla(\tilde{K}*K*w)(t,\cdot)\right\|_{L^{\infty}({\mathbb{R}^{d}})}$
$\displaystyle\quad\leq
C_{3}\left\|\nabla(K*u)\right\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}))}\left\|w(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}.$
(3.75)

For the second term of (3.6), we use that $v\in L^{\infty}(Q_{T})\cap
L^{\infty}(0,T;L^{1}({\mathbb{R}^{d}}))\Rightarrow v\in
L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}))$ and apply Cauchy’s inequality:

$\displaystyle\left|\int_{\mathbb{R}^{d}}v\nabla(K*w)\nabla(\tilde{K}*K*w){\rm
d}x\right|$
$\displaystyle\quad\leq\left\|v(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}\left\|\nabla
K*w(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}\left\|\nabla\tilde{K}*K*w(t,\cdot)\right\|_{L^{\infty}({\mathbb{R}^{d}})}$
$\displaystyle\quad\leq
C_{3}\left\|v\right\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}))}\left(\varepsilon\left\|\nabla
K*w(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}+\frac{1}{\varepsilon}\left\|w(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}\right).$
(3.76)

Choosing
$\varepsilon=\frac{D}{2C_{3}\left\|v\right\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}))}}$,
we conclude that

$\displaystyle\frac{1}{2}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}\left|K*w\right|^{2}\mathop{}\\!\mathrm{d}x+\frac{D}{2}\int_{\mathbb{R}^{d}}\left|\nabla
K*w\right|^{2}\mathop{}\\!\mathrm{d}x$ $\displaystyle\leq
C_{4}\left\|w(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2},$ (3.77)

where $C_{4}<\infty$.

We now combine estimates (3.73) and (3.77). To this end, we multiply estimate
(3.73) by $\frac{D}{4C_{2}}$ and add the result to (3.77):

$\displaystyle\frac{1}{2}\frac{{\rm d}}{{\rm
d}t}\left(\frac{D}{4C_{3}}\int_{\mathbb{R}^{d}}w^{2}\mathop{}\\!\mathrm{d}x+\int_{\mathbb{R}^{d}}\left|K*w\right|^{2}\mathop{}\\!\mathrm{d}x\right)+\frac{D^{2}}{8C_{3}}\int_{\mathbb{R}^{d}}\left|\nabla
w\right|^{2}\mathop{}\\!\mathrm{d}x+\frac{D}{4}\int_{\mathbb{R}^{d}}\left|\nabla
K*w\right|^{2}\mathop{}\\!\mathrm{d}x$
$\displaystyle\quad\leq(C_{1}(t)+C_{4})\int_{\mathbb{R}^{d}}w^{2}\mathop{}\\!\mathrm{d}x\leq(C_{1}(t)+C_{4})\int_{\mathbb{R}^{d}}\left(w^{2}+\left|K*w\right|^{2}\right){\rm
d}x.$ (3.78)

Grönwall’s lemma implies that $w\equiv 0$ a.e. in $Q_{T}$, and uniqueness
follows. ∎

###  3.7. Existence of unique strong and classical solutions (proof of
Theorems 2.3-2.4)

We are now ready to prove Theorems 2.3-2.4.

###### Proof of Theorem 2.3.

By Lemma 3.7, the global weak solution obtained in either Theorem 2.1 (under
the addition of hypothesis (H4)) or Theorem 2.2 belongs to the class

$\displaystyle u$ $\displaystyle\in L^{\infty}(Q_{T})\cap
L^{\infty}(0,T;L^{1}({\mathbb{R}^{d}}));$ $\displaystyle\nabla u$
$\displaystyle\in L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}));$
$\displaystyle\Delta u$ $\displaystyle\in L^{2}(Q_{T}).$

By Theorem 3.9 the solution is unique. As argued in the proof of Theorems
2.1-2.2, the improved regularity of the weak solution allows one to take
$p=\infty$ in (3.4) to conclude that $\frac{\partial u}{\partial t}\in
L^{2}(0,T;H^{-1}(\mathbb{R}^{d}))$, and moreover $u\in
L^{2}(0,T;H^{1}(\mathbb{R}^{d}))$. Consequently, $u\in
C(0,T;L^{2}(\mathbb{R}^{d}))$ and so we conclude that
$\left\|u(t,\cdot)-u_{0}(\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}\to 0$ as $t\to
0^{+}$. ∎

###### Proof of Theorem 2.4.

Working with the approximate solutions $u^{\varepsilon}$ as in (3.40), we
expand the right hand side

$\frac{\partial u^{\varepsilon}}{\partial t}-D\Delta u^{\varepsilon}=\nabla
u^{\varepsilon}\cdot\nabla(K*\omega^{\varepsilon}*u^{\varepsilon})+u^{\varepsilon}\Delta(K*\omega^{\varepsilon}*u^{\varepsilon}).$

By Lemma 3.7, $\\{u^{\varepsilon}\\}_{\varepsilon>0}$ is bounded in
$L^{\infty}(Q_{T})$; moreover, by Lemma 3.8, $\\{\nabla
u^{\varepsilon}\\}_{\varepsilon>0}$ is bounded in
$L^{\infty}(0,T;L^{p}(\mathbb{R}^{d}))$ for any $p\geq 2$. Therefore,
$\\{\nabla
u^{\varepsilon}\cdot\nabla(K*\omega^{\varepsilon}*u^{\varepsilon})\\}_{\varepsilon>0}$
is bounded in $L^{\infty}(0,T;L^{p}(\mathbb{R}^{d}))$ for any $p\geq 2$.
Similarly, writing
$\Delta(K*\omega^{\varepsilon}*u^{\varepsilon})=\nabla(K*\omega^{\varepsilon})*\nabla
u^{\varepsilon}$, Hypothesis (H2) allows one to conclude that
$\\{u^{\varepsilon}\nabla(K*\omega^{\varepsilon})*\nabla
u^{\varepsilon}\\}_{\varepsilon>0}$ is also bounded in
$L^{\infty}(0,T;L^{p}(\mathbb{R}^{d}))$ for any $p\geq 2$.

Hence, $L^{p}$-estimates for parabolic equations (see, e.g., [62, Theorem 6])
imply that for any smooth, compact set $\Omega^{\prime}\subset\mathbb{R}^{d}$,
$\\{u^{\varepsilon}\\}_{\varepsilon>0}$ is bounded in
$W^{1,2}_{p}((0,T)\times\Omega^{\prime})$, the space of functions twice weakly
differentiable in space, once weakly differentiable in time, with derivatives
belonging to $L^{p}((0,T)\times\Omega^{\prime})$. Therefore, choosing $p>d+2$,
the Sobolev embedding (see, e.g., [64, Theorem 1.4.1]) implies that in fact
$\\{u^{\varepsilon}\\}_{\varepsilon>0}$ is bounded in
$C^{\sigma,1+\sigma}([0,T]\times\overline{\Omega^{\prime}})$ for any
$\sigma\in(0,1)$. Since $\Omega^{\prime}$ is arbitrary, we have that
$u^{\varepsilon}\in C^{\sigma,1+\sigma}_{loc}(Q_{T})$ uniformly in
$\varepsilon$.

We now set $w_{i}^{\varepsilon}:=\tfrac{\partial u^{\varepsilon}}{\partial
x_{i}}$ for each $i=1,\ldots,d$. Working with each component, we again expand
the right hand side of the equation solved by $w_{i}^{\varepsilon}$:

$\displaystyle\frac{\partial w^{\varepsilon}_{i}}{\partial t}-D\Delta
w^{\varepsilon}_{i}=$ $\displaystyle\,\nabla
w^{\varepsilon}_{i}\cdot\nabla(K*\omega^{\varepsilon}*u^{\varepsilon})+\nabla
u^{\varepsilon}\cdot\nabla(K*\omega^{\varepsilon}*w_{i}^{\varepsilon})$
$\displaystyle\,+w^{\varepsilon}_{i}\Delta(K*\omega^{\varepsilon}*u^{\varepsilon})+u^{\varepsilon}\Delta(K*\omega^{\varepsilon}*w^{\varepsilon}_{i}).$
(3.79)

Since $u^{\varepsilon}$ is bounded in $W^{1,2}_{p,loc}(Q_{T})$, we have that
$\nabla w_{i}^{\varepsilon}\in L^{p}_{loc}(Q_{T})$ for any $p\geq 2$, for each
$i=1,\ldots,d$. Moreover, by the classical differentiability obtained,
$w_{i}^{\varepsilon}$ is uniformly bounded over $Q_{T}$ for each
$i=1,\ldots,d$. Therefore, exploiting the compact support of $K$, the first
three terms on the right hand side of (3.7) are uniformly bounded in
$L^{p}_{loc}(Q_{T})$ for any $p\geq 2$. Then, using Hypothesis (H2), the
boundedness of $u^{\varepsilon}$ and the compact support of $K$, we also have
that $u^{\varepsilon}\Delta(K*\omega^{\varepsilon}*w^{\varepsilon}_{i})$ is
uniformly bounded in $L^{p}_{loc}(Q_{T})$ for any $p\geq 2$. Using once more
$L^{p}$-estimates for parabolic equations and the Sobolev embedding, we
conclude that $w_{i}^{\varepsilon}=\tfrac{\partial u^{\varepsilon}}{\partial
x_{i}}\in C^{\sigma,1+\sigma}([0,T]\times\overline{\Omega^{\prime}})$ for any
$\sigma\in(0,1)$, for any smooth, compact set
$\Omega^{\prime}\subset\mathbb{R}^{d}$. Consequently, we conclude that
$\\{u^{\varepsilon}\\}_{\varepsilon>0}$ is bounded in
$C^{\sigma,2+\sigma}_{loc}(Q_{T})$ for any $\sigma\in(0,1)$. In particular,
$\nabla
u^{\varepsilon}\cdot\nabla(K*\omega^{\varepsilon}*u^{\varepsilon})+u^{\varepsilon}K*\omega^{\varepsilon}*\Delta
u^{\varepsilon}\in C^{\sigma}_{loc}(Q_{T})$ for any $\sigma\in(0,1)$.

Returning once more to the right hand side of the equation satisfied by
$u^{\varepsilon}$, we have shown that $u_{t}^{\varepsilon}-D\Delta
u^{\varepsilon}\in C^{\sigma}_{loc}(Q_{T})$. By Schauder estimates for
parabolic equations (see, e.g., [49, Theorem 5.1.8]),
$\\{u^{\varepsilon}\\}_{\varepsilon>0}$ is bounded in
$C^{1+\sigma/2,2+\sigma}_{loc}(Q_{T})$ for any $\sigma\in(0,1)$. Consequently,
there exists a subsequence (still labelled by $\varepsilon$) such that
$u^{\varepsilon}\to u\in
C^{1+\sigma^{\prime}/2,2+\sigma^{\prime}}_{loc}(Q_{T})$ as $\varepsilon\to
0^{+}$ for any $\sigma^{\prime}<\sigma$, and $u$ is the classical solution to
problem (1.1). By the assumed regularity of the initial data, we conclude that
$u(t,x)\to u_{0}(x)$ in $C(\mathbb{R}^{d})$ as $t\to 0^{+}$.

We conclude by showing that the solution is strictly positive over
$\mathbb{R}^{d}$ for all $t>0$. The approach is standard once we have the
boundedness of $\Delta(K*u)$. Consider the auxiliary function $v:=e^{\lambda
t}u$ for $\lambda\in\mathbb{R}$ to be chosen later. Then $v$ is nonnegative,
nonconstant, and satisfies

$\frac{\partial v}{\partial t}-D\Delta v-\nabla
v\cdot\nabla(K*u)-v[\Delta(K*u)+\lambda]=0.$ (3.80)

By the smoothness of $u$ and Hypothesis (H6), we have that $K*\Delta u\in
L^{\infty}((0,T)\times\Omega^{\prime})$ for any compact
$\Omega^{\prime}\subset\mathbb{R}^{d}$ compact, and any $T>0$ fixed. Suppose
$\Omega^{\prime}$ is connected and that there exists a point
$(t_{0},x_{0})\in(0,T)\times\Omega^{\prime}$ such that $v(t_{0},x_{0})=0$.
Choosing $\lambda$ large enough so that $\lambda-\left|K*\Delta u\right|\geq
0$ over $(0,T)\times\Omega^{\prime}$, we find that $v_{t}-D\Delta
v-\nabla(K*u)\cdot\nabla v\geq 0$ over $(0,T)\times\Omega^{\prime}$. By the
strong maximum principle, we conclude that $v$ is constant over
$(0,t_{0})\times\Omega^{\prime}$, a contradiction.

Since $\Omega^{\prime}$, $T$ were arbitrary, we conclude that $v$ cannot
attain a minimum (of zero) in any compact subset of $\mathbb{R}^{d}$ and must
be positive everywhere in $(0,T)\times\mathbb{R}^{d}$. ∎

##  4\. The General $n$-Species System

In this section we establish the following theorems generalizing the scalar
cases to the $n$-species system.

######  Theorem 4.1 (Existence of global weak solution for $n$-species system
with small mass).

Assume (H1) holds and that $c_{i,n}>0$ for each $i=1,\ldots,n$, where
$c_{i,n}$ is defined as

$\displaystyle
c_{i,n}:=D_{i}-\frac{1}{2}\sum_{j=1}^{n}(m_{i}\left\|K_{ij}\right\|_{L^{\infty}({\mathbb{R}^{d}})}+m_{j}\left\|K_{ji}\right\|_{L^{\infty}({\mathbb{R}^{d}})})$
(4.1)

Then, there exists a global weak solution $\mathbf{u}\geq 0$ in $Q_{T}$
solving problem (1.2) in the sense of Definition 3.1. Moreover, $u_{i}\in
L^{\infty}(0,T;L^{p}({\mathbb{R}^{d}}))$ for any $1\leq p<\infty$, and $\nabla
u_{i}\in L^{2}(0,T;L^{2}({\mathbb{R}^{d}}))$ for each $i=1,\ldots,n$.

It is not difficult to see that when $n=1$, $c_{1,1}>0$ is equivalent to
condition (2.4) for the single species case.

######  Theorem 4.2 (Existence of global weak solution for $n$-species system
with arbitrary mass).

Assume (H1)-(H3) and (H5) hold. Then for any initial mass there exists a
global weak solution $\mathbf{u}\geq 0$ in $Q_{T}$ solving problem (1.2) in
the sense of Definition (3.1). Moreover, $u_{i}\in
L^{\infty}(0,T;L^{p}({\mathbb{R}^{d}}))$ for any $1\leq p<\infty$, and $\nabla
u_{i}\in L^{2}(0,T;L^{2}({\mathbb{R}^{d}}))$ for each $i=1,\ldots,n$.

######  Theorem 4.3 (Existence of unique strong solution for $n$-species
system).

Assume each component of the initial data satisfies $\nabla u_{i0}\in
L^{2}(\mathbb{R}^{d})$ and $K_{ij}*u_{i0}\in W^{2,p}(\mathbb{R}^{d})$ for some
$p>2$, for each $j=1,\ldots,n$. Under the assumptions of Theorem 4.2, or under
the assumptions of Theorem 4.1 together with (H4), the obtained global weak
solution is the unique, global strong solution solving problem (1.2) in the
sense of Definition 3.2.

######  Theorem 4.4 (Existence of unique classical solution for $n$-species
system).

Assume in addition to the hypothesis of Theorem 4.3 that (H6) holds and each
component of the initial data satisfies $u_{i0}\in C^{3}(\mathbb{R}^{d})$ as
well as $\nabla u_{i0}\in L^{p}(\mathbb{R}^{d})$ for some $p>d+2$. Then, the
obtained unique global strong solution is the unique, global classical
solution solving system 1.2 in the sense of Definition 3.3. Moreover, each
component $u_{i}$ is strictly positive over $\mathbb{R}^{d}$ for all $t>0$.

As in the single species case, we first treat the small mass case under
hypothesis (H1), and then treat the arbitrary mass case under the additional
hypotheses (H2)-(H3) and (H5). For the small mass case, the addition of
hypotheses (H4) will give that the solution is in fact unique and classical.
As much of the technical arguments are the same as in the scalar case, we
present the key details when a careful manipulation of the indices of $K_{ij}$
is required.

###  4.1. Apriori estimates

As in the scalar case, we first establish some apriori estimates under the
assumption that we have a positive smooth solution
$\mathbf{u}:=(u_{1},\ldots,u_{n})$ solving problem (1.2). To this end, fix
$n\geq 2$ and define the following $n$-population counterparts to the entropy,
total interaction energy, and free energy of the scalar system.

We introduce the entropy (sometimes referred to as the weighted Shannon
entropy [44]):

${\bf H}[\mathbf{u}(t)]:=\sum_{i=1}^{n}D_{i}\pi_{i}H[u_{i}(t)].$

Then the total interaction energy is given by

${\mathbfcal{K}}[\mathbf{u}(t)]:=\frac{1}{2}\sum_{i,j=1}^{n}\int_{\mathbb{R}^{d}}\pi_{i}u_{i}(K_{ij}*u_{j})\,\mathop{}\\!\mathrm{d}x,$

and so the free energy for a system in detailed balance (hypothesis (H5)) is
given as

${\mathbfcal{F}}[\mathbf{u}(t)]:=\sum_{i=1}^{n}\int_{\mathbb{R}^{d}}D_{i}\pi_{i}u_{i}\log
u_{i}\,\mathop{}\\!\mathrm{d}x+\frac{1}{2}\sum_{i,j=1}^{n}\int_{\mathbb{R}^{d}}\pi_{i}u_{i}(K_{ij}*u_{j})\,\mathop{}\\!\mathrm{d}x={\bf
H}[\mathbf{u}(t)]+{\mathbfcal{K}}[\mathbf{u}(t)]$

In the small mass regime, we use the following auxiliary functional, the sum
of the individual entropies of the species:

${\bf H}_{S}[\mathbf{u}(t)]:=\sum_{i=1}^{n}H[u_{i}(t)].$

We first prove the following, an $n$-species analogue to Lemma 3.4.

######  Lemma 4.5 (Apriori estimates under (H1) and small-mass condition).

Fix $T>0$. Assume (H1) and suppose $c_{i,n}>0$ for each $i=1,\ldots,n$ where
$c_{i,n}$ is as defined in the statement of Theorem 4.1. Then for any smooth,
positive solution $\mathbf{u}$ to problem (1.2) there holds

$\displaystyle\sup_{t\in(0,T)}I[u_{i}(t)]$ $\displaystyle\leq C;$ (4.2)
$\displaystyle\left\|\nabla\sqrt{u_{i}}\right\|_{L^{2}(Q_{T})}^{2}$
$\displaystyle\leq C;$ (4.3) $\displaystyle\left\|\sqrt{u_{i}}\nabla
K_{ij}*u_{j}\right\|_{L^{2}(Q_{T})}$ $\displaystyle\leq C,$ (4.4)

for each $i,j=1,\ldots,n$, where
$C=C(D_{i},T,c_{i,n}^{-1},\left\|K_{ij}\right\|_{L^{\infty}(\mathbb{R}^{d})},d,m_{i},I[u_{i0}],H[u_{i0}])\text{
for all }i,j=1,\ldots,n)$.

We also prove the following, an $n$-species analogue to Lemma 3.5.

######  Lemma 4.6 (Apriori estimates under (H1)-(H3), (H5)).

Fix $T>0$. Assume (H1)-(H3) and (H5). Then for any smooth, positive solution
$\mathbf{u}$ solving problem (1.2) there holds

$\displaystyle\sup_{t\in(0,T)}\int_{\mathbb{R}^{d}}u_{i}\left|x\right|^{2}\mathop{}\\!\mathrm{d}x\leq
C;$ (4.5) $\displaystyle\sup_{t\in(0,T)}\int_{\mathbb{R}^{d}}u_{i}\left|\log
u_{i}\right|\mathop{}\\!\mathrm{d}x+\frac{1}{2}\left\|f_{i}\right\|_{L^{2}(Q_{T})}^{2}\leq
C;$ (4.6) $\displaystyle\left\|\nabla
u_{i}\right\|_{L^{2}(0,T;L^{1}(\mathbb{R}^{d}))}\leq C;$ (4.7)
$\displaystyle\left\|\sqrt{u_{i}}\nabla
K_{ij}*u_{j}\right\|_{L^{2}(Q_{T})}\leq C;$ (4.8)
$\displaystyle\left\|\nabla\sqrt{u_{i}}\right\|_{L^{2}(Q_{T})}\leq C,$ (4.9)

where $f_{i}=\sqrt{u_{i}}(D_{i}\log u_{i}+\sum_{j=1}^{n}K_{ij}*u_{j})$ for
each $i=1,\ldots,n$ and
$C=C(D_{i},T,\left\|K_{ij}\right\|_{L^{\infty}(\mathbb{R}^{d})},$
$\pi_{i},d,m_{i},I[u_{i0}],H[u_{i0}])\text{ for all }i,j=1,\ldots,n)$.

###### Proof of Lemma 4.5.

We first consider the quantity ${\bf H}_{S}[\mathbf{u}(t)]$. Taking the
derivative of ${\bf H}_{S}[\mathbf{u}(t)]$ with respect to time, using the
conservation of mass, and integrating by parts yields

$\displaystyle\frac{{\rm d}}{{\rm d}t}{\bf H}_{S}[\mathbf{u}(t)]$
$\displaystyle=-\sum_{i=1}^{n}\int_{\mathbb{R}^{d}}\frac{\nabla
u_{i}}{u_{i}}\left(D_{i}\nabla
u_{i}+u_{i}\sum_{j=1}^{n}\nabla(K_{ij}*u_{j})\right)\mathop{}\\!\mathrm{d}x$
$\displaystyle=-4\sum_{i=1}^{n}D_{i}\left\|\nabla\sqrt{u_{i}}(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}-\sum_{i,j=1}^{n}\int_{\mathbb{R}^{d}}\nabla
u_{i}\nabla(K_{ij}*u_{j})\mathop{}\\!\mathrm{d}x.$ (4.10)

By using the same estimates of the scalar case, (3.2) and (3.16), we obtain

$\displaystyle\left|\int_{\mathbb{R}^{d}}\nabla u_{i}\nabla(K_{ij}*u_{j}){\rm
d}x\right|$ $\displaystyle\leq
4\,\left\|K_{ij}\right\|_{L^{\infty}(\mathbb{R}^{d})}\,\sqrt{m_{i}}\,\sqrt{m_{j}}\,\left\|\nabla\sqrt{u_{i}}(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}\,\left\|\nabla\sqrt{u_{j}}(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle\leq
2\left\|K_{ij}\right\|_{L^{\infty}(\mathbb{R}^{d})}\left(m_{i}\left\|\nabla\sqrt{u_{i}}(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}+m_{j}\left\|\nabla\sqrt{u_{j}}(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}\right)$
(4.11)

Summing over $i,j=1,\ldots,n$, and carefully reindexing sums, (4.1) becomes

$\displaystyle\frac{{\rm d}}{{\rm d}t}{\bf H}_{S}[\mathbf{u}(t)]$
$\displaystyle\leq-4\sum_{i=1}^{n}\left[D_{i}-\frac{m_{i}}{2}\sum_{j=1}^{n}\left(\left\|K_{ij}\right\|_{L^{\infty}(\mathbb{R}^{d})}+\left\|K_{ji}\right\|_{L^{\infty}(\mathbb{R}^{d})}\right)\right]\left\|\nabla\sqrt{u_{i}}(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle=-4\sum_{i=1}^{n}c_{i,n}\left\|\nabla\sqrt{u_{i}}(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}.$
(4.12)

Integrating both sides from $0$ to $t$ yields

$\displaystyle{\bf
H}_{S}[\mathbf{u}(t)]+4\sum_{i=1}^{n}c_{i,n}\left\|\nabla\sqrt{u_{i}}\right\|_{L^{2}(Q_{t})}^{2}\leq{\bf
H}_{S}[\mathbf{u}_{0}].$ (4.13)

Similarly as in (3.2), we obtain

$\displaystyle\sum_{i,j=1}^{n}\left\|\sqrt{u_{i}}\,\nabla(K_{ij}*u_{j})(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle\leq
4\sum_{i,j=1}^{n}\left\|K_{ij}\right\|_{L^{\infty}(\mathbb{R}^{d})}^{2}m_{i}\,m_{j}\left\|\nabla\sqrt{u_{j}}(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle\leq
C_{2}\sum_{i=1}^{n}\left\|\nabla\sqrt{u_{i}}(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2},$
(4.14)

where $C_{2}=C_{2}(\left\|K_{ij}\right\|_{L^{\infty}(\mathbb{R}^{d})},m_{i})$
for all $i,j=1,\ldots,n$.

As in the proof of Lemma 3.4, we can control the second moment of all species
as

$\displaystyle\sum_{i=1}^{n}\frac{{\rm d}}{{\rm d}t}I[u_{i}(t)]\leq$
$\displaystyle\,\varepsilon^{-1}\sum_{i=1}^{n}I[u_{i}(t)]+4\varepsilon\sum_{i=1}^{n}D_{i}^{2}\left\|\nabla\sqrt{u_{i}}(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle\,+\varepsilon
n\sum_{i,j=1}^{n}\left\|\sqrt{u_{i}}\nabla(K_{ij}*u_{j})(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle\leq$
$\displaystyle\,\varepsilon^{-1}I[u_{i}(t)]+\varepsilon(4\overline{D}^{2}+C_{2}n)\sum_{i=1}^{n}\left\|\nabla\sqrt{u_{i}}(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2},$
(4.15)

where $\overline{D}=\max_{i}D_{i}$. Thus, combining estimates (4.13)-(4.1), as
done in (3.2)-(3.25) for the scalar case, we find that for $\varepsilon$
sufficiently small there holds

$\displaystyle\sum_{i=1}^{n}$
$\displaystyle\left(\int_{\mathbb{R}^{d}}u_{i}\left|\log
u_{i}\right|\,\mathop{}\\!\mathrm{d}x+2c_{i,n}\left\|\nabla\sqrt{u_{i}}\right\|_{L^{2}(Q_{t})}^{2}+I[u_{i}(t)]\right)\leq
2nC_{1}$ $\displaystyle\,+{\bf
H}_{S}[\mathbf{u}_{0}]+3\varepsilon^{-1}\sum_{i=1}^{n}\int_{0}^{t}I[u_{i}(t)]{\rm
d}s)$ (4.16)

where the control of the negative part of $u_{i}\log u_{i}$ for each $i$ is
done as in (3.2). Grönwall’s lemma yields estimate (4.2), from which estimates
(4.3)-(4.4) follow. ∎

###### Proof of Lemma 4.6.

As in the scalar case, the main idea is to write the dissipation for the
$i^{\text{th}}$ component:

$f_{i}:=\sqrt{u_{i}}\,\nabla\left(D_{i}\log
u_{i}+\sum_{j=1}^{n}K_{ij}*u_{j}\right).$

Then, $(u_{i})_{t}=\nabla\cdot(\sqrt{u_{i}}f_{i})$ for each component $u_{i}$.
Under the detailed balance condition hypothesis (H5) there holds

$\sum_{i,j=1}^{n}\int_{\mathbb{R}^{d}}\pi_{i}u_{i}K_{ij}*(u_{j})_{t}\mathop{}\\!\mathrm{d}x=\sum_{i,j=1}^{n}\int_{\mathbb{R}^{d}}\pi_{i}(u_{i})_{t}K_{ij}*u_{j}{\rm
d}x$

and so direct computation gives

$\displaystyle\frac{{\rm d}}{{\rm
d}t}{\mathbfcal{F}}[\mathbf{u}(t)]+\sum_{i=1}^{n}\int_{\mathbb{R}^{d}}\pi_{i}\left|f_{i}\right|^{2}\mathop{}\\!\mathrm{d}x=0\,.$
(4.17)

Integrating from $0$ to $t$, and using the conservation of mass for each
component then yields

$\displaystyle{\mathbfcal{F}}[\mathbf{u}(t)]+\sum_{i=1}^{n}\pi_{i}\left\|f_{i}\right\|_{L^{2}(Q_{t})}^{2}\leq\sum_{i,j=1}^{n}\pi_{i}\left\|K_{ij}\right\|_{L^{\infty}(\mathbb{R}^{d})}m_{i}m_{j}.$
(4.18)

As it has been the case, we now control the negative part of $u_{i}\log u_{i}$
in terms of the second moment. It is not difficult to see that

$\displaystyle\sum_{i=1}^{n}I[u_{i}(t)]\leq\varepsilon^{-1}\int_{0}^{t}I[u_{i}(s)]{\rm
d}s+\varepsilon\sum_{i=1}^{n}\left\|f_{i}\right\|_{L^{2}(Q_{t})}^{2}+\sum_{i=1}^{n}I[u_{i0}],$
(4.19)

where $\varepsilon$ is to be chosen sufficiently small. Using the same
procedure as in (4.1)-(4.1) yields estimates (4.5)-(4.6) for each
$i=1,\ldots,n$.

Next, we write $D_{i}\nabla
u_{i}=\sqrt{u_{i}}f_{i}-u_{i}\sum_{j=1}^{n}\nabla(K_{ij}*u_{j})$ to see that

$\displaystyle\sum_{i=1}^{n}D_{i}\left\|\nabla
u_{i}\right\|_{L^{2}(0,T;L^{1}(\mathbb{R}^{d}))}\leq\sum_{i=1}^{n}\sqrt{m_{i}}\left\|f_{i}\right\|_{L^{2}(Q_{T})}+\sum_{i,j=1}^{n}\left\|u_{i}\nabla(K_{ij}*u_{j})\right\|_{L^{2}(0,T;L^{1}(\mathbb{R}^{d}))}.$
(4.20)

Then, as in (3.2) we consider for each $i$ the sets
$\underline{\Omega}_{i}=\\{(t,x):u_{i}\leq\ell\\}$ and
$\overline{\Omega}_{i}=\\{(t,x):u_{i}>\ell\\}$ to conclude that for any
$\ell>1$ there holds

$\displaystyle\left\|u_{i}\nabla(K_{ij}*u_{j})\right\|_{L^{2}(0,T;L^{1}(\mathbb{R}^{d}))}\leq
C_{K}\ell m_{j}\sqrt{T}+\frac{C_{3}}{\log(\ell)}\left\|\nabla
u_{j}\right\|_{L^{2}(0,T;L^{1}(\mathbb{R}^{d}))},$ (4.21)

where $C_{K}:=\max_{i,j}C_{K_{ij}}$, the largest constant coming from
hypothesis (H2) across all components $K_{ij}$, and $C_{3}$ depends only on
$\max_{i,j}\left\|K_{ij}\right\|_{L^{\infty}(\mathbb{R}^{d})}$ and
$\max_{i}\left\|u_{i}\left|\log
u_{i}\right|\right\|_{L^{\infty}(0,T;L^{1}(\mathbb{R}^{d}))}$. Summing (4.21)
over $i,j=1,\ldots,n$, combining with estimate (4.20) and choosing $\ell$
sufficiently large allows one to conclude that

$\displaystyle\sum_{i=1}^{n}\left\|\nabla
u_{i}\right\|_{L^{2}(0,T;L^{1}(\mathbb{R}^{d}))}\leq
C_{4}\sum_{i=1}^{n}m_{i}\left\|f_{i}\right\|_{L^{2}(Q_{T})}+C_{K}n^{2}\ell\sqrt{T}\max_{i}m_{i}$
(4.22)

and so estimate (4.7) follows.

As in estimate (3.2) for the scalar case, estimate (4.7) yields estimate
(4.8). Estimate (4.9) then follows from the $L^{2}(Q_{T})$ estimates on
$f_{i}$ and $\sqrt{u_{i}}\nabla K_{ij}*u_{j}$ for each $i,j=1,\ldots,n$. ∎

###  4.2. Improved estimates and existence of weak solutions

As in the scalar case, we improve these estimates to $u_{i}\in
L^{\infty}(0,T;L^{p}({\mathbb{R}^{d}})$ for any $2\leq p<\infty$, and $\nabla
u_{i}$ in $L^{2}(0,T;L^{2}({\mathbb{R}^{d}}))$ for each $i=1,\ldots,n$.

######  Lemma 4.7 (Improved estimates with no further assumptions for
$n$-species case).

Assume the conditions of Lemma 4.5 (Lemma 4.6) are satisfied. Then, for any
$p\in[2,\infty)$ the following estimates hold for each $1,\ldots,n$:

$\displaystyle\left\|u_{i}\right\|_{L^{\infty}(0,T;L^{p}({\mathbb{R}^{d}}))}$
$\displaystyle\leq
e^{C\max_{i,j}\left\|K_{ij}\right\|_{L^{\infty}(\mathbb{R}^{d})}^{2}p/{D_{i}}}\left\|u_{i0}\right\|_{L^{p}({\mathbb{R}^{d}})};$
(4.23) $\displaystyle\left\|\nabla u_{i}\right\|_{L^{2}(Q_{T})}$
$\displaystyle\leq
D_{i}^{-1}\tilde{C}\left\|u_{i0}\right\|_{L^{2}({\mathbb{R}^{d}})},$ (4.24)

where $\tilde{C}=\tilde{C}(C)$ and $C$ is given in Lemma 4.5 (Lemma 4.6).

###### Proof.

The proof is identical to Lemma 3.6 applied to each component $u_{i}$. ∎

###### Proof of Theorems 4.1-4.2.

The proof follows the blueprint of the scalar case. First, one can regularize
each equation of system (1.2) for $\varepsilon>0$:

$\frac{\partial u_{i}^{\varepsilon}}{\partial t}=D_{i}\Delta
u_{i}^{\varepsilon}+\nabla\cdot(u_{i}^{\varepsilon}\nabla
K_{ij}*\omega^{\varepsilon}*u_{i}^{\varepsilon})$

where $\omega^{\varepsilon}$ is the standard mollifier. Existence of a unique
classical solution follows from, e.g., [38], or by generalizing the arguments
used in [22, Theorem 2.2] from the scalar case to the system case. As in the
scalar case, if $c_{i,n}>0$ is satisfied for each $i=1,\ldots,n$ where
$c_{i,n}$ is defined in (4.1), then it is also satisfied by
$K_{ij}*\omega^{\varepsilon}$ and we remain in the small mass regime.

By either Lemma 4.5 or Lemma 4.6, we have the following estimates for all
$i,j=1,\ldots,n$:

  1. (A)

$\\{\nabla\sqrt{u_{i}^{\varepsilon}}\\}_{\varepsilon>0}$ is bounded in
$L^{2}(Q_{T})$;

  2. (B)

$\\{\sqrt{u_{i}^{\varepsilon}}\nabla
K_{ij}*{\omega}^{\varepsilon}*u_{j}^{\varepsilon}\\}_{\varepsilon>0}$ is
bounded in $L^{2}(Q_{T})$;

  3. (C)

$\\{\sup_{t\in(0,T)}\int_{\mathbb{R}^{d}}u^{\varepsilon}_{i}\left|x\right|^{2}\mathop{}\\!\mathrm{d}x\\}_{\varepsilon>0}$
is bounded.

Notice that the estimates we have are exactly the same as in the scalar case.
Following identical steps, one can finalize the proof of Theorems 4.1-4.2. ∎

###  4.3. Higher order estimates and uniqueness

We now obtain improved estimates as in Lemmas 3.7-3.8 for the $n$-species case
under the additional hypotheses (H4).

######  Lemma 4.8 (Higher order estimates for $n$-species).

Assume that each component of the initial data satisfies $\nabla u_{i}\in
L^{2}$ and $K*u_{i0}\in W^{2,p}(\mathbb{R}^{d})$ for some $p>2$. Suppose
hypotheses (H4) hold in addition to the conditions of Lemma 4.5 (Lemma 4.6).
Then, there holds

$\displaystyle\left\|\nabla
u_{i}\right\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}))}$
$\displaystyle\leq\hat{C};$ (4.25) $\displaystyle\left\|\Delta
u_{i}\right\|_{L^{2}(Q_{T})}$ $\displaystyle\leq\hat{C};$ (4.26)
$\displaystyle\left\|u_{i}\right\|_{L^{\infty}(Q_{T})}$
$\displaystyle\leq\hat{C},$ (4.27)

for each $i=1,\ldots,n$, where
$\hat{C}=\hat{C}(D_{i},T,C,\tilde{C},\max_{i,j}\left\|\nabla(\tilde{K}_{ji}*K_{ji})\right\|_{L^{2}(\mathbb{R}^{d})})$
and $C$, $\tilde{C}$ are given in Lemmas 4.5 and 4.7 (Lemmas 4.6 and 4.7).

###### Proof.

Again, the proof is very similar to the proof of Lemma 3.7 so we provide only
key details. First, testing the equation for $u_{i}$ against
$\tilde{K}_{ji}*K_{ji}*\Delta u_{i}$ we find

$\displaystyle\frac{1}{2}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}\left|\nabla(K_{ji}*u_{i})\right|^{2}\mathop{}\\!\mathrm{d}x\,+$
$\displaystyle\,D_{i}\int_{\mathbb{R}^{d}}\left|\Delta(K_{ji}*u_{i})\right|^{2}\mathop{}\\!\mathrm{d}x=$
$\displaystyle\sum_{l=1}^{n}\int_{\mathbb{R}^{d}}\Delta(K_{ji}*u_{i})K_{ji}*(\nabla
u_{i}\cdot\nabla(K_{il}*u_{l}))\mathop{}\\!\mathrm{d}x$ $\displaystyle+$
$\displaystyle\sum_{l=1}^{n}\int_{\mathbb{R}^{d}}u_{i}\,\Delta(K_{il}*u_{l})\tilde{K}_{ji}*\Delta(K_{ji}*u_{i})\mathop{}\\!\mathrm{d}x=:I_{1}^{i}+I_{2}^{i}.$

Cauchy’s inequality and Young’s convolution inequality yields for each $i$

$\displaystyle\left|I_{1}^{i}\right|\leq\frac{D_{i}}{4}\left\|\Delta(K_{ji}*u_{i})\right\|^{2}_{L^{2}(\mathbb{R}^{d})}+\underline{D}^{-1}na^{2}(t)\sum_{l=1}^{n}\left\|\nabla(K_{il}*u_{l})(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2},$

where $\underline{D}=\min_{i}D_{i}$,
$a(t)=\max_{i,j}\left\|K_{ij}\right\|_{L^{2}(\mathbb{R}^{d})}\left\|\nabla
u_{i}(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}$ and $a^{2}(t)\in L^{1}(0,T)$
by Lemma 4.7. Similarly, Hölder’s inequality and Cauchy’s inequality yields
for each $i$

$\displaystyle\left|I_{2}^{i}\right|\leq\sum_{l=1}^{n}\frac{D_{l}}{4n}\left\|\Delta(K_{il}*u_{l})\right\|_{L^{2}(\mathbb{R}^{d})}^{2}+\underline{D}^{-1}b^{2}(t),$

where
$b(t)=n\max_{i,j}\left\|\nabla(\tilde{K}_{ji}*K_{ji})\right\|_{L^{2}(\mathbb{R}^{d})}\left\|u_{i}\right\|_{L^{\infty}(0,T;L^{2}(\mathbb{R}^{d})}\left\|\nabla
u_{i}(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}$, and in particular,
$b^{2}(t)\in L^{1}(0,T)$. Hence, we have for each $i$

$\displaystyle\frac{1}{2}\frac{{\rm d}}{{\rm d}t}\int_{\mathbb{R}^{d}}$
$\displaystyle\left|\nabla(K_{ji}*u_{i})\right|^{2}\mathop{}\\!\mathrm{d}x+\frac{3D_{i}}{4}\int_{\mathbb{R}^{d}}\left|\Delta(K_{ji}*u_{i})\right|^{2}\mathop{}\\!\mathrm{d}x\leq$
$\displaystyle\underline{D}^{-1}na^{2}(t)\sum_{l=1}^{n}\left\|\nabla(K_{il}*u_{l})(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}+\sum_{l=1}^{n}\frac{D_{l}}{4n}\left\|\Delta(K_{il}*u_{l})\right\|_{L^{2}(\mathbb{R}^{d})}^{2}+\underline{D}^{-1}b^{2}(t),$

and so if we sum both sides over $i,j=1,\ldots,n$ and reindex the sums on the
right hand side we obtain

$\displaystyle\sum_{i,j=1}^{n}\left(\frac{1}{2}\frac{{\rm d}}{{\rm
d}t}\right.$
$\displaystyle\left.\int_{\mathbb{R}^{d}}\left|\nabla(K_{ji}*u_{i})\right|^{2}\mathop{}\\!\mathrm{d}x+\frac{D_{i}}{2}\int_{\mathbb{R}^{d}}\left|\Delta(K_{ji}*u_{i})\right|^{2}\mathop{}\\!\mathrm{d}x\right)$
$\displaystyle\leq\underline{D}^{-1}n^{2}\left(a^{2}(t)\sum_{i,j=1}^{n}\left\|\nabla(K_{ji}*u_{i})(t,\cdot)\right\|_{L^{2}(\mathbb{R}^{d})}^{2}+b^{2}(t)\right)$
(4.28)

Grönwall’s lemma gives that $K_{ij}*u_{j}\in
L^{\infty}(0,T;L^{2}(\mathbb{R}^{d}))$ and $K_{ij}*\Delta u_{j}\in
L^{2}(Q_{T})$ for all $i,j=1,\ldots,n$.

By an identical argument to that made in the proof of Lemma 3.7 using maximal
regularity of the heat equation, we conclude that estimates (4.25)-(4.26)
hold, completing the first part of the proof.

The uniform estimates on each component $u_{i}$ follows in an identical
fashion to the scalar case, using the previously obtained estimates and the
quantity $\max_{i,j}\left\|K_{ij}*\nabla u_{j}\right\|_{L^{\infty}(Q_{T})}$
replacing the constant $C_{0}=\left\|K*\nabla u\right\|_{L^{\infty}(Q_{T})}$
in the proof of Lemma 3.7.

∎

As in the scalar case, estimates obtained in Lemma 4.8 give rise to improved
integrability of the gradient of each component of the solution. The proof is
identical to the proof of Lemma 4.8, working with the sum across all
components, and so we omit the details.

######  Lemma 4.9 (Improved integrability of gradient).

Suppose each component of the initial data satisfies $\nabla u_{i0}\in
L^{p}(\mathbb{R}^{d})$. Under the conditions of Lemma 4.8, for each $p\geq 2$
there exists a constant $\tilde{C}_{0}$ such that there holds

$\displaystyle\left\|\nabla
u_{i}\right\|_{L^{\infty}(0,T;L^{p}(\mathbb{R}^{d}))}\leq\hat{C}_{0},$ (4.29)

for each $i=1,\ldots,d$, where
$\hat{C}_{0}=\hat{C}_{0}(\hat{C},p,\left\|\nabla
u_{i0}\right\|_{L^{p}(\mathbb{R}^{d})})$, and $\hat{C}$ is given in Lemma 4.8.

######  Theorem 4.10.

Suppose (H4) holds. Then any nonnegative solution
$\mathbf{u}=(u_{1},\ldots,u_{n})$ solving problem (1.2) belonging to the class

$\displaystyle u_{i}$ $\displaystyle\in L^{\infty}(Q_{T})\cap
L^{\infty}(0,T;L^{1}({\mathbb{R}^{d}}));$ $\displaystyle\nabla u_{i}$
$\displaystyle\in L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}));$
$\displaystyle\Delta u_{i}$ $\displaystyle\in L^{2}(Q_{T}),$

for each $i=1,\ldots,n$, is unique.

###### Proof.

Set $\underline{D}:=\min_{i}D_{i}>0$. We begin as in the proof of Theorem 3.9:
assume that there are two solutions, $\mathbf{u}$ and $\mathbf{v}$. Denote by
$w_{i}:=u_{i}-v_{i}$. Then, for each $i=1,\ldots,n$, $w_{i}$ solves

$\displaystyle(w_{i})_{t}-D_{i}\Delta
w_{i}=\nabla\cdot\left(w_{i}\sum_{j=1}^{n}\nabla(K_{ij}*u_{j})\right)+\nabla\cdot\left(v_{i}\sum_{j=1}^{n}\nabla(K_{ij}*w_{j})\right).$
(4.30)

Testing against $w_{i}$ for each $i=1,\ldots,n$, summing over $i=1,\ldots,n$
and following an identical procedure to that of the proof of Theorem 3.9
yields

$\displaystyle\frac{1}{2}\sum_{i=1}^{n}\left(\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}w_{i}^{2}\mathop{}\\!\mathrm{d}x+D_{i}\int_{\mathbb{R}^{d}}\left|\nabla
w_{i}\right|^{2}\mathop{}\\!\mathrm{d}x\right)\leq$
$\displaystyle\,C_{1}(t)\sum_{i=1}^{n}\int_{\mathbb{R}^{d}}w_{i}^{2}\mathop{}\\!\mathrm{d}x$
$\displaystyle+C_{2}\sum_{i,j=1}^{n}\int_{\mathbb{R}^{d}}\left|\nabla(K_{ij}*w_{j})\right|^{2}\mathop{}\\!\mathrm{d}x,$
(4.31)

where $C_{1}(t)=\max_{i,j}\left\|K_{ij}*\Delta
u_{j}(t,\cdot)\right\|_{L^{\infty}({\mathbb{R}^{d}})}\in L^{1}(0,T)$ and
$C_{2}=\max_{i}\left\|v_{i}\right\|_{L^{\infty}(Q_{T})}/\underline{D}$.

Now we seek to estimate the second term on the right hand side of (4.3). To
this end, we consider equation (4.30) by first reindexing $j\mapsto l$
followed by $i\mapsto j$. We then test with $\tilde{K}_{ij}*K_{ij}*w_{j}$ and
sum the result over $j=1,\ldots,n$ to obtain

$\displaystyle\sum_{j=1}^{n}\left(\frac{1}{2}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}\left|K_{ij}*w_{j}\right|^{2}\mathop{}\\!\mathrm{d}x\right.$
$\displaystyle\left.+\,D_{j}\int_{\mathbb{R}^{d}}\left|\nabla(K_{ij}*w_{j})\right|^{2}\mathop{}\\!\mathrm{d}x\right)$
$\displaystyle=$
$\displaystyle\,-\sum_{j,l=1}^{n}\int_{\mathbb{R}^{d}}\nabla(\tilde{K}_{ij}*K_{ij}*w_{j})\cdot
w_{j}\nabla(K_{jl}*u_{l}))\mathop{}\\!\mathrm{d}x$
$\displaystyle\,-\sum_{j,l=1}^{n}\int_{\mathbb{R}^{d}}\nabla(\tilde{K}_{ij}*K_{ij}*w_{j})\cdot
v_{j}\nabla(K_{jl}*w_{l})\mathop{}\\!\mathrm{d}x.$ (4.32)

We may easily estimate the first term on the right hand side of (4.3) as done
in the proof of Theorem 3.9 to obtain

$\displaystyle\left|-\sum_{j,l=1}^{n}\int_{\mathbb{R}^{d}}\nabla(\tilde{K}_{ij}*K_{ij}*w_{j})\cdot
w_{j}\nabla(K_{jl}*u_{l})){\rm d}x\right|\leq
C_{3}n\sum_{j=1}^{n}\left\|w_{j}(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2},$
(4.33)

where $C_{3}$ depends on
$\left\|\nabla(\tilde{K}_{ij}*K_{ij})\right\|_{L^{2}(\mathbb{R}^{d})}$ and
$\left\|\nabla(K_{jl}*u_{l}))\right\|_{L^{\infty}(0,T;L^{2}(\mathbb{R}^{d}))}$,
finite by hypothesis (H4) and by the assumed regularity of each component
$u_{i}$ respectively, for each $i,j,l=1,\ldots,n$.

The second term on the right hand side of (4.3) requires a bit more care.
First, for fixed $i$, $j$, $l$, we can estimate as in the proof of Theorem 3.9
to deduce

$\displaystyle\Big{|}\int_{\mathbb{R}^{d}}\nabla(\tilde{K}_{ij}*K_{ij}*w_{j})\cdot
v_{j}$ $\displaystyle\nabla(K_{jl}*w_{l}){\rm d}x\Big{|}$
$\displaystyle\leq\frac{C_{5}}{2}\left(\varepsilon_{l}\left\|\nabla
K_{jl}*w_{l}(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}+\varepsilon_{l}^{-1}\left\|w_{j}(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}\right),$
(4.34)

where $C_{5}$ depends on
$\left\|\nabla(\tilde{K}_{ij}*K_{ij})\right\|_{L^{2}({\mathbb{R}^{d}})}$,
$\left\|v_{j}\right\|_{L^{\infty}(Q_{T})}$, and $\varepsilon_{l}>0$ is to be
determined. Notice carefully that the right hand side does not depend on $i$
as this dependence is absorbed into the condition
$\nabla(\tilde{K}_{ij}*K_{ij})\in L^{2}({\mathbb{R}^{d}})$. Summing (4.3) over
$j,l=1,\ldots,n$ then gives

$\displaystyle\Big{|}\sum_{j,l=1}^{n}\int_{\mathbb{R}^{d}}\nabla(\tilde{K}_{ij}\,*$
$\displaystyle\,K_{ij}*w_{j})\cdot v_{j}\nabla(K_{jl}*w_{l}){\rm d}x\Big{|}$
$\displaystyle\leq\frac{C_{5}}{2}\sum_{j,l=1}^{n}\left(\varepsilon_{l}\left\|\nabla(K_{jl}*w_{l})(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}+\varepsilon_{l}^{-1}\left\|w_{j}(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}\right).$
(4.35)

Combining estimates (4.33) and (4.3) we arrive at

$\displaystyle\sum_{j=1}^{n}\left(\frac{1}{2}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}\right.$
$\displaystyle\left.\left|K_{ij}*w_{j}\right|^{2}\mathop{}\\!\mathrm{d}x+D_{j}\int_{\mathbb{R}^{d}}\left|\nabla(K_{ij}*w_{j})\right|^{2}\mathop{}\\!\mathrm{d}x\right)\leq
C_{3}n\sum_{j=1}^{n}\left\|w_{j}(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}$
$\displaystyle+\frac{C_{5}}{2}\sum_{j,l=1}^{n}\left(\varepsilon_{l}\left\|\nabla(K_{jl}*w_{l})(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}+\varepsilon_{l}^{-1}\left\|w_{j}(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2}\right).$
(4.36)

Notice again that the right hand side does not depend on $i$. Hence, we choose
$\varepsilon_{l}=\frac{D_{l}}{C_{5}n}$ for each $l=1,\ldots,n$, sum both sides
from $i=1,\ldots,n$, and then reindex the right hand side as $j\mapsto i$
followed by $l\mapsto j$ to conclude that

$\displaystyle\frac{1}{2}\sum_{i,j=1}^{n}\left(\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}\left|K_{ij}*w_{j}\right|^{2}\mathop{}\\!\mathrm{d}x+D_{j}\int_{\mathbb{R}^{d}}\left|\nabla(K_{ij}*w_{j})\right|^{2}\mathop{}\\!\mathrm{d}x\right)\leq
C_{6}(t)\sum_{i=1}^{n}\left\|w_{j}(t,\cdot)\right\|_{L^{2}({\mathbb{R}^{d}})}^{2},$
(4.37)

for some $C_{6}(t)\in L^{1}(0,T)$.

To conclude, we multiply both sides of estimate (4.3) by
$\frac{\underline{D}}{4C_{2}}$ and combine the result with estimate (4.37) to
obtain

$\displaystyle\frac{\underline{D}}{8C_{2}}\sum_{i=1}^{n}\frac{{\rm d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}w_{i}^{2}\mathop{}\\!\mathrm{d}x+\frac{1}{2}\sum_{i,j=1}^{n}\frac{{\rm
d}}{{\rm
d}t}\int_{\mathbb{R}^{d}}\left|K_{ij}*w_{j}\right|^{2}\mathop{}\\!\mathrm{d}x\leq
C_{7}(t)\sum_{i=1}^{n}\int_{\mathbb{R}^{d}}w_{i}^{2}\mathop{}\\!\mathrm{d}x,$
(4.38)

where $C_{7}(t)\in L^{1}(0,T)$. The conclusion of the theorem follows from
Grönwall’s lemma. ∎

###  4.4. Existence of unique, global classical solutions

We conclude with the proof of Theorems 4.3-4.4.

###### Proof of Theorem 4.3.

By Lemma 4.8, the global weak solution $\mathbf{u}$ obtained in either Theorem
4.2, or Theorem 4.1 in addition to hypothesis (H4), satisfies for each
component $u_{i}$:

$\displaystyle u_{i}$ $\displaystyle\in L^{\infty}(Q_{T})\cap
L^{\infty}(0,T;L^{1}({\mathbb{R}^{d}}));$ $\displaystyle\nabla u_{i}$
$\displaystyle\in L^{\infty}(0,T;L^{2}({\mathbb{R}^{d}}));$
$\displaystyle\Delta u_{i}$ $\displaystyle\in L^{2}(Q_{T}).$

Arguing as in the scalar case, each component $u_{i}$ satisfies
$\left\|u_{i}(t,\cdot)-u_{i0}\right\|_{L^{2}(\mathbb{R}^{d})}\to 0$ as $t\to
0^{+}$, and so $\mathbf{u}=(u_{1},\ldots,u_{n})$ is a strong solution solving
problem 1.2 in the sense of Definition 3.2. ∎

We conclude with the proof of Theorem 4.4. The ingredients are essentially
identical to the proof of Theorem 2.4 and so we omit most details.

###### Proof of Theorem 4.4.

The argument is identical to that of Theorem 2.4. First, the improved
integrability of the gradient of each component of the solution $u_{i}$ allows
one to conclude by $L^{p}$-theory of parabolic equations and the Sobolev
embedding that $u_{i}\in C^{\sigma,1+\sigma}_{loc}(Q_{T})$ for any
$\sigma\in(0,1)$. Working with each component $w_{i,j}:=\frac{\partial
u_{i}}{\partial x_{j}}$, we find that $w_{i,j}\in
C^{\sigma,1+\sigma}_{loc}(Q_{T})$ for each $i,j=1,\ldots,d$. Consequently,
each component $u_{i}\in C^{\sigma,2+\sigma}_{loc}(Q_{T})$ and Schauder
estimates imply that $u_{i}\in C^{1+\sigma/2,2+\sigma}_{loc}(Q_{T})$ for each
$i=1,\ldots,n$, for any $\sigma\in(0,1)$. This additional regularity paired
with the strong maximum principle yields the strict positivity of each
component $u_{i}$. ∎

##  5\. Applications & Simulations

In this section we present some typical examples in one spatial dimension and
discuss how they relate to the theory developed in the previous sections.
Using a finite-volume method proposed in [20, 4], we solve the problem in a
fixed one-dimensional domain with no-flux boundary conditions for $1$\- and
$2$-species cases as an approximation to the problem on the whole space. The
scheme is second order accurate in space, and we use 100 cells per spatial
unit, i.e., given a domain of length $2L$, $200^{*}L$ cells are used to solve
the problem over $(-L,L)$. Time stepping is performed using a third order
strong stability preserving Runge-Kutta method.

###  5.1. The Scalar Equation

Several models studied in mathematical ecology feature a spatial convolution
with a tophat kernel directing movement towards or away from some stimulus
(e.g., a resource gradient [34], scent marks on a landscape [59], remembered
locations of previous interactions [57, 58], or detection of the population
density itself [59, 61]). We consider the following prototypical scenario with
tophat kernel:

$\displaystyle\begin{cases}\frac{\partial u}{\partial
t}=\frac{\partial}{\partial x}\left(D\frac{\partial u}{\partial
x}+u\frac{\partial K_{\text{tophat}}*u}{\partial x}\right)\quad\text{ in
}(0,T)\times\mathbb{R},\\\
u(0,x)=\chi_{(-\ell,\ell)}(x),\quad\quad\quad\quad\quad\quad\quad\text{ in
}\mathbb{R},\end{cases}$ (5.1)

where $D>0$ and $\chi_{(-\ell,\ell)}(x)$ is the characteristic function of the
symmetric interval of radius $\ell>0$, normalized such that
$\left\|\chi_{(-\ell,\ell)}\right\|_{L^{1}(\mathbb{R})}=1$. Here,
$K_{\text{tophat}}$ denotes the top-hat kernel with a strength parameter
$\alpha$ and perceptual radius $R$ introduced in (1.4). Members of population
$u$ are able to detect a nonlocal average density $R$ units about the location
of detection, biasing the movement towards/away from areas of high/low density
regions depending on the sign and magnitude of the parameter $\alpha$. The
convention in (1.4) is such that it is an attractive kernel, hence $\alpha>0$
corresponds to attraction to areas of high density whereas $\alpha<0$
corresponds to repulsion from high density areas.

As previously noted, the top-hat kernel satisfies hypotheses (H1)-(H3) and
(H6). In this example it is informative to note that in one dimension the
problem becomes local in the sense that the nonlocal interaction term can be
written as the difference of Diracs:

$\frac{\partial K_{\text{tophat}}*u}{\partial
x}(t,x)=-\alpha\left(\frac{u(t,x+R)-u(t,x-R)}{2R}\right).$

Therefore, for any given smooth $\phi$ there holds

$\left\|\frac{\partial}{\partial
x}(K_{\text{tophat}}*\phi)\right\|_{L^{1}(\mathbb{R})}\leq\frac{\left|\alpha\right|}{R}\left\|\phi\right\|_{L^{1}(\mathbb{R})},$

and so (H2) is satisfied with $C_{K}=\left|\alpha\right|/R$. By the comments
made in the introduction, this is sufficient to conclude that hypothesis (H4)
is also satisfied. Hypothesis (H3) and (H6) are obviously satisfied.

By Theorem 2.4, there exists a unique, global classical solution solving
problem (5.1). Strictly speaking, the initial data does not satisfy the
hypotheses of the Theorem; in this sense, the solution is only classical on
the open time interval $(0,T)$ and the initial data is satisfied in the sense
of a weak solution.

The numerical approximation of (5.1) is done on a finite spatial interval of
length $2L>0$ with no-flux boundary conditions. The size of the simulated
domain $L$ is chosen large enough to approximate the problem on the whole
$\mathbb{R}$ if the long time behavior leads to a fast-decaying at infinity
steady state. If the diffusion is dominating then the no-flux boundary
condition will change the long time asymptotic behavior of the equation. Our
numerical experiments indicate that if there are steady solutions of (5.1),
they are supported on the whole domain as one expects from the effect of the
linear diffusion term.

In Figures 1-4, we present simulations for four exemplary cases of the
dynamical behaviour. In all cases, we fix $D=0.25$, $R=1.0$, $\ell=4.0$, and
$L$ is indicated in each of the figure captions. In each figure, we change
only the strength parameter $\alpha$. In the upper panel (A), we display a
contour plot of the solution profile and plot the free energy with respect to
time; in the lower panel (B), we display cross-sections of the solution
profile at select times. The solution profile is the solid black line, while
the quantity $\left|\xi\right|$ is the dashed black line, where $\xi$ is given
by

$\xi(t,x):=D\log(u)+K_{\text{tophat}}*u,$

which should be constant on the support of the solution $u$ when a (numerical)
steady-state is achieved. Studying this quantity provides evidence that we are
indeed at a (numerical) steady-state. Of course, by Theorem 2.4 we know that
the solution is strictly positive everywhere; therefore, the grey columns
indicate regions for which the solution $u$ is smaller than $10^{-4}$.

In Figure 1, we have a small amount of self-attraction ($\alpha=2$). In this
case, the diffusive forces dominate and the solution settles down to the
constant state (i.e., $u\equiv 1/2L$). The free energy decreases very slowly.
We observe similar behaviour for small amounts of self-repulsion. Notice that
the expected behavior on the whole line is different for large times, as the
solution will continue spreading as for the heat equation. This has been
checked by taking larger and larger values of $L$.

In Figure 2, we have a large amount of self-attraction ($\alpha=30$). In
contrast to Figure 1, the attractive forces dominate and the solution
concentrates a majority of its mass within a single region about $x=0$. Within
the grey column, we observe at time $t=100$ that $\xi$ is constant. For longer
times, $\xi$ eventually approaches a constant over the entire domain (see
Figure 3).

In contrast to the single peak observed for strong attraction, in Figure 3 we
observe two distinct peaks when the amount of self-attraction is moderate
($\alpha=20$). In very short times, we see that $\xi$ is constant over the
regions of concentration; running until time $t=200$, we observe that $\xi$ is
constant over the entire domain. The solution, during the time spanned from
$t=2.7$ to $t=200$, is slowly concentrating around the two peaks dissolving
the mass that is still located at $x=0$ and $t=2.7$. We have checked this
behavior by plotting the densities in log-scale.

Finally, in Figure 4 we have a large amount of self-repulsion ($\alpha=-20$).
In this case, the repulsive forces produce a patterned state, at least for
small times; however, for larger times we begin to observe a very slow decay
to a constant steady state. Similar to Figure 1, we observe the (numerical)
solution to approach $1/2L$. In such cases, we do not believe a no-flux
boundary condition will appropriately approximate the solution on the whole
space for fixed numerical domain lengths.

(a)

(b)

Figure 1. Simulation of problem (5.1) with $\alpha=2.0$ and numerical domain
size of $24$ ($L=12$). A supplementary video of this simulation is hosted on
figshare found here: https://doi.org/10.6084/m9.figshare.25942969.v1

(a)

(b)

Figure 2. Simulation of problem (5.1) with $\alpha=30$ and numerical domain
size of $12$ ($L=6$). A supplementary video of this simulation is hosted on
figshare found here: https://doi.org/10.6084/m9.figshare.25934431.v1

(a)

(b)

Figure 3. Simulation of problem (5.1) with $\alpha=20$ and numerical domain
size of $24$ ($L=12$). A supplementary video of this simulation is hosted on
figshare found here: https://doi.org/10.6084/m9.figshare.25934425.v1

(a)

(b)

Figure 4. Simulation of problem (5.1) with $\alpha=-20$ and numerical domain
size $32$ ($L=16$). A supplementary video of this simulation is hosted on
figshare found here: https://doi.org/10.6084/m9.figshare.25934428.v1

###  5.2. The $n$-Species System

Different from the scalar equation case, the $n$-species system leaves more
room to explore the differences between Theorems 4.1 and 4.2 due to the
increased complexity of the detailed balance hypothesis (H5) for several
interacting populations. For simplicity, we first consider the following
general $2$-species aggregation-diffusion system

$\displaystyle\begin{cases}\frac{\partial u}{\partial
t}=\frac{\partial}{\partial x}\left(D\frac{\partial u}{\partial
x}+u\frac{\partial}{\partial x}(K_{11}*u+K_{12}*v)\right),\quad\text{ in
}(0,T)\times\mathbb{R},\\\ \frac{\partial v}{\partial
t}=\frac{\partial}{\partial x}\left(D\frac{\partial v}{\partial
x}+v\frac{\partial}{\partial x}(K_{21}*u+K_{22}*v)\right),\quad\text{ in
}(0,T)\times\mathbb{R},\\\
u(0,x)=v(0,x)=\chi_{(-\ell,\ell)}(x),\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text{
in }\mathbb{R}.\end{cases}$ (5.2)

In these simulations we again fix $D=0.25$, $R=1.0$, and $\ell=4$ with the
initial data is normalized so that $m_{i}=1$ for $i=1,2$. We again use no-flux
boundary conditions on a domain of length $2L>0$ specified in each of the
figure captions.

Under hypothesis (H1), Theorem 4.1 guarantees that there exists a global weak
solution $(u,v)$ solving problem (5.2) so long as there holds

$\frac{1}{2}\left(\left\|K_{12}\right\|_{L^{\infty}(\mathbb{R}^{d})}+\left\|K_{21}\right\|_{L^{\infty}(\mathbb{R}^{d})}\right)<\min_{i=1,2}\left\\{D-\left\|K_{ii}\right\|_{L^{\infty}(\mathbb{R}^{d})}\right\\}.$

Notice that if the small mass condition (2.4) from the scalar case is violated
for either population $u$ or $v$, we cannot guarantee the existence of a
solution from Theorem 4.1; when the small mass condition is satisfied, a
global weak solution exists for the $2$-species system so long as the cross-
interaction between the two populations (described by $K_{12}$ and $K_{21}$)
is not too strong. It is interesting to note that this requires no relation
between the kernels of different populations: the kernel $K_{12}$ may be a
completely different form than $K_{21}$, such as featuring different detection
radii $R$ between populations. We do not explore such possibilities here,
however. We instead consider as a concrete example the typical case where

$\displaystyle K_{ij}(\cdot):=\alpha_{ij}K_{\text{tophat}}(\cdot),$ (5.3)

for some coefficients $\alpha_{ij}\in\mathbb{R}$, $i,j=1,2$. In this way, both
populations have identical detection abilities, differing only in their
relative rates of attraction/repulsion. With our chosen parameters, a
necessary condition for the existence of a weak solution via Theorem 4.1 is

$\left|\alpha_{ii}\right|<2DR=\frac{1}{2},\quad i=1,2.$

Then, if the cross interaction is such that $\left|\alpha_{ij}\right|\ll 1$,
$i\neq j$, Theorem 4.1 ensures the existence of a global weak solution; since
(H4) is also satisfied, Theorem 4.3 ensures that this solution is in fact the
unique, global classical solution for a rather large class of interaction
kernels with no symmetry requirements of any kind. In such cases, numerical
simulation reveals behaviour similar to that of a single equation: the cross-
interaction cannot be too strong, and so the dynamics are primarily governed
by diffusion and self-interaction (i.e., $\alpha_{ii}$).

On the other hand, while Theorem 4.2 allows for strong cross-interactions, the
addition of hypothesis (H5) requires some additional structure on the
coefficients $\alpha_{ij}$. We explore this briefly below.

For $n=2$ or $n=3$ interacting populations, it is not difficult to verify that
the condition

$\pi_{i}\alpha_{ij}=\pi_{j}\alpha_{ji},\quad\quad\pi_{i}>0,\,i=1,\ldots,n,$

results in a consistent system so long as the cross-interaction terms
$\alpha_{ij}$, $i\neq j$, are of the same sign (in the case of $2$ interacting
populations, it is an over-determined system and one simply takes
$\pi_{1}=1$). Therefore, Theorem 4.2 ensures the existence of a global weak
solution to problem (5.2) (or to its $n=3$ species counterpart) for any
initial mass $m_{i}>0$, any $D_{i}>0$ fixed, and any kernels of the form given
in (5.3) so long as there holds

$\operatorname{sign}\\{\alpha_{ij}\\}=\operatorname{sign}\\{\alpha_{ji}\\},$

for each $i,j=1,\ldots,n$. Moreover, hypothesis (H4) is again satisfied by the
top-hat detection function and so the solution obtained is in fact the unique,
classical solution.

In Figure 5 we present the solution profiles for the two-species system with
repulsive symmetric cross-interaction, i.e. $\alpha_{12}=\alpha_{21}<0$. We
choose $\alpha_{11}=20$ so that the expected solution profile for $u$ is that
of Figure 3 in the absence of $v$; we then choose $\alpha_{22}=2$ (an order of
magnitude less than $\alpha_{11}$), so that the expected solution profile of
$v$ is that of Figure 1 in the absence of $u$. It is interesting to note that
the solution behaviour is roughly as expected for each single equation with
the no-flux boundary conditions, where the steady-state for $u$ concentrates
into two peaks as indicated by the flatness of $\xi_{u}$ on the essential
support of $u$ and the solution profile for $v$ stabilizing towards a steady
state as much as $u$ by noticing the constant behavior of $\xi_{v}$ on its
essential support. We observe the segregation of the mass of both species as
well as the decay of the free energy functional in time. We remark that we
expect the long time behavior of the solutions in the whole space completely
different from the no-flux boundary conditions. Clarifying this point would
need further numerical exploration.

In Figure 6, we run a simulation for a scenario that is not covered by the
theory presented here. We fix $\alpha_{ii}=20$, $i=1,2$, so that a patterned
state of the form in Figure 3 for each population in the absence of the other
is expected. We then choose $\alpha_{12}=-10<\alpha_{21}=5$ so that the
detailed balance condition of Hypothesis (H5) is violated. Over short times
scales some complex rearrangement of the two population densities occurs, but
eventually the solution profile settles down to a form found in Figures 2-3.
The flatness of $\xi_{u}$, $\xi_{v}$ in the grey regions suggests this is
indeed the steady-state, and this continues to flatten over a longer
simulation time. Of course, this case no longer features a gradient flow
structure, and so we observe non-monotone behaviour of the “free energy" of