Ex I C symbol=C, format=0 EE symbol=, format=0 R symbol=R, format=0 SS
symbol=, format=0 T symbol=T, format=0 U symbol=U, format=0 UU symbol=,
format=0 W symbol=W, format=0 WW symbol=, format=0 Rho symbol=ρ, format=0

# Local well-posedness for a novel nonlocal model for cell-cell adhesion via
receptor binding

Mabel Lizzy Rajendran and Anna Zhigun School of Mathematics, Watson Building,
University of Birmingham, Edgbaston, Birmingham B15 2TT, UK,
<EMAIL_ADDRESS>of Mathematics and Physics, Queen’s University
Belfast, University Road, Belfast BT7 1NN, Northern Ireland, UK,
<EMAIL_ADDRESS>
###### Abstract

Local well-posedness is established for a highly nonlocal nonlinear diffusion-
adhesion system for bounded initial values with small support. Macroscopic
systems of this kind were previously obtained by the authors through upscaling
in [32] and can account for the effect of microscopic receptor binding
dynamics in cell-cell adhesion. The system analysed here couples an integro-
PDE featuring degenerate diffusion of the porous media type and nonlocal
adhesion with a novel nonlinear integral equation. The approach is based on
decoupling the system and using Banach’s fixed point theorem to solve each of
the two equations individually and subsequently the entire system. The main
challenge of the implementation lies in selecting a suitable framework. One of
the key results is the local well-posedness for the integral equation with a
Radon measure as a parameter. The analysis of this equation utilizes the
Kantorovich-Rubinstein norm, marking the first application of this norm in
handling a nonlinear integral equation.

Keywords: aggregation, degenerate diffusion, integro-PDE, Kantorovich-
Rubinstein norm, nonlinear integral equation, nonlocal cell-cell adhesion,
Wasserstein metric
MSC 2020: 35K65 35Q92 45G10 45K05 47G10 92C17

##  1 Introduction

In this paper we study the coupling of the Cauchy problem for the integro-PDE

$\displaystyle\partial_{t}u=\nabla\cdot\left(2u\nabla u-v\chi(u)(\nabla H\star
u)\right)$ $\displaystyle\quad\text{in }(0,T]\times\mathbb{R}^{d},$ (1.1a)
$\displaystyle u(0,\cdot)=u_{0}$ $\displaystyle\quad\text{in }\mathbb{R}^{d},$
(1.1b)

with the integral equation

$\displaystyle
v=\frac{{\mathcal{G}}^{+}(u-v)}{{\mathcal{G}}^{+}(u-v)+{\mathcal{G}}^{-}v}u\qquad\text{in
}(0,T]\times\mathbb{R}^{d},$ (1.2)

where

$\displaystyle(u,v):[0,T]\times\mathbb{R}^{d}\to\\{(z_{1},z_{2})\in\mathbb{R}^{2}:\quad
0\leq z_{2}\leq z_{1}\\}$

is a pair of unknowns, $\star$ denotes convolution in the spatial variable
$x\in\mathbb{R}^{d}$, $d\in\mathbb{N}$, $T>0$, and

$\displaystyle\chi\in C^{1}_{b}([0,\infty)),\qquad\chi(0)=0,$ (1.3a)
$\displaystyle H(x):=-\int_{0}^{|x|}\mathbf{1}_{[0,1]}F(s)\,ds,$ (1.3b)
$\displaystyle F\in C^{1}([0,\infty)),$ (1.3c)
$\displaystyle{\mathcal{G}}^{\pm}f(t,x):=\int_{\mathbb{R}^{d}}G^{\pm}(t,x,y)f(y)\,dy,$
(1.3d) $\displaystyle
G^{\pm}(t,x,y):=\varphi^{\pm}(|x-y|)K^{\pm}\left(t,\frac{x+y}{2}\right),$
(1.3e)
$\displaystyle\varphi^{\pm}(s):=\begin{cases}\left(1-d^{b^{\pm}}\right)^{\pm
a^{\pm}}&\text{for }d\in[0,1),\\\ 0&\text{for }d\in[1,\infty),\end{cases}$
(1.3f) $\displaystyle K^{\pm}\in C^{2}([0,T]\times\mathbb{R}^{d};(0,\infty)),$
(1.3g) $\displaystyle a^{\pm}>0,\ b^{\pm}\geq 2.$ (1.3h)

System 1.1-1.2 is a modification of a novel system that the authors recently
derived using a multiscale approach, see (4.35) in [32]. Equation 1.1a models
the evolution of density $u=u(t,x)$, $t$ being time and $x$ position in space,
of a cell population dispersing through diffusion and nonlocal cell-cell
adhesion. Advection direction due to adhesion is given by the nonlocal term
$\nabla H\star u$, often called ’adhesion velocity’ [3]. The additional
dependent variable $v$ describes the amount of formed bonds and modulates
’adhesion sensitivity’. More precisely,

$\displaystyle w:=\frac{v}{u}\in[0,1]$

is the average fraction of bounded receptors. Variables $u$ and $v$ are
further connected through a nonlinear nonlocal equation of a new type, 1.2
that describes the equilibration of the receptor binding dynamics. We refer to
[32] for further details regarding this effect and its modelling leading to
1.2.

Unlike the original model in [32], system 1.1-1.2 includes the porous media-
type (PM) diffusion

$\displaystyle\Delta(u^{2})$ (1.4)

rather than the linear myopic diffusion

$\displaystyle\nabla\nabla^{T}:({\mathbb{D}}u),\qquad{\mathbb{D}}={\mathbb{D}}(x)\in\mathbb{R}^{d\times
d}.$ (1.5)

This modification is mainly motivated by the analytical challenges that are
discussed in Section 2 below. Still, from the modelling perspective, such a
term is justifiable and, in fact, often arises in derivations of models for
cell motion due to a combination of short- and long-range interactions between
population members. For example, in [26] a nonlocal equation was derived that
corresponds to 1.1a for constant $\chi$ and $w$. Yet another distinction
between 1.1a and the original equation is that the adhesion sensitivity
coefficient $\chi$ is no longer an arbitrary function of $t$ and $x$. Instead,
it is assumed to be a function of $u$. This choice enables the incorporation
of the influence of local density on adhesion sensitivity.

Since $0\leq v\leq u$, it is convenient to work with the pair

$\displaystyle(u,w):[0,T]\times\mathbb{R}^{d}\to[0,\infty)\times[0,1]$

rather than $(u,v)$, thus keeping the codomains of the unknowns independent.
Restated for these variables, equations 1.1-1.2 take the form

$\displaystyle\partial_{t}u=\nabla\cdot\left(2u\nabla u-wu\chi(u)(\nabla
H\star u)\right)$ $\displaystyle\quad\text{in }(0,T]\times\mathbb{R}^{d},$
(1.6a) $\displaystyle u(0,\cdot)=u_{0}$ $\displaystyle\quad\text{in
}\mathbb{R}^{d},$ (1.6b)

and

$\displaystyle w={\mathcal{Y}}(u,w)\qquad\text{in }\\{u>0\\},$ (1.7)

where

$\displaystyle{\mathcal{Y}}(u,w):=\frac{{\mathcal{G}}^{+}((1-w)u)}{{\mathcal{G}}^{+}((1-w)u)+{\mathcal{G}}^{-}(wu)}.$
(1.8)

We mostly deal with this reformulation of the original system. In this work we
make the first step in its study. Here we prove a local well-posedness result
for the special case where $u$ remains compactly supported in a sufficiently
small ball. In particular, the support diameter needs to be smaller than $1$,
the radius of the sampling domain in the integral operators
${\mathcal{G}}^{\pm}$. The smallness of the support of the cell density can be
interpreted as an initial stage of population dispersal, e.g. when a small
tumour just starts invading the surrounding tissue.

The result is local in the following sense. The solution exists for finite
times provided $u_{0}$ is close to a $\mu_{0}$ such that for some $w_{0}$ the
pair $(\mu_{0},w_{0})$ solves 1.7 and satisfies a non-degeneracy condition. It
is unique as long as it stays close to this pair.

The rest of the paper is organised as follows. In Section 2, we present the
main challenges encountered in the analysis of system 1.6-1.7 and explain how
we handle them. After collecting some notation and useful facts in Section 3,
we fully set up our problem and state the main results in Section 4. The
analysis of the individual equations 1.7 and 1.6 is presented in Sections 5
and 7, respectively. In the intermediate Section 6 we study an auxiliary
parabolic PDE including PM diffusion and a fixed advection direction. The
results for this equation feed into the analysis in Section 7. The final
Section 8 is devoted to the proof of a well-posedness result for the whole
system 1.6-1.7.

##  2 Analytical challenges of 1.6-1.7 and their handling

System 1.6-1.7 combines two equations of a very different nature. In this
situation, a standard approach, which we also adopt in this paper, is first to
decouple the system and establish solvability of the two equations separately,
assuming that $w$ and $u$ are fixed in 1.6 and 1.7, respectively, and then to
obtain a joint solution by means of some fixed-point argument. The non-
standard nonlinear integral equation 1.7 puts multiple obstacles in the way.

  1. (IE1)

As with every new equation, it would be desirable to have some explicit
solutions of 1.7 at hand, at least for some simple time-independent $u$. Not
only would it help to understand what to expect from its solutions in general,
but it could also allow to prove existence of other solutions that are close
to already known ones. This is even more important in the case of system
1.6-1.7 where $u$ solves an initial value problem, so that one needs a $w_{0}$
corresponding to a given initial value $u_{0}$, to start with. Yet already for
constant functions $u$ it is not evident how, if at all, a matching $w$ could
be determined. If, however, we allow $u$ and $w$ to be singular measures,
accordingly reinterpreting the involved integral operators
${\mathcal{G}}^{\pm}$ as integrals of kernels $G^{\pm}$ against such a measure
$u$ (see Subsection 5.1), then the equation can be resolved, e.g. in the
special case of $u$ a point mass (see Example 5.12).

  2. (IE2)

Based on the observations made in the previous item, one is led to try to
establish solvability of 1.7 for such $u$ that are close to a point mass but
are functions rather than singular measures. This would be desirable for its
own sake but also because equation 1.6a includes nonlinear functions of $u$,
hence does not allow for measure-valued solutions. As is well-known, singular
measures cannot be approximated by Lebesgue integrable functions if the
distance between them is measured by the metric induced by the variation norm,
with which the whole space of signed Radon measures is equipped. To circumvent
this, weaker norms restricted to suitable subspaces are used instead. A
popular choice is the Kantorovich-Rubinstein (KR) norm [21] (see Subsection
3.2) that we also utilise in this work. The KR metric generated by the KR norm
is a special case of the Wasserstein metric. This metric has proven to be a
valuable tool for measuring distances between probability measures in various
contexts, including optimal transportation problems [29, 2], analysis of
evolution PDEs with a gradient flow structure [2], upscaling of mean field
equations [14], and more recently in statistical machine learning [24]. To the
best of our knowledge, the KR metric has so far not been used in analysis of
nonlinear integral equations. As it turns out, it offers a convenient
framework for such equations as 1.7. In particular, balls in this metric
centred at a point mass contain bounded compactly supported functions of any
shape (see Example 8.3). This provides an ample source of suitable initial
values $u_{0}$.

  3. (IE3)

The trade-off associated with using the KR norm in the context of the linear
integral operators such as ${\mathcal{G}}^{\pm}$ is that it requires their
kernels to be sufficiently smooth in order to be able to derive the bounds we
need. In the case of $G^{-}$ we have to avoid integration in the
neighbourhoods of its singularities. For each $x$, the corresponding
singularity set is a circle of radius $1$ centred at $x$. To maintain a
positive distance from these sets, we restrict our study of 1.7 to measures
$u$ with support diameter smaller than $1$.

  4. (IE4)

The nonlinear operator ${\mathcal{Y}}$ on the right-hand side of 1.7 is of the
form

$\displaystyle{\mathcal{Y}}=\frac{a}{a+b},$ (2.1)

where the applications of the integral operators ${\mathcal{G}}^{+}$ and
${\mathcal{G}}^{-}$ that are inserted into $a$ and $b$, respectively, are both
zero at points that are a distance farther than $1$ from the support of $u$.
The function on the right-hand side of 2.1 cannot be extended to $(0,0)$ in a
continuous fashion. Hence, the denominator of ${\mathcal{Y}}$ is a potential
source of singularities of this operator. Since we aim at solving 1.7 for
$u(t,\cdot)$, $t\in[0,T]$ for some $T>0$, we strengthen the condition from the
previous item, requiring the supports of all members of this family to be
confined to a single ball of diameter less than $1$. Then, for each $t$, 1.7
only needs to be solved in this ball, and there ${\mathcal{Y}}$ has good
properties (see Subsection 5.2).

  5. (IE5)

When solving an equation with a parameter, some version of the implicit
mapping theorem is often the method of choice. For instance, the Banach space
adaptation, [31, Chapter 4, §4.7, Theorem 4.B], provides solvability in a
small ball in the parameter space centred at a parameter value for which the
existence of a solution is known. Having chosen a setting for 1.7 based on the
discussion in (IE1), (IE2), (IE3), and (IE4), we cannot apply that theorem
directly because the set of admissible parameters does not contain a ball.
Still, since it is convex, it is possible to construct a similar argument and
thereby derive some useful estimates that ensure a locally stable dependence
of $w$ upon $u$ (see Theorem 4.5 and its proof given in Subsection 5.3).

Having decided on a suitable framework for 1.7, we turn to 1.6.

  1. (PDE1)

As already mentioned in the Introduction, the original derivation in [32]
yielded a diffusion-advection equation with the linear myopic diffusion 1.5.
The few available works on analysis of such equations specifically focused on
local taxis [30, 15, 10]. This type of advection occurs along the gradient of
an external stimulus and not as a result of nonlocal self-adhesion.

Assume that ${\mathbb{D}}$ is a smooth positive semi-definite matrix. If it is
also uniformly positive definite, then 1.5 does not pose considerable
additional analytical difficulties compared to the basic constant diffusion
case covered in [16] for diffusion-adhesion models. However, a nondegenerate
diffusion, whether myopic or not, immediately destroys compactness of support,
hence is unsuitable for the present setting, see (IE4). To evade this unwanted
effect, diffusion has to be degenerate. In this work, we selected the PM
diffusion 1.4. This choice is primarily motivated by the need to guarantee
that for small times the support of $u$ is confined to a small ball, as
advocated in (IE4). At the same time, it allows us to benefit from a range of
analytical tools already available for this prototypical degenerate diffusion
[28], which is much better understood than a myopic one.

On the whole, perhaps somewhat paradoxically, degeneracy enables the analysis
of 1.6-1.7.

  2. (PDE2)

Analysis of equations that combine quasilinear degenerate diffusion such as
1.4 and nonlocal advection with velocity

$\displaystyle\nabla H\star u=\nabla(H\star u),$

corresponding to

$\displaystyle\chi\equiv$ $\displaystyle 1,\qquad$ $\displaystyle w\equiv$
$\displaystyle 1,$

has attracted much attention. For smooth kernels $H$, well-posedness, which is
what is relevant for us here, initially in entropy and then also in weak
senses was established in [7] and [4], respectively. Subsequent works have
primarily focused on kernels singular only at the origin where they tend to
infinity, see [8] and references therein. The situation is different for the
self-adhesion kernel 1.3b. Its gradient is bounded and supported in the unit
ball centred at the origin, yet generally has discontinuities in the centre
and on the boundary of the ball. As a result, $\Delta H$ is, in most cases, a
singular Radon measure with singular parts supported on the ball’s boundary
and, in dimension one, also in the origin (see Lemma 7.1). Still, because this
measure is finite, the standardly required estimates in $L^{p}$ norms for its
convolution with $u$ can be obtained (see in the proof of Lemma 7.2).

Two recent studies [9, 13] incorporated a non-constant coefficient function
$\chi$. In a one-dimensional setting, weak solutions were obtained there as
limits of first-order deterministic many-particle systems. In these works,
$\chi$ was assumed to be nonnegative and decreasing, as well as satisfy
certain other conditions. As to the kernel, it was not mandated to be
compactly supported but allowed to have a Lipschitz singularity solely at the
origin [9].

Unlike [9, 13], we impose 1.3a, thus ensuring that $\chi$ is subordinate to
the diffusion coefficient at $u=0$. This assumption is sufficient for
uniqueness of weak solutions $u$ to 1.6 for fixed $w$ and is convenient for
several other steps in our analysis that includes well-posedness as well as
other properties such as support control and stable dependence on $w$ (see
Subsection 7.2).

We emphasise that our objective here is not to have the least restrictive
assumptions on the parameters in 1.6a. We have therefore chosen conditions
that ensure solvability of the entire system 1.6-1.7, while at the same time
being reasonable from the modelling perspective.

##  3 Preliminaries

###  3.1 Miscellaneous notation

For convenience, in the few cases of a numerical fraction that has a finite
numerator $a$ and a potentially zero denominator the interpretation is

$\displaystyle\frac{a}{0}:=\infty.$

The spacial domains that we mostly consider are either the whole space
$\mathbb{R}^{d}$, $d\in\mathbb{N}$, or the ball $B_{\rho}$ in $\mathbb{R}^{d}$
that is centred at the origin and has radius $\rho>0$, its boundary being the
sphere $S_{\rho}$. More generally, $B_{\rho}(x)$ stands for the $\rho$-ball
centred at $x$. The $(d-1)$-dimensional unit ball centred at the origin is
denoted by $B_{1}^{d-1}$.

For $T>0$, we define the time-space cylinder
$E_{T}:=(0,T)\times\mathbb{R}^{d}$.

Depending on the context, $|\cdot|$ can stand for: the absolute value of a
number, the Euclidean norm of a vector in $\mathbb{R}^{d}$, the
$d$-dimensional Lebesgue measure of a subset in $\mathbb{R}^{d}$, and the
total variation of a Radon measure.

For any set $A$, we denote by $\mathbf{1}_{A}$ its characteristic function.
Another standard function we use is

$\displaystyle\operatorname{sign}(x):=\begin{cases}\frac{x}{|x|}&\text{for
}x\in\mathbb{R}^{k}\backslash\\{0\\},\\\ 0&\text{for }x=0,\end{cases}\qquad
k\in\\{1,d\\}.$

For a real variable $z$, e.g. such as $t$ or $x_{i}$, we write $\partial_{z}$
for the usual partial derivative with respect to $z$. Further, $\nabla$,
$\nabla\cdot$, and $\Delta$ stand for the spatial gradient, divergence, and
Laplace operators, respectively.

When referring to the Gateaux partial differentials in an infinite-dimensional
setting, we always mean the right-side Gateaux partial semi-differentials. For
example, an operator $l=l(\mu,w)$ in a Banach space possesses the Gateaux
partial differential at $(\mu_{0},w_{0})$ with respect to $w$ in direction $h$
if the following limit exists:

$\displaystyle\partial_{w}l(\mu_{0},w_{0})h:=\underset{s\to
0+}{\lim}\frac{1}{s}\left(l(\mu_{0},w_{0}+sh)-l(\mu_{0},w_{0})\right).$

The same notation is used for the Fréchet partial derivatives. Which of the
two notations is used depends on the context and is explicitly mentioned.

For the (continuous) dual of a normed space $X$ we use the notation $X^{*}$.
Symbols $\rightharpoonup$ and $\overset{*}{\rightharpoonup}$ represent the
weak and weak-$*$ convergence, respectively. By $<\cdot,\cdot>$ we denote the
duality paring.

For $X$, $Y$, and $Z$ normed spaces, we write $L(X;Z)$ for the space of
bounded linear maps from $X$ into $Z$ and $B(X\times Y;Z)$ for the space of
bounded bilinear maps from $X\times Y\to Z$.

If $X,Y\subset Z$, we set

$\displaystyle\|\cdot\|_{X\cap Y}:=\max\\{\|\cdot\|_{X},\|\cdot\|_{Y}\\}.$

Let $k\in\mathbb{N}_{0}\cup\\{\infty\\}$ and $E\subset\mathbb{R}^{N}$ for some
$N\in\mathbb{N}$. As usual, $C^{k}(E)$ ($C(E):=C^{0}(E)$) is the notation for
the space of $k$ times continuously differentiable real-valued functions in
$E$. The subspaces of bounded and compactly supported functions in this class
are denoted by $C_{b}^{k}(E)$ and $C_{0}^{k}(E)$, respectively. When dealing
with vector- or Banach-valued function, but, also, when we want to specify the
codomain that is a particular subset of $\mathbb{R}$, we use, e.g. the
notation $C^{k}(E;F)$ for a $C^{k}$-mapping from $E$ into $F$. For $F$ a
Banach space, $C_{w}(E;F)$ denotes the space of weakly continuous maps between
these sets.

We use standard Lebesgue spaces $L^{p}(\Omega)$, Sobolev spaces
$W^{s,p}(\Omega)$ and $H^{s}(\Omega):=W^{s,2}(\Omega)$ for various
$p\in[1,\infty]$ and $s\in\mathbb{Z}$, as well as Bochner spaces of functions
taking values in such spaces. In particular, we denote

$\displaystyle L^{p,q}(E_{T}):=L^{p}(0,T;L^{q}(\mathbb{R}^{d})),\qquad
L^{p}(E_{T}):=L^{p,p}(E_{T})\qquad\text{for }p,q\in[1,\infty].$

We assume the reader to be familiar with the standard properties of the above
mentioned functional spaces. We also need to deal with subsets of the space of
finite measures. The related notation and required facts are provided in
Subsection 3.2 below.

Throughout the paper, $C_{i}$, $T_{i}$, and $\rho_{i}$ are either positive
constants or positive-valued mappings, whereas $U_{i}$ and $W_{i}$ are sets or
families of sets. In many instances we do not track the dependence of these
entities on some or all of parameters $d$, $\chi$, $F$, $a^{\pm}$, $b^{\pm}$,
$K^{\pm}$, $u_{0}$, as well as $w_{0}$, an initial state of $w$. When a change
occurs, e.g. because we stop tracking certain dependences, a remark on
notation is provided at the beginning of the corresponding section/subsection.

###  3.2 Radon measures and the Kantorovich-Rubinstein norm

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain. We denote by
${\mathcal{M}}(\overline{\Omega})$ (${\mathcal{M}}^{+}(\overline{\Omega})$)
the space of finite signed (positive) Radon measures in $\overline{\Omega}$
and by $|\mu|$ the total variation of
$\mu\in{\mathcal{M}}(\overline{\Omega})$. As is well-known,
$({\mathcal{M}}(\overline{\Omega}),|\cdot|)$ is a non-separable Banach space.
In particular, singular measures, such as point masses, cannot be obtained as
limits of Lebesgue integrable functions in this norm. This issue is relevant
in our case, see the discussion in Section 2(IE2). To counteract it, we also
use the Kantorovich-Rubinstein (KR) norm associated with the Euclidean metric
on $\mathbb{R}^{d}$. Since $\overline{\Omega}$ is compact, this norm is well-
defined on the subspace

${\mathcal{M}}_{KR}(\overline{\Omega}):=\\{\mu\in{\mathcal{M}}(\overline{\Omega}):\
\mu(\overline{\Omega})=0\\}.$

By the KR duality theorem [21, Chapter VIII, §4.5, Theorem 1], this norm is
given by

$\displaystyle\|\mu\|_{{\mathcal{M}}_{KR}(\overline{\Omega})}:=\sup\left\\{\int_{\overline{\Omega}}\varphi\,d\mu(y):\quad\varphi\in
W^{1,\infty}(\Omega)\text{ and }\|\nabla\varphi\|_{L^{\infty}(\Omega)}\leq
1\right\\}\quad\text{for all }\mu\in{\mathcal{M}}_{KR}(\overline{\Omega}).$
(3.1)

It is strictly weaker than the total variation and satisfies

$\displaystyle\|\mu\|_{{\mathcal{M}}_{KR}(\overline{\Omega})}\leq\frac{\operatorname{diam}(\Omega)}{2}|\mu|\qquad\text{for
all }\mu\in{\mathcal{M}}_{KR}(\overline{\Omega}).$ (3.2)

This norm gives rise to the Wasserstein $1$-distance on the set of probability
measures [29] or, more generally, on the set of positive measures that have
the same total variation. Moreover, when restricted to an arbitrary closed
ball in the variational norm, the convergence with respect to the Wasserstein
metric is equivalent to the weak-$*$ convergence [21, Chapter VIII, §4.6,
Theorem 3]. As a result, this metric turns such a ball into a metric space
which is compact and, hence, separable. We refer, e.g. to [21, Chapter VIII,
§4] for further details.

For $w:\overline{\Omega}\to\mathbb{R}$ Borel and bounded and
$\mu\in{\mathcal{M}}^{+}(\overline{\Omega})$ we use the standard definition
for their product:

$\displaystyle w\mu(B):=\int_{B}w\,d\mu\quad\text{for any Borel set
}B\subset\overline{\Omega}.$

It is well-known that $w\mu\in{\mathcal{M}}(\overline{\Omega})$, and for any
bounded Borel $f:\overline{\Omega}\to\mathbb{R}$ it holds that

$\displaystyle\int_{\overline{\Omega}}f\,d(w\mu)=\int_{\overline{\Omega}}fw\,d\mu.$

The notation $\mu\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}A$ means the restriction of a
measure $\mu$ to a set $A$.

By ${\mathcal{L}}^{d}$ and ${\cal H}^{d-1}$ we denote the $d$-dimensional
Lebesgue and the $(d-1)$-dimensional Hausdorff measures, respectively. We
identify $u$ and $u{\mathcal{L}}^{d}$ for $u\in L^{1}$.

Unless stated otherwise, a.e. means ’almost everywhere’ with respect to the
Lebesgue measure with dimension that is standardly adopted for a set in
question.

##  4 Problem setting and main results

We define solutions to individual equations 1.1, 1.6, 1.2, and 1.7 and the
corresponding systems as follows.

######  Definition 4.1 (Solutions to 1.1 and 1.6).

Let $T>0$, 1.3b, 1.3a, and 1.3c, and

$\displaystyle 0\leq u_{0}\in L^{\infty}(\mathbb{R}^{d})\cap
L^{1}(\mathbb{R}^{d})$ (4.1)

be satisfied.

  1. (1)

We call a pair of functions
$(u,v):[0,T]\times\mathbb{R}^{d}\to[0,\infty)\times[0,\infty)$ a solution to
1.1 in $[0,T]\times\mathbb{R}^{d}$ if:

    1. (i)

$u\in C_{w}\left([0,T];L^{\infty}(\mathbb{R}^{d})\cap
L^{1}(\mathbb{R}^{d})\right)\cap L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{d}))$;

    2. (ii)

$u^{\frac{3}{2}}\in L^{2}\left(0,T;H^{1}(\mathbb{R}^{d})\right)$;

    3. (iii)

$\partial_{t}u\in L^{2}\left(0,T;H^{-1}(\mathbb{R}^{d})\right)$,
$\partial_{t}u\in
L^{\infty}\left(0,T;\left(W^{2,p}\left(\mathbb{R}^{d}\right)\right)^{\prime}\right)$
for all $p\in[1,\infty]$;

    4. (iv)

$v\in L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{d}))$;

    5. (v)

$(u,v)$ satisfies 1.1 in a weak,

$\displaystyle\left<\partial_{t}u,\varphi\right>=-\int_{\mathbb{R}^{d}}\left(\nabla
u^{2}-v\chi(u)(\nabla H\star u)\right)\cdot\nabla\varphi\,dx\qquad\text{a.e.
in }\left(0,T\right)\text{ for all }\varphi\in H^{1}(\mathbb{R}^{d}),$ and a
very weak,
$\displaystyle\left<\partial_{t}u,\varphi\right>=\int_{\mathbb{R}^{d}}u^{2}\Delta\varphi+v\chi(u)(\nabla
H\star u)\cdot\nabla\varphi\,dx\quad\text{a.e. in }\left(0,T\right)\text{ for
all }\varphi\in\bigcup\limits_{p=1}^{\infty}W^{2,p}(\mathbb{R}^{d}),$

senses;

    6. (vi)

$u\left(0,\cdot\right)=u_{0}$ in $L^{\infty}(\mathbb{R}^{d})\cap
L^{1}(\mathbb{R}^{d})$.

  2. (2)

We call a pair of functions
$(u,w):[0,T]\times\mathbb{R}^{d}\to[0,\infty)\times[0,\infty)$ a solution to
1.6 in $[0,T]\times\mathbb{R}^{d}$ if $(u,wu)$ is a solution to 1.1 in
$[0,T]\times\mathbb{R}^{d}$ in the above sense.

######  Definition 4.2 (Time-independent solutions to 1.2 and 1.7).

Let assumptions 1.3e, 1.3f, 1.3g, and 1.3h be satisfied.

  1. (1)

We call a pair of measures $(\mu,\nu)$ a solution to 1.2 if:

    1. (i)

$\mu\in{\mathcal{M}}^{+}(\mathbb{R}^{d})$;

    2. (ii)

${\mathcal{G}}^{-}\mu<\infty$;

    3. (iii)

$\nu\in{\mathcal{M}}^{+}(\mathbb{R}^{d})$;

    4. (iv)

$\nu\leq\mu$ in $\mathbb{R}^{d}$;

    5. (v)

$(\mu,\nu)$ satisfies 1.2 in the following sense:

$\displaystyle\nu=\frac{{\mathcal{G}}^{+}(\mu-\nu)}{{\mathcal{G}}^{+}(\mu-\nu)+{\mathcal{G}}^{-}\nu}\mu\qquad\text{in
}{\mathcal{M}}^{+}(\mathbb{R}^{d}),$ (4.3)

where $0/0$ in the fraction is interpreted as $1$;

  2. (2)

We call a pair of a measure and a function $(\mu,w)$ a solution to 1.7 if:

    1. (i)

$w:\mathbb{R}^{d}\to[0,1]$ is Borel;

    2. (ii)

$(\mu,w\mu)$ is a solution to 1.2 in the above sense.

######  Remark 4.3.

  1. (1)

In consequence of Lemma 5.3 below, the fraction on the right-hand side of 4.3
is well-defined and Borel. Therefore, Definition 4.2 makes sense.

  2. (2)

If $(\mu,w)$ has the regularity required by Definition 4.2 and $\nu=w\mu$,
then 4.3 is obviously equivalent to

$\displaystyle w={\mathcal{Y}}(\mu,w)\qquad\mu-\text{a.e.}$

Thus, through 4.3 we have a rigorous interpretation of 1.7.

######  Definition 4.4 (Solutions to 1.1-1.2 and 1.6-1.7).

Let $T>0$ and 1.3b, 1.3a, and 1.3c, 1.3e, 1.3f, 1.3g, and 1.3h, and 4.1 be
satisfied. We call a pair of functions $(u,v)$ ($(u,w)$) a solution to system
1.1-1.2 (1.6-1.7) in $[0,T]\times\mathbb{R}^{d}$ if:

  1. (1)

$(u,v)$ ($(u,w)$) solves 1.1 (1.6) in $[0,T]\times\mathbb{R}^{d}$ in the sense
of Definition 4.1;

  2. (2)

for all $t\in[0,T]$, $(u(t,\cdot),v(t,\cdot))$ ($(u(t,\cdot),w(t,\cdot))$)
solves 1.1 (1.6) in the sense of Definition 4.2.

The main results of this paper are the following Theorems on local well-
posedness of equation 1.7 and system 1.6-1.7.

######  Theorem 4.5 (Local well-posedness of 1.7).

Let

$\rho\in\left(0,\frac{1}{2}\right)$

and $(\mu_{0},w_{0})$ satisfy

$\displaystyle\mu_{0}\in({\mathcal{M}}^{+}(\overline{B_{\rho}}))\backslash\\{0\\},$
(4.4a) $\displaystyle w_{0}\in W^{1,\infty}(B_{\rho};(0,1)),$ (4.4b)
$\displaystyle w_{0}={\mathcal{Y}}(\mu_{0},w_{0})\qquad\text{in
}\overline{B_{\rho}},$ (4.4c)
$\displaystyle{\mathcal{X}}:=id-\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0})\text{
is invertible in }L(W^{1,\infty}(B_{\rho})),$ (4.4d)

where the partial derivative $\partial_{w}$ is taken in the Fréchet sense.
Define sets

$\displaystyle\Cl[U]{SU2}:=$
$\displaystyle\Big{\\{}\mu\in{\mathcal{M}}^{+}(\overline{B_{\rho}}):$
$\displaystyle\quad|\mu|=|\mu_{0}|\ \text{ and }\
\|\mu-\mu_{0}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}\leq\Cl{R2}\Big{\\}},$
(4.5a) $\displaystyle\Cl[W]{W3}:=$ $\displaystyle\Big{\\{}w\in
W^{1,\infty}(B_{\rho};[0,1]):$
$\displaystyle\quad\|w-w_{0}\|_{W^{1,\infty}(B_{\rho})}\leq\Cl{R1}\Big{\\}}.$
(4.5b)

Then, there exist constants $\Cr{R2},\Cr{R1},\Cl{F-contin-mu}>0$ that depend
only on the parameters from 1.3e, 1.3f, 1.3g, and 1.3h as well as

$\displaystyle|\mu_{0}|,\ \rho,\ \|\nabla
w_{0}\|_{L^{\infty}\left(B_{\rho}\right)},\ \min_{\overline{B_{\rho}}}w_{0},\
\max_{\overline{B_{\rho}}}w_{0},\
\left\|{\mathcal{X}}^{-1}\right\|_{{L\left(W^{1,\infty}\left(B_{\rho}\right)\right)}},$

such that for all $\mu\in\Cr{SU2}$ there exists a unique $w\in\Cr{W3}$ for
which

$\displaystyle w={\mathcal{Y}}(\mu,w)\qquad\text{in }\overline{B_{\rho}},$
(4.6)

and the solution map

$\displaystyle\Cl[WW]{OpW1}:\Cr{SU2}\to\Cr{W3}\cap
C^{2}(\overline{B_{\rho}};(0,1)),\qquad\mu\mapsto w,$ (4.7)

is well-defined and Lipschitz continuous in the following sense:

$\displaystyle\|\Cr{OpW1}(\mu_{1})-\Cr{OpW1}(\mu_{2})\|_{W^{1,\infty}(B_{\rho})}\leq\Cr{F-contin-
mu}\|\mu_{1}-\mu_{2}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}\qquad\text{for
all }\mu_{1},\mu_{2}\in\Cr{SU2}.$ (4.8)

######  Remark 4.6.

The Fréchet partial differentiability of ${\mathcal{Y}}$ with respect to $w$
in the case of a spatial domain being a ball of radius smaller then $1/2$ is
established in Lemma 5.10.

######  Remark 4.7.

By Lemma 5.8, every solution to 4.6 satisfies $w\in
C^{2}(\overline{B_{\rho}};(0,1))$. Hence, the requirement $w\in\Cr{W3}\cap
C^{2}(\overline{B_{\rho}};(0,1))$ in 4.7 only restricts the distance to
$w_{0}$.

The proof of Theorem 4.5 is given in Subsection 5.3.

######  Theorem 4.8 (Local well-posedness of 1.6-1.7).

Let $d\in\mathbb{N}$, 1.3b, 1.3a, and 1.3c and 1.3e, 1.3f, 1.3g, and 1.3h
hold, and be given: numbers $m,m_{\infty},\rho>0$ such that

$\displaystyle\rho<\min\left\\{\frac{1}{2},\frac{d+2}{m\|\chi^{\prime}\|_{L^{\infty}((0,\infty))}\|F\|_{L^{\infty}(0,1)}}\right\\},$
(4.9)

and a pair $(\mu_{0},w_{0})$ that satisfies assumption 4.4 of Theorem 4.5 and

$\displaystyle|\mu_{0}|=m.$

Define

$\displaystyle\Cl[U]{U01}:=$ $\displaystyle\Big{\\{}0\leq u_{0}\in
L^{\infty}(\mathbb{R}^{d}):$
$\displaystyle\quad\operatorname{supp}(u_{0})\subset\overline{B_{\frac{1}{4}\rho}},$
$\displaystyle\quad\|u_{0}\|_{L^{\infty}(\mathbb{R}^{d})}\leq m_{\infty},$
$\displaystyle\quad\|u_{0}\|_{L^{1}(\mathbb{R}^{d})}=m,$
$\displaystyle\quad\|u_{0}-\mu_{0}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}\leq\frac{1}{2}\Cr{R2}\Big{\\}}.$

Then, there exists a number $\Cl[T]{Tstar}$ that depends only on the
parameters from 1.3 as well as

$\displaystyle m,\ m_{\infty},\ \rho,\ \|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})},\ \min_{\overline{B_{\rho}}}w_{0},\
\max_{\overline{B_{\rho}}}w_{0},\
\left\|{\mathcal{X}}^{-1}\right\|_{{L\left(W^{1,\infty}(B_{\rho})\right)}},$
(4.10)

such that for any $u_{0}\in\Cr{U01}$ there exists a pair of functions $(u,w)$
that solves 1.6-1.7 in $[0,\Cr{Tstar}]\times\mathbb{R}^{d}$ in the sense of
Definition 4.4 and satisfies

$\displaystyle\operatorname{supp}(u(t,\cdot))\subset B_{\rho}\qquad\text{for
all }t\in[0,\Cr{Tstar}],$ (4.11a)
$\displaystyle\|u-u_{0}\|_{C([0,\Cr{Tstar}],{\mathcal{M}}_{KR}(\overline{B_{\rho}}))}\leq\Cr{R2},$
(4.11b)

and

$\displaystyle w(t,\cdot)\in C^{2}(\overline{B_{\rho}};(0,1))\qquad\text{for
all }t\in[0,\Cr{Tstar}],$ (4.12a) $\displaystyle w\in
C([0,\Cr{Tstar}];W^{1,\infty}(\mathbb{R}^{d})),$ (4.12b)
$\displaystyle\|w-w_{0}\|_{C([0,\Cr{Tstar}];W^{1,\infty}(B_{\rho}))}\leq\Cr{R1}.$
(4.12c)

In the above, constants $\Cr{R2}$ and $\Cr{R1}$ are from Theorem 4.5.

The solution is locally unique in the following sense: if for some
$T\in(0,\Cr{Tstar}]$ another solution $(\widehat{u},\widehat{w})$ in
$[0,T]\times\mathbb{R}^{d}$ satisfies

$\displaystyle\widehat{w}\in C([0,T];W^{1,\infty}(\mathbb{R}^{d})),$ (4.13a)
$\displaystyle\|\widehat{w}-w_{0}\|_{C([0,T];W^{1,\infty}(B_{\rho}))}\leq\Cr{R1},$
(4.13b)

then for all $t\in[0,T]$ it holds that

$\displaystyle u(t,\cdot)=\widehat{u}(t,\cdot)$ $\displaystyle\qquad\text{
a.e. in }\mathbb{R}^{d},$ $\displaystyle w(t,\cdot)=\widehat{w}(t,\cdot)$
$\displaystyle\qquad\text{ in }\overline{B_{\rho}}.$

Finally, a Lipschitz property holds: for every
$u_{0},\widehat{u}_{0}\in\Cr{U01}$ the corresponding solutions $(u,w)$ and
$(\widehat{u},\widehat{w})$ in $[0,\Cr{Tstar}]\times\mathbb{R}^{d}$ satisfy

$\displaystyle\max_{[0,\Cr{Tstar}]}\left\|u-\widehat{u}\right\|_{L^{1}(B_{\rho})}+\left\|w-\widehat{w}\right\|_{C([0,\Cr{Tstar}];W^{1,\infty}(B_{\rho}))}\leq\Cl{CLipS}\left\|u_{0}-\widehat{u}_{0}\right\|_{L^{1}\left(B_{\frac{\rho}{4}}\right)},$
(4.14)

where constant $\Cr{CLipS}$ only depends on parameters from 1.3 and 4.10.

This Theorem is proved in Section 8.

Solutions to 1.2 and 1.1-1.2 can be recovered from the corresponding solutions
to 1.7 and 1.6-1.7, respectively. Moreover, there are no other solutions, as
the next two theorems imply.

######  Theorem 4.9 (Time-independent solutions to 1.2 vs. 1.7).

If $(\mu,\nu)$ is a time-independent solution to 1.2, then
$\left(\mu,\frac{d\nu}{d\mu}\right)$ is a time-independent solution to 1.7.

######  Theorem 4.10 (Solutions to 1.1-1.2 vs. 1.6-1.7).

If $(u,v)$ is a solution to 1.1-1.2 in $[0,T]\times\mathbb{R}^{d}$, then
$\left(u,w\right)$ is a solution to 1.6-1.7 for some $w$ that for all
$t\in[0,T]$ satisfies

$\displaystyle w(t,\cdot)\text{ is Borel},$ $\displaystyle
w(t,\cdot)=\frac{u}{v}(t,\cdot)\qquad\text{a.e. in }\\{u>0\\}.$

These Theorems are proved in Subsections 5.3 and 8, respectively.

##  5 Well-posedness of 1.7 for fixed $u(=:\mu)$

The main goal of this Section is to prove Theorem 4.5, a result on local well-
posedness of equation 4.6 for given $\mu$ from a suitable subset of a space of
measures and sufficiently close to some $\mu_{0}$ for which a solution is
assumed to exist. Throughout the Section we assume 1.3d, 1.3e, 1.3f, 1.3g, and
1.3h to hold and also that all involved functions/measures are time-
independent. To highlight the fact that the first argument of ${\mathcal{Y}}$
can be a measure, we denote it by $\mu$ rather than $u$.

We begin by properly defining operator ${\mathcal{Y}}$ and studying its
properties, first in general domains in Subsection 5.1 and then in a small
ball in Subsection 5.2. Some of the presented results are not needed in
Subsection 5.3, where we prove Theorem 4.5. Still, they provide useful
insights into the nature of this new operator.

###  5.1 Operator ${\mathcal{Y}}$

Throughout this Subsection we assume

$\displaystyle\Omega\subset\mathbb{R}^{d}\text{ a domain}.$

Here we study operator ${\mathcal{Y}}$ from 1.8 for measures in
$\overline{\Omega}$ as its first argument. We start with operators
${\mathcal{G}}^{\pm}$. Set

$\displaystyle{\mathcal{G}}^{\pm}\mu(x):=\int_{\overline{\Omega}}G^{\pm}(x,y)\,d\mu(y),$

whenever the integral exists, and

$\displaystyle{\mathcal{M}}_{G}(\overline{\Omega}):=\left\\{\mu\in{\mathcal{M}}(\overline{\Omega}):\
{\mathcal{G}}^{-}|\mu|<\infty\ \text{in }\overline{\Omega}\right\\},$ (5.1a)
$\displaystyle{\mathcal{M}}_{G}^{+}(\overline{\Omega}):={\mathcal{M}}_{G}(\overline{\Omega})\cap{\mathcal{M}}^{+}(\overline{\Omega}).$
(5.1b)

Our first Lemma ensures that set ${\mathcal{M}}_{G}(\overline{\Omega})$ and
operators ${\mathcal{G}}^{\pm}$ applied to its elements are well-defined.

######  Lemma 5.1.

  1. (1)

Let $\mu\in{\mathcal{M}}^{+}(\overline{\Omega})$. Then
${\mathcal{G}}^{\pm}\mu:\overline{\Omega}\to[0,\infty]$ are well-defined Borel
functions.

  2. (2)

${\mathcal{M}}_{G}(\overline{\Omega})$ and
${\mathcal{M}}_{G}^{+}(\overline{\Omega})$ are well-defined linear subspace
and convex subset, respectively, of ${\mathcal{M}}(\overline{\Omega})$.

  3. (3)

Let $\mu\in{\mathcal{M}}_{G}(\overline{\Omega})$. Then
${\mathcal{G}}^{\pm}\mu:\overline{\Omega}\to[0,\infty)$ are well-defined Borel
functions.

###### Proof.

Being superpositions of continuous functions and the characteristic function
of an interval,

$G^{\pm}:\overline{\Omega}\times\overline{\Omega}\to[0,\infty)$

are Borel. By Fubini’s theorem,

${\mathcal{G}}^{\pm}\mu_{\pm}:\overline{\Omega}\to[0,\infty]$

are well-defined Borel functions for every
$\mu\in{\mathcal{M}}(\overline{\Omega})$.

Further, due to continuity of $K^{+}$ and $\varphi^{+}$ and compactness of the
support of the latter, $G^{+}(x,\cdot)$ is bounded for all
$x\in\overline{\Omega}$. Hence, it is integrable against every
$\mu\in{\mathcal{M}}(\overline{\Omega})$.

The above observations combined readily imply that (1) and (3) hold and sets
5.1 are well-defined.

Finally, it is obvious that ${\mathcal{M}}_{G}(\overline{\Omega})$ is a linear
subspace of ${\mathcal{M}}(\overline{\Omega})$ and
${\mathcal{M}}_{G}^{+}(\overline{\Omega})$ is its convex subset. ∎

Next, we define operator ${\mathcal{Y}}$ for measures. Let

$\displaystyle\psi:[0,\infty)\times[0,\infty)\to[0,1],\quad\psi(a,b):=\begin{cases}\frac{a}{a+b}&\text{for
}(a,b)\neq(0,0),\\\ 1&\text{for }(a,b)=(0,0).\end{cases}$ (5.2)

The following Lemma can be easily verified by induction.

######  Lemma 5.2.

Let $\psi$ be as defined in 5.2. Then $\psi\in
C^{\infty}(([0,\infty)\times[0,\infty))\backslash\\{(0,0)\\})$, and for all
$0\leq k_{1}\leq k$ it holds that

$\displaystyle\partial^{k}_{a^{k_{1}}b^{k-k_{1}}}\psi(a,b)=\frac{C_{k_{1},k,1}a+C_{k_{1},k,2}b}{(a+b)^{k+1}}\quad\text{for
all }(a,b)\in([0,\infty)\times[0,\infty))\backslash\\{(0,0)\\}$

and

$\displaystyle\left|\partial^{k}_{a^{k_{1}}b^{k-k_{1}}}\psi(a,b)\right|\leq
C_{k}(a+b)^{-k}\quad\text{for all
}(a,b)\in([0,\infty)\times[0,\infty))\backslash\\{(0,0)\\}$ (5.3)

for some constants $C_{k_{1},k,1},C_{k_{1},k,2}\in\mathbb{R}$ and $C_{k}>0$.

For $\mu\in{\mathcal{M}}_{G}^{+}(\overline{\Omega})$ and a Borel function
$w:\overline{\Omega}\to[0,1]$, we set

$\displaystyle{\mathcal{Y}}(\mu,w):=\psi({\mathcal{G}}^{+}((1-w)\mu),{\mathcal{G}}^{-}(w\mu)).$

The subsequent Lemma infers that ${\mathcal{Y}}$ is well-defined and
measurable.

######  Lemma 5.3.

Let $\mu\in{\mathcal{M}}_{G}^{+}(\overline{\Omega})$,
$\nu\in{\mathcal{M}}^{+}(\overline{\Omega})$, and $\nu\leq\mu$. Then

$\psi({\mathcal{G}}^{+}(\mu-\nu),{\mathcal{G}}^{-}\nu):\overline{\Omega}\to[0,1]$

is a well-defined Borel function.

###### Proof.

Due to the assumptions on $\mu$ and $\nu$, we have
$\mu-\nu,\nu\in{\mathcal{M}}_{G}^{+}(\overline{\Omega})$, so Lemma 5.1 applies
and yields that
${\mathcal{G}}^{+}(\mu-\nu),{\mathcal{G}}^{-}\nu:\overline{\Omega}\to[0,\infty)$
are well-defined Borel functions. Further, $\psi$ is continuous at every point
of its domain of definition apart from $(0,0)$. Hence it is Borel.

Altogether, we have
$\psi({\mathcal{G}}^{+}(\mu-\nu),{\mathcal{G}}^{-}\nu):\overline{\Omega}\to[0,1]$
is well-defined and, as a composition of Borel functions, it is also Borel. ∎

To study directional derivatives of ${\mathcal{Y}}$, we introduce sets of
’admissible’ directions

$\displaystyle{\mathcal{H}}_{\mu}:=\\{h_{\mu}\in{\mathcal{M}}_{KR}(\overline{\Omega}):$
$\displaystyle\quad\mu+s_{0}h_{\mu}\in{\mathcal{M}}^{+}(\overline{\Omega})$
$\displaystyle\ \text{ for some }s_{0}>0\\}$ $\displaystyle\qquad\text{for
}\mu\in{\mathcal{M}}^{+}(\overline{\Omega}),$
$\displaystyle{\mathcal{H}}_{w}:=\\{h:\overline{\Omega}\to[-1,1]\text{
Borel}:$ $\displaystyle\quad 0\leq w+s_{0}h\leq 1\text{ in }\overline{\Omega}$
$\displaystyle\ \text{ for some }s_{0}>0\\}$ $\displaystyle\qquad\text{for
}w:\overline{\Omega}\to[0,1]\text{ Borel}.$

Our next Lemma establishes Gateaux differentiability at a spatial point $x$.

######  Lemma 5.4.

Assume $\mu_{0}\in{\mathcal{M}}_{G}^{+}(\overline{\Omega})$,
$w_{0}:\overline{\Omega}\to[0,1]$ Borel, and $x\in\mathbb{R}^{d}$, such that

$\displaystyle\mu_{0}(\overline{\Omega}\cap B_{1}(x))>0.$ (5.4)

Then,

$\displaystyle{\mathcal{G}}^{+}((1-{w_{0}})\mu_{0})(x)+{\mathcal{G}}^{-}({w_{0}}\mu_{0})(x)>0,$
(5.5)

and functional

$(\mu,w)\mapsto{\mathcal{Y}}(\mu,w)(x)$

possesses Gateaux partial differentials of any order with respect to $\mu$ and
$w$ at $(\mu_{0},w_{0})$ along the directions from
${\mathcal{H}}_{\mu_{0}}\cap{\mathcal{M}}_{G}(\overline{\Omega})$ and
${\mathcal{H}}_{w_{0}}$, respectively. In particular, we have the following
formulas:

$\displaystyle(\partial_{w}{\mathcal{Y}}(\mu_{0},{w_{0}})(x))h_{1}=$
$\displaystyle
D\psi({\mathcal{G}}^{+}((1-{{w_{0}}})\mu_{0}),{\mathcal{G}}^{-}({{w_{0}}}\mu_{0}))\cdot(-{\mathcal{G}}^{+}(h_{1}\mu_{0}),{\mathcal{G}}^{-}(h_{1}\mu_{0}))(x),$
(5.6a)
$\displaystyle(\partial_{\mu}{\mathcal{Y}}(\mu_{0},{w_{0}})(x))h_{\mu_{0}}=$
$\displaystyle
D\psi({\mathcal{G}}^{+}((1-{{w_{0}}})\mu_{0}),{\mathcal{G}}^{-}({{w_{0}}}\mu_{0}))\cdot({\mathcal{G}}^{+}((1-{{w_{0}}})h_{\mu_{0}}),{\mathcal{G}}^{-}({w_{0}}h_{\mu_{0}}))(x),$
(5.6b) and
$\displaystyle(\partial_{w}(\partial_{w}{\mathcal{Y}}(\mu_{0},{w_{0}})(x))h_{1})h_{2}=$
$\displaystyle
D^{2}\psi({\mathcal{G}}^{+}((1-{{w_{0}}})\mu_{0}),{\mathcal{G}}^{-}({{w_{0}}}\mu_{0}))(-{\mathcal{G}}^{+}(h_{1}\mu_{0}),{\mathcal{G}}^{-}(h_{1}\mu_{0}))^{T}$
$\displaystyle\cdot(-{\mathcal{G}}^{+}(h_{2}\mu_{0}),{\mathcal{G}}^{-}(h_{2}\mu_{0}))^{T}(x),$
(5.6c)
$\displaystyle(\partial_{\mu}(\partial_{w}{\mathcal{Y}}(\mu_{0},{w_{0}})(x))h_{1})h_{\mu_{0}}=$
$\displaystyle
D^{2}\psi({\mathcal{G}}^{+}((1-{{w_{0}}})\mu_{0}),{\mathcal{G}}^{-}({{w_{0}}}\mu_{0}))(-{\mathcal{G}}^{+}(h_{1}\mu_{0}),{\mathcal{G}}^{-}(h_{1}\mu_{0}))^{T}$
$\displaystyle\cdot({\mathcal{G}}^{+}((1-{{w_{0}}})h_{\mu_{0}}),{\mathcal{G}}^{-}({w_{0}}h_{\mu_{0}}))^{T}(x)$
$\displaystyle+D\psi({\mathcal{G}}^{+}((1-{{w_{0}}})\mu_{0}),{\mathcal{G}}^{-}({{w_{0}}}\mu_{0}))\cdot(-{\mathcal{G}}^{+}(h_{1}h_{\mu_{0}}),{\mathcal{G}}^{-}(h_{1}h_{\mu_{0}}))(x)$
(5.6d)

for all
$h_{\mu_{0}}\in{\mathcal{H}}_{\mu_{0}}\cap{\mathcal{M}}_{G}(\overline{\Omega})$
and $h_{1},h_{2}\in{\mathcal{H}}_{w_{0}}$.

###### Proof.

Let
$h_{\mu_{0}}\in{\mathcal{H}}_{\mu_{0}}\cap{\mathcal{M}}_{G}(\overline{\Omega})$
and let $s_{0}>0$ be a number such that
$\mu_{0}+s_{0}h_{\mu_{0}}\in{\mathcal{M}}_{G}^{+}(\overline{\Omega})$. Since
$0\in{\mathcal{H}}_{\mu_{0}}$ and ${\mathcal{M}}_{G}^{+}(\overline{\Omega})$
is convex, we have
$\mu_{0}+sh_{\mu_{0}}\in{\mathcal{M}}_{G}^{+}(\overline{\Omega})$ for all
$s\in[0,s_{0}]$. Similarly, for $h\in{\mathcal{H}}_{w_{0}}$ we obtain $0\leq
w_{0}+sh\leq 1$ for all $s\in[0,{s_{0}}]$. Therefore, taking the limits
necessary to compute partial directional derivatives along such directions is
possible.

Next, we observe that 5.4 implies 5.5. This is because, firstly,
$\varphi^{\pm}$ and $K^{\pm}$ are strictly positive in $B_{1}(x)$ (compare
1.3g and 1.3f), so that the kernels $G^{\pm}(x,\cdot)$ are strictly positive
there, and secondly, $1-w_{0},w_{0}\geq 0$, and
$(1-w_{0})\mu_{0}+w_{0}\mu_{0}=\mu_{0}$. Consequently,
${\mathcal{G}}^{+}((1-{w_{0}})\mu_{0})(x)\geq 0$ and
${\mathcal{G}}^{-}({w_{0}}\mu_{0})(x)\geq 0$ cannot be zero at the same time.

Thus, $\psi$ is infinitely many times differentiable at
$({\mathcal{G}}^{+}((1-{w_{0}})\mu_{0}),{\mathcal{G}}^{-}({w_{0}}\mu_{0}))(x)$.
This, the fact that $(\mu,w)\mapsto{\mathcal{G}}^{\pm}(w\mu)$ is bilinear, and
the chain rule together imply the infinite directional differentiability of
$(\mu,w)\to{\mathcal{Y}}(\mu,w)(x)$ at $(\mu_{0},w_{0})$ and formulas 5.6.

∎

###  5.2 Properties of ${\mathcal{Y}}$ for ${\Omega}$ a small ball

In this Subsection we assume

$\displaystyle\Omega=B_{\rho}\qquad\text{for some
}\rho\in\left(0,\frac{1}{2}\right).$ (5.7)

######  Notation 5.5.

To simplify the notation, we do not explicitly mention the dependence on
$\rho$ for constants, mappings, and sets that we introduce in this and
subsequent Subsections.

The properties of operator ${\mathcal{Y}}$ that we derive below for this
special case allow to establish local existence of 4.6 in the subsequent
Subsection 5.3.

As previously observed in Section 2(IE3), the smallness of the diameter of
domain $\Omega$ allows to avoid singularities in $G^{-}$, as well as zeros in
the denominator of ${\mathcal{Y}}$, ensuring, as we see below in Lemma 5.7
that property 1.3e holds in the whole of $\overline{\Omega}$. We formulate a
Lemma that addresses the properties of $G^{\pm}$ in this case.

######  Lemma 5.6.

Let $\rho\in\left(0,{1}/{2}\right)$ hold. Then,

$\displaystyle G^{\pm}\in
C^{2}(\overline{B_{\rho}}\times\overline{B_{\rho}}),$ (5.8a)
$\displaystyle\|G^{\pm}\|_{C^{2}(\overline{B_{\rho}}\times\overline{B_{\rho}})}\leq\Cl{Cro},$
(5.8b) $\displaystyle\Cl{CGlb}:=\min\left\\{G^{\pm}(x,y):\
(x,y)\in\overline{B_{\rho}}\times\overline{B_{\rho}}\right\\}>0.$ (5.8c)

###### Proof.

Since $\rho\in\left(0,{1}/{2}\right)$, we have with 1.3e, 1.3f, and 1.3h that

$\displaystyle|x-y|<1\qquad\text{for all
}(x,y)\in\overline{B_{\rho}}\times\overline{B_{\rho}}.$ (5.9)

Combined with assumptions $0<K^{\pm}\in C^{2}$ and $0<\varphi^{\pm}\in
C^{2}[0,1)$, 5.9 yields properties 5.8.

∎

Our next Lemma collects several consequences of 5.7 for operators
${\mathcal{G}}^{\pm}$ and ${\mathcal{Y}}$.

######  Lemma 5.7.

Let $\rho\in\left(0,{1}/{2}\right)$. Then:

  1. (1)

${\mathcal{M}}(\overline{B_{\rho}})={\mathcal{M}}_{G}(\overline{B_{\rho}})$;

  2. (2)

if $\mu\in{\mathcal{M}}^{+}(\overline{B_{\rho}})$, then

$\displaystyle\mu(\overline{B_{\rho}}\cap B_{1}(\cdot))=|\mu|\qquad\text{in
}\overline{B_{\rho}}$ (5.10)

and for all $w:\overline{B_{\rho}}\to[0,1]$ Borel

$\displaystyle{\mathcal{G}}^{+}((1-w)\mu)+{\mathcal{G}}^{-}(w\mu)\geq$
$\displaystyle\Cl{B3}|\mu|\qquad\text{in }\overline{B_{\rho}};$ (5.11)

  3. (3)

${\mathcal{G}}^{\pm}\in
L({\mathcal{M}}(\overline{B_{\rho}});C^{2}(\overline{B_{\rho}}))$ and

$\displaystyle\|{\mathcal{G}}^{\pm}\|_{L({\mathcal{M}}(\overline{B_{\rho}});C^{2}(\overline{B_{\rho}}))}\leq\Cr{Cro};$
(5.12)

  4. (4)

${\mathcal{G}}^{\pm}((\cdot)(\cdot))\in
B(W^{1,\infty}(B_{\rho})\times{\mathcal{M}}_{KR}(\overline{B_{\rho}});C^{1}(\overline{B_{\rho}}))$
and

$\displaystyle\|{\mathcal{G}}^{\pm}((\cdot)(\cdot))\|_{B(W^{1,\infty}(B_{\rho})\times{\mathcal{M}}_{KR}(\overline{B_{\rho}});C^{1}(\overline{B_{\rho}}))}\leq\Cl{CroG3};$
(5.13)

  5. (5)

${\mathcal{Y}}:(({\mathcal{M}}^{+}(\overline{B_{\rho}}))\backslash\\{0\\})\times\\{w:\overline{B_{\rho}}\to[0,1]\
\text{Borel}\\}\to C^{2}(\overline{B_{\rho}};[0,1])$.

###### Proof.

Thanks to 5.8a, function $G^{-}(x,\cdot)$ is continuous on
$\overline{B_{\rho}}$. Hence, ${\mathcal{G}}^{-}|\mu|<\infty$ for any
$\mu\in{\mathcal{M}}(\overline{B_{\rho}})$, so (1) holds. If
$\mu\in{\mathcal{M}}^{+}(\overline{B_{\rho}})$, then assumption
$\rho\in(0,1/2)$ implies that

$\displaystyle\overline{B_{\rho}}\subset B_{1}(x)\qquad\text{for all
}x\in\overline{B_{\rho}},$

yielding 5.10. As to 5.11, it obviously follows with 5.8c, 5.10, and 1.3d.

Next, we observe each $\mu\in{\mathcal{M}}(\overline{B_{\rho}})$ produces
functions ${\mathcal{G}}^{\pm}\mu$ that inherit the smoothness that $G^{\pm}$
has and that differentiation of ${\mathcal{G}}^{\pm}\mu$ is interchangeable
with integration in the definitions of $G^{\pm}$. This is a direct consequence
of [23, Chapter 6, Theorems 6.27-6.28] applied to each coordinate $x_{k}$
separately. In particular, 5.8a implies that $G^{\pm}\mu\in
C^{2}(\overline{B_{\rho}})$. Further, since
$h\mu\in{\mathcal{M}}(\overline{B_{\rho}})$ for $h\in W^{1,\infty}(B_{\rho})$
and $\mu\in{\mathcal{M}}(\overline{B_{\rho}})$, we have that
${\mathcal{G}}^{\pm}(h\mu)\in C^{2}(\overline{B_{\rho}})$. Also, since
${\mathcal{G}}^{\pm}$ is linear, ${\mathcal{G}}^{\pm}((\cdot)(\cdot))$ is
bilinear. Hence the linear and bilinear operators in (3) and (4) are well-
defined between the required spaces. Checking 5.12 and thus boundedness of
${\mathcal{G}}^{\pm}$ is straightforward, given that integration and
differentiation can be interchanged. Combining 5.8b and 3.1 with the product
rule, we can estimate as follows:

$\displaystyle\underset{x\in\overline{B_{\rho}}}{\max}\,|{\mathcal{G}}^{\pm}(h\mu)(x)|\leq$
$\displaystyle\underset{x\in\overline{B_{\rho}}}{\max}\,\|\nabla_{y}(G^{+}(x,\cdot)h)\|_{(L^{\infty}(B_{\rho}))^{n}}\|\mu\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}$
$\displaystyle\leq$
$\displaystyle\|G^{\pm}\|_{C^{1}(\overline{B_{\rho}}\times\overline{B_{\rho}})}\|h\|_{W^{1,\infty}(B_{\rho})}\|\mu\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}$
$\displaystyle\leq$
$\displaystyle\Cr{Cro}\|h\|_{W^{1,\infty}(B_{\rho})}\|\mu\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}$

and

$\displaystyle\underset{x\in\overline{B_{\rho}}}{\max}\,|\nabla_{x}{\mathcal{G}}^{\pm}(h\mu)(x)|\leq$
$\displaystyle\underset{x\in\overline{B_{\rho}}}{\max}\,\left\|\nabla_{y}(\nabla_{x}^{T}G^{+}(x,\cdot)h)\right\|_{(L^{\infty}(B_{\rho}))^{n}}\|\mu\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}$
$\displaystyle\leq$
$\displaystyle\|G^{\pm}\|_{C^{2}(\overline{B_{\rho}}\times\overline{B_{\rho}})}\|h\|_{W^{1,\infty}(B_{\rho})}\|\mu\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}$
$\displaystyle\leq$
$\displaystyle\Cr{Cro}\|h\|_{W^{1,\infty}(B_{\rho})}\|\mu\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}$

for all $h\in W^{1,\infty}(B_{\rho})$ and
$\mu\in{\mathcal{M}}_{KR}(\overline{B_{\rho}})$, yielding 5.13.

By (3), ${\mathcal{G}}^{+}((1-w)\mu),{\mathcal{G}}^{-}(w\mu)\in
C^{2}(\overline{B_{\rho}})$ for $\mu\in{\mathcal{M}}^{+}(\overline{B_{\rho}})$
and $w:\overline{B_{\rho}}\to[0,1]$ Borel. Thanks to 5.11 the sum of these
functions is bounded below by a positive number if $\mu\neq 0$. Together with
the chain rule and the smoothness of $\psi$ away from $(0,0)$, this implies
(5) and completes the proof. ∎

Our next Lemma shows that under the assumption $\rho\in\left(0,{1}/{2}\right)$
any solution to 4.6 cannot attain values $0$ and $1$ anywhere in
$\overline{B_{\rho}}$.

######  Lemma 5.8.

Let $\rho\in\left(0,{1}/{2}\right)$. Let a pair of
$\mu\in({\mathcal{M}}^{+}(\overline{B_{\rho}}))\backslash\\{0\\}$ and a Borel
$w:\overline{B_{\rho}}\to[0,1]$ be a solution to 4.6 at every point
$x\in\overline{B_{\rho}}$. Then,

$\displaystyle w\in C^{2}(\overline{B_{\rho}}),$ $\displaystyle
0<\min_{\overline{B_{\rho}}}\leq\max_{\overline{B_{\rho}}}w<1.$

###### Proof.

By Lemma 5.7(5), $w={\mathcal{Y}}(\mu,w)\in C^{2}(\overline{B_{\rho}})$. In
particular, $w$ is continuous, so the required inequalities are satisfied
provided $w\in(0,1)$ in $\overline{B_{\rho}}$. Since

$\displaystyle\\{\psi=0\\}=\\{(a,b)\in[0,\infty)\times[0,\infty):\quad
a=0\\},$

we have

$\displaystyle\\{w=0\\}=$ $\displaystyle\\{{\mathcal{Y}}(\mu,w)=0\\}$
$\displaystyle=$ $\displaystyle\\{{\mathcal{G}}^{+}((1-w)\mu)=0\\}.$ (5.14)

Since $(1-w)\mu\in{\mathcal{M}}^{+}(\overline{B_{\rho}})$ and kernel $G^{+}$
is strictly positive in $\overline{B_{\rho}}\times\overline{B_{\rho}}$ (see
Lemma 5.6), 5.14 yields that either $\\{w=0\\}$ is empty or $w\equiv 0$ in
$\overline{B_{\rho}}$ and $(1-w)\mu=0$. The latter combination is impossible
since $\mu\neq 0$, so $w$ is nowhere zero. In the very same way one verifies
that $w$ is nowhere equal one.

∎

Next, we combine Lemmas 5.4 and 5.7 in order to obtain the following result on
Gateaux differentiability at every spatial point $x$.

######  Lemma 5.9.

Let $\rho\in\left(0,{1}/{2}\right)$. Let
$\mu_{0}\in({\mathcal{M}}^{+}(\overline{B_{\rho}}))\backslash\\{0\\}$ and
$w_{0}\in W^{1,\infty}(B_{\rho};(0,1))$. Then, functional
$(\mu,w)\mapsto{\mathcal{Y}}(\mu,w)(x)$ possesses Gateaux partial
differentials of any order with respect to $\mu$ and $w$ at $(\mu_{0},w_{0})$
along the directions from ${\mathcal{H}}_{\mu_{0}}$ and
$W^{1,\infty}(B_{\rho})$, respectively, for every $x\in\overline{B_{\rho}}$.
As functions of $x$, these Gateaux differentials belong to
$C^{2}(\overline{B_{\rho}})$ and formulas 5.6 are satisfied together with
inequalities

$\displaystyle\|(\partial_{w}{\mathcal{Y}}(\mu_{0},{w_{0}})(\cdot))h_{1}\|_{{W^{1,\infty}(B_{\rho})}}\leq$
$\displaystyle\Cr{CCro}\|h_{1}\|_{L^{\infty}(B_{\rho})},$ (5.15a)
$\displaystyle\|(\partial_{\mu}{\mathcal{Y}}(\mu_{0},{w_{0}})(\cdot))h_{\mu_{0}}\|_{{W^{1,\infty}(B_{\rho})}}\leq$
$\displaystyle\Cr{CCro}|\mu_{0}|^{-1}\left(1+\|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})}\right)\|h_{\mu_{0}}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})},$
(5.15b)
$\displaystyle\|(\partial_{w}(\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0})(\cdot))h_{1})h_{2}\|_{{W^{1,\infty}(B_{\rho})}}\leq$
$\displaystyle\Cr{CCro}\|h_{1}\|_{L^{\infty}(B_{\rho})}\|h_{2}\|_{L^{\infty}(B_{\rho})},$
(5.15c)
$\displaystyle\|(\partial_{\mu}(\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0})(\cdot))h_{1})h_{\mu_{0}}\|_{{W^{1,\infty}(B_{\rho})}}\leq$
$\displaystyle\Cr{CCro}|\mu_{0}|^{-1}\left(1+\|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})}\right)\|h_{1}\|_{W^{1,\infty}(B_{\rho})}\|h_{\mu_{0}}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}.$
(5.15d)

###### Proof.

Since $w_{0}\in W^{1,\infty}(B_{\rho};(0,1))\subset
C(\overline{B_{\rho}};(0,1))$, we have

$0<\min_{\overline{B_{\rho}}}w_{0}\leq\max_{\overline{B_{\rho}}}w_{0}<1,$

hence ${\mathcal{H}}_{w_{0}}=W^{1,\infty}(B_{\rho})$. Further, Lemma 5.7(1)
and (2) yield that $\mu_{0}\in{\mathcal{M}}_{G}^{+}(\overline{B_{\rho}})$,
5.10 holds, and
${\mathcal{H}}_{\mu_{0}}\cap{\mathcal{M}}_{G}(\overline{B_{\rho}})={\mathcal{H}}_{\mu_{0}}$.
These observations allow us to apply Lemma 5.4, yielding existence of the
Gateaux partial differentials of any order for functional
$(\mu,w)\to{\mathcal{Y}}(\mu,w)(x)$ and bounds 5.6 along the directions from
${\mathcal{H}}_{\mu_{0}}\times W^{1,\infty}(B_{\rho})$ for all
$x\in\overline{B_{\rho}}$. Furthermore, the resulting differentials are two
times continuously differentiable functions of $x$ due to the chain rule, the
smoothness of $\psi$ away from $(0,0)$, and the regularity provided by Lemma
5.7(3).

It remains to establish estimates 5.15. First, using 5.12 and 5.13 and the
assumptions on $w_{0},\mu_{0}$, and $h_{i}$, we find that

$\displaystyle\|{\mathcal{G}}^{+}((1-w_{0})\mu_{0})\|_{{W^{1,\infty}(B_{\rho})}}\leq$
$\displaystyle\Cr{Cro}\|1-w_{0}\|_{L^{\infty}(B_{\rho})}|\mu_{0}|$
$\displaystyle\leq$ $\displaystyle\Cr{Cro}|\mu_{0}|,$ (5.16a)
$\displaystyle\|{\mathcal{G}}^{-}(w_{0}\mu_{0})\|_{{W^{1,\infty}(B_{\rho})}}\leq$
$\displaystyle\Cr{Cro}\|w_{0}\|_{L^{\infty}(B_{\rho})}|\mu_{0}|$
$\displaystyle\leq$ $\displaystyle\Cr{Cro}|\mu_{0}|,$ (5.16b)
$\displaystyle\|{\mathcal{G}}^{\pm}(h_{i}\mu_{0})\|_{{W^{1,\infty}(B_{\rho})}}\leq$
$\displaystyle\Cr{Cro}\|h_{i}\|_{L^{\infty}(B_{\rho})}|\mu_{0}|,\qquad
i\in\\{1,2\\},$ (5.16c)
$\displaystyle\|{\mathcal{G}}^{+}((1-w_{0})h_{\mu_{0}})\|_{{W^{1,\infty}(B_{\rho})}}\leq$
$\displaystyle\Cr{CroG3}\|1-w_{0}\|_{W^{1,\infty}(B_{\rho})}\|h_{\mu_{0}}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}$
$\displaystyle\leq$ $\displaystyle\Cr{CroG3}\left(1+\|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})}\right)\|h_{\mu_{0}}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})},$
(5.16d)
$\displaystyle\|{\mathcal{G}}^{-}(w_{0}h_{\mu_{0}})\|_{{W^{1,\infty}(B_{\rho})}}\leq$
$\displaystyle\Cr{CroG3}\|w_{0}\|_{W^{1,\infty}(B_{\rho})}\|h_{\mu_{0}}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})},$
(5.16e)
$\displaystyle\|{\mathcal{G}}^{\pm}(h_{1}h_{\mu_{0}})\|_{{W^{1,\infty}(B_{\rho})}}\leq$
$\displaystyle\Cr{CroG3}\|h_{1}\|_{W^{1,\infty}(B_{\rho})}\|h_{\mu_{0}}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}.$
(5.16f)

Combining 5.3, 5.11, and 5.16 and using the product and chain rules where
necessary, we can estimate the directional derivatives from 5.6 in
$\overline{B_{\rho}}$ as follows:

$\displaystyle|(\partial_{w}{\mathcal{Y}}(\mu_{0},{w_{0}})(\cdot))h_{1}|$
$\displaystyle\leq$
$\displaystyle|D\psi({\mathcal{G}}^{+}((1-{{w_{0}}})\mu_{0}),{\mathcal{G}}^{-}({{w_{0}}}\mu_{0}))||(-{\mathcal{G}}^{+}(h_{1}\mu_{0}),{\mathcal{G}}^{-}(h_{1}\mu_{0}))|$
$\displaystyle\leq$ $\displaystyle\Cr{CCro}\|h_{1}\|_{L^{\infty}(B_{\rho})},$
(5.17a)
$\displaystyle|\nabla_{x}((\partial_{w}{\mathcal{Y}}(\mu_{0},{w_{0}})(\cdot))h_{1})|$
$\displaystyle\leq$
$\displaystyle|D^{2}\psi({\mathcal{G}}^{+}((1-{{w_{0}}})\mu_{0}),{\mathcal{G}}^{-}({{w_{0}}}\mu_{0}))|(|\nabla_{x}{\mathcal{G}}^{+}((1-w_{0})\mu_{0})|+|\nabla_{x}{\mathcal{G}}^{-}(w_{0}\mu_{0})|)|(-{\mathcal{G}}^{+}(h_{1}\mu_{0}),{\mathcal{G}}^{-}(h_{1}\mu_{0}))|$
$\displaystyle+|D\psi({\mathcal{G}}^{+}((1-{{w_{0}}})\mu_{0}),{\mathcal{G}}^{-}({{w_{0}}}\mu_{0}))||(-\nabla_{x}{\mathcal{G}}^{+}(h_{1}\mu_{0}),\nabla_{x}{\mathcal{G}}^{-}(h_{1}\mu_{0}))|$
$\displaystyle\leq$ $\displaystyle\Cr{CCro}\|h_{1}\|_{L^{\infty}(B_{\rho})},$
(5.17b)
$\displaystyle|\partial_{\mu}{\mathcal{Y}}(\mu_{0},{w_{0}})(\cdot))h_{\mu_{0}}|$
$\displaystyle\leq$
$\displaystyle|D\psi({\mathcal{G}}^{+}((1-{{w_{0}}})\mu_{0}),{\mathcal{G}}^{-}({{w_{0}}}\mu_{0}))||({\mathcal{G}}^{+}((1-{{w_{0}}})h_{\mu_{0}}),{\mathcal{G}}^{-}({w_{0}}h_{\mu_{0}}))|$
$\displaystyle\leq$ $\displaystyle\Cr{CCro}|\mu_{0}|^{-1}\left(1+\|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})}\right)\|h_{\mu_{0}}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})},$
(5.17c)
$\displaystyle|\nabla_{x}(\partial_{\mu}{\mathcal{Y}}(\mu_{0},{w_{0}})(\cdot))h_{\mu_{0}})|$
$\displaystyle\leq$
$\displaystyle|D^{2}\psi({\mathcal{G}}^{+}((1-{{w_{0}}})\mu_{0}),{\mathcal{G}}^{-}({{w_{0}}}\mu_{0}))|(|\nabla_{x}{\mathcal{G}}^{+}((1-w_{0})\mu_{0})|+|\nabla_{x}{\mathcal{G}}^{-}(w_{0}\mu_{0})|)$
$\displaystyle\cdot|({\mathcal{G}}^{+}((1-{{w_{0}}})h_{\mu_{0}}),{\mathcal{G}}^{-}({w_{0}}h_{\mu_{0}}))|$
$\displaystyle+|D\psi({\mathcal{G}}^{+}((1-{{w_{0}}})\mu_{0}),{\mathcal{G}}^{-}({{w_{0}}}\mu_{0}))||(\nabla_{x}{\mathcal{G}}^{+}((1-{{w_{0}}})h_{\mu_{0}}),\nabla_{x}{\mathcal{G}}^{-}({w_{0}}h_{\mu_{0}}))|$
$\displaystyle\leq$ $\displaystyle\Cr{CCro}|\mu_{0}|^{-1}\left(1+\|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})}\right)\|h_{\mu_{0}}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})},$
(5.17d)
$\displaystyle|(\partial_{w}(\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0})(\cdot))h_{1})h_{2}|$
$\displaystyle\leq$
$\displaystyle|D^{2}\psi({\mathcal{G}}^{+}((1-w_{0})\mu_{0}),{\mathcal{G}}^{-}(w_{0}\mu_{0}))||(-{\mathcal{G}}^{+}(h_{1}\mu_{0}),{\mathcal{G}}^{-}(h_{1}\mu_{0}))||(-{\mathcal{G}}^{+}(h_{2}\mu_{0}),{\mathcal{G}}^{-}(h_{2}\mu_{0}))|$
$\displaystyle\leq$
$\displaystyle\Cl{CCro}\|h_{1}\|_{L^{\infty}(B_{\rho})}\|h_{2}\|_{L^{\infty}(B_{\rho})},$
(5.17e)
$\displaystyle|\nabla_{x}((\partial_{w}(\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0})(\cdot))h_{1})h_{2})|$
$\displaystyle\leq$
$\displaystyle|D^{3}\psi({\mathcal{G}}^{+}((1-w_{0})\mu_{0}),{\mathcal{G}}^{-}(w_{0}\mu_{0}))|(|\nabla_{x}{\mathcal{G}}^{+}((1-w_{0})\mu_{0})|+|\nabla_{x}{\mathcal{G}}^{-}(w_{0}\mu_{0})|)|(-{\mathcal{G}}^{+}(h_{1}\mu_{0}),{\mathcal{G}}^{-}(h_{1}\mu_{0}))|$
$\displaystyle\cdot|(-{\mathcal{G}}^{+}(h_{2}\mu_{0}),{\mathcal{G}}^{-}(h_{2}\mu_{0}))|+|D^{2}\psi({\mathcal{G}}^{+}((1-w_{0})\mu_{0}),{\mathcal{G}}^{-}(w_{0}\mu_{0}))|$
$\displaystyle\cdot(|(-{\mathcal{G}}^{+}(h_{1}\mu_{0}),{\mathcal{G}}^{-}(h_{1}\mu_{0}))||(-\nabla_{x}{\mathcal{G}}^{+}(h_{2}\mu_{0}),\nabla_{x}{\mathcal{G}}^{-}(h_{2}\mu_{0}))|$
$\displaystyle\phantom{\cdot(}+|(-\nabla_{x}{\mathcal{G}}^{+}(h_{1}\mu_{0}),\nabla_{x}{\mathcal{G}}^{-}(h_{1}\mu_{0}))||(-{\mathcal{G}}^{+}(h_{2}\mu_{0}),{\mathcal{G}}^{-}(h_{2}\mu_{0}))|)$
$\displaystyle\leq$
$\displaystyle\Cr{CCro}\|h_{1}\|_{L^{\infty}(B_{\rho})}\|h_{2}\|_{L^{\infty}(B_{\rho})},$
(5.17f)
$\displaystyle|(\partial_{\mu}(\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0})(\cdot))h_{1})h_{\mu_{0}}|$
$\displaystyle\leq$
$\displaystyle|D^{2}\psi({\mathcal{G}}^{+}((1-w_{0})\mu_{0}),{\mathcal{G}}^{-}(w_{0}\mu_{0}))||(-{\mathcal{G}}^{+}(h_{1}\mu_{0}),{\mathcal{G}}^{-}(h_{1}\mu_{0}))||({\mathcal{G}}^{+}((1-w_{0})h_{\mu_{0}}),{\mathcal{G}}^{-}(wh_{\mu_{0}}))|$
$\displaystyle+|D\psi({\mathcal{G}}^{+}((1-w_{0})\mu_{0}),{\mathcal{G}}^{-}(w_{0}\mu_{0}))||(-{\mathcal{G}}^{+}(h_{1}h_{\mu_{0}}),{\mathcal{G}}^{-}(h_{1}h_{\mu_{0}}))|$
$\displaystyle\leq$ $\displaystyle\Cr{CCro}|\mu_{0}|^{-1}\left(1+\|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})}\right)\|h_{1}\|_{W^{1,\infty}(B_{\rho})}\|h_{\mu_{0}}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})},$
(5.17g) and
$\displaystyle|\nabla_{x}((\partial_{\mu}(\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0})(\cdot))h_{1})h_{\mu_{0}})|$
$\displaystyle\leq$
$\displaystyle|D^{3}\psi({\mathcal{G}}^{+}((1-w_{0})\mu_{0}),{\mathcal{G}}^{-}(w_{0}\mu_{0}))|(|\nabla_{x}{\mathcal{G}}^{+}((1-w_{0})\mu_{0})|+|\nabla_{x}{\mathcal{G}}^{-}(w_{0}\mu_{0})|)|(-{\mathcal{G}}^{+}(h_{1}\mu_{0}),{\mathcal{G}}^{-}(h_{1}\mu_{0}))|$
$\displaystyle\cdot|{\mathcal{G}}^{+}((1-w_{0})h_{\mu_{0}}),{\mathcal{G}}^{-}(wh_{\mu_{0}})|+|D^{2}\psi({\mathcal{G}}^{+}((1-w_{0})\mu_{0}),{\mathcal{G}}^{-}(w_{0}\mu_{0}))|$
$\displaystyle\cdot(|(-{\mathcal{G}}^{+}(h_{1}\mu_{0}),{\mathcal{G}}^{-}(h_{1}\mu_{0}))||(\nabla_{x}{\mathcal{G}}^{+}((1-w_{0})h_{\mu_{0}}),\nabla_{x}{\mathcal{G}}^{-}(wh_{\mu_{0}}))|$
$\displaystyle\phantom{\cdot(}+|(-\nabla_{x}{\mathcal{G}}^{+}(h_{1}\mu_{0}),\nabla_{x}{\mathcal{G}}^{-}(h_{1}\mu_{0}))||({\mathcal{G}}^{+}((1-w_{0})h_{\mu_{0}}),{\mathcal{G}}^{-}(wh_{\mu_{0}}))|)$
$\displaystyle+|D\psi({\mathcal{G}}^{+}((1-w_{0})\mu_{0}),{\mathcal{G}}^{-}(w_{0}\mu_{0}))||(-\nabla_{x}{\mathcal{G}}^{+}(h_{1}h_{\mu_{0}}),\nabla_{x}{\mathcal{G}}^{-}(h_{1}h_{\mu_{0}}))|$
$\displaystyle+|D^{2}\psi({\mathcal{G}}^{+}((1-w_{0})\mu_{0}),{\mathcal{G}}^{-}(w_{0}\mu_{0}))|(|\nabla_{x}{\mathcal{G}}^{+}((1-w_{0})\mu_{0})|+|\nabla_{x}{\mathcal{G}}^{-}(w_{0}\mu_{0})|)$
$\displaystyle\cdot|(-{\mathcal{G}}^{+}(h_{1}h_{\mu_{0}}),{\mathcal{G}}^{-}(h_{1}h_{\mu_{0}}))|$
$\displaystyle\leq$ $\displaystyle\Cr{CCro}|\mu_{0}|^{-1}\left(1+\|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})}\right)\|h_{1}\|_{W^{1,\infty}(B_{\rho})}\|h_{\mu_{0}}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}.$
(5.17h)

Combining 5.17, we arrive at 5.15.

∎

The next Lemma deals with the Fréchet partial derivative of ${\mathcal{Y}}$
with respect to $w$. In particular, we establish an estimate (see 5.19 below)
that we use in Subsection 5.3 in order to prove our local existence result,
Theorem 4.5.

######  Lemma 5.10.

Let $\rho\in\left(0,{1}/{2}\right)$. Then, operator ${\mathcal{Y}}$ is
continuously partially Fréchet differentiable with respect to $w$ at each
$(\mu_{0},w_{0})\in(({\mathcal{M}}^{+}(\overline{B_{\rho}}))\backslash\\{0\\})\times
W^{1,\infty}(B_{\rho};(0,1))$. The Fréchet partial derivative is a compact
operator given by

$\displaystyle\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0})h=$
$\displaystyle(\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0})(\cdot))h\qquad{\text{in
}\overline{B_{\rho}}}\qquad\text{for }h\in W^{1,\infty}(B_{\rho})$ (5.18)

and satisfies the following estimate: for all

$\displaystyle w_{1},w_{2}\in W^{1,\infty}(B_{\rho};(0,1)),$
$\displaystyle\mu_{1},\mu_{2}\in({\mathcal{M}}^{+}(\overline{B_{\rho}}))\backslash\\{0\\}\qquad\text{such
that }|\mu_{1}|=|\mu_{2}|=|\mu_{0}|,$ $\displaystyle h\in
W^{1,\infty}(B_{\rho})$

it holds that

$\displaystyle\left\|(\partial_{w}{\mathcal{Y}}(\mu_{1},w_{1})-\partial_{w}{\mathcal{Y}}(\mu_{2},w_{2}))h\right\|_{{W^{1,\infty}(B_{\rho})}}$
$\displaystyle\leq$ $\displaystyle\Cr{CCro}|\mu_{0}|^{-1}\left(1+\|\nabla
w_{1}\|_{L^{\infty}(B_{\rho})}\right)\|h\|_{W^{1,\infty}(B_{\rho})}\|\mu_{1}-\mu_{2}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}+\Cr{CCro}\|h\|_{L^{\infty}(B_{\rho})}\|w_{1}-w_{2}\|_{W^{1,\infty}(B_{\rho})}.$
(5.19)

###### Proof.

######  Step 1.

Let $h\in W^{1,\infty}(B_{\rho})$ be such that
$\|h\|_{L^{\infty}(B_{\rho})}<\min\\{\min w_{0},1-\max w_{0}\\}$. Thanks to
Lemma 5.9, function $s\mapsto{\mathcal{Y}}(\mu_{0},w_{0}+sh)(x)$ is two times
continuously differentiable on $[0,1]$. By Taylor’s theorem for real
functions, we then have

$\displaystyle{\mathcal{Y}}(\mu_{0},w_{0}+h)(x)-{\mathcal{Y}}(\mu_{0},w_{0})(x)-(\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0})(x))h$
$\displaystyle=$
$\displaystyle{\mathcal{Y}}(\mu_{0},w_{0}+h)(x)-{\mathcal{Y}}(\mu_{0},w_{0})(x)-\frac{d}{ds}{\mathcal{Y}}(\mu_{0},w_{0}+sh)(x)|_{s=0}$
$\displaystyle=$
$\displaystyle\int_{0}^{1}(1-s)\frac{d^{2}}{ds^{2}}{\mathcal{Y}}(\mu_{0},w_{0}+sh)(x)\,ds$
$\displaystyle=$
$\displaystyle\int_{0}^{1}(1-s)(\partial_{w}(\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0}+sh)(x))h)h\,ds.$
(5.20)

Using 5.6c, we compute the integrand on the right-hand side of 5.20:

$\displaystyle(1-s)(\partial_{w}(\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0}+sh)(x))h)h$
$\displaystyle=$
$\displaystyle(1-s)D^{2}\psi({\mathcal{G}}^{+}((1-{(w_{0}+sh)})\mu_{0}),{\mathcal{G}}^{-}((w_{0}+sh)\mu_{0}))(-{\mathcal{G}}^{+}(h\mu_{0}),{\mathcal{G}}^{-}(h\mu_{0}))^{T}$
$\displaystyle\cdot(-{\mathcal{G}}^{+}(h\mu_{0}),{\mathcal{G}}^{-}(h\mu_{0}))^{T}(x).$
(5.21)

Exploiting the regularity of $\psi$ and the linearity and regularity of
${\mathcal{G}}^{\pm}$, we conclude that the right-hand side of 5.21 and its
spatial gradient are continuous functions of $x$ and $s$. Consequently, the
spatial gradient of the right-hand side of 5.20 is the integral of the
gradient of the integrand. Together with 5.15c and 5.20 this implies

$\displaystyle\left\|{\mathcal{Y}}(\mu_{0},w_{0}+h)-{\mathcal{Y}}(\mu_{0},w_{0})-(\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0})(\cdot))h\right\|_{{W^{1,\infty}(B_{\rho})}}$
$\displaystyle=$
$\displaystyle\left\|\int_{0}^{1}(1-s)(\partial_{w}(\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0}+sh)(\cdot))h)h\,ds\right\|_{{W^{1,\infty}(B_{\rho})}}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}(1-s)\left\|(\partial_{w}(\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0}+sh)(\cdot))h)h\right\|_{{W^{1,\infty}(B_{\rho})}}\,ds$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}(1-s)\Cr{CCro}\|h\|_{L^{\infty}(B_{\rho})}^{2}\,ds$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\Cr{CCro}\|h\|_{L^{\infty}(B_{\rho})}^{2}$
$\displaystyle=$ $\displaystyle o\left(\|h\|_{W^{1,\infty}(B_{\rho})}\right).$
(5.22)

From 5.6a it is clear that
$h\mapsto(\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0})(\cdot))h$ is a linear
operator. It is also continuous by 5.15a. By 5.22, this operator is then the
Fréchet derivative of $(\mu_{0},w_{0})$ with respect to $w$ in
$W^{1,\infty}(B_{\rho})$.

Furthermore, formula 5.6a shows that
$\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0})$ is a superposition of
multiplication by a smooth function and an integral operator with a smooth
kernel. As in several instances above, this is due to smoothness of $G^{\pm}$
and $\psi$ by Lemmas 5.6 and 5.2, respectively. Therefore, it is a
superposition of bounded and compact operators, and as such, it is compact.

######  Step 2.

Finally, we derive estimate 5.19. Using smoothness in the Gateaux sense at
every point $x\in\overline{B_{\rho}}$ as provided by Lemma 5.9 and once again
Taylor’s theorem for real functions, we compute for every
$x\in\overline{B_{\rho}}$ that

$\displaystyle(\partial_{w}{\mathcal{Y}}(\mu_{1},w_{1}){(x)}-\partial_{w}{\mathcal{Y}}(\mu_{2},w_{2}){(x)})h$
$\displaystyle=$
$\displaystyle(\partial_{w}{\mathcal{Y}}(\mu_{1},w_{1}){(x)}-\partial_{w}{\mathcal{Y}}(\mu_{2},w_{1}){(x)})h+(\partial_{w}{\mathcal{Y}}(\mu_{2},w_{1}){(x)}-\partial_{w}{\mathcal{Y}}(\mu_{2},w_{2}){(x)})h$
$\displaystyle=$
$\displaystyle\int_{0}^{1}(\partial_{\mu}(\partial_{w}{\mathcal{Y}}(\mu(s),w_{1}){(x)})h)h_{\mu}\,ds+\int_{0}^{1}(\partial_{w}(\partial_{w}{\mathcal{Y}}(\mu_{2},w(s)){(x)})h)h_{w}\,ds,$
(5.23)

where

$\displaystyle h_{w}=w_{1}-w_{2},$ $\displaystyle\qquad
h_{\mu}=\mu_{1}-\mu_{2},$ $\displaystyle w(s)=(1-s)w_{1}+sw_{2},$
$\displaystyle\qquad\mu(s)=(1-s)\mu_{1}+s\mu_{2}.$

Combining 5.23 with 5.15c and 5.15d, we can estimate as follows:

$\displaystyle\left\|(\partial_{w}{\mathcal{Y}}(\mu_{1},w_{1})(\cdot)-\partial_{w}{\mathcal{Y}}(\mu_{2},w_{2})(\cdot))h\right\|_{{W^{1,\infty}(B_{\rho})}}$
$\displaystyle\leq$
$\displaystyle\left\|\int_{0}^{1}(\partial_{\mu}(\partial_{w}{\mathcal{Y}}(\mu(s),w_{1})(\cdot))h)h_{\mu}\,ds\right\|_{{W^{1,\infty}(B_{\rho})}}+\left\|\int_{0}^{1}(\partial_{w}(\partial_{w}{\mathcal{Y}}(\mu_{2},w(s))(\cdot))h)h_{w}\,ds\right\|_{{W^{1,\infty}(B_{\rho})}}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}\Cr{CCro}|\mu(s)|^{-1}\left(1+\|\nabla
w_{1}\|_{L^{\infty}(B_{\rho})}\right)\|h\|_{W^{1,\infty}(B_{\rho})}\|h_{\mu}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}{\,ds}+\Cr{CCro}\|h\|_{L^{\infty}(B_{\rho})}\|h_{w}\|_{L^{\infty}(B_{\rho})}$
$\displaystyle=$ $\displaystyle\Cr{CCro}|\mu_{0}|^{-1}\left(1+\|\nabla
w_{1}\|_{L^{\infty}(B_{\rho})}\right)\|h\|_{W^{1,\infty}(B_{\rho})}\|h_{\mu}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}+\Cr{CCro}\|h\|_{L^{\infty}(B_{\rho})}\|h_{w}\|_{L^{\infty}(B_{\rho})},$

so 5.19 holds. It remains to observe that this estimate directly implies
continuity of the Fréchet derivative $\partial_{w}{\mathcal{Y}}$.

∎

We conclude this Subsection with a result for the case of a spatial domain
extension.

######  Lemma 5.11.

Let $0<\rho_{0}<\rho<{1}/{2}$. Let $(\mu_{0},w_{0})$ be such that

$\displaystyle\mu_{0}\in({\mathcal{M}}^{+}(\overline{B_{\rho_{0}}}))\backslash\\{0\\},$
$\displaystyle w_{0}\in W^{1,\infty}(B_{\rho_{0}};(0,1)),$
$\displaystyle(\mu_{0},w_{0})\text{ satisfies
\lx@cref{creftype~refnum}{LambdaEqmu} pointwise in }\overline{B_{\rho_{0}}}.$
(5.24)

Define extensions

$\displaystyle\mu_{0}:=$ $\displaystyle 0\qquad\text{in
}\overline{B_{\rho}}\backslash\overline{B_{\rho_{0}}},$ (5.25a)
$\displaystyle\overline{w}_{0}:=$ $\displaystyle\begin{cases}w_{0}&\text{in
}\overline{B_{\rho_{0}}},\\\ 0&\text{in
}\overline{B_{\rho}}\backslash\overline{B_{\rho_{0}}},\end{cases}$ (5.25b)
$\displaystyle w_{0}:=$
$\displaystyle{\mathcal{Y}}_{\rho}(\mu_{0},\overline{w}_{0})\qquad\text{in
}\overline{B_{\rho}}\backslash\overline{B_{\rho_{0}}}$ (5.25c)

and operators

$\displaystyle{\mathcal{X}}_{r}:=$ $\displaystyle
id-\partial_{w}{\mathcal{Y}}_{r}(\mu_{0},w_{0})\qquad\text{in
}W^{1,\infty}(B_{r})\qquad\text{for }r\in\\{\rho,\rho_{0}\\},$

where ${\mathcal{Y}}_{r}$ denotes the realisation of ${\mathcal{Y}}$
corresponding to the spatial domain $\overline{B_{r}}$. Then, $w_{0}\in
C^{2}(\overline{B_{\rho}})$ and equation 4.6 is satisfied pointwise in
$\overline{B_{\rho}}$. Furthermore, ${\mathcal{X}}_{\rho}$ is invertible if
and only if ${\mathcal{X}}_{\rho_{0}}$ is invertible.

###### Proof.

Since $\rho<1/2$, ${\mathcal{Y}}(\mu_{0},\overline{w}_{0})\in
C^{2}(\overline{B_{\rho}})$ by Lemma 5.7(5). With 5.25c, 5.24, and 5.25b we
deduce that 4.6 is satisfied in $\overline{B_{\rho}}$.

It remains to verify the equivalence of invertibility in
$W^{1,\infty}(B_{\rho})$ and $W^{1,\infty}(B_{\rho_{0}})$. By Lemma 5.10,
${\mathcal{X}}_{\rho_{0}}$ and ${\mathcal{X}}_{\rho}$ are Fredholm. Hence, it
suffices to check

$\displaystyle\text{Ker}_{W^{1,\infty}(B_{\rho})}({\mathcal{X}}_{\rho})\neq\\{0\\}\qquad\Leftrightarrow\qquad\text{Ker}_{W^{1,\infty}(B_{\rho_{0}})}({\mathcal{X}}_{\rho_{0}})\neq\\{0\\}.$
(5.26)

Combining 5.6a, 5.18, and 5.25a, we obtain for all $h,h_{1},h_{2}\in
W^{1,\infty}(B_{\rho})$ with $h_{1}=h_{2}$ in $\overline{B_{\rho_{0}}}$ that

$\displaystyle\partial_{w}{\mathcal{Y}}_{\rho}(\mu_{0},w_{0})h=$
$\displaystyle\partial_{w}{\mathcal{Y}}_{\rho_{0}}(\mu_{0},w_{0})h$
$\displaystyle\qquad\text{in }\overline{B_{\rho_{0}}},$ (5.27a)
$\displaystyle\partial_{w}{\mathcal{Y}}_{\rho}(\mu_{0},w_{0})h_{1}=$
$\displaystyle\partial_{w}{\mathcal{Y}}_{\rho}(\mu_{0},w_{0})h_{2}$
$\displaystyle\qquad\text{in }\overline{B_{\rho}}.$ (5.27b)

With 5.27a it follows directly that if
$h\in\text{Ker}_{W^{1,\infty}(B_{\rho})}({\mathcal{X}}_{\rho})$, then
$h\in\text{Ker}_{W^{1,\infty}(B_{\rho_{0}})}({\mathcal{X}}_{\rho_{0}})$ as
well. Conversely, let

$\displaystyle
h_{0}\in\text{Ker}_{W^{1,\infty}(B_{\rho_{0}})}({\mathcal{X}}_{\rho_{0}}).$
(5.28)

Choose $\overline{h}_{0}\in W^{1,\infty}(B_{\rho})$ such that it coincides
with $h_{0}$ on $\overline{B_{\rho_{0}}}$ and set

$\displaystyle\widetilde{h}_{0}:=$
$\displaystyle\partial_{w}{\mathcal{Y}}_{\rho}(\mu_{0},w_{0})\overline{h}_{0}\qquad\text{in
}\overline{B_{\rho}}.$ (5.29)

Together, 5.27a, 5.29, and 5.28 imply

$\displaystyle\widetilde{h}_{0}=$
$\displaystyle\partial_{w}{\mathcal{Y}}_{\rho}(\mu_{0},w_{0})\overline{h}_{0}$
$\displaystyle=$
$\displaystyle\partial_{w}{\mathcal{Y}}_{\rho_{0}}(\mu_{0},w_{0})h_{0}$
$\displaystyle=$ $\displaystyle h_{0}$ $\displaystyle=$
$\displaystyle\overline{h}_{0}$ $\displaystyle\qquad\text{in
}\overline{B_{\rho_{0}}}.$ (5.30)

Finally, 5.30, 5.27b, and 5.29 yield

$\displaystyle\widetilde{h}_{0}=$
$\displaystyle\partial_{w}{\mathcal{Y}}_{\rho}(\mu_{0},w_{0})\overline{h}_{0}$
$\displaystyle=$
$\displaystyle\partial_{w}{\mathcal{Y}}_{\rho}(\mu_{0},w_{0})\widetilde{h}_{0}$
$\displaystyle\qquad\text{in }\overline{B_{\rho}},$

meaning that
$\widetilde{h}_{0}\in\text{Ker}_{W^{1,\infty}(B_{\rho})}({\mathcal{X}}_{\rho})$.
Thus, we have proved that 5.26 holds. ∎

###  5.3 Local well-posedness of 4.6 for ${\Omega}$ a small ball (proof of
Theorem 4.5)

In this Subsection we use the findings in Subsections 5.2 and 5.1 to prove
Theorem 4.5 on local well-posedness of 4.6. Our proof is based on an argument
similar to the one used in the proof of the implicit mapping theorem, see the
discussion in Section 2(IE5).

###### Proof of Theorem 4.5.

Define constants

$\displaystyle\Cl{Cw0}:=$
$\displaystyle\frac{1}{2}\min\left\\{\min_{\overline{B_{\rho}}}w_{0},1-\max_{\overline{B_{\rho}}}w_{0}\right\\},$
(5.31a) $\displaystyle\Cl{CC6}:=$
$\displaystyle\left\|{\mathcal{X}}^{-1}\right\|_{{L(W^{1,\infty}(B_{\rho}))}}\Cr{CCro},$
(5.31b) $\displaystyle\Cr{R1}:=$
$\displaystyle\min\left\\{\left(1+2\Cr{CC6}\right)^{-1},\Cr{Cw0}\right\\},$
(5.31c) $\displaystyle\Cr{R2}:=$
$\displaystyle\frac{1}{2}\Cr{CC6}^{-1}\Cr{R1}|\mu_{0}|\left(1+\|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})}\right)^{-1},$ (5.31d)
$\displaystyle\Cr{F-contin-mu}:=$
$\displaystyle\frac{1}{2}\Cr{CC6}|\mu_{0}|^{-1}\left(1+\|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})}+\Cr{R1}\right).$ (5.31e)

Consider operator

$\displaystyle\Cl[WW]{OpWu}(\mu,w):=w-{\mathcal{X}}^{-1}(w-{\mathcal{Y}}(\mu,w)).$

Obviously, solutions of 4.6 are exactly the solutions of

$\displaystyle w=\Cr{OpWu}(\mu,w).$ (5.32)

In order to solve 5.32, we prove that $\Cr{OpWu}$ is a contraction in its
second variable and subsequently use Banach’s fixed point theorem.

######  Step 1.

To begin with, we establish the following estimates for all
$\mu,\mu_{1},\mu_{2}\in\Cr{SU2}$ and $w,w_{1},w_{2}\in\Cr{W3}$:

$\displaystyle\|\Cr{OpWu}(\mu,w_{2})-\Cr{OpWu}(\mu,w_{1})\|_{W^{1,\infty}(B_{\rho})}\leq$
$\displaystyle\frac{1}{2}\|w_{1}-w_{2}\|_{W^{1,\infty}(B_{\rho})},$ (5.33a)
$\displaystyle\|\Cr{OpWu}(\mu_{1},w)-\Cr{OpWu}(\mu_{2},w)\|_{W^{1,\infty}(B_{\rho})}\leq$
$\displaystyle\frac{1}{2}\Cr{F-contin-
mu}\|\mu_{1}-\mu_{2}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})},$ (5.33b)
$\displaystyle\|\Cr{OpWu}(\mu,w)-w_{0}\|_{W^{1,\infty}(B_{\rho})}\leq$
$\displaystyle\Cr{R1}.$ (5.33c)

Using the generalised Taylor’s theorem [31, Chapter 4, §4.6, Theorem 4.A] and
chain rule [31, Chapter 4, §4.3, Proposition 4.10], Lemma 5.10, and the
definition of $\Cr{OpWu}$, we compute

$\displaystyle\Cr{OpWu}(\mu,w_{2})-\Cr{OpWu}(\mu,w_{1})=$
$\displaystyle\int_{0}^{1}\partial_{w}\Cr{OpWu}(\mu,w(s_{1}))h_{1}\,ds_{1}$
$\displaystyle=$
$\displaystyle\int_{0}^{1}{\mathcal{X}}^{-1}(\partial_{w}{\mathcal{Y}}(\mu,w(s_{1}))-\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0}))h_{1}\,ds_{1}\qquad\text{in
}W^{1,\infty}(B_{\rho}),$ (5.34a)
$\displaystyle\Cr{OpWu}(\mu_{1},w)-\Cr{OpWu}(\mu_{2},w)=$
$\displaystyle\int_{0}^{1}\partial_{\mu}\Cr{OpWu}(\mu(s_{1}),w)h_{\mu}\,ds_{1}$
$\displaystyle=$
$\displaystyle\int_{0}^{1}{\mathcal{X}}^{-1}\partial_{\mu}{\mathcal{Y}}(\mu(s_{1}),w)h_{\mu}\,ds_{1}\qquad\text{in
}W^{1,\infty}(B_{\rho}),$ (5.34b)

where

$\displaystyle h_{1}:=w_{2}-w_{1},\qquad$ $\displaystyle
h_{\mu}:=\mu_{1}-\mu_{2},$ (5.35a) $\displaystyle
w(s_{1}):=(1-s_{1})w_{1}+s_{1}w_{2},\qquad$
$\displaystyle\mu(s_{1}):=(1-s_{1})\mu_{1}+\mu_{2}.$ (5.35b)

Combining 5.19, 5.35, 5.31, and 4.5, we can estimate the right-hand side of
5.34a using, in particular, convexity of $\Cr{W3}$ and $\Cr{SU2}$, and obtain

$\displaystyle\|\Cr{OpWu}(\mu,w_{2})-\Cr{OpWu}(\mu,w_{1})\|_{W^{1,\infty}(B_{\rho})}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}\left\|{\mathcal{X}}^{-1}\right\|_{{L(W^{1,\infty}(B_{\rho}))}}\|(\partial_{w}{\mathcal{Y}}(\mu,w(s_{1}))-\partial_{w}{\mathcal{Y}}(\mu_{0},w_{0}))h_{1}\|_{W^{1,\infty}(B_{\rho})}\,ds_{1}$
$\displaystyle\leq$ $\displaystyle\Cr{CC6}|\mu_{0}|^{-1}\left(1+\|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})}\right)\|w_{1}-w_{2}\|_{W^{1,\infty}(B_{\rho})}\|\mu-\mu_{0}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}$
$\displaystyle+\Cr{CC6}\|w_{1}-w_{2}\|_{{L^{\infty}(B_{\rho})}}\int_{0}^{1}\|w(s_{1})-w_{0}\|_{W^{1,\infty}(B_{\rho})}\,ds_{1}$
$\displaystyle\leq$ $\displaystyle\Cr{CC6}\left(|\mu_{0}|^{-1}\left(1+\|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})}\right)\Cr{R2}+\Cr{R1}\right)\|w_{1}-w_{2}\|_{W^{1,\infty}(B_{\rho})}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\|w_{1}-w_{2}\|_{W^{1,\infty}(B_{\rho})},$

so 5.33a holds. In a similar fashion, this time relying on 5.15a and 5.18
rather than 5.19, we estimate the right-hand side of 5.34b, concluding that

$\displaystyle\|\Cr{OpWu}(\mu_{1},w)-\Cr{OpWu}(\mu_{2},w)\|_{W^{1,\infty}(B_{\rho})}$
$\displaystyle\leq$
$\displaystyle\left\|{\mathcal{X}}^{-1}\right\|_{{L(W^{1,\infty}(B_{\rho}))}}\int_{0}^{1}\|\partial_{\mu}{\mathcal{Y}}(\mu(s_{1}),w)h_{\mu}\|_{W^{1,\infty}(B_{\rho})}\,ds_{1}$
$\displaystyle\leq$
$\displaystyle\Cr{CC6}\int_{0}^{1}|\mu(s_{1})|^{-1}\,ds_{1}\left(1+\|\nabla
w\|_{L^{\infty}(B_{\rho})}\right)\|h_{\mu}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}$
$\displaystyle=$ $\displaystyle\Cr{CC6}|\mu_{0}|^{-1}\left(1+\|\nabla
w\|_{L^{\infty}(B_{\rho})}\right)\|\mu_{1}-\mu_{2}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}$
(5.36) $\displaystyle\leq$
$\displaystyle\Cr{CC6}|\mu_{0}|^{-1}\left(1+\|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})}+\Cr{R1}\right)\|\mu_{1}-\mu_{2}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\Cr{F-contin-
mu}\|\mu_{1}-\mu_{2}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})},$

so 5.33b holds too. Finally, we obtain 5.33c using 5.33a, 5.36, and 5.31d and
the assumption that $(\mu_{0},w_{0})$ solves 4.6:

$\displaystyle\|\Cr{OpWu}(\mu,w)-w_{0}\|_{W^{1,\infty}(B_{\rho})}$
$\displaystyle=$
$\displaystyle\|\Cr{OpWu}(\mu,w)-\Cr{OpWu}(\mu_{0},w_{0})\|_{W^{1,\infty}(B_{\rho})}$
$\displaystyle\leq$
$\displaystyle\|\Cr{OpWu}(\mu,w)-\Cr{OpWu}(\mu,w_{0})\|_{W^{1,\infty}(B_{\rho})}+\|\Cr{OpWu}(\mu,w_{0})-\Cr{OpWu}(\mu_{0},w_{0})\|_{W^{1,\infty}(B_{\rho})}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\|w-w_{0}\|_{W^{1,\infty}(B_{\rho})}+\Cr{CC6}|\mu_{0}|^{-1}\left(1+\|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})}\right)\|\mu-\mu_{0}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\Cr{R1}+\Cr{CC6}|\mu_{0}|^{-1}\left(1+\|\nabla
w_{0}\|_{L^{\infty}(B_{\rho})}\right)\Cr{R2}$ $\displaystyle=$
$\displaystyle\Cr{R1}.$

######  Step 2.

Let $\mu\in\Cr{SU2}$. By Lemma 5.7(5), ${\mathcal{Y}}$ and, hence,
$\Cr{OpWu}(\mu,\cdot)$, are well-defined in $\Cr{W3}$. Furthermore, due to
5.33c, 5.31a, and 5.31c, $\Cr{OpWu}(\mu,\cdot)$ is a self-map and, due to
5.33a, a contraction in this closed ball of $W^{1,\infty}(B_{\rho})$. Applying
Banach’s fixed point theorem, we infer the existence of a unique fixed point
$w$. As mentioned at the beginning of the proof, $(\mu,w)$ is then a solution
to 4.6. Invoking Lemma 5.8, we find that $w\in
C^{2}(\overline{B_{\rho}};(0,1))$. Altogether, we conclude that the solution
map

$\displaystyle\Cr{OpW1}:\Cr{SU2}\to\Cr{W3}\cap
C^{2}(\overline{B_{\rho}};(0,1)),\qquad\Cr{OpW1}(\mu):=w$

is well-defined.

It remains to verify 4.8. Consider $\mu_{1},\mu_{2}\in\Cr{SU2}$ and
$w_{1}=\Cr{OpW1}(\mu_{1})$ and $w_{2}=\Cr{OpW1}(\mu_{2})$. Combining 5.33a and
5.33b, we obtain

$\displaystyle\|\Cr{OpW1}(\mu_{1})-\Cr{OpW1}(\mu_{2})\|_{W^{1,\infty}(B_{\rho})}$
$\displaystyle=$
$\displaystyle\|\Cr{OpWu}(\mu_{1},w_{1})-\Cr{OpWu}(\mu_{2},w_{2})\|_{W^{1,\infty}(B_{\rho})}$
$\displaystyle\leq$
$\displaystyle\|\Cr{OpWu}(\mu_{1},w_{1})-\Cr{OpWu}(\mu_{1},w_{2})\|_{W^{1,\infty}(B_{\rho})}+\|\Cr{OpWu}(\mu_{1},w_{2})-\Cr{OpWu}(\mu_{2},w_{2})\|_{W^{1,\infty}(B_{\rho})}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\|w_{1}-w_{2}\|_{W^{1,\infty}(B_{\rho})}+\frac{1}{2}\Cr{F-contin-
mu}\|\mu_{1}-\mu_{2}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})}.$

This gives

$\displaystyle\|\Cr{OpW1}(\mu_{1})-\Cr{OpW1}(\mu_{2})\|_{W^{1,\infty}(B_{\rho})}\leq$
$\displaystyle\Cr{F-contin-
mu}\|\mu_{1}-\mu_{2}\|_{{\mathcal{M}}_{KR}(\overline{B_{\rho}})},$

so estimate 4.8 holds.

∎

We conclude this Subsection with a proof of Theorem 4.9.

###### Proof of Theorem 4.9.

Since $\nu\leq\mu$, the Radon-Nikodym theorem provides the existence of a
Radon-Nykodym derivative $\frac{d\nu}{d\mu}$, whereas the Lebesgue
differentiation theorem implies that $w$ takes values in $[0,1]$, $\mu$-a.e.
Since $(\mu,\nu)$ solves 1.2, $(\mu,\frac{d\nu}{d\mu})$ then solves 1.7 by
definition.

∎

###  5.4 Solvability of 4.6 for $\mu_{0}$ a point mass

Let us consider a special case when $\mu_{0}$ is a point mass. As seen below,
in this situation, equation 4.6 is explicitly solvable and, moreover, the
solution pair $(\mu_{0},w_{0})$ satisfies 4.4.

######  Example 5.12 ($\mu_{0}$ point mass).

Let $\rho\in(0,1)$, $m>0$,

$\displaystyle\mu_{0}:=m\delta_{0},$

and $w_{0}:\mathbb{R}^{d}\to[0,1]$ be Borel. Using

$\displaystyle\operatorname{supp}(\mu_{0})=\\{0\\}$

and the linearity of ${\mathcal{G}}^{\pm}$, we compute

$\displaystyle{\mathcal{Y}}(\mu_{0},w_{0})=$
$\displaystyle\frac{(1-{w_{0}}{(0)}){\mathcal{G}}^{+}\delta_{0}}{(1-{w_{0}}{(0)}){\mathcal{G}}^{+}\delta_{0}+{w_{0}}{(0)}{\mathcal{G}}^{-}\delta_{0}}\qquad\text{in
}\overline{B_{\rho}},$ (5.37)

where

$\displaystyle{\mathcal{G}}^{\pm}\delta_{0}=G^{\pm}(\cdot,0)>0.$

due to assumptions 1.3e, 1.3f, 1.3g, and 1.3h and $\rho<1$. In particular,

$\displaystyle{\mathcal{Y}}(\mu_{0},w_{0}){(0)}=$
$\displaystyle\frac{(1-{w_{0}}{(0)})\gamma^{+}}{(1-{w_{0}}{(0)})\gamma^{+}+{w_{0}}{(0)}\gamma^{-}},$

where

$\displaystyle\gamma^{\pm}:=$ $\displaystyle G^{\pm}(0,0)>0.$

Thus, $w_{0}$ is a solution to 1.7 if and only if

$\displaystyle{w_{0}}{(0)}=$
$\displaystyle\frac{(1-{w_{0}}{(0)})\gamma^{+}}{(1-{w_{0}}{(0)})\gamma^{+}+{w_{0}}{(0)}\gamma^{-}}.$
(5.38)

Since $0\leq{w_{0}}{(0)}\leq 1$, the only solution that 5.38 has is

$\displaystyle
w_{0}(0)=\frac{\sqrt{\gamma^{+}}}{\sqrt{\gamma^{+}}+\sqrt{\gamma^{-}}}\in(0,1).$
(5.39)

Combining 5.39 and 5.37, we obtain the unique solution to 4.6 in the form

$\displaystyle{w_{0}}=$
$\displaystyle\frac{\sqrt{\gamma^{-}}G^{+}}{\sqrt{\gamma^{-}}G^{+}+\sqrt{\gamma^{+}}G^{-}}(\cdot,0)\qquad\text{in
}\overline{B_{\rho}}.$

Let $x\in\overline{B_{\rho}}$. Using formula 5.6a, we compute

$\displaystyle\partial_{w}({\mathcal{Y}}(\mu_{0},w_{0})(x))h_{1}=$
$\displaystyle\frac{1}{m}\xi(x)h_{1}{(0)},$ (5.40)

where

$\displaystyle\xi:=$ $\displaystyle
D\psi((1-w_{0}(0))G^{+}(\cdot,0),w_{0}(0)G^{-}(\cdot,0))\cdot(-G^{+}(\cdot,0),G^{-}(\cdot,0))$
$\displaystyle\in$ $\displaystyle W^{1,\infty}({B_{\rho}}).$

In particular,

$\displaystyle\xi{(0)}=$ $\displaystyle
D\psi((1-w_{0}(0))\gamma^{+},w_{0}(0)\gamma^{-})\cdot(-\gamma^{+},\gamma^{-})$
$\displaystyle=$ $\displaystyle
D\psi\left(\frac{\sqrt{\gamma^{-}}}{\sqrt{\gamma^{+}}+\sqrt{\gamma^{-}}}\gamma^{+},\frac{\sqrt{\gamma^{+}}}{\sqrt{\gamma^{+}}+\sqrt{\gamma^{-}}}\gamma^{-}\right)\cdot(-\gamma^{+},\gamma^{-})$
$\displaystyle=$ $\displaystyle-1.$ (5.41)

Combining 5.41, 5.40, and 5.18, we conclude that operator

$\displaystyle{\mathcal{X}}h_{1}=$ $\displaystyle
h_{1}-\partial_{w}{\mathcal{Y}}(\delta_{0},w_{0})h_{1}$ $\displaystyle=$
$\displaystyle h_{1}-\frac{1}{m}\xi h_{1}{(0)}$

is invertible in $W^{1,\infty}({B_{\rho}})$, its inverse being

$\displaystyle{\mathcal{X}}^{(-1)}g=g+\frac{1}{m+1}\xi g{(0)}.$

Thus, $(\mu_{0},w_{0})$ satisfies all conditions in 4.4.

##  6 Well-posedness of a PDE with PM diffusion and fixed advection direction

In this Section we consider the Cauchy problem for a PDE with the PM diffusion
1.4 and a fixed advection direction $V$:

$\displaystyle\partial_{t}u=\Delta u^{2}-\nabla\cdot\left(Vu\chi(u)\right)$
$\displaystyle\quad\text{in }(0,T]\times\mathbb{R}^{d},$ (6.1a) $\displaystyle
u(0,\cdot)=u_{0}$ $\displaystyle\quad\text{in }\mathbb{R}^{d}.$ (6.1b)

Later in Section 7 we take

$\displaystyle V=w\left(\nabla H\star u\right),$

which turns 6.1a into 1.6a.

For $V\equiv 0$, 6.1a is a special case of the standard porous media equation
that was covered in depth in [28]. Many works also addressed 6.1a for either
constant $V$ or $\chi\equiv 1$, see e.g. references in the survey [20]. More
recent studies have concentrated on the case $\chi\equiv 1$ and $V$ that
belongs to a possibly anisotropic $L^{p,q}$-space. We refer the interested
reader to the detailed explorations in [19, 18] as well as references therein.

In our setting, both $\chi$ and $V$ are non-constant. Since we were unable to
locate the necessary facts specifically for this particular scenario and
moreover, need to account for dependencies on the parameters in the estimates
for its solutions for the purpose of our analysis in Sections 8 and 7, we
provide all required results. For the reader’s convenience, we include proofs
in most cases even though they follow standard patterns.

The rest of the Section is decomposed into two parts. We start with developing
well-posedness in Subsection 6.1. In Subsection 6.2, we deal with the
situation when the initial value $u_{0}$ is compactly supported. We verify
that the support of the corresponding solution cannot explode for small times.

###  6.1 Well-posedness of 6.1

In this Subsection we aim at establishing well-posedness of 6.1. For $V$ and
$u_{0}$ we consider the following set of assumptions: for some $T>0$

$\displaystyle V$ $\displaystyle\in\left(L^{\infty}(E_{T})\right)^{d},$ (6.2a)
$\displaystyle V$ $\displaystyle\in L^{\infty,1}(E_{T}),$ (6.2b)
$\displaystyle\nabla\cdot V$ $\displaystyle\in L^{\infty}(E_{T}),$ (6.2c)
$\displaystyle\nabla\cdot V$ $\displaystyle\in L^{\infty,1}(E_{T});$ (6.2d)

and

$\displaystyle 0\leq$ $\displaystyle u_{0}\in L^{\infty}(\mathbb{R}^{d})\cap
L^{2}(\mathbb{R}^{d}),$ (6.3a) $\displaystyle u_{0}\in L^{1}(\mathbb{R}^{d}).$
(6.3b)

Not all of them are required for each result.

We introduce the following notion of (sub/super-) solution to 6.1.

######  Definition 6.1 ((Sub/super-)solutions to 6.1).

Let $T>0$ and 1.3a, 6.2a, and 6.3a be satisfied. We call a function
$u:[0,T]\times\mathbb{R}^{d}\to[0,\infty)$ a solution to 6.1 in
$[0,T]\times\mathbb{R}^{d}$ if:

  1. (1)

$u\in L^{\infty}\left(0,T;L^{\infty}(\mathbb{R}^{d})\cap
L^{2}(\mathbb{R}^{d})\right)$;

  2. (2)

$u^{\frac{3}{2}}\in L^{2}\left(0,T;H^{1}(\mathbb{R}^{d})\right)$;

  3. (3)

$\partial_{t}u\in L^{2}\left(0,T;H^{-1}(\mathbb{R}^{d})\right)$,
$\partial_{t}u\in
L^{\infty}\left(0,T;\left(W^{2,p}(\mathbb{R}^{d})\right)^{\prime}\right)$ for
all $p\in[1,\infty]$;

  4. (4)

$u$ satisfies 6.1a in a weak,

$\displaystyle\left<\partial_{t}u,\varphi\right>=-\int_{\mathbb{R}^{d}}\left(\nabla
u^{2}-Vu\chi(u)\right)\cdot\nabla\varphi\,dx\qquad\text{a.e. in }(0,T)\text{
for all }\varphi\in H^{1}(\mathbb{R}^{d}),$ (6.4a) and a very weak,
$\displaystyle\left<\partial_{t}u,\varphi\right>=\int_{\mathbb{R}^{d}}u^{2}\Delta\varphi+Vu\chi(u)\cdot\nabla\varphi\,dx\qquad\text{a.e.
in }(0,T)\text{ for all
}\varphi\in\bigcup\limits_{p=1}^{\infty}W^{2,p}(\mathbb{R}^{d}),$ (6.4b)

senses;

  5. (5)

$u(0,\cdot)=u_{0}$ in $H^{-1}(\mathbb{R}^{d})$.

We call $u$ a subsolution (supersolution) to 6.1 in
$[0,T]\times\mathbb{R}^{d}$ if $=$ is replaced by $\leq$ ($\geq$) in 6.4a and
6.4b.

Our first Lemma deals with uniqueness of solutions to 6.1.

######  Lemma 6.2.

Under the assumptions of Definition 6.1 solutions to 6.1 are unique.

###### Proof.

We use a version of the duality argument that is standard when dealing with
weak solutions of an equation with PM diffusion. The nonlinearity in the
advection term is handled similar to [12, Section 2], [11, Chapter 9, Section
2].

Let $u$ and $\widehat{u}$ be two solutions corresponding to $u_{0}$ and $V$.
Subtracting the very weak formulations 6.4b for these solutions, we obtain for
their difference

$\displaystyle\left<\partial_{t}\left(u-\widehat{u}\right),\varphi\right>=$
$\displaystyle\int_{\mathbb{R}^{d}}\left(u^{2}-\widehat{u}^{2}\right)\Delta\varphi\,dx+\int_{\mathbb{R}^{d}}V\left(u\chi(u)-\widehat{u}\chi\left(\widehat{u}\right)\right)\cdot\nabla\varphi\,dx$
(6.5)

for $\varphi\in H^{2}(\mathbb{R}^{d})$ a.e. in $(0,T)$. Consider the equation

$\displaystyle-\Delta\varphi+\varphi=u-\widehat{u}.$ (6.6)

As is well-known,
$-\Delta+\operatorname{id}:H^{k+2}\left(\mathbb{R}^{d}\right)\to
H^{k}\left(\mathbb{R}^{d}\right)$ is an isometric isomorphism for all
$k\in\mathbb{Z}$. Since $u,{\widehat{u}}\in{L^{\infty,2}}(E_{T})$ and
$\partial_{t}u,{\partial_{t}\widehat{u}}\in
L^{2}\left(0,T;H^{-1}(\mathbb{R}^{d})\right)$, we have

$\displaystyle{\varphi\in
L^{\infty}\left((0,T);H^{2}(\mathbb{R}^{d})\right)\cap
C\left([0,T];H^{1}(\mathbb{R}^{d})\right),\qquad\partial_{t}\varphi\in
L^{2}\left(0,T;H^{1}(\mathbb{R}^{d})\right)},$
$\displaystyle\|\varphi\|_{H^{1}(\mathbb{R}^{d})}=\left\|u-\widehat{u}\right\|_{H^{-1}(\mathbb{R}^{d})}\qquad\text{in
}[0,T].$

Plugging $\varphi$ into 6.5 and integrating over $\left(0,t\right)$ for
$t\in(0,T)$, we obtain using standard calculus of functions with values in a
Hilbert space

$\displaystyle\frac{1}{2}\left[\int_{\mathbb{R}^{d}}|\nabla\varphi|^{2}+\varphi^{2}\,dx\right]_{0}^{t}=$
$\displaystyle\int_{0}^{t}\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^{d}}|\nabla\varphi|^{2}+\varphi^{2}\,dx\,ds$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\left<\partial_{t}\left(-\Delta\varphi+\varphi\right),\varphi\right>\,ds$
$\displaystyle=$
$\displaystyle-\int_{0}^{t}\int_{\mathbb{R}^{d}}\left(u^{2}-\widehat{u}^{2}\right)\left(u-\widehat{u}\right)\,dx\,ds+\int_{\mathbb{R}^{d}}\left(u^{2}-\widehat{u}^{2}\right)\varphi\,dx\,ds$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{d}}V\left(u\chi(u)-\widehat{u}\chi\left(\widehat{u}\right)\right)\cdot\nabla\varphi\,dx\,ds\qquad\text{in
}[0,T].$ (6.7)

Due to the assumptions on $\chi$, we have

$\displaystyle|z_{1}\chi\left(z_{1}\right)-z_{2}\chi\left(z_{2}\right)|=$
$\displaystyle\left|\int_{z_{1}}^{z_{2}}s\chi^{\prime}\left(s\right)+\chi\left(s\right)\,ds\right|$
$\displaystyle\leq$ $\displaystyle
2\|\chi^{\prime}\|_{L^{\infty}((0,\infty))}\left|\int_{z_{1}}^{z_{2}}s\,ds\right|$
$\displaystyle=$
$\displaystyle\|\chi^{\prime}\|_{L^{\infty}((0,\infty))}\left|z_{1}^{2}-z_{2}^{2}\right|.$
(6.8)

Combining 6.7 and 6.8 with the basic algebraic inequalities

$\displaystyle\left(z_{1}^{2}-z_{2}^{2}\right)\left(z_{1}-z_{2}\right)\geq\Cl{Ctriv0}\left|z_{1}^{\frac{3}{2}}-z_{2}^{\frac{3}{2}}\right|^{2},$
$\displaystyle\left|z_{1}^{2}-z_{2}^{2}\right|\leq\Cl{Ctriv1}\left(\max\\{z_{1},z_{2}\\}\right)^{\frac{1}{2}}\left|z_{1}^{\frac{3}{2}}-z_{2}^{\frac{3}{2}}\right|$

and using Hölder’s and Young’s inequalities, we estimate as follows:

$\displaystyle\frac{1}{2}\left[\int_{\mathbb{R}^{d}}|\nabla\varphi|^{2}+\varphi^{2}\,dx\right]_{0}^{t}$
$\displaystyle\leq$
$\displaystyle-\Cr{Ctriv0}\int_{0}^{t}\int_{\mathbb{R}^{d}}\left|u^{\frac{3}{2}}-\widehat{u}^{\frac{3}{2}}\right|^{2}\,dx\,ds+\Cr{Ctriv1}\|\left(u,\widehat{u}\right)\|_{L^{\infty}\left(0,T;\left(L^{\infty}(\mathbb{R}^{d})\right)^{2}\right)}^{\frac{1}{2}}\int_{0}^{t}\int_{\mathbb{R}^{d}}\left|u^{\frac{3}{2}}-\widehat{u}^{\frac{3}{2}}\right||\varphi|\,dx\,ds$
$\displaystyle+\Cl{Ctr1}\||V|\|_{\left(L^{\infty}(E_{T})\right)^{d}}{\|\left(u,\widehat{u}\right)\|_{L^{\infty}\left(0,T;\left(L^{\infty}(\mathbb{R}^{d})\right)^{2}\right)}^{\frac{1}{2}}}\int_{0}^{t}\int_{\mathbb{R}^{d}}\left|u^{\frac{3}{2}}-\widehat{u}^{\frac{3}{2}}\right||\nabla\varphi|\,dx\,ds$
$\displaystyle\leq$
$\displaystyle\Cl{Ctr2}\int_{0}^{t}\int_{\mathbb{R}^{d}}|\nabla\varphi|^{2}+|\varphi|^{2}\,dx\,ds\qquad\text{in
}[0,T],$ (6.9)

where $\Cr{Ctr2}$ depends on norms of $u,\widehat{u}$, $\chi$, and $V$.
Applying Grönwall’s inequality to 6.9 yields

$\displaystyle\|\varphi\left(t\right)\|_{H^{1}(\mathbb{R}^{d})}^{2}\leq$
$\displaystyle
e^{t2\Cr{Ctr2}}\|\varphi(0)\|_{H^{1}(\mathbb{R}^{d})}^{2}\qquad\text{in
}[0,T].$ (6.10)

Combining 6.6 and 6.10 and using $\varphi(0,\cdot)=u_{0}-u_{0}=0$ yields
uniqueness.

∎

As is customary in such cases, we seek to obtain solutions to 6.1 as limits of
solutions $\left(u_{\epsilon m}\right)$,

$\displaystyle\varepsilon\in(0,1)\text{ and }m\in\mathbb{N},$

to initial-boundary value problems (IBVPs) in balls for non-degenerate
diffusion-advection equations with regular coefficients,

$\displaystyle\partial_{t}u_{\varepsilon m}=\varepsilon\Delta u_{\varepsilon
m}+\Delta u_{\varepsilon m}^{2}-\nabla\cdot\left(V_{\varepsilon}u_{\varepsilon
m}\chi_{\varepsilon}\left(u_{\varepsilon m}\right)\right)$
$\displaystyle\qquad\text{in }(0,T]\times B_{m},$ (6.11a) $\displaystyle
u_{\varepsilon m}=0$ $\displaystyle\qquad\text{in }(0,T]\times S_{m},$ (6.11b)
$\displaystyle u_{\varepsilon m}(0,\cdot)=u_{\varepsilon 0}$
$\displaystyle\qquad\text{in }\overline{B_{m}},$ (6.11c)

extending them to the whole space by

$\displaystyle u_{\varepsilon m}=0\qquad\text{in
}[0,T]\times\left(\mathbb{R}^{d}\backslash\overline{B_{m}}\right).$

We make the following assumptions on the coefficient functions
$\chi_{\varepsilon}$ and $V_{\varepsilon}$ and the initial data
$u_{\varepsilon 0}$:

$\displaystyle\chi_{\varepsilon}\in
C^{2}_{b}\left([0,\infty)\right),\chi(0)=0,$ (6.12a) $\displaystyle
V_{\varepsilon}\in
C^{1,2}_{b}\left([0,T]\times\mathbb{R}^{d};\mathbb{R}^{d}\right),$ (6.12b)
$\displaystyle 0\leq u_{\varepsilon 0}\in
C^{3}_{0}\left(\mathbb{R}^{d}\right).$ (6.12c)

The main purpose of the subsequent Lemmas is to collect useful estimates for
solutions to 6.11. Most of them continue to hold for solutions to 6.1 and are
used in the proofs in Sections 7 and 8.

######  Lemma 6.3.

Let 6.12 be satisfied and let
$m\geq\operatorname{diam}\left(\operatorname{supp}\left(u_{\varepsilon
0}\right)\right)$. Then, the IBVP 6.11 possesses a unique classical solution
$u_{\varepsilon m}\in C^{1,2}\left([0,T]\times\overline{B_{m}}\right)$. The
solution satisfies the following set of estimates:

$\displaystyle 0\leq u_{\varepsilon
m}\leq\Cl{D1}\left(T,\chi_{\varepsilon},V_{\varepsilon},u_{\varepsilon
0}\right)\qquad\text{in }[0,T]\times\overline{B_{m}},$ (6.13a)
$\displaystyle\|u_{\varepsilon
m}\|_{L^{\infty}\left(0,T;L^{2}\left(B_{m}\right)\right)}^{2}+\varepsilon\||\nabla
u_{\varepsilon
m}|\|_{L^{2}\left(0,T;L^{2}\left(B_{m}\right)\right)}^{2}+\left\|\left|\nabla
u_{\varepsilon
m}^{\frac{3}{2}}\right|\right\|_{L^{2}\left(0,T;L^{2}\left(B_{m}\right)\right)}^{2}\leq\Cl{D2}\left(T,\chi_{\varepsilon},V_{\varepsilon},u_{\varepsilon
0}\right),$ (6.13b) $\displaystyle\|\partial_{t}u_{\varepsilon
m}\|_{L^{2}\left(0,T;H^{-1}\left(B_{m}\right)\right)}^{2}\leq\Cl{D3}\left(T,\chi_{\varepsilon},V_{\varepsilon},u_{\varepsilon
0}\right),$ (6.13c)

where

$\displaystyle\Cr{D1}\left(T,\chi_{\varepsilon},V_{\varepsilon},u_{\varepsilon
0}\right)$ $\displaystyle:=\|u_{\varepsilon
0}\|_{L^{\infty}(\mathbb{R}^{d})}e^{T\|\chi_{\varepsilon}\|_{L^{\infty}((0,\infty))}\|\nabla\cdot
V_{\varepsilon}\|_{L^{\infty}(E_{T})}},$ (6.14a)
$\displaystyle\Cr{D2}\left(T,\chi_{\varepsilon},V_{\varepsilon},u_{\varepsilon
0}\right)$ $\displaystyle:=\|u_{\varepsilon
0}\|_{L^{2}(\mathbb{R}^{d})}^{2}e^{T\Cl{Ce1}\|\chi_{\varepsilon}^{\prime}\|_{L^{\infty}((0,\infty))}^{2}\||V_{\varepsilon}|\|_{{L^{\infty}(E_{T})}}^{2}\Cr{D1}\left(T,\chi_{\varepsilon},V_{\varepsilon},u_{\varepsilon
0}\right)},$ (6.14b)
$\displaystyle\Cr{D3}\left(T,\chi_{\varepsilon},V_{\varepsilon},u_{\varepsilon
0}\right)$
$\displaystyle:=\Cr{D2}\left(T,\chi_{\varepsilon},V_{\varepsilon},u_{\varepsilon
0}\right)\Cl{C26}\max\left\\{1,\Cr{D1}\left(T,\chi_{\varepsilon},V_{\varepsilon},u_{\varepsilon
0}\right),T\|\chi_{\varepsilon}\|_{L^{\infty}((0,\infty))}^{2}\||V_{\varepsilon}|\|_{L^{\infty}(E_{T})}^{2}\right\\}.$
(6.14c)

###### Proof.

Since 6.11a is a non-degenerate quasilinear parabolic PDE, standard theory
(see, e.g. [25, Chapter V, §6, Theorem 6.1]) implies that the IBVP 6.11
possesses a unique classical solution $u_{\varepsilon m}$ provided it is a
priori bounded. That is the case if 6.13a holds. Thus, to complete the proof
of the Lemma it remains to prove that estimates 6.13 are a priori valid.

Estimates 6.13a are a direct application of the maximal principle for non-
degenerate parabolic PDEs.

Further, multiplying equation 6.11a by $\varphi\in
H^{1}_{0}\left(B_{m}\right)$ and integrating by parts using 6.11b, we obtain
the following weak reformulation:

$\displaystyle\left<\partial_{t}u_{\varepsilon
m},\varphi\right>=-\int_{B_{m}}\varepsilon\nabla u_{\varepsilon
m}\cdot\nabla\varphi+\left(\nabla u_{\varepsilon
m}^{2}-V_{\varepsilon}u_{\varepsilon m}\chi_{\varepsilon}\left(u_{\varepsilon
m}\right)\right)\cdot\nabla\varphi\,dx\qquad\text{in }(0,T)\ \text{for
}\varphi\in H^{1}_{0}\left(B_{m}\right).$ (6.15)

Taking $\varphi=u_{\varepsilon m}$ in 6.15 and using assumptions on
$\chi_{\varepsilon}$, the chain rule, and Hölder’s and Young’s inequalities,
we deduce

$\displaystyle\frac{1}{2}\frac{d}{dt}\|u_{\varepsilon
m}\|_{L^{2}\left(B_{m}\right)}^{2}+\varepsilon\||\nabla u_{\varepsilon
m}|\|_{L^{2}\left(B_{m}\right)}^{2}$ $\displaystyle=$
$\displaystyle-\frac{8}{9}\left\|\left|\nabla u_{\varepsilon
m}^{\frac{3}{2}}\right|\right\|_{L^{2}\left(B_{m}\right)}^{2}+\frac{2}{3}\int_{B_{m}}V_{\varepsilon}u_{\varepsilon
m}^{\frac{1}{2}}\chi_{\varepsilon}\left(u_{\varepsilon m}\right)\cdot\nabla
u_{\varepsilon m}^{\frac{3}{2}}\,dx$ $\displaystyle\leq$
$\displaystyle-\frac{1}{2}\left\|\left|\nabla u_{\varepsilon
m}^{\frac{3}{2}}\right|\right\|_{L^{2}\left(B_{m}\right)}^{2}+\Cr{Ce1}\|\chi_{\varepsilon}^{\prime}\|_{L^{\infty}((0,\infty))}^{2}\||V_{\varepsilon}|\|_{{L^{\infty}(E_{T})}}^{2}\|u_{\varepsilon
m}\|_{L^{\infty}\left(0,T;L^{\infty}\left(B_{m}\right)\right)}\|u_{\varepsilon
m}\|_{L^{2}\left(B_{m}\right)}^{2}.$ (6.16)

Applying Grönwall’s inequality to 6.16 yields

$\displaystyle\|u_{\varepsilon
m}(t,\cdot)\|_{L^{2}\left(B_{m}\right)}^{2}+2\varepsilon\||\nabla
u_{\varepsilon
m}|\|_{L^{2}\left(0,T;L^{2}\left(B_{m}\right)\right)}^{2}+\left\|\left|\nabla
u_{\varepsilon
m}^{\frac{3}{2}}\right|\right\|_{L^{2}\left(0,T;L^{2}\left(B_{m}\right)\right)}^{2}$
$\displaystyle\leq$ $\displaystyle\|u_{\varepsilon
0}\|_{L^{2}(\mathbb{R}^{d})}^{2}e^{t2\Cr{Ce1}\|\chi_{\varepsilon}^{\prime}\|_{L^{\infty}((0,\infty))}^{2}\||V_{\varepsilon}|\|_{{L^{\infty}(E_{T})}}^{2}\|u_{\varepsilon
m}\|_{L^{\infty}\left(0,T;L^{\infty}\left(B_{m}\right)\right)}},\qquad\text{for
}t\in[0,T].$ (6.17)

Together, 6.13a and 6.17 yield 6.13b.

Next, we estimate the right-hand side of 6.15 in $H^{-1}\left(B_{m}\right)$:

$\displaystyle\|\partial_{t}u_{\varepsilon
m}\|_{H^{-1}\left(B_{m}\right)}\leq$ $\displaystyle\varepsilon\||\nabla
u_{\varepsilon m}|\|_{L^{2}\left(B_{m}\right)}$
$\displaystyle+\frac{4}{3}\|u_{\varepsilon
m}\|_{L^{\infty}\left(0,T;L^{\infty}\left(B_{m}\right)\right)}^{\frac{1}{2}}\left\|\left|\nabla
u_{\varepsilon m}^{\frac{3}{2}}\right|\right\|_{L^{2}\left(B_{m}\right)}$
$\displaystyle+\|\chi_{\varepsilon}\|_{L^{\infty}((0,\infty))}\||V_{\varepsilon}|\|_{L^{\infty}(E_{T})}\|u_{\varepsilon
m}\|_{L^{2}\left(B_{m}\right)}\qquad\text{in }(0,T).$ (6.18)

Taking the $L^{2}(0,T)$ norm on both sides of 6.18, we deduce

$\displaystyle\|\partial_{t}u_{\varepsilon
m}\|_{L^{2}\left(0,T;H^{-1}\left(B_{m}\right)\right)}\leq$
$\displaystyle\varepsilon\||\nabla u_{\varepsilon
m}|\|_{L^{2}\left(0,T;L^{2}\left(B_{m}\right)\right)}$
$\displaystyle+\frac{4}{3}\|u_{\varepsilon
m}\|_{L^{\infty}\left(0,T;L^{\infty}\left(B_{m}\right)\right)}^{\frac{1}{2}}\left\|\left|\nabla
u_{\varepsilon
m}^{\frac{3}{2}}\right|\right\|_{L^{2}\left(0,T;L^{2}\left(B_{m}\right)\right)}$
$\displaystyle+T^{\frac{1}{2}}\|\chi_{\varepsilon}\|_{L^{\infty}((0,\infty))}\||V_{\varepsilon}|\|_{L^{\infty}(E_{T})}\|u_{\varepsilon
m}\|_{L^{\infty}\left(0,T;L^{2}\left(B_{m}\right)\right)}.$ (6.19)

Combining 6.19 with 6.13a and 6.13b yields 6.13c. ∎

Our next Lemma deals with stability in $L^{1}$.

######  Lemma 6.4.

Let $\chi_{\varepsilon}$ satisfy 6.12a,
$V_{\varepsilon},\widehat{V}_{\varepsilon}$ satisfy 6.12b, and $u_{\varepsilon
0},\widehat{u}_{\varepsilon 0}$ satisfy 6.12c, as well as

$\displaystyle
m\geq\max\\{\operatorname{diam}\left(\operatorname{supp}\left(u_{\varepsilon
0}\right)\right),\operatorname{diam}\left(\operatorname{supp}\left(\widehat{u}_{\varepsilon
0}\right)\right)\\}.$

Then, the corresponding pair of classical solutions $u_{\varepsilon m}$ and
$\widehat{u}_{\varepsilon m}$ to the IBVP 6.11 satisfies for $t\in[0,T]$

$\displaystyle\left\|\left(u_{\varepsilon m}-\widehat{u}_{\varepsilon
m}\right)(t,\cdot)\right\|_{L^{1}\left({B_{m}}\right)}$ $\displaystyle\leq$
$\displaystyle\left\|u_{\varepsilon 0}-\widehat{u}_{\varepsilon
0}\right\|_{L^{1}(\mathbb{R}^{d})}$
$\displaystyle+\Cl{D4_{1}}\left(T,\chi_{\varepsilon},V_{\varepsilon},\widehat{u}_{\varepsilon
0}\right)\left(\left\|u_{\varepsilon
m}\nabla\cdot\left(V_{\varepsilon}-\widehat{V}_{\varepsilon}\right)\right\|_{L^{1}((0,T);L^{1}(B_{m}))}+\left\|u_{\varepsilon
m}^{\frac{1}{2}}\left|V_{\varepsilon}-\widehat{V}_{\varepsilon}\right|\right\|_{L^{2}(0,T;L^{2}(B_{m}))}\right),$
(6.20)

where

$\displaystyle\Cr{D4_{1}}\left(T,\chi_{\varepsilon},V_{\varepsilon},\widehat{u}_{\varepsilon
0}\right):=\frac{4}{3}\|\chi_{\varepsilon}\|_{C_{b}^{1}\left([0,\infty)\right)}\max\left\\{1,\Cr{D2}^{\frac{1}{2}}\left(T,\chi_{\varepsilon},V_{\varepsilon},\widehat{u}_{\varepsilon
0}\right)\right\\}$ (6.21)

and $\Cr{D2}(\dots)$ is as defined in 6.14.

###### Proof.

The proof follows a standard scheme and relies on a version of Kato’s
inequality. Let $u_{\varepsilon m}$ and $\widehat{u}_{\varepsilon m}$ be two
classical solutions to the IBVP 6.11 for given $V_{\varepsilon}$ and
$\widehat{V}_{\varepsilon}$, respectively. Their difference

$\displaystyle U_{\varepsilon m}:=\widehat{u}_{\varepsilon m}-u_{\varepsilon
m}$

satisfies

$\displaystyle\partial_{t}U_{\varepsilon m}=$
$\displaystyle\Delta\left(a_{0}U_{\varepsilon
m}\right)-\nabla\cdot\left(\widetilde{V_{\varepsilon}}U_{\varepsilon
m}\right)-b_{0}\left(u_{\varepsilon
m}\right)\nabla\cdot\left(\widehat{V}_{\varepsilon}-V_{\varepsilon}\right)-\left(\widehat{V}_{\varepsilon}-V_{\varepsilon}\right)b_{1}\left(u_{\varepsilon
m}\right)\nabla u_{\varepsilon m}^{\frac{3}{2}}\ \ \text{in }(0,T)\times
B_{m},$ (6.22)

where

$\displaystyle a_{0}:=$ $\displaystyle\varepsilon+u_{\varepsilon
m}+\widehat{u}_{\varepsilon m}>0,$ (6.23a) $\displaystyle b_{0}(z):=$
$\displaystyle z\chi_{\varepsilon}(z),$ (6.23b) $\displaystyle
b_{1}\left(z\right):=$
$\displaystyle\frac{2}{3}z^{-\frac{1}{2}}b_{0}^{\prime}\left(z\right)$
$\displaystyle=$
$\displaystyle\frac{2}{3}z^{\frac{1}{2}}\left(\chi_{\varepsilon}^{\prime}\left(z\right)+\int_{0}^{1}\chi_{\varepsilon}^{\prime}\left(sz\right)\,ds\right),$
(6.23c) $\displaystyle\widetilde{V}:=$
$\displaystyle\widehat{V}_{\varepsilon}\left(b_{0}\left(\widehat{u}_{\varepsilon
m}\right)-b_{0}(u_{\varepsilon m})\right)$ $\displaystyle=$
$\displaystyle\widehat{V}_{\varepsilon}\int_{0}^{1}b_{0}^{\prime}\left(s\widehat{u}_{\varepsilon
m}+\left(1-s\right)u_{\varepsilon m}\right)\,ds.$ (6.23d)

Multiplying 6.22 by $\operatorname{sign}\left(U_{\varepsilon m}\right)$,
integrating over $B_{m}$, by parts where necessary, and using 6.23, the chain
rule, and Kato’s and Hölder’s inequalities, we can estimate as follows in
$[0,T]$:

$\displaystyle\frac{d}{dt}\int_{B_{m}}|U_{\varepsilon m}|\,dx$
$\displaystyle=$
$\displaystyle\int_{B_{m}}\operatorname{sign}\left({a_{0}}U_{\varepsilon
m}\right)\Delta\left(a_{0}U_{\varepsilon
m}\right)\,dx-\int_{B_{m}}\nabla\cdot\left(\widetilde{V}|U_{\varepsilon
m}|\right)\,dx$ (6.24)
$\displaystyle-\int_{B_{m}}\operatorname{sign}\left(U_{\varepsilon
m}\right)b_{0}\left(u_{\varepsilon
m}\right)\nabla\cdot\left(\widehat{V}_{\varepsilon}-V_{\varepsilon}\right)\,dx-\int_{B_{m}}\operatorname{sign}\left(U_{\varepsilon
m}\right)\left(\widehat{V}_{\varepsilon}-V_{\varepsilon}\right)\cdot
b_{1}\left(u_{\varepsilon m}\right)\nabla u_{\varepsilon
m}^{\frac{3}{2}}\,dx,$ $\displaystyle\leq$
$\displaystyle\int_{B_{m}}\left|b_{0}\left(u_{\varepsilon
m}\right)\nabla\cdot\left(\widehat{V}_{\varepsilon}-V_{\varepsilon}\right)\right|\,dx+\int_{B_{m}}\left|\left(\widehat{V}_{\varepsilon}-V_{\varepsilon}\right)b_{1}\left(u_{\varepsilon
m}\right)\nabla u_{\varepsilon m}^{\frac{3}{2}}\right|\,dx$
$\displaystyle\leq$
$\displaystyle\|\chi_{\varepsilon}\|_{L^{\infty}((0,\infty))}\left\|u_{\varepsilon
m}\nabla\cdot\left(\widehat{V}_{\varepsilon}-V_{\varepsilon}\right)\right\|_{L^{1}(B_{m})}+\frac{4}{3}\|\chi_{\varepsilon}^{\prime}\|_{L^{\infty}((0,\infty))}\left\|\left|\nabla
u_{\varepsilon
m}^{\frac{3}{2}}\right|\right\|_{L^{2}\left(B_{m}\right)}\left\|u_{\varepsilon
m}^{\frac{1}{2}}\left|V_{\varepsilon}-\widehat{V}_{\varepsilon}\right|\right\|_{L^{2}(B_{m})}.$
(6.25)

Here we have used a version of Kato’s inequality [22] provided by a
combination of [6, Theorems 1.2-1.3]: for all $f\in W^{1,1}_{0}(B_{m})$ such
that $\Delta u\in L^{1}(B_{m})$ we have $\partial_{\nu}f\in L^{1}(S_{m}(0))$
and

$\displaystyle\int_{B_{m}}\operatorname{sign}(f)\Delta f\,dx\leq$
$\displaystyle-\int_{S_{m}(0)}|\partial_{\nu}f|\,dx,$ $\displaystyle\leq$
$\displaystyle 0$ (6.26)

where $\partial_{\nu}f$ denotes the outer normal derivative of $f$ in the
trace sense. It is inequality 6.26 that allows to estimate the first term on
the right-hand side of 6.24 from above by zero.

Integrating 6.25 over $[0,T]$ and using Hölder’s inequality yields

$\displaystyle\|U_{\varepsilon
m}(t,\cdot)\|_{L^{1}\left(B_{m}\right)}-\|U_{\varepsilon
m}(0,\cdot)\|_{L^{1}\left(B_{m}\right)}$ $\displaystyle\leq$
$\displaystyle\|\chi_{\varepsilon}\|_{L^{\infty}((0,\infty))}\left\|u_{\varepsilon
m}\nabla\cdot\left(\widehat{V}_{\varepsilon}-V_{\varepsilon}\right)\right\|_{L^{1}((0,T);L^{1}(B_{m}))}$
$\displaystyle+\frac{4}{3}\|\chi_{\varepsilon}^{\prime}\|_{L^{\infty}((0,\infty))}\left\|\left|\nabla
u_{\varepsilon
m}^{\frac{3}{2}}\right|\right\|_{L^{2}(0,T;L^{2}(B_{m}))}\left\|u_{\varepsilon
m}^{\frac{1}{2}}\left|V_{\varepsilon}-\widehat{V}_{\varepsilon}\right|\right\|_{L^{2}(0,T;L^{2}(B_{m}))}$
$\displaystyle\leq$
$\displaystyle\frac{4}{3}\|\chi_{\varepsilon}\|_{C_{b}^{1}\left([0,\infty)\right)}\max\left\\{1,\left\|\nabla
u_{\varepsilon
m}^{\frac{3}{2}}\right\|_{L^{2}\left(0,T;L^{2}\left(B_{m}\right)\right)}\right\\}$
$\displaystyle\cdot\left(\left\|u_{\varepsilon
m}\nabla\cdot\left(\widehat{V}_{\varepsilon}-V_{\varepsilon}\right)\right\|_{L^{1}((0,T);L^{1}(B_{m}))}+\left\|u_{\varepsilon
m}^{\frac{1}{2}}\left|V_{\varepsilon}-\widehat{V}_{\varepsilon}\right|\right\|_{L^{2}(0,T;L^{2}(B_{m}))}\right)\qquad\text{in
}[0,T].$ (6.27)

Finally, combining 6.27 with 6.13b, we arrive at 6.20. ∎

A comparison property in $L^{1}$ holds as well.

######  Lemma 6.5.

Let $\chi_{\varepsilon}$ satisfy 6.12a, $V_{\varepsilon}$ satisfy 6.12b, and
$u_{\varepsilon 0},\widehat{u}_{\varepsilon 0}$ satisfy 6.12c as well as

$\displaystyle
m\geq\max\\{\operatorname{diam}\left(\operatorname{supp}\left(u_{\varepsilon
0}\right)\right),\operatorname{diam}\left(\operatorname{supp}\left(\widehat{u}_{\varepsilon
0}\right)\right)\\}.$

Then, any corresponding pair of classical subsolution $u_{\varepsilon m}$ and
supersolution $\widehat{u}_{\varepsilon m}$ to the IBVP 6.11 satisfies

$\displaystyle\left\|\left(u_{\varepsilon m}-\widehat{u}_{\varepsilon
m}\right)_{+}(t,\cdot)\right\|_{L^{1}(\mathbb{R}^{d})}\leq\left\|\left(u_{\varepsilon
m}-\widehat{u}_{\varepsilon
m}\right)_{+}(0,\cdot)\right\|_{L^{1}(\mathbb{R}^{d})}\qquad\text{for
}t\in[0,T].$

We omit the proof of Lemma 6.5 since it can be carried out very similar to
that of Lemma 6.9.

With the results from Lemmas 6.3 and 6.4 at hand, we can now establish
existence and estimates for 6.1.

######  Lemma 6.6.

Let 1.3a, 6.2a, 6.2c, and 6.3a be satisfied. Then, the Cauchy problem 6.1
possesses a unique solution $u$ in terms of Definition 6.1. The solution
satisfies the following set of estimates:

$\displaystyle\|u\|_{L^{\infty}(E_{T})}\leq\Cr{D1}\left(T,\chi,V,u_{0}\right),$
(6.28a) $\displaystyle\|u\|_{L^{\infty,2}(E_{T})}^{2}+\left\|\left|\nabla
u^{\frac{3}{2}}\right|\right\|_{L^{2}(E_{T})}^{2}\leq\Cr{D2}\left(T,\chi,V,u_{0}\right),$
(6.28b)
$\displaystyle\|\partial_{t}u\|_{L^{2}\left(0,T;H^{-1}\left(\mathbb{R}^{d}\right)\right)}\leq\Cr{D3}\left(T,\chi,V,u_{0}\right),$
(6.28c)

where $C_{i}$s are as in 6.14.

###### Sketch of the proof.

Lemma 6.2 provides uniqueness, so we only need to prove existence. Since the
arguments are standard, we omit most details. To begin with, we choose
families of regularised coefficient functions $\chi_{\varepsilon}$ and
$V_{\varepsilon}$ and initial data $u_{\varepsilon m0}$ for

$\varepsilon\in(0,1)\text{ and }m\geq 2$

as follows:

$\displaystyle V_{\varepsilon}:=\zeta_{d+1,\varepsilon}\star V,\qquad$
$\displaystyle V:=0\text{ in }(-\infty,0)\times\mathbb{R}^{d},$ (6.29)
$\displaystyle\chi_{\varepsilon}:=\zeta_{1,\varepsilon}\star\left((1-\eta_{1,2\varepsilon})\chi\right),\qquad$
$\displaystyle\chi:=0\text{ in }(-\infty,0),$ (6.30) $\displaystyle
u_{\varepsilon
m0}:=\zeta_{d,\varepsilon}\star\left(\eta_{d,\frac{m}{2}}u_{0}\right),\qquad$
(6.31)

where

$\displaystyle\zeta_{k,\varepsilon}(x):=\varepsilon^{-k}\zeta_{1}\left(\varepsilon^{-1}|x|\right)\text{
in }\mathbb{R}^{k},\qquad$
$\displaystyle\zeta_{1}(s):=\begin{cases}C^{-1}e^{\frac{1}{s^{2}-1}}&\text{for
}|s|<1,\\\ 0&\text{for }|s|\geq 1,\end{cases}\qquad
C:=\int_{-1}^{1}e^{\frac{1}{s^{2}-1}}\,ds,$ (6.32)
$\displaystyle\eta_{k,R}(x):=\eta_{1}\left(R^{-1}|x|\right)\text{ in
}\mathbb{R}^{k},$ $\displaystyle\eta_{1}\in
C_{0}^{\infty}([0,1);{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}[0,1]}),\
\eta_{1}=1\text{ in }\left[0,\frac{1}{2}\right].$ (6.33)

This choice and standard properties of convolution ensure that 6.12 is
satisfied and for every $\varepsilon\in(0,1)$ and $m\geq 2$ it holds that

$\displaystyle V_{\varepsilon}\underset{\varepsilon\rightarrow
0}{\overset{*}{\rightharpoonup}}V$ $\displaystyle\qquad\text{in
}\left(L^{\infty}(E_{T})\right)^{d},$ (6.34a) $\displaystyle
V_{\varepsilon}\underset{\varepsilon\rightarrow 0}{\rightarrow}V$
$\displaystyle\qquad\text{in }\left(L^{p}(E_{T})\right)^{d},\ p\in[1,\infty),$
(6.34b) $\displaystyle\nabla\cdot
V_{\varepsilon}\underset{\varepsilon\rightarrow
0}{\overset{*}{\rightharpoonup}}\nabla\cdot V$ $\displaystyle\qquad\text{in
}L^{\infty}(E_{T}),$ (6.34c) $\displaystyle\nabla\cdot
V_{\varepsilon}\underset{\varepsilon\rightarrow 0}{\rightarrow}\nabla\cdot V$
$\displaystyle\qquad\text{in }L^{1}(E_{T}),$ (6.34d)
$\displaystyle\||V_{\varepsilon}|\|_{L^{\infty}(E_{T})}\leq$
$\displaystyle\||V|\|_{L^{\infty}(E_{T})},$ (6.35a)
$\displaystyle\|\nabla\cdot V_{\varepsilon}\|_{L^{\infty}(E_{T})}\leq$
$\displaystyle\|\nabla\cdot V\|_{L^{\infty}(E_{T})},$ (6.35b)
$\displaystyle\chi_{\varepsilon}\to\chi\qquad\text{ in
}C^{1}_{b}\left(\mathbb{R}^{+}_{0}\right),$ (6.36)
$\displaystyle\operatorname{supp}(u_{\varepsilon m0})\subset B_{m},$ (6.37a)
$\displaystyle\|u_{\varepsilon
m0}\|_{L^{\infty}(\mathbb{R}^{d})}\leq\|u_{0}\|_{L^{\infty}(\mathbb{R}^{d})}$
, (6.37b) $\displaystyle u_{\varepsilon m0}\underset{\varepsilon\to
0}{\to}\underset{m\to\infty}{\to}u_{0}$ $\displaystyle\qquad\text{in
}L^{2}(\mathbb{R}^{d}).$ (6.37c)

Since all estimates from Lemma 6.3 hold for any $\varepsilon\in(0,1)$ and
$2\leq m\in\mathbb{N}$, a standard argument based on compactness and a limit
procedure yields the existence of a solution $u$ (defined in terms of
Definition 6.1) for the Cauchy problem 6.1 such that

$\displaystyle
u_{\varepsilon_{k}m_{l}}\underset{k\to\infty}{\to}\underset{l\to\infty}{\to}u\qquad\text{a.e.
in }E_{T}$ (6.38)

for some sequences $(\varepsilon_{k})$ and $(m_{l})$. The solution satisfies
estimates 6.28, which can be obtained by taking the limit inferior on both
sides of each estimate in 6.13 and using the weak-$*$ continuity of norms, as
well as 6.37, 6.36, and 6.35. ∎

Out next two Lemmas contain some results on continuity with respect to time.

######  Lemma 6.7.

Let the assumptions of Lemma 6.6 be satisfied. Assume additionally that
$u_{0}\in L^{1}(\mathbb{R}^{d})$. Then, the solution $u$ to the Cauchy problem
6.1 satisfies

$\displaystyle u\in C_{w}\left([0,T];L^{\infty}(\mathbb{R}^{d})\cap
L^{1}(\mathbb{R}^{d})\right)$

and

$\displaystyle\|u\|_{L^{1}(\mathbb{R}^{d})}\equiv\|u_{0}\|_{L^{1}(\mathbb{R}^{d})}\qquad\text{in
}[0,T].$ (6.39)

###### Proof.

Consider ${\eta_{d,n}}$, $n\in\mathbb{N}$, as defined by 6.33. Then,

$\displaystyle{\eta_{d,n}}\underset{n\to\infty}{\to}1$
$\displaystyle\qquad\text{in }\mathbb{R}^{d},$ (6.40a)
$\displaystyle\eta_{d,n}\underset{n\to\infty}{\overset{*}{\rightharpoonup}}1$
$\displaystyle\qquad\text{in }W^{2,\infty}(\mathbb{R}^{d}).$ (6.40b)

Using $\partial_{t}u\in
L^{\infty}\left(0,T;\left(W^{2,\infty}(\mathbb{R}^{d})\right)^{\prime}\right)$,
6.4b for $\varphi\equiv 1$, 6.40b, and integrability of $u_{0}$, we deduce
that

$\displaystyle\underset{n\to\infty}{\lim}\int_{\mathbb{R}^{d}}u(t,\cdot){\eta_{d,n}}\,dx=$
$\displaystyle\left<u(t,\cdot),1\right>$ $\displaystyle=$
$\displaystyle\|u_{0}\|_{L^{1}(\mathbb{R}^{d})}\qquad\text{ for }t\in[0,T].$
(6.41)

With 6.40a and 6.41 and Fatou’s lemma it follows that

$\displaystyle\int_{\mathbb{R}^{d}}u(t,\cdot)\,dx\leq$
$\displaystyle\underset{n\to\infty}{\lim}\int_{\mathbb{R}^{d}}u(t,\cdot){\eta_{d,n}}\,dx$
(6.42a) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}u_{0}\,dx\qquad\text{for }t\in[0,T],$
(6.42b)

so that $u\in L^{\infty,1}(E_{T})$. Since $\eta_{d,n}\in[0,1]$, we conclude
with 6.42 that

$\displaystyle\|u(t,\cdot)\|_{L^{1}(\mathbb{R}^{d})}=\|u_{0}\|_{L^{1}(\mathbb{R}^{d})}\qquad\text{
for }t\in[0,T].$

Finally, we confirm weak continuity. Since

$\displaystyle u\in C\left([0,T];H^{-1}(\mathbb{R}^{d})\right)\cap
L^{\infty}\left(0,T;L^{\infty}(\mathbb{R}^{d})\cap
L^{1}(\mathbb{R}^{d})\right),$ $\displaystyle L^{\infty}(\mathbb{R}^{d})\cap
L^{1}(\mathbb{R}^{d})\subset L^{2}(\mathbb{R}^{d})\subset
H^{-1}(\mathbb{R}^{d}),$

[27, Chapter 3, §1, Lemma 1.4] applies and yields

$u\in C_{w}\left([0,T];L^{\infty}(\mathbb{R}^{d})\cap
L^{1}(\mathbb{R}^{d})\right),$

as required. ∎

The next Lemma provides an estimate for the solution change over time.

######  Lemma 6.8.

Let the assumptions of Lemma 6.6 be satisfied. Then, the solution $u$ to the
Cauchy problem 6.1 satisfies for all $0\leq\tau<t\leq T$, $R>0$, and

$\displaystyle\varphi\text{
Lipschitz},\quad\varphi(0)=0,\quad\|\nabla\varphi\|_{L^{\infty}(\mathbb{R}^{d})}\leq
1$ (6.43)

the estimate

$\displaystyle\left|\int_{\mathbb{R}^{d}}(u(t,\cdot)-u(\tau,\cdot))\eta_{d,{R}}\varphi\,dx\right|\leq(t-\tau)^{\frac{1}{2}}{R}^{\frac{d}{2}}\Cl{C37}\Cr{D3}\left(T,\chi,V,u_{0}\right),$
(6.44)

where $\eta_{d,{R}}$ is the cut-off function from 6.33 and $\Cr{D3}(\dots)$ is
as defined in 6.14.

###### Proof.

With 6.33 and 6.43, the chain rule, and Hölder’s inequality we have

$\displaystyle\left\|\nabla\left(\eta_{d,{R}}\varphi\right)\right\|_{L^{2}(\mathbb{R}^{d})}=$
$\displaystyle\left\|\eta_{d,{R}}\nabla\varphi+\varphi\nabla\eta_{d,{R}}\right\|_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle\leq$
$\displaystyle\left\|\nabla\varphi\right\|_{L^{\infty}(\mathbb{R}^{d})}\left\|\eta_{d,{R}}\right\|_{L^{2}(\mathbb{R}^{d})}+R\left\|\nabla\varphi\right\|_{L^{\infty}(\mathbb{R}^{d})}\left\|\nabla\eta_{d,{R}}\right\|_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle\leq$
$\displaystyle{R}^{\frac{d}{2}}\sqrt{2}\left\|\eta_{d,1}\right\|_{H^{1}(\mathbb{R}^{d})}$
$\displaystyle=$ $\displaystyle{R}^{\frac{d}{2}}\Cr{C37}.$ (6.45)

Combining 6.45 and 6.28c and Hölder’s inequality, we obtain

$\displaystyle\left|\int_{\mathbb{R}^{d}}(u(t,\cdot)-u(\tau,\cdot))\eta_{d,{R}}\varphi\,dx\right|\leq$
$\displaystyle\left|\int_{\tau}^{t}\left<\partial_{t}u,\eta_{d,{R}}\varphi\right>\,ds\right|$
$\displaystyle\leq$
$\displaystyle(t-\tau)^{\frac{1}{2}}\left\|\nabla\left(\eta_{d,{R}}\varphi\right)\right\|_{L^{2}(\mathbb{R}^{d})}\|\partial_{t}u\|_{L^{2}\left(0,T;H^{-1}(\mathbb{R}^{d})\right)}$
$\displaystyle\leq$
$\displaystyle(t-\tau)^{\frac{1}{2}}{R}^{\frac{d}{2}}\Cr{C37}\Cr{D3}\left(T,\chi,V,u_{0}\right),$

as required. ∎

Next, we establish $L^{1}$ stability for 6.1.

######  Lemma 6.9.

Let $\chi$ satisfy 1.3a, $V,\widehat{V}$ satisfy 6.2, and
$u_{0},\widehat{u}_{0}$ satisfy 6.3. Then, the corresponding solutions $u$ and
$\widehat{u}$ to the Cauchy problem 6.1 satisfy

$\displaystyle\underset{[0,T]}{\max}\left\|u-\widehat{u}\right\|_{L^{1}(\mathbb{R}^{d})}$
$\displaystyle\leq$
$\displaystyle\left\|u_{0}-\widehat{u}_{0}\right\|_{L^{1}(\mathbb{R}^{d})}$
$\displaystyle+\Cl{D4}\left(T,\chi,V,u_{0}\right)\left(T\left\|\mathbf{1}_{\\{u>0\\}}\nabla\cdot\left(V-\widehat{V}\right)\right\|_{L^{\infty,1}(E_{T})}+T^{\frac{1}{2}}\left\|\mathbf{1}_{\\{u>0\\}}\left|V-\widehat{V}\right|\right\|_{L^{\infty}(E_{T})\cap
L^{\infty,1}(E_{T})}\right),$ (6.46)

where

$\displaystyle\Cr{D4}\left(T,\chi,V,u_{0}\right):=\Cr{D4_{1}}\max\left\\{\Cr{D1},\Cr{D1}^{\frac{1}{2}}\right\\}\left(T,\chi,V,u_{0}\right)$
(6.47)

and $\Cr{D1}$ and $\Cr{D4_{1}}$ are as defined in 6.14a and 6.21,
respectively.

###### Proof.

Since solutions to 6.1 are unique, they can be obtained as limits of
approximations such as introduced in 6.29, 6.30, 6.31, 6.32, and 6.33 in the
proof of Lemma 6.6. In particular,

$\displaystyle\left\|u_{0}-\widehat{u}_{0}\right\|_{L^{1}(\mathbb{R}^{d})}=\underset{m\to\infty}{\lim}\underset{\varepsilon\to
0}{\lim}\left\|u_{\varepsilon m0}-\widehat{u}_{\varepsilon
m0}\right\|_{L^{1}(\mathbb{R}^{d})}.$ (6.48)

The corresponding solution families $(u_{\varepsilon m})$ and
$(\widehat{u}_{\varepsilon m})$ satisfy estimates 6.13 from Lemma 6.3.
Therefore, a standard application of the Lions-Aubin lemma, a diagonal
argument, and the above mentioned uniqueness of the limit solutions together
imply that

$\displaystyle\left\|u-\widehat{u}\right\|_{L^{1}(\mathbb{R}^{d})}=$
$\displaystyle\sup_{R>0}\left\|u-\widehat{u}\right\|_{L^{1}(B_{R})}$
$\displaystyle=$
$\displaystyle\sup_{R>0}\underset{m\to\infty}{\lim}\underset{\varepsilon\to
0}{\lim}\left\|u_{\varepsilon m}-\widehat{u}_{\varepsilon
m}\right\|_{L^{1}(B_{R})}$ $\displaystyle\leq$
$\displaystyle\underset{m\to\infty}{\lim\inf}\underset{\varepsilon\to
0}{\lim\inf}\left\|u_{\varepsilon m}-\widehat{u}_{\varepsilon
m}\right\|_{L^{1}(B_{m})}\qquad\text{a.e. in }(0,T).$

Further, since assumptions of Lemma 6.4 are satisfied, $(u_{\varepsilon m})$
and $(\widehat{u}_{\varepsilon m})$ satisfy estimate 6.20. Passing to the
limit inferior on both sides of 6.20 and using 6.48, 6.49, 6.34b, 6.34d, 6.36,
6.37b, 6.13a, and 6.38 and the dominated convergence theorem, we arrive at the
inequality

$\displaystyle\left\|u-\widehat{u}\right\|_{L^{1}(\mathbb{R}^{d})}\leq$
$\displaystyle\left\|u_{0}-\widehat{u}_{0}\right\|_{L^{1}(\mathbb{R}^{d})}+\Cr{D4_{1}}\left(T,\chi,V,u_{0}\right)\left(\left\|u\nabla\cdot\left(V-\widehat{V}\right)\right\|_{L^{1}(E_{T})}+\left\|u^{\frac{1}{2}}\left|V-\widehat{V}\right|\right\|_{L^{2}(E_{T})}\right)\quad\text{a.e.}$
(6.49)

Using Hölder’s inequality and interpolation inequalities in $L^{p}$ spaces on
the right-hand side of 6.49 and the fact that $u,\widehat{u}\in
C_{w}([0,T];L^{1}(\mathbb{R}^{d}))$ finally yields 6.46. ∎

A comparison property in $L^{1}$ also holds.

######  Lemma 6.10.

Let $\chi$ satisfy 1.3a, $V$ satisfy 6.2, and $u_{0},\widehat{u}_{0}$ satisfy
6.3. Then, any corresponding pair of subsolution $u$ and supersolution
$\widehat{u}$ to the Cauchy problem 6.1 satisfies

$\displaystyle\left\|\left(u-\widehat{u}\right)_{+}(t,\cdot)\right\|_{L^{1}(\mathbb{R}^{d})}\leq\left\|\left(u-\widehat{u}\right)_{+}(0,\cdot)\right\|_{L^{1}(\mathbb{R}^{d})}\qquad\text{for
}t\in[0,T].$

We omit the proof of Lemma 6.10 since it can be carried out very similar to
that of Lemma 6.9.

###  6.2 Non-explosion of support for small times

In this Subsection we show that the support of a solution to 6.1 that has a
sufficiently small support at the initial time cannot explode on a short time
interval. More precisely, our result is as follows.

######  Lemma 6.11.

Let $\chi$ satisfy 1.3a, $V$ satisfy 6.2, and $u_{0}$ satisfy 6.3. Define

$\displaystyle\Cl{Dsupp1}(T,\chi,V):=\frac{1}{4(d+2)}\|(\nabla\cdot
V)_{-}\|_{L^{\infty}(E_{T})}\|\chi\|_{L^{\infty}((0,\infty))},$ (6.50a)
$\displaystyle\Cl[Rho]{rho-u0V}(T,\chi,V):=\frac{d+2}{\||V|\|_{L^{\infty}(E_{T})}\|\chi^{\prime}\|_{L^{\infty}((0,\infty))}},$
(6.50b)
$\displaystyle\Cl[Rho]{rho4}\left(T,\chi,V,a\right):=\left(1-\frac{\Cr{Dsupp1}(T,\chi,V)}{a}\right)\Cr{rho-u0V}(T,\chi,V),$
(6.50c)
$\displaystyle\Cl[T]{T-usuppV}(T,\chi,V,\rho,a,b):=\frac{1}{8a}\ln\left(\frac{1}{b}\min\left\\{\rho,\Cr{rho4}\left(T,\chi,V,a\right)\right\\}\right),$
(6.50d) $\displaystyle\rho_{a,b}(t):=be^{t4a},$ (6.50e) where constants $a$,
$b$, $\delta$, and $\rho$ are such that $\displaystyle
0<\rho\leq\Cr{rho-u0V}(T,\chi,V),$ (6.50f) $\displaystyle\delta\in(0,1),$
(6.50g) $\displaystyle b\in(\delta\rho,\rho),$ (6.50h) $\displaystyle
a>\frac{\Cr{Dsupp1}(T,\chi,V)}{1-\frac{b}{\Cr{rho-u0V}(T,\chi,V)}},$ (6.50i)
$\displaystyle
a\geq\frac{\|u_{0}\|_{L^{\infty}(\mathbb{R}^{d})}}{b^{2}-(\delta\rho)^{2}}.$
(6.50j)

Then,

$\displaystyle\operatorname{supp}(u_{0})\subset\overline{B_{\delta\rho}}$

implies for any corresponding supersolution
$u:[0,T]\times\overline{\Omega}\to[0,\infty)$ (in terms of Definition 6.1) to
6.1 that

$\displaystyle\operatorname{supp}(u(t,\cdot))\subset\overline{B_{\rho_{a,b}(t)}},\qquad\rho_{a,b}(t)\in(0,\rho)$

for all $t\in[0,\min\left\\{T,\Cr{T-usuppV}(T,\chi,V,\rho,a,b)\right\\})$.

###### Proof.

Following a common approach for equations with PM diffusion (see, e.g. [28,
Chapter 14]), we construct a supersolution to 6.1a that is supported in
$B_{\rho}$ and is no smaller than $u$ for $t\in[0,\Cr{T-usuppV}]$.

Direct computation shows that

$\displaystyle U_{a,b}(t,x):=$ $\displaystyle
a\left(b^{2}e^{t8a}-|x|^{2}\right)_{+}\qquad\text{for
}(t,x)\in\mathbb{R}\times\mathbb{R}^{d}$ (6.51)

is a strong and, hence, a weak and very weak solution to

$\displaystyle\partial_{t}U=\Delta U^{2}+4(d+2)aU\quad\text{in
}\mathbb{R}\times\mathbb{R}^{d}$ (6.52)

and has the following properties:

$\displaystyle\operatorname{supp}(U_{a,b}(t,\cdot))=\overline{B_{\rho_{a,b}(t)}},\qquad\rho_{a,b}(t):=be^{t4a}\qquad\text{for
}t\in\mathbb{R},$ (6.53)

and

$\displaystyle U_{a,b}(t,\cdot)\in W^{1,\infty}(\mathbb{R}^{d}),\qquad\|\nabla
U_{a,b}(t,\cdot)\|_{L^{\infty}(\mathbb{R}^{d})}=2abe^{t4a}\qquad\text{for
}t\in\mathbb{R}.$ (6.54)

Due to 6.53 we have

$\displaystyle\rho_{a,b}(t)<\rho\qquad\text{for
}t\in\left[0,\min\left\\{T,\frac{1}{4a}\ln\left(\frac{\rho}{b}\right)\right\\}\right),$
(6.55)

whereas 6.50h, 6.51, and 6.50j imply

$\displaystyle\underset{B_{\delta\rho}}{\inf}U_{a,b}(0,\cdot)=$ $\displaystyle
a\left(b^{2}-(\delta\rho)^{2}\right)\geq\|u_{0}\|_{L^{\infty}(\mathbb{R}^{d})}.$
(6.56)

We are going to show that $U_{a,b}$ is a supersolution to 6.1a in
$[0,\Cl[T]{T_{A}B}]\times\mathbb{R}^{d}$ if $\Cr{T_{A}B}\in(0,T]$ is
sufficiently small. Since $U_{a,b}$ satisfies equation 6.52, being a
supersolution to 6.1a is equivalent to

$\displaystyle
4(d+2)aU_{a,b}+\nabla\cdot\left(VU_{a,b}\chi\left(U_{a,b}\right)\right)\geq
0\qquad\text{a.e. in }(0,\Cr{T_{A}B}]\times\mathbb{R}^{d}.$

Using 6.54, 6.50a, and 6.50b and the assumptions on $V$ and $\chi$, we can
estimate as follows:

$\displaystyle
4(d+2)aU_{a,b}+\nabla\cdot\left(VU_{a,b}\chi\left(U_{a,b}\right)\right)$
$\displaystyle=$ $\displaystyle 4(d+2)aU_{a,b}+(\nabla\cdot
V)U_{a,b}\chi\left(U_{a,b}\right)+V\left(U_{a,b}\chi^{\prime}\left(U_{a,b}\right)+\chi\left(U_{a,b}\right)\right)\nabla
U_{a,b}$ $\displaystyle\geq$ $\displaystyle 4(d+2)aU_{a,b}-\|(\nabla\cdot
V)_{-}\|_{L^{\infty}(E_{T})}\|\chi\|_{L^{\infty}((0,\infty))}U_{a,b}$
$\displaystyle-2\||V|\|_{L^{\infty}(E_{T})}\|\chi^{\prime}\|_{L^{\infty}((0,\infty))}\|\nabla
U_{a,b}\|_{L^{\infty}(\mathbb{R}^{d})}U_{a,b}$ $\displaystyle\geq$
$\displaystyle 4(d+2)aU_{a,b}\left(1-\frac{1}{4(d+2)a}\|(\nabla\cdot
V)_{-}\|_{L^{\infty}(E_{T})}\|\chi\|_{L^{\infty}((0,\infty))}\right.$
$\displaystyle\phantom{4(d+2)aU_{a,b}\left(1\right.}\left.-\frac{1}{d+2}\||V|\|_{L^{\infty}(E_{T})}\|\chi^{\prime}\|_{L^{\infty}((0,\infty))}be^{t4a}\right)$
$\displaystyle\geq$ $\displaystyle
4(d+2)aU_{a,b}\left(1-\frac{\Cr{Dsupp1}(T,\chi,V)}{a}-\frac{b}{{\Cr{rho-u0V}(T,\chi,V)}}e^{t4a}\right)$
$\displaystyle\geq$ $\displaystyle 0\qquad\text{a.e. in
}{(0,\min\\{T,\Cr{T_{A}B}({\chi,V},{a,b})\\}]}\times\mathbb{R}^{d},$

where

$\displaystyle\Cr{T_{A}B}({\chi,V},{a,b}):=$
$\displaystyle\frac{1}{4a}\ln\left(\frac{\Cr{rho4}\left(T,\chi,V,a\right)}{b}\right)>0$

due to 6.50i and 6.50c. Consequently, $U_{a,b}$ is indeed a supersolution to
6.1 on $[0,\Cr{T_{A}B}(\dots)]\times\mathbb{R}^{d}$.

Altogether, we have shown that
$U_{a,b}:[0,\min\left\\{T,\Cr{T-usuppV}(\dots)\right\\}]\times\mathbb{R}^{d}\to[0,\infty)$
is a weak supersolution to 6.1a, $U_{a,b}(0,\cdot)\geq u_{0}$ holds a.e. due
to 6.56, and $U_{a,b}(\cdot)$ is supported in $B_{\rho_{a,b}(\cdot)}$ due to
6.55. Now we can apply Lemma 6.10 and conclude that $u\leq U_{a,b}$ a.e. in
$[0,\min\left\\{T,\Cr{T-usuppV}(\dots)\right\\}]\times\mathbb{R}^{d}$. Hence,
$\operatorname{supp}(u(t,\cdot))\subset\overline{B_{\rho_{a,b}(t)}}$ for all
$t\in[0,\min\left\\{T,\Cr{T-usuppV}(\dots)\right\\}]$, as required.

∎

##  7 Well-posedness of 1.6 for fixed $w$

In this Section we study the Cauchy problem 1.6 for a fixed $w$. We refer to
Section 2(PDE1) and (PDE2) for the discussion of this equation. Our proofs
here rely on the results collected for the Cauchy problem 6.1 in Section 6. In
order to produce a fixed advection direction as in 6.1a, we take

$\displaystyle V:={\mathcal{V}}(w,u):=w(\nabla H\star u).$ (7.1)

In Subsection 7.1 we establish some useful properties of such ${\mathcal{V}}$
needed in order to analyse 1.6 in Subsection 7.2.

###  7.1 Properties of $H$ and ${\mathcal{V}}$

In this Subsection we establish some properties of kernel $H$ from 1.3b and
the bilinear operator ${\mathcal{V}}$ from 7.1.

######  Lemma 7.1 (Properties of $H$).

Let $F\in C^{1}([0,\infty))$, and $H$ as defined in 1.3b. Then,

  1. (1)

$H\in W^{1,\infty}(\mathbb{R}^{d})$, $\nabla H$ is supported in
$\overline{B_{1}}$,

$\displaystyle\nabla
H=-\mathbf{1}_{[0,1]}F(|\cdot|)\operatorname{sign}\qquad\text{ in
}B_{1}\backslash\\{0\\},$ (7.2)

and

$\displaystyle\|\nabla H\|_{(L^{\infty}(\mathbb{R}^{d}))^{d}}=$
$\displaystyle\max_{[0,1]}|F|,$ (7.3a) $\displaystyle\|\nabla
H\|_{(L^{1}(\mathbb{R}^{d}))^{d}}=$ $\displaystyle
2d\left|B^{d-1}_{1}\right|\int_{0}^{1}|F(s)|s^{d-1}\,ds;$ (7.3b)

  2. (2)

for $d\geq 2$, it holds that
$\nabla\nabla^{T}H\in({\mathcal{M}}(\mathbb{R}^{d}))^{d\times d}$,

$\displaystyle\nabla\nabla^{T}H=$