C symbol=C, format=0 F symbol=F, format=0 G symbol=G, format=0 reset=section M
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affil0affil0affiliationtext:  Queen’s University Belfast, School of
Mathematics and Physics
University Road, Belfast BT7 1NN, Northern Ireland, UK
<EMAIL_ADDRESS>
# Global entropy solutions to a degenerate parabolic-parabolic chemotaxis
system for flux-limited dispersal

Anna Zhigun

###### Abstract

Existence of global finite-time bounded entropy solutions to a parabolic-
parabolic system proposed in [15] is established in bounded domains under no-
flux boundary conditions for nonnegative bounded initial data. This
modification of the classical Keller-Segel model features degenerate diffusion
and chemotaxis that are both subject to flux-saturation. The approach is based
on Schauder’s fixed point theorem and calculus of functions of bounded
variation.

Keywords: flux-limited diffusion; flux-limited chemotaxis; entropy solutions.
MSC 2020: 26B30 35K55 35K65 35M30 35Q92 92C17

##  1 Introduction

In this paper we study the initial-boundary value problem (IBVP) for the
chemotaxis system

$\displaystyle\partial_{t}c=\nabla\cdot\left(D_{c}\frac{c\nabla
c}{\sqrt{c^{2}+\frac{D_{c}^{2}}{C^{2}}|\nabla c|^{2}}}-\chi c\frac{\nabla
v}{\sqrt{1+|\nabla v|^{2}}}\right)+f_{c}(c,v),$ (1.1a)
$\displaystyle\partial_{t}v=D_{v}\Delta v+f_{v}(c,v)$ (1.1b)

equipped with no-flux boundary conditions. Equations 1.1 were derived in [15]
and serve to overcome several issues arising in connection with the classical
Keller-Segel model for chemotaxis [31, 32]

$\displaystyle\partial_{t}c=\nabla\cdot(D_{c}\nabla c-\chi c\nabla v),$ (1.2a)
$\displaystyle\partial_{t}v=D_{v}\Delta v-\gamma v+\alpha c.$ (1.2b)

Both systems describe spatio-temporal evolution of cell density, $c$ and
concentration of chemoattractant, $v$. Functions $f_{c}$ and $f_{v}$ and
constants $C,D_{c},D_{v},\chi,\alpha,\gamma>0$ are assumed given. Each of the
$c$-equations 1.2a and 1.1a can be rewritten in the form of a continuity
equation

$\displaystyle\partial_{t}c=-\nabla\cdot(c(V_{d}+V_{c})),$

where $V_{d}$ and $V_{c}$ are velocities due to diffusion and chemotaxis,
respectively. Whereas for 1.2a

$\displaystyle V_{d}=-D_{C}\frac{\nabla c}{c},\qquad V_{c}=\chi\nabla v$

need not be finite everywhere, for 1.1a we have

$\displaystyle V_{d}=-D_{c}\frac{\nabla
c}{\sqrt{c^{2}+\frac{D_{c}^{2}}{C^{2}}|\nabla c|^{2}}},\qquad
V_{c}=\chi\frac{\nabla v}{\sqrt{1+|\nabla v|^{2}}},$ (1.3)

so that $V_{d}$ and $V_{c}$ are a priori bounded by $C$ and $\chi$,
respectively. Thus, unlike 1.2a, both diffusion and chemotaxis in 1.1a are
flux-limited (FL). In this work, we concentrate solely on the analysis of 1.1
and refer the interested reader to [15] as well as our review [45] for details
on the biological motivation behind equations with FL diffusion and/or taxis
and their derivation.

While an equation with a flux-limited diffusion may be attractive for
modelling, the presence of such a term leads to substantial analytical
difficulties. In particular, spatial derivatives of solutions that are
nonnegative but not everywhere positive are, in general, Radon measures, and
not some integrable functions. Dealing with such solutions requires using the
calculus of functions of bounded variation (BV functions).

Equations with FL diffusion, also with $V_{d}$ different from that in 1.3,
however without chemotaxis, i.e. for $\chi=0$, are already well-understood:
well-posedness and other qualitative properties under various boundary
conditions and in the whole space were established in [12, 5, 9, 8, 10, 7, 11,
4, 21], see also the review paper [20] for further references. Well-posedness
of the Cauchy problem for a FL reaction-diffusion equation was obtained in [6]
and the corresponding travelling waves were studied in [23]. As to FL
diffusion-chemotaxis systems, only parabolic-elliptic modifications, i.e. such
where 1.1b is replaced by some elliptic equation, have been analysed so far.
Existence and boundedness/blow-up were addressed in the radial-symmetric case
for strictly positive initial values of $c$ in [16, 17, 38, 24, 43, 44]. The
travelling waves analysis in [14, 19, 22] allowed for biologically relevant
not necessarily strictly positive wave profiles. A qualitative study of a
fully parabolic system such as 1.1 has been lacking. In the present work we
make a step in this direction by establishing for

$0<\chi<C$

global existence for the IBVP associated with 1.1 subject to no-flux boundary
conditions in a certain class of entropy solutions. Our main Theorem 3.7 (see
below) holds in general spatial domains and for general nonnegative and
bounded initial data. No radial symmetry or initial positivity of $c$ is thus
required.

System 1.1 raises analytical challenges that differ from those for 1.2.
Indeed, since the latter is a regular upper-triangular parabolic system,
standard theory (see, e.g. [2]) ensures global well-posedness in the classical
sense provided that solutions are a priori bounded for finite times. The study
of such systems thus mostly concentrates upon the issue of boundedness/blow-
up. In the case of system 1.1 sufficiently regular weak solutions are (see
Lemma 4.1 below) a priori bounded over finite times if $0<\chi<C$ and $f_{c}$
satisfies appropriate assumptions. This holds thanks to the imposed limitation
of the chemotactic flux. Notwithstanding, saturation in the diffusion term
precludes the application of standard results for parabolic equations and
systems. In [12, 5, 9, 8, 10, 7, 11], where a solvability theory for single
parabolic FL diffusion equations was developed, one first constructed a mild
solution and then verified that it is actually the unique entropy solution. In
the case of a strongly coupled system such as 1.1 it seems that neither it is
possible to set up a semigroup for the system, nor to prove uniqueness of
entropy solutions, mainly due to the generally poor regularity of $c$. To
overcome this issue, we first decouple the system, treat solvability of the
two equations separately, and finally obtain a joint solution with the help of
Schauder’s fixed point theorem. In order to solve equation 1.1a for a fixed
$v$ we adapt the aforementioned theory [12, 5, 9, 8, 10, 7, 11].

The rest of the paper is organised as follows. We begin with the preliminary
Section 2 where we collect some notation and useful facts. In particular, we
recall some machinery which helps to operate with BV functions. In Section 3
we fully set up the problem and state the main result, the existence Theorem
3.7. Sections 4 and 5 are devoted to the solvability of 1.1a and 1.1b for
fixed $v$ and $c$, respectively. The most extensive Section of this work,
Section 4, begins with Section 4.1 where some a priori bounds are derived for
sufficiently regular solutions. It is succeeded by a study of some elliptic
operators related to the $c$-equation in Section 4.2. Next, in Section 4.3, we
consider a family of regularised time discretisations of 1.1a for a fixed $v$.
This allows us to pass to a limit and thus obtain an entropy solution to 1.1a
in Section 4.4. Finally, in Section 6 we establish the existence of entropy
solutions to the whole system 1.1.

##  2 Preliminaries

Throughout this paper, $\Omega\subset\mathbb{R}^{d}$, $d\in\mathbb{N}$, is a
smooth bounded domain and $O\subset\mathbb{R}^{N}$, $N\in\mathbb{N}$, is any
domain.

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.

###  2.1 Spaces of scalar functions

Let $k\in\mathbb{N}_{0}$. As usual, $C^{k}(\overline{O})$
($C(\overline{O}):=C^{0}(\overline{O})$) denotes the space of $k$ times
continuously differentiable real-valued functions in $\overline{O}$ equipped
with their standard respective norms. By $D(O)$ we denote the space of test
functions in $O$. The closure of $D(O)$ in $C^{k}(\overline{O})$
($C(\overline{O})$) is denoted by $C_{0}^{k}(\overline{O})$
($C_{0}(\overline{O})$). Similarly, we write, e.g. $C(E;F)$ for the space of
continuous functions with domain $E$ and values in $F$.

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 $s\in\mathbb{R}$. We assume the reader to be familiar
with their standard properties.

Index $\nu$ added into the notation of a space of functions defined in
$\overline{\Omega}$ means that this is the subspace of functions with zero
normal derivative, understood in the trace sense where necessary, on the
boundary of $\Omega$.

###  2.2 Measures and BV functions

In this Subsection we introduce some notation and recall a few facts about
Radon measures and BV functions. A detailed treatment of BV functions can be
found, e.g. in [3]. By ${\cal M}(\Omega)$ ($({\cal M}(\Omega))^{d}$) we denote
the space of finite real (vector, with values in $\mathbb{R}^{d}$) Radon
measures in $\Omega$. The norms in these spaces are given by the total
variation. We follow a standard convention and identify these measures with
the corresponding continuous lineal functionals on $C_{0}(\overline{\Omega})$
defined via integration. We write $|\mu|$/$\mu^{ac}$/$\mu^{s}$ for the
variation/absolutely continuous part/singular part of a Radon measure $\mu$,
respectively. We also identify the absolutely continuous part with the
corresponding density (with respect to the Lebesgue measure, that is). By
$\frac{\mu_{1}}{\mu_{2}}$ we denote the Radon-Nikodym derivative of a measure
$\mu_{1}$ with respect to a positive measure $\mu_{2}$.

By $|\cdot|$ we denote the $d$-dimensional Lebesgue measure and by ${\cal
H}^{d-1}$ the $(d-1)$-dimensional Hausdorff measure in $\mathbb{R}^{d}$.

Unless specified otherwise, a.e. (a.a.), meaning almost everywhere (almost
all), is to be understood with respect to the Lebesgue measure of the
dimension that is standardly used for a set in question.

As usual, $BV(\Omega)$ stands for the subspace of $L^{1}(\Omega)$ which
consists of functions $w$ of bounded variation. The latter means that the
distributional gradient $Dw$ is given by a Radon measure of a finite total
variation. We write $\nabla w/D^{s}w/D^{c}w/D^{j}w$ for the absolutely
continuous/singular/Cantor/jump parts of $Dw$, respectively. If $w\in
W^{1,1}(\Omega)$, then, of course, $Dw=\nabla w$. We denote by $S_{w}$ the
approximate discontinuity set and by $J_{w}$ the set of approximative jump
points of $w$. The approximate limit of $w$ at $x\in\Omega\backslash S_{w}$ is
denoted by $\widetilde{w}(x)$. For $x\in J_{w}$, the approximate limits of $w$
arising in directions $\nu_{w}(x):=\frac{Dw}{|Dw|}(x)$ and $-\nu_{w}(x)$ are
denoted by $w^{+}(x)$ and $w^{-}(x)$, respectively.

As in [12, 5, 9, 8, 10, 7, 11], we need pairings defined via the ’partial
integration formula’, in our case for a bounded domain:

$\displaystyle\left<(Z,Dw),\rho\right>:=\int_{\Omega}-\rho w\nabla\cdot
Z-Zw\cdot\nabla\rho\,dx\qquad\text{for all }\rho\in
C_{0}(\overline{\Omega})\cap C^{1}(\overline{\Omega})$ (2.1)

for $w\in BV(\Omega)$ and $Z\in X_{\infty}(\Omega)$, where

$\displaystyle X_{\infty}(\Omega):=\\{Z\in(L^{\infty}(\Omega))^{d}:\
\nabla\cdot Z\in L^{\infty}(\Omega)\\}.$ (2.2)

We make use of the following properties originating from [13, 33].

  1. (1)

The mapping $(\cdot,D\cdot):X_{\infty}(\Omega)\times
BV(\Omega)\rightarrow{{\cal M}(\Omega)}$ is well-defined, bilinear, bounded,
and satisfies

$\displaystyle\|(Z,Dw)\|_{{{\cal
M}(\Omega)}}\leq\|z\|_{L^{\infty}(\Omega)}\|Dw\|_{{{\cal
M}(\Omega)}}\qquad\text{for all }Z\in X_{\infty}(\Omega)\text{ and }w\in
BV(\Omega).$

  2. (2)

Paring $(\cdot,\cdot)$ extends product of functions and multiplication of a
finite measure by a continuous function:

$\displaystyle(Z,Dw)=Z\cdot\nabla w$ $\displaystyle\qquad\text{for all }Z\in
X_{\infty}(\Omega),\ w\in W^{1,1}(\Omega),$
$\displaystyle\left<(Z,Dw),\rho\right>=\left<Dw,Z\rho\right>$
$\displaystyle\qquad\text{for all }Z\in(C(\overline{\Omega}))^{d},\
\rho\in{C_{0}(\overline{\Omega})}.$

  3. (3)

Measure $(Z,Dw)$ is absolutely continuous with respect to $|Dw|$ and has with
respect to it a bounded density.

  4. (4)

For the absolutely continuous part of $(Z,Dw)$ we have the explicit formula

$\displaystyle(Z,Dw)^{ac}=Z\cdot\nabla w\qquad\text{a.e. in }\Omega.$ (2.3a)

  5. (5)

The singular part $(Z,Dw)^{s}=(Z,Dw)-Z\cdot\nabla w$ is a finite measure which
is absolutely continuous with respect to $|D^{s}_{x}w|$ and satisfies

$\displaystyle|(Z,Dw)^{s}|\leq\|Z\|_{L^{\infty}(\Omega)}|D^{s}w|\qquad\text{in
}{\cal M}(\Omega).$ (2.3b)

###  2.3 Spaces of functions and measures with values in Banach spaces

In this Subsection, $I\subset\mathbb{R}_{0}^{+}$ is an interval,
$p\in[1,\infty]$, and $X$ is a Banach space with the dual space $X^{*}$.

We denote: by $C(I;X)$ ($C^{1}(I;X)$) the space of continuous (continuously
differentiable) functions $u:I\rightarrow X$, by $C_{w}(I;X^{*})$ the space of
weakly continuous functions $u:I\rightarrow X^{*}$, by $L^{p}(I;X)$ the space
of strongly measurable functions $u:I\rightarrow X$ with $\|u\|_{X}\in
L^{p}(I)$, and by $L^{p}_{w}(I;X)$ ($L^{p}_{w-*}(I;X^{*})$) the space of
weakly (weakly-$*$) measurable functions $u:I\to X$ ($u:I\rightarrow X^{*}$)
with $\|u\|_{X}\in L^{p}(I)$ ($\|u\|_{X^{*}}\in L^{p}(I)$). Further,
$W^{1,p}(I;X)$ stands for the Sobolev space of weakly differentiable functions
$u\in L^{p}(I;X)$ such that $\partial_{t}u\in L^{p}(I;X)$.

Let $Y$ be a space that contains $X$. If $u$ is a $Y$-valued distribution on
$I$ and such that it can be extended to a function with values in $X$, then we
identify $u$ with that extension.

Since we only deal here with the cases where $X$ is a space of functions or
measures, we typically adopt the notation $u(t,\cdot)$ rather than $u(t)$ for
$t\in I$.

For a function $u:I\to X^{d}$, $d\in\mathbb{N}$, we often write
$\|u(t,\cdot)\|_{X}$ instead of $\||u(t,\cdot)|\|_{X}$, where $|\cdot|$
denotes the Euclidean norm in $\mathbb{R}^{d}$.

If $loc$ appears in the index of a space notation, then the corresponding
defining property is assumed to hold on every compact subset of $I$.

If $f:I\to{\cal M}(\Omega)$, then we set
$(f^{ac})(t,\cdot):=(f(t,\cdot))^{ac}$ and
$(f^{s})(t,\cdot):=(f(t,\cdot))^{s}$ for $t\in I$.

By ${\cal M}(I;X^{*})$ we denote the space of $X^{*}$-valued countably
additive regular vector measures in $I$ that are of bounded variation. The
reader is referred to [28, Chapter 1 §1] for the definitions of the involved
terms. We equip this vector measure space with the variation over $I$ as norm.
An extension of the Riesz-Markov-Kakutani representation theorem holds for
compact $I$[41] (see also [29, Chapter III §19]):

$\displaystyle{\cal M}(I;X^{*})\cong(C(I;X))^{*}.$ (2.4)

In view of 2.4 and a representation result given by Lemma B.1 from Appendix B
we identify $f\in L^{1}_{w-*}((0,T);X^{*})$ with the vector measure defined by

$\displaystyle\left<\mu_{f}(E),\varphi\right>:=\int_{E}\left<f(t),\varphi\right>dt\qquad\text{for
all }\varphi\in X\text{ and Lebesgue measurable }E\subset I$

as well as with the corresponding element of $(C(I;X))^{*}$.

###  2.4 Interpolation and discretisation of vector-valued functions

For $n\in\mathbb{N}$ set

$\displaystyle\frac{1}{n}\mathbb{N}_{0}:=\left\\{\frac{k}{n}:\
k\in\mathbb{N}_{0}\right\\}.$

Let $X$ be some Banach space. We use two standard ways of interpolating a
function $u:\frac{1}{n}\mathbb{N}_{0}\cap[0,T]\to X$: piecewise constant

$\displaystyle\underline{\Pi}_{n}{u}(t):=u\left(\frac{k-1}{n}\right)$
$\displaystyle\qquad\text{in
}\left(\frac{k-1}{n},\frac{k}{n}\right)\qquad\text{ for
}k\in\\{1,2,\dots,nT\\},$ $\displaystyle{u}(t):=u\left(\frac{k}{n}\right)$
$\displaystyle\qquad\text{in
}\left(\frac{k-1}{n},\frac{k}{n}\right)\qquad\text{ for
}k\in\\{1,2,\dots,nT\\},$

and piecewise linear

$\displaystyle\Lambda_{n}{u}(t):=$
$\displaystyle\frac{u\left(\frac{k}{n}\right)-u\left(\frac{k-1}{n}\right)}{\frac{1}{n}}\left(t-\frac{k}{n}\right)+u\left(\frac{k}{n}\right)$
$\displaystyle=$
$\displaystyle\frac{\frac{k}{n}-t}{\frac{1}{n}}u\left(\frac{k-1}{n}\right)+\frac{t-\frac{k-1}{n}}{\frac{1}{n}}u\left(\frac{k}{n}\right)$
$\displaystyle=$
$\displaystyle\frac{\frac{k}{n}-t}{\frac{1}{n}}\underline{\Pi}_{n}{u}(t)+\frac{t-\frac{k-1}{n}}{\frac{1}{n}}{u}(t)\qquad\text{in
}\left(\frac{k-1}{n},\frac{k}{n}\right)\qquad\text{ for
}k\in\\{1,2,\dots,nT\\}.$ (2.5)

For $\widetilde{u}\in L^{1}_{loc}([0,T];X)$, its $n$-th discretisation via
Steklov averages is given by

$\displaystyle
P_{n}\widetilde{u}\left(\frac{k}{n}\right):=n\int_{\frac{k-1}{n}}^{\frac{k}{n}}\widetilde{u}(t)\,dt\qquad\text{for
}k\in\\{1,2,\dots,nT\\}.$

Further, we set

$\displaystyle
P_{n}\widetilde{u}(0):=P_{n}\widetilde{u}\left(\frac{1}{n}\right).$

Clearly, the introduced interpolation and discretisation operators are linear
and preserve non-negativity: for a functional space $X$

$\displaystyle u\geq 0\text{ in }\frac{1}{n}\mathbb{N}_{0}\cap[0,T]$
$\displaystyle\qquad\Rightarrow\qquad\underline{\Pi}_{n}u,u,\Lambda_{n}u\geq
0\text{ in }[0,T],$ $\displaystyle\widetilde{u}\geq 0\text{ in }[0,T]$
$\displaystyle\qquad\Rightarrow\qquad P_{n}\widetilde{u}\geq 0\text{ in
}\frac{1}{n}\mathbb{N}_{0}\cap[0,T].$

Several other useful properties of these operators are provided in Appendix A.

###  2.5 Further notation

We make use of functions

$\displaystyle T_{m,M}(x):=$ $\displaystyle\begin{cases}m&\text{ for }x\leq
m,\\\ x&\text{ for }x\in(m,M),\\\ M&\text{ for }x\geq
M,\end{cases}\qquad-\infty<m<M<\infty,$ $\displaystyle T_{\delta}:=$
$\displaystyle T_{-\delta,\delta},$
$\displaystyle\operatorname{sign}_{\delta}(x):=$
$\displaystyle\begin{cases}-1&\text{ for }x<-\delta,\\\
\frac{1}{\delta}x&\text{ for }x\in[-\delta,\delta],\\\ 1&\text{ for
}x>\delta,\end{cases}\qquad\delta>0,$ $\displaystyle x_{+}:=$
$\displaystyle\begin{cases}x&\text{ for }x\geq 0\\\ 0&\text{ for
}x<0.\end{cases}$

For $h:\mathbb{R}^{k}\times\mathbb{R}^{d}\to\mathbb{R}$, $k,d\in\mathbb{N}$,
we define its recession function with respect to the second variable as in
[26]:

$\displaystyle h^{\infty}(y,\xi):=\underset{s\to
0}{\lim}sh\left(y,\frac{\xi}{s}\right)\qquad\text{for }y\in\mathbb{R}^{k},\
\xi\in\mathbb{R}^{d}.$

Finally, we make the following conventions.

  * •

For indices $i\in\mathbb{N}$, $C_{i}$ denotes a positive constant or,
alternatively, a positive non-decreasing function of its arguments. The
dependence upon the parameters of the problem (see Section 3 below), i.e. the
space dimension $d$, domain $\Omega$, constants $\chi$, $p_{0}$, and
$\alpha_{0}$, functions $f_{c}$ and $f_{v}$, the involved norms of the initial
data, etc. is mostly omitted.

  * •

Further, $g:\mathbb{R}_{0}^{+}\to\mathbb{R}$ stands for an arbitrary locally
Lipschitz function. We do not track dependence upon $g$ in the
constants/functions $C_{i}$.

##  3 Problem setting and main result

Given a smooth bounded domain $\Omega\subset\mathbb{R}^{d}$, $d\in\mathbb{N}$,
with the unit outer normal $\nu$, we consider the system

$\displaystyle\partial_{t}c=\nabla\cdot\left(a(c,\nabla c)-\chi c\Phi(\nabla
v)\right)+f_{c}(c,v)$ $\displaystyle\qquad\text{in }(0,\infty)\times\Omega,$
(3.1a) $\displaystyle\left(a(c,\nabla c)-\chi c\Phi(\nabla
v)\right)\cdot\nu=0$ $\displaystyle\qquad\text{in
}(0,\infty)\times\partial\Omega,$ (3.1b) $\displaystyle c(0)=c_{0}$
$\displaystyle\qquad\text{in }\\{0\\}\times\Omega,$ (3.1c)

and

$\displaystyle\partial_{t}v=D_{v}\Delta v+f_{v}(c,v)$
$\displaystyle\qquad\text{in }(0,\infty)\times\Omega,$ (3.2a)
$\displaystyle\partial_{\nu}v=0$ $\displaystyle\qquad\text{in
}(0,\infty)\times\partial\Omega,$ (3.2b) $\displaystyle v(0)=v_{0}$
$\displaystyle\qquad\text{in }\\{0\\}\times\Omega.$ (3.2c)

We next list our assumptions on the parameters along with some of their
properties:

$\displaystyle\chi\in(0,1),\qquad D_{v}>0;$ (3.3) $\displaystyle
a(c,\xi):=\begin{cases}\frac{c\xi}{\sqrt{c^{2}+|\xi|^{2}}}&\text{for }c>0,\\\
0&\text{for }c=0,\end{cases}\qquad\xi\in\mathbb{R}^{d},$ (3.4)

so that

$\displaystyle a\in C(\mathbb{R}_{0}^{+}\times\mathbb{R}^{d};\mathbb{R}^{d}),$
(3.5a) $\displaystyle|a(c,\xi)|\leq c$ $\displaystyle\qquad\text{for }c\geq
0,\ \xi\in\mathbb{R}^{d},$ (3.5b)
$\displaystyle|a(c_{1},\xi)-a(c_{2},\xi)|\leq|c_{1}-c_{2}|$
$\displaystyle\qquad\text{for }c_{1},c_{2}\geq 0,\ \xi\in\mathbb{R}^{d},$
(3.5c) $\displaystyle a(c,\xi)\cdot\xi\geq c(|\xi|-c)$
$\displaystyle\qquad\text{for }c\geq 0,\ \xi\in\mathbb{R}^{d},$ (3.5d)
$\displaystyle(a(c,\xi_{1})-a(c,\xi_{2}))\cdot(\xi_{1}-\xi_{2})\geq 0$
$\displaystyle\qquad\text{for }c\geq 0,\ \xi_{1},\xi_{2}\in\mathbb{R}^{d},$
(3.5e) $\displaystyle(a(c,\cdot)\cdot(\cdot))^{\infty}(\xi)=c|\xi|$
$\displaystyle\qquad\text{for }c\geq 0,\ \xi\in\mathbb{R}^{d};$ (3.5f)
$\displaystyle\Phi(\xi):=\frac{\xi}{\sqrt{1+|\xi|^{2}}}\qquad\text{for
}\xi\in\mathbb{R}^{d},$ (3.6)

so that

$\displaystyle\Phi\in C^{\infty}_{b}(\mathbb{R}^{d};\mathbb{R}^{d}),$ (3.7a)
$\displaystyle|\Phi(\xi)|\leq 1$ $\displaystyle\qquad\text{for
}\xi\in\mathbb{R}^{d};$ (3.7b) $\displaystyle f_{c}\in
C^{1}(\mathbb{R}_{0}^{+}\times\mathbb{R}_{0}^{+};\mathbb{R}),$ (3.8a)
$\displaystyle f_{c}(c,v)\leq\Cl[F]{fc1}c$ $\displaystyle\qquad\text{for
}c,v\geq 0,$ (3.8b) $\displaystyle f_{c}(0,v)=0$ $\displaystyle\qquad\text{for
}v\geq 0,$ (3.8c)

where $\Cr{fc1}\in\mathbb{R}$;

$\displaystyle f_{v}\in
C^{1}(\mathbb{R}_{0}^{+}\times\mathbb{R}_{0}^{+};\mathbb{R}),$ (3.9a)
$\displaystyle f_{v}(c,v)\leq\Cl[F]{fv1}(c)(v+\Cl[F]{fv2}(c))$
$\displaystyle\qquad\text{for }c,v\geq 0,$ (3.9b) $\displaystyle
f_{v}(c,0)\geq 0$ $\displaystyle\qquad\text{for }c\geq 0,$ (3.9c)

where $\Cr{fv1},\Cr{fv2}\in C(\mathbb{R}_{0}^{+};\mathbb{R}_{0}^{+})$;

$\displaystyle 0\leq c_{0}\in L^{\infty}(\Omega),$ (3.10) $\displaystyle 0\leq
v_{0}\in W^{2\left(1-\frac{1}{p_{0}}\right),p_{0}}_{\nu}\qquad\text{for some
}p_{0}>d+2.$ (3.11)

######  Remark 3.1.

  1. 1.

The specific choice of $a$ in 3.4 corresponds to $D_{c}=C=1$ in 1.1a. There is
no loss of generality here because such an $a$ can be achieved through
introducing rescaled time and space variables

$\displaystyle t^{\prime}:=\left(\frac{C}{D_{c}}\right)^{\frac{1}{2}}t,\qquad
x^{\prime}:=\frac{C}{D_{c}}x.$

Moreover, our analysis is not tied to a FL diffusion such as in 1.1a and,
similarly to, e.g. [8], extends to more general FL diffusion terms.

  2. 2.

Less restrictive conditions on other parameters could also be considered. For
instance, one could relax the one-sided sublinear growth condition 3.8a on
$f_{c}$ or assume $\chi$ to be a function rather than a constant. To
streamline the exposition, we abstained from such generalisations here.

Note that the logistic growth $f_{c}=f_{c}(c)=c(1-c)$, such as previously
adopted in [6], is admissible under 3.8.

We define solutions to 3.1-3.2 as follows.

######  Definition 3.2 (Weak and entropy solutions to 3.1).

Let $a$ and $\Phi$ be as defined in 3.4 and 3.6, respectively, $f_{c}$ satisfy
3.8, and $c_{0}$ satisfy 3.10. We call a pair of functions
$(c,v):\mathbb{R}_{0}^{+}\times\overline{\Omega}\rightarrow\mathbb{R}_{0}^{+}\times\mathbb{R}_{0}^{+}$
a global weak solution to 3.1-3.2 if $c$ and $v$ satisfy:

$\displaystyle c\in
L^{\infty}_{loc}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega))\cap
L_{w,loc}^{1}(\mathbb{R}_{0}^{+};BV(\Omega))\cap
C_{w}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega)),$ (3.12a) $\displaystyle c\in
W^{1,\infty}_{loc}(\mathbb{R}_{0}^{+},(W^{1,1}(\Omega))^{*}),$ (3.12b)
$\displaystyle v\in C(\mathbb{R}_{0}^{+};C^{1}_{\nu}(\overline{\Omega})),$
(3.13)

the weak formulation

$\displaystyle\left<\partial_{t}c,\varphi\right>=\int_{\Omega}-J\cdot\nabla\varphi+f_{c}(c,v)\varphi\,dx\qquad\text{a.e.
in }(0,\infty)\qquad\text{for }\varphi\in
L^{1}_{loc}(\mathbb{R}_{0}^{+};W^{1,1}(\Omega)),$ (3.14)

where

$\displaystyle J:=a(c,\nabla c)-\chi c\Phi(\nabla v),$

and the initial condition 3.1c in $L^{\infty}(\Omega)$.

We call $(c,v)$ a global entropy solution to 3.1-3.2 if it is a global weak
solution and additionally satisfies for all $T\in\mathbb{N}$ and $0\leq
m<M<\infty$:

$\displaystyle\partial_{t}\int_{0}^{c}T_{m,M}(u)\,du+J\cdot\nabla
T_{m,M}(c)-\nabla\cdot(JT_{m,M}(c))-f_{c}(c,v)T_{m,M}(c)\in
L^{1}_{w-*,loc}(\mathbb{R}_{0}^{+};{\cal M}(\Omega))$

and the entropy inequalities

$\displaystyle-\Cl{CBNDchi}(T)|D^{s}T_{m,M}(c)|$ $\displaystyle\leq$
$\displaystyle\partial_{t}\int_{0}^{c}T_{m,M}(u)\,du+J\cdot\nabla
T_{m,M}(c)-\nabla\cdot(JT_{m,M}(c))-f_{c}(c,v)T_{m,M}(c)$ (3.15a)
$\displaystyle\leq$
$\displaystyle-\frac{1}{2}|D^{s}T_{m,M}^{2}(c)|+\frac{\chi}{2}D^{s}T_{m,M}^{2}(c)\cdot\Phi(\nabla
v)\qquad\text{in }{\cal M}(\Omega)\text{ a.e. in }(0,T),$ (3.15b) where
$\displaystyle\Cr{CBNDchi}(T):=\Cl{CBNDchi1}(1+T)e^{T\Cr{fc1}}\|c_{0}\|_{L^{\infty}(\Omega)}$
(3.15c) for some constant $\Cr{CBNDchi1}$ that depends only on $d$, $\Omega$,
and $\chi$.

######  Remark 3.3.

  1. 1.

The one-sided entropy inequalities 3.15b are reminiscent of those in, e.g. [9,
Definition 6]. In contrast, inequalities adopted in, e.g. [8, Definition 4.4]
each contain a pair of piecewise linear transformations of the form $T_{m,M}$.
Such more delicate entropy inequalities, but also further conditions on the
time derivative of $c$ and/or boundary traces (compare [9, Definition 6]) can
be considered for 3.1 as well. Since we do not aim at uniqueness, we chose to
restrict ourselves to 3.15b.

  2. 2.

In contrast to, e.g. [9], an estimate from below, 3.15a, is included as well.
Each weak solution to 3.1-3.2 satisfies it, see Remark 4.11. Together, 3.15a
and 3.15b underscore the fact that the measure on the right-hand side of 3.15a
is singular.

######  Definition 3.4 (Strong solution to 3.2).

Let $f_{v}$ satisfy 3.9 and $v_{0}$ satisfy 3.11. We call a pair of functions
$(c,v):\mathbb{R}_{0}^{+}\times\overline{\Omega}\rightarrow\mathbb{R}_{0}^{+}\times\mathbb{R}_{0}^{+}$
a global strong solution to 3.2 if $c$ and $v$ have the regularity

$\displaystyle c\in L^{\infty}_{loc}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega))$
(3.16)

and

$\displaystyle v\in
L_{loc}^{p_{0}}(\mathbb{R}_{0}^{+};W^{2,p_{0}}_{\nu}(\Omega))\cap
W^{1,p_{0}}_{loc}(\mathbb{R}_{0}^{+};L^{p_{0}}(\Omega)),$ (3.17)

respectively, and satisfy 3.2a strongly in
$L_{loc}^{p_{0}}(\mathbb{R}_{0}^{+};L^{p_{0}}(\Omega))$ and 3.2b and 3.2c
pointwise.

######  Remark 3.5.

Combining the trace theorem [1, Chapter 2 Theorem 4.10.2] and the standard
results on interpolation and embedding of Sobolev spaces and using assumption
$p_{0}>d+2$, we have the continuous embeddings

$\displaystyle W^{1,p_{0}}((0,T);L^{p_{0}}(\Omega))\cap
L^{p_{0}}((0,T);W^{2,p_{0}}_{\nu}(\Omega))\subset$ $\displaystyle
C\left([0,T];W^{2\left(1-\frac{1}{p_{0}}\right),p_{0}}_{\nu}\right)$ (3.18a)
$\displaystyle\subset$ $\displaystyle
C([0,T];C^{1}_{\nu}(\overline{\Omega})).$ (3.18b)

Thus, the $v$-component of a strong solution to 3.2 satisfies

$\displaystyle v\in
C\left([0,T];W^{2\left(1-\frac{1}{p_{0}}\right),p_{0}}_{\nu}\right)\subset
C([0,T];C^{1}_{\nu}(\overline{\Omega})),$ (3.19)

so that 3.2b and 3.2c can indeed be stated pointwise.

######  Definition 3.6 (Solution to 3.1-3.2).

We call a pair of functions $(c,v)$ a global solution to 3.1-3.2 if it is a
global entropy solution to 3.1 and a global strong solution to 3.2.

Our main result now reads:

######  Theorem 3.7 (Existence for 3.1-3.2).

Let assumptions 3.3, 3.4, 3.6, 3.8, 3.9, 3.10, and 3.11 be satisfied. Then
there exists a pair of functions $(c,v)$ that solves 3.1-3.2 in the sense of
Definition 3.6.

The proof of Theorem 3.7 requires some preparation and is therefore postponed
to the final Section 6. It is based on solving 3.1 and 3.2 for fixed $v$ and
$c$, respectively, and then obtaining a joint solution to the whole system
3.1-3.2 by means of Schauder’s fixed point theorem. Smoothness of $v$ allows
to establish the solvability of 3.1 in the entropy sense using an approach
that resembles the one developed and pursued in [12, 5, 9, 8, 10, 7, 11].

######  Remark 3.8.

  1. 1.

Our existence proof is valid only for $\chi\in(0,1)$. For $\chi>1$, the study
in [17] of a parabolic-elliptic version of 3.1-3.2 in the radial symmetric
case indicated that a local solution may cease to exist globally due to a
blow-up in finite time, and that for any initial mass
$m_{0}:=\int_{\Omega}c_{0}\,dx$ in higher dimensions, and, for sufficiently
large $m_{0}$, even in the case of dimension one.

  2. 2.

Due to a possibly low regularity of the $c$-component and the strong coupling
of the equations the uniqueness of solutions remains a challenging open issue.

##  4 Existence of entropy solutions to 3.1 for fixed $v$

In this Section we establish the existence of entropy solutions to 3.1 for
given $v$. To start, we derive some a priori bounds for the $c$-component in
Section 4.1. Next, we study several spatial differential operators that arise
in the context of 3.1a-3.1b in Section 4.2. These steps serve as preparation
for the proof of a result on well-posedness of a regularised time-discrete
approximation of 3.1 that we give in Section 4.3. In Section 4.4 we are then
finally able to deduce the existence of entropy solutions to 3.1 that depend
on $v$ in a stable fashion.

###  4.1 A priori bounds for $c$

This Subsection is devoted to the derivation of a priori estimates for a weak
solution $c$ of 3.1 under the additional regularity assumption

$\displaystyle c\in L^{1}_{loc}(\mathbb{R}_{0}^{+};W^{1,1}(\Omega)).$ (4.1)

In the proofs, estimates that we obtain for time-dependent quantities are to
be understood to hold a.e., at least.

######  Lemma 4.1.

Let $\chi\in(0,1)$ and $(c,v)$ be a weak solution to 3.1 such that 4.1 is
satisfied. Then for all $T>0$

$\displaystyle\|c\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\leq\Cl{CBND_{1}}(1+T)e^{T\Cr{fc1}}\|c_{0}\|_{L^{\infty}(\Omega)}=:\Cl{CBND}(T),$
(4.2)

where $\Cr{CBND_{1}}$ depends only on $d$, $\Omega$, and $\chi$.

###### Proof.

Throughout this proof, constants $C_{i}$ depend on $d$, $\Omega$, and $\chi$
alone.

We use the standard method of propagation of $L^{p}$ bounds. To begin with,
taking $\varphi\equiv 1$ in 3.14 and using 3.8b and the Gronwall lemma one
readily verifies

$\displaystyle\|c(t,\cdot)\|_{L^{1}(\Omega)}\leq
e^{t\Cr{fc1}}\|c_{0}\|_{L^{1}(\Omega)}\qquad\text{for all }t\geq 0.$ (4.3)

In particular, if $c_{0}=0$ a.e., then by 4.3 we have $c=0$ a.e. This proves
4.2 for this case. Therefore, from now on we assume

$\displaystyle|\\{c_{0}>0\\}|>0.$

In 3.14, choose

$\varphi:={p}c^{{p}-1},\ {p}\geq 2.$

Due to 3.12a and 4.1 we have $\varphi\in
L^{1}_{loc}(\mathbb{R}_{0}^{+};W^{1,1}(\Omega))$, as needed for 3.14, and so
by the weak chain rule

$\displaystyle\frac{d}{dt}\int_{\Omega}c^{p}dx=$
$\displaystyle\int_{\Omega}-{p}({p}-1)\left(a(c,\nabla c)-\chi c\Phi(\nabla
v)\right)\cdot c^{{p}-2}\nabla c+{p}c^{{p}-1}f_{c}(c,v)\,dx.$ (4.4)

Using 4.1, the essential boundedness of $c$ and $v$, and properties of $a$,
$\Phi$, and $f_{c}$, we conclude from 4.4 that

$\displaystyle\int_{\Omega}c^{p}dx\in
W^{1,\infty}_{loc}(\mathbb{R}_{0}^{+})\subset C(\mathbb{R}_{0}^{+}).$

Further, assumption $\chi\in(0,1)$, properties of $a$ and $\Phi$, and the weak
chain rule together allow to estimate the right-hand side of 4.4 to obtain

$\displaystyle\frac{d}{dt}\|c^{p}\|_{L^{1}(\Omega)}\leq$
$\displaystyle\int_{\Omega}-{p}({p}-1)\left((1-\chi)|\nabla
c|-c\right)c^{{p}-1}+\Cr{fc1}{p}c^{p}\,dx$ $\displaystyle=$
$\displaystyle-(1-\chi)({p}-1)\left\|\nabla
c^{p}\right\|_{L^{1}(\Omega)}+{p}({p}-1)\|c^{p}\|_{L^{1}(\Omega)}+\Cr{fc1}p\|c^{p}\|_{L^{1}(\Omega)}$
$\displaystyle\leq$ $\displaystyle-\Cl{Cd1}p\left\|\nabla
c^{p}\right\|_{L^{1}(\Omega)}+\Cl{Cd2}p^{2}\|c^{p}\|_{L^{1}(\Omega)}+\Cr{fc1}p\|c^{p}\|_{L^{1}(\Omega)}.$
(4.5)

Next, we combine a version of the Gagliardo-Nirenberg interpolation inequality
in the form provided by [36, Lemma 2.3], choosing there $r:=p:=1$ and
$q:=s:=1/2$, with Young’s inequality to obtain

$\displaystyle\|w\|_{L^{1}(\Omega)}\leq$ $\displaystyle\varepsilon\|\nabla
w\|_{L^{1}(\Omega)}+\Cl{Cint}\left(1+\varepsilon^{-\frac{\theta}{1-\theta}}\right)\left\|w^{\frac{1}{2}}\right\|_{L^{1}(\Omega)}^{2}\qquad\text{for
all }\varepsilon>0.$ (4.6)

For

$w:=c^{p}\qquad\text{and}\qquad\varepsilon:=\frac{\Cr{Cd1}}{2\Cr{Cd2}}p^{-1}$

inequality 4.6 leads to

$\displaystyle\|c^{p}\|_{L^{1}(\Omega)}\leq$
$\displaystyle\frac{\Cr{Cd1}}{2\Cr{Cd2}}p^{-1}\left\|\nabla
c^{p}\right\|_{L^{1}(\Omega)}+\Cr{Cint}\left(1+\left(\frac{\Cr{Cd1}}{2\Cr{Cd2}}\right)^{-\frac{\theta}{1-\theta}}{p}^{\frac{\theta}{1-\theta}}\right)\left\|c^{\frac{p}{2}}\right\|_{L^{1}(\Omega)}^{2}$
$\displaystyle\leq$

Combining 4.5 and LABEL:interp1, we conclude that

$\displaystyle\frac{d}{dt}\|c^{p}\|_{L^{1}(\Omega)}\leq$
$\displaystyle\Cl{Cint1}{p}^{\Cl{ckappa}}\left\|c^{\frac{p}{2}}\right\|_{L^{1}(\Omega)}^{2}+\Cr{fc1}p\|c^{p}\|_{L^{1}(\Omega)}.$
(4.7)

Applying Gronwall’s lemma to 4.7 and dividing throughout by $e^{-t\Cr{fc1}p}$
yields

$\displaystyle\left\|\left(e^{-t\Cr{fc1}}c(t,\cdot)\right)^{p}\right\|_{L^{1}(\Omega)}\leq\|c_{0}^{p}\|_{L^{1}(\Omega)}+\Cr{Cint1}{p}^{\Cr{ckappa}}\int_{0}^{t}\left\|\left(e^{-s\Cr{fc1}}c(s,\cdot)\right)^{\frac{p}{2}}\right\|_{L^{1}(\Omega)}^{2}\,ds\qquad\text{for
}t\geq 0,\ p\geq 2.$ (4.8)

Let

$\displaystyle
B_{p}(t):=\frac{\left\|\left(e^{-t\Cr{fc1}}c(t,\cdot)\right)^{p}\right\|_{L^{1}(\Omega)}}{\|c_{0}^{p}\|_{L^{1}(\Omega)}}\qquad\text{for
}t\geq 0,\ p\geq 1.$ (4.9)

Obviously,

$\displaystyle\|B_{p}\|_{L^{\infty}((0,T))}\geq B_{p}(0)=1\qquad\text{for all
}T>0,\ p\geq 1.$ (4.10)

Due to 4.3 is also holds that

$\displaystyle\|B_{1}\|_{L^{\infty}((0,\infty))}=B_{1}(0)=1.$ (4.11)

Combining 4.8, 4.9, and 4.10 and Hölder’s inequality, we have for $p\geq 2$

$\displaystyle\|B_{p}\|_{L^{\infty}(0,T)}\leq$ $\displaystyle
1+T\frac{\left\|c_{0}^{\frac{p}{2}}\right\|_{L^{1}(\Omega)}^{2}}{\|c_{0}^{p}\|_{L^{1}(\Omega)}}\Cr{Cint1}{p}^{\Cr{ckappa}}\left\|B_{\frac{p}{2}}\right\|_{L^{\infty}(0,T)}^{2}$
$\displaystyle\leq$ $\displaystyle
1+T|\Omega|\Cr{Cint1}{p}^{\Cr{ckappa}}\left\|B_{\frac{p}{2}}\right\|_{L^{\infty}(0,T)}^{2}$
$\displaystyle\leq$
$\displaystyle(1+T\Cl{C57}){p}^{\Cr{ckappa}}\left\|B_{\frac{p}{2}}\right\|_{L^{\infty}(0,T)}^{2}.$
(4.12)

Choosing ${p}:=2^{n+1}$, $n\in\mathbb{N}_{0}$, in 4.12, we arrive at the
sequence of recursive inequalities

$\displaystyle\left\|B_{2^{n+1}}\right\|_{L^{\infty}((0,T))}\leq(1+T\Cr{C57})\left(2^{\Cr{ckappa}}\right)^{n}\left\|B_{2^{n}}\right\|_{L^{\infty}((0,T))}^{2}\qquad\text{for
}n\in\mathbb{N}_{0}.$ (4.13)

Combining 4.13 and 4.11 with [34, Chapter II Lemma 5.6], we deduce

$\displaystyle\frac{e^{-t\Cr{fc1}}\|c(t,\cdot)\|_{L^{\infty}(\Omega)}}{\|c_{0}\|_{L^{\infty}(\Omega)}}\leq$
$\displaystyle\frac{\underset{s\in[0,T]}{\max}e^{-s\Cr{fc1}}\|c(s,\cdot)\|_{L^{\infty}(\Omega)}}{\|c_{0}\|_{L^{\infty}(\Omega)}}$
$\displaystyle=$
$\displaystyle\underset{n\to\infty}{\lim}\left\|B_{2^{n}}\right\|_{L^{\infty}((0,T))}^{2^{-n}}$
$\displaystyle\leq$
$\displaystyle\underset{n\to\infty}{\lim}((1+T\Cr{C57}))^{1-2^{-n}}2^{\Cr{ckappa}(1-2^{-n}(n+1))}\left\|B_{1}\right\|_{L^{\infty}((0,T))}$
$\displaystyle=$ $\displaystyle(1+T\Cr{C57})2^{\Cr{ckappa}}$
$\displaystyle\leq$ $\displaystyle(1+T)\Cr{CBND_{1}}\qquad\text{for all }0\leq
t\leq T,$

yielding 4.2. ∎

######  Lemma 4.2.

Under the assumptions of Lemma 4.1 it holds for all $T>0$ that

$\displaystyle\|\nabla c\|_{L^{1}((0,T);L^{1}(\Omega))}\leq$
$\displaystyle\Cl{Cc0}+T\Cl{C111}\left(\Cr{CBND}(T),\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\right)$
$\displaystyle=:$
$\displaystyle\Cl{CD}\left(T,\Cr{CBND}(T),\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\right).$
(4.14)

###### Proof.

Set

$\varphi:=\ln(\max\\{c,\omega\\}),\ \omega\in(0,1).$

Due to 3.12a and 4.1 we have $\varphi\in
L^{1}_{loc}(\mathbb{R}_{0}^{+};W^{1,1}(\Omega))$, as needed in 3.14. Plugging
$\varphi$ into 3.14, integrating over $(0,T)$, and using the weak chain rule,
properties of $a,\Phi$, and $f_{c}$, estimate 4.2, and boundedness of $v$, we
obtain

$\displaystyle\int_{\Omega}\int_{c_{0}}^{c(T,\cdot)}\ln(\max\\{s,\omega\\})\,dsdx$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\\{c>\omega\\}}-\left(a(c,\nabla c)-\chi
c\Phi(\nabla v)\right)\cdot\nabla\ln
c\,dx+\int_{\Omega}f_{c}(c,v)\ln(\max\\{c,\omega\\})\,dxdt$
$\displaystyle\leq$
$\displaystyle\int_{0}^{T}\int_{\\{c>\omega\\}}-(1-\chi)\left|\nabla
c\right|+c\,dxdt+\int_{0}^{T}\int_{\Omega}\left|\frac{f_{c}(c,v)-f_{c}(0,v)}{c}\right|\max\\{c,\omega\\}|\ln(\max\\{c,\omega\\})|\,dxdt$
$\displaystyle\leq$
$\displaystyle-(1-\chi)\int_{0}^{T}\int_{\\{c>\omega\\}}\left|\nabla
c\right|\,dxdt+T\Cl{C111\left(\|c\|_{L^{\infty}((0,T);L^{\infty}(\Omega))},\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\right).}$
(4.15)

Thanks to the monotone convergence theorem and the fact that $\nabla w=0$ a.e.
in $w=0$ for any $w\in W^{1,1}(\Omega)$ we have

$\displaystyle\int_{0}^{T}\int_{\\{c>\omega\\}}\left|\nabla
c\right|\,dxdt\underset{\omega\to 0}{\to}\int_{0}^{T}\int_{\Omega}\left|\nabla
c\right|\,dxdt.$ (4.16)

Further, since $\int_{0}^{c}\ln(\max\\{s,\omega\\})\,ds-\int_{0}^{c}\ln s\,ds$
converges to zero uniformly on finite intervals and $c(T,\cdot)\in
L^{\infty}(\Omega)$ due to 3.12a,

$\displaystyle\int_{\Omega}\int_{c_{0}}^{c(T,\cdot)}\ln(\max\\{s,\omega\\})\,dsdx\underset{\omega\to
0}{\to}\int_{\Omega}(s\ln s-s)|_{s=c_{0}}^{c(T,\cdot)}dx.$ (4.17)

Passing to the limit in 4.15 and using 4.16 and 4.17, we obtain

$\displaystyle\int_{\Omega}(s\ln s-s)|_{s=c_{0}}^{c(T,\cdot)}dx\leq$
$\displaystyle-(1-\chi)\int_{0}^{T}\int_{\Omega}\left|\nabla
c\right|\,dxdt+T\Cr{C111\left(\|c\|_{L^{\infty}((0,T);L^{\infty}(\Omega))},\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\right).}$
(4.18)

Rearranging 4.18 and using $s\ln s-s\geq-1$ for all $s>0$, we arrive at the
estimate

$\displaystyle(1-\chi)\int_{0}^{T}\int_{\Omega}\left|\nabla
c\right|\,dxdt\leq$ $\displaystyle 1+\int_{\Omega}c_{0}\ln
c_{0}-c_{0}\,dx+T\Cr{C111\left(\|c\|_{L^{\infty}((0,T);L^{\infty}(\Omega))},\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\right)}$
$\displaystyle\leq$
$\displaystyle\C+T\Cr{C111\left(\Cr{CBND}(T),\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\right).}$
(4.19)

Together, 4.2 and 4.19 and assumption $\chi<1$ yield 4.14. ∎

###  4.2 Auxiliary operators

In preparation for the analysis of the IBVP 3.1, we introduce and study
several operators defined on spaces of functions which depend on $x$ alone. We
start with a family of operators ${\cal A}[v]$ parametrised by $v$ and given
by

$\displaystyle\left<{\cal A}[v](c),\varphi\right>:=$
$\displaystyle-\left<a(|c|,\nabla c)-\chi c\Phi(\nabla
v),\nabla\varphi\right>.$ (4.20)

The proof of the following lemma is straightforward, so we omit it.

######  Lemma 4.3.

Let $a$ and $\Phi$ be as defined in 3.4 and 3.6, respectively. For all $v\in
C^{1}(\overline{\Omega})$ the corresponding operator ${\cal A}[v]$ is well-
defined, continuous, and bounded between

  1. (1)

$H^{1}(\Omega)$ and $(H^{1}(\Omega))^{*}$;

  2. (2)

$W^{1,1}(\Omega)\cap L^{\infty}(\Omega)$ and $(W^{1,1}(\Omega))^{*}$.

Our next Lemma establishes a one-sided stability property of ${\cal
A}[\cdot]$.

######  Lemma 4.4.

Let

$\displaystyle 0\leq c_{1},c_{2}\in W^{1,1}(\Omega)\cap L^{\infty}(\Omega),$
$\displaystyle v_{1},v_{2}\in W^{2,1}(\Omega)\cap
C^{1}_{\nu}(\overline{\Omega}).$

Then the following estimate holds:

$\displaystyle\underset{\delta\rightarrow 0}{\lim\sup}\left<{\cal
A}[v_{1}](c_{1})-{\cal
A}[v_{2}](c_{2}),\operatorname{sign}_{\delta}(c_{1}-c_{2})\right>$
$\displaystyle\leq$ $\displaystyle\Cr{C19}\left(\|\nabla
c_{2}\|_{L^{1}(\Omega)}\|v_{1}-v_{2}\|_{C^{1}(\overline{\Omega})}+\|c_{2}\|_{L^{\infty}(\Omega)}\|v_{1}-v_{2}\|_{W^{2,1}(\Omega)}\right).$
(4.21)

###### Proof.

For any $v\in W^{2,1}_{\nu}(\Omega)$ and $c,\varphi\in W^{1,1}(\Omega)\cap
L^{\infty}(\Omega)$ we can integrate by parts in the second summand in 4.20 to
obtain

$\displaystyle\left<{\cal A}[v](c),\varphi\right>=$
$\displaystyle-\int_{\Omega}a(c,\nabla
c)\cdot\nabla\varphi\,dx-\chi\int_{\Omega}\nabla\cdot(c\Phi(\nabla
v))\varphi\,dx.$ (4.22)

Since $c_{1},c_{2}\in W^{1,1}(\Omega)$, we have

$\displaystyle\varphi:=\operatorname{sign}_{\delta}(c_{1}-c_{2})\in
W^{1,1}(\Omega)\cap L^{\infty}(\Omega)$ (4.23)

for every $\delta>0$. Subtracting 4.22 for $(c,v)=(c_{1},v_{1})$ from 4.22 for
$(c,v)=(c_{2},v_{2})$ and testing with $\varphi$ as defined in 4.23, we obtain
using properties of $a$, $\Phi$, and $\operatorname{sign}_{\delta}$, the weak
chain rule, and Hölder’s inequality that

$\displaystyle\left<{\cal A}[v_{1}](c_{1})-{\cal
A}[v_{2}](c_{2}),\operatorname{sign}_{\delta}(c_{1}-c_{2})\right>$
$\displaystyle=$ $\displaystyle-\int_{\Omega}(a(c_{1},\nabla
c_{1})-a(c_{2},\nabla
c_{2}))\cdot\nabla\operatorname{sign}_{\delta}(c_{1}-c_{2})-\int_{\Omega}\nabla\cdot\left(c_{1}\Phi(\nabla
v_{1})-c_{2}\Phi(\nabla
v_{2})\right)\operatorname{sign}_{\delta}(c_{1}-c_{2})\,dx$ $\displaystyle=$
$\displaystyle-\int_{\\{0<|c_{1}-c_{2}|<\delta\\}}(a(c_{1},\nabla
c_{1})-a(c_{1},\nabla c_{2}))\cdot\frac{1}{\delta}(\nabla c_{1}-\nabla
c_{2})\,dx$ $\displaystyle-\int_{\\{0<|c_{1}-c_{2}|<\delta\\}}(a(c_{1},\nabla
c_{2})-a(c_{2},\nabla c_{2}))\cdot\frac{1}{\delta}\nabla(c_{1}-c_{2})\,dx$
$\displaystyle-\int_{\Omega}\nabla\cdot((c_{1}-c_{2})\Phi(\nabla
v_{1}))\operatorname{sign}_{\delta}(c_{1}-c_{2})\,dx-\int_{\Omega}\nabla\cdot\left(c_{2}(\Phi(\nabla
v_{1})-\Phi(\nabla
v_{2}))\right)\operatorname{sign}_{\delta}(c_{1}-c_{2})\,dx$
$\displaystyle\leq$
$\displaystyle\int_{\\{0<|c_{1}-c_{2}|<\delta\\}}|a(c_{1},\nabla
c_{2})-a(c_{2},\nabla c_{2})|\frac{1}{\delta}|\nabla(c_{1}-c_{2})|\,dx$
$\displaystyle-\int_{\Omega}\nabla\cdot((c_{1}-c_{2})\Phi(\nabla
v_{1}))\operatorname{sign}_{\delta}(c_{1}-c_{2})\,dx+\int_{\Omega}|\nabla\cdot\left(c_{2}(\Phi(\nabla
v_{1})-\Phi(\nabla v_{2}))\right)|\,dx$ $\displaystyle\leq$
$\displaystyle\int_{\\{0<|c_{1}-c_{2}|<\delta\\}}|\nabla(c_{1}-c_{2})|\,dx-\int_{\Omega}\nabla\cdot((c_{1}-c_{2})\Phi(\nabla
v_{1}))\operatorname{sign}_{\delta}(c_{1}-c_{2})\,dx$
$\displaystyle+\Cl{C19}\left(\|\nabla
c_{2}\|_{L^{1}(\Omega)}\|v_{1}-v_{2}\|_{C^{1}(\overline{\Omega})}+\|c_{2}\|_{L^{\infty}(\Omega)}\|v_{1}-v_{2}\|_{W^{2,1}(\Omega)}\right).$
(4.24)

Passing to the limit in 4.24 as $\delta\rightarrow 0$ and using the dominated
convergence theorem, the weak chain rule, and the partial integration formula
combined with $\partial_{\nu}v_{1}=0$, so that $\Phi(\nabla v)\cdot\nu=0$ on
$\partial\Omega$, we arrive at

$\displaystyle\underset{\delta\rightarrow 0}{\lim\sup}\left<{\cal
A}[v_{1}](c_{1})-{\cal
A}[v_{2}](c_{2}),\operatorname{sign}_{\delta}(c_{1}-c_{2})\right>$
$\displaystyle\leq$
$\displaystyle-\int_{\Omega}\nabla\cdot((c_{1}-c_{2})\Phi(\nabla
v_{1}))\operatorname{sign}(c_{1}-c_{2})\,dx$
$\displaystyle+\Cr{C19}\left(\|\nabla
c_{2}\|_{L^{1}(\Omega)}\|v_{1}-v_{2}\|_{C^{1}(\overline{\Omega})}+\|c_{2}\|_{L^{\infty}(\Omega)}\|v_{1}-v_{2}\|_{W^{2,1}(\Omega)}\right)$
$\displaystyle=$
$\displaystyle-\int_{\Omega}\nabla\cdot(|c_{1}-c_{2}|\Phi(\nabla v_{1}))\,dx$
$\displaystyle+\Cr{C19}\left(\|\nabla
c_{2}\|_{L^{1}(\Omega)}\|v_{1}-v_{2}\|_{C^{1}(\overline{\Omega})}+\|c_{2}\|_{L^{\infty}(\Omega)}\|v_{1}-v_{2}\|_{W^{2,1}(\Omega)}\right)$
$\displaystyle=$ $\displaystyle\Cr{C19}\left(\|\nabla
c_{2}\|_{L^{1}(\Omega)}\|v_{1}-v_{2}\|_{C^{1}(\overline{\Omega})}+\|c_{2}\|_{L^{\infty}(\Omega)}\|v_{1}-v_{2}\|_{W^{2,1}(\Omega)}\right).$

∎

A direct consequence of Lemma 4.4 for $v_{1}=v_{2}=:v$ is the following
Corollary.

######  Corollary 4.5 (Inequality of Kato type).

Let

$\displaystyle 0\leq c_{1},c_{2}\in W^{1,1}(\Omega)\cap L^{\infty}(\Omega),$
$\displaystyle v\in W^{2,1}(\Omega)\cap C^{1}_{\nu}(\overline{\Omega}).$

Then it holds that

$\displaystyle\underset{\delta\rightarrow 0}{\lim\sup}\left<{\cal
A}[v](c_{1})-{\cal
A}[v](c_{2}),\operatorname{sign}_{\delta}(c_{1}-c_{2})\right>\leq$
$\displaystyle 0.$

Next, we introduce regularised perturbations of ${\cal A}[v]$ that include a
source term. For $m,n\in\mathbb{N}$ and $L>0$ let

$\displaystyle\left<{\cal B}_{m}[v](c),\varphi\right>:=$
$\displaystyle\left<{\cal
A}[v](c),\varphi\right>+\int_{\Omega}-\frac{1}{m}\nabla
c\cdot\nabla\varphi+f_{c}(c,v)\varphi\,dx,$ (4.25a) $\displaystyle\left<{\cal
B}_{L,m}[v](c),\varphi\right>:=$ $\displaystyle\left<{\cal
A}[v](c),\varphi\right>+\int_{\Omega}-\frac{1}{m}\nabla
c\cdot\nabla\varphi+f_{c}(T_{0,L}(c),v)\varphi\,dx,$ (4.25b)
$\displaystyle\left<{\cal B}_{n,L,m}[v](c),\varphi\right>:=$
$\displaystyle\left<{\cal
B}_{L,m}[v](c),\varphi\right>-n\int_{\Omega}c\varphi\,dx.$ (4.25c)

In 4.25b, the truncation by means of $T_{0,L}$ serves to turn the source term
into a Lipschitz continuous mapping.

######  Lemma 4.6.

Let $a$ and $\Phi$ be as defined in 3.4 and 3.6, respectively, $f_{c}$ satisfy
3.8, $m,n\in\mathbb{N}$, $L>0$, and $v\in C^{1}(\overline{\Omega})$. Then
operators

${\cal B}_{L,m}[v],{\cal
B}_{n,L,m}[v]:H^{1}(\Omega)\rightarrow(H^{1}(\Omega))^{*}$

are well-defined, continuous, and bounded. If

$n>(\Cr{fc1})_{+}+1$

then $-{\cal B}_{n,L,m}[v]$ is pseudo-monotone and coercive.

###### Proof.

Given that ${\cal A}[v]$ is well-defined, continuous, and bounded and
$f_{c}(T_{0,L}(\cdot),\cdot)$ is Lipschitz, it is evident that ${\cal
B}_{L,m}[v]$ and ${\cal B}_{n,L,m}[v]$ are well-defined, continuous, and
bounded. We turn to the remaining properties of $-{\cal B}_{n,L,m}[v]$. Set

$\displaystyle A_{1}(x,c,\xi):=$ $\displaystyle\frac{1}{m}\xi+a(|c|,\xi)-\chi
c\Phi(\nabla v(x))$ $\displaystyle A_{0}(x,c):=$ $\displaystyle nc-
f_{c}(T_{0,L}(c),v(x)).$

Using assumption $\chi<1$ and properties of $v$, $a$, $\Phi$, and $f_{c}$, we
obtain

$\displaystyle A_{1}\in
C(\overline{\Omega}\times\mathbb{R}\times\mathbb{R}^{d};\mathbb{R}^{d}),$
(4.26a) $\displaystyle A_{0}\in C(\overline{\Omega}\times\mathbb{R});$ (4.26b)

for all $x\in\Omega$, $c\in\mathbb{R}$, and $\xi\in\mathbb{R}^{d}$

$\displaystyle|A_{1}(x,c,\xi)|\leq$
$\displaystyle\frac{1}{m}\left|\xi\right|+(1+\chi)|c|,$ (4.27a)
$\displaystyle|A_{0}(x,c)|\leq$ $\displaystyle
n|c|+\Cl{fcL1}\left(L,\|v\|_{L^{\infty}(\Omega)}\right);$ (4.27b) for all
$x\in\Omega$, $c\in\mathbb{R}$, and $\xi_{1},\xi_{2}\in\mathbb{R}^{d}$,
$\xi_{1}\neq\xi_{2}$,
$\displaystyle\left(A_{1}(x,c,\xi_{1})-A_{1}(x,c,\xi_{2})\right)\cdot(\xi_{1}-\xi_{2})\geq\frac{1}{m}|\xi_{1}-\xi_{2}|^{2}>0;$
(4.28)

and for $x\in\Omega$, $c\in\mathbb{R}$, and $\xi\in\mathbb{R}^{d}$

$\displaystyle A_{1}(x,c,\xi)\cdot\xi+A_{0}(x,c)c\geq$
$\displaystyle\frac{1}{m}|\xi|^{2}+|c|((1-\chi)|\xi|-|c|)+(n-\Cr{fc1})c^{2}$
$\displaystyle\geq$ $\displaystyle\frac{1}{m}|\xi|^{2}+(n-1-\Cr{fc1})c^{2}.$
(4.29)

Observe that

$\displaystyle-\left<{\cal
B}_{n,L,m}[v](c),\varphi\right>=\int_{\Omega}A_{1}(\cdot,c,\nabla
c)\cdot\nabla\varphi+A_{0}(\cdot,c)\varphi\,dx\qquad\text{for all
}c,\varphi\in H^{1}(\Omega).$

Let $n>(\Cr{fc1})_{+}+1$. Due to properties 4.26, 4.27, 4.29, and 4.28
operator $-{\cal B}_{n,L,m}[v](c)$ satisfies the assumptions of [18, Theorem
1] for $m=1$, $p=2$, so that it is pseudo-monotone by that result. Finally,
coercivity is the direct consequence of 4.29.

∎

###  4.3 Well-posedness of time-discrete approximations of 3.1

In preparation for the construction of solutions to 3.1 for a fixed $v$ we
study a family of local time-discrete approximations of this problem. Let
$n,m\in\mathbb{N}$. We introduce system

$\displaystyle c_{nm}(0,\cdot)=c_{0},$ (4.30a)
$\displaystyle\frac{c_{nm}\left(\frac{k}{n},\cdot\right)-c_{nm}\left(\frac{k-1}{n},\cdot\right)}{\frac{1}{n}}={\cal
B}_{m}\left[P_{n}v\left(\frac{k}{n},\cdot\right)\right]\left(c_{nm}\left(\frac{k}{n},\cdot\right)\right)\qquad\text{for
}k\in\mathbb{N},$ (4.30b)

with ${\cal B}_{m}$ defined in 4.25a. In the notation from Section 2.4
equations 4.30b can also be rewritten in the form of a single equation
involving interpolations,

$\displaystyle\partial_{t}\Lambda_{n}{c}_{nm}={\cal
B}_{m}\left[\overline{\Pi}_{n}P_{n}v\right]\left(\overline{\Pi}_{n}c_{nm}\right)\qquad\text{for
}t\in(0,\infty)\backslash\left(\frac{1}{n}\mathbb{N}_{0}\right).$ (4.31)

The definition of bounded weak solutions to 4.30 is as follows.

######  Definition 4.7 (Weak solution to 4.30).

Let $a$ and $\Phi$ be as defined in 3.4 and 3.6, respectively, $f_{c}$ satisfy
3.8, $m,n,T\in\mathbb{N}$, and

$\displaystyle 0\leq c_{0}\in L^{\infty}(\Omega)\cap H^{1}(\Omega),$ (4.32a)
$\displaystyle 0\leq v\in C([0,T];C^{1}_{\nu}(\overline{\Omega})).$ (4.32b)

We call a function
$c_{nm}:\left(\frac{1}{n}\mathbb{N}_{0}\cap[0,T]\right)\times\overline{\Omega}\rightarrow\mathbb{R}_{0}^{+}$
a weak solution to 4.30 if it satisfies

$\displaystyle c_{nm}(t,\cdot)\in L^{\infty}(\Omega)\cap
H^{1}(\Omega)\qquad\text{for all }t\in\frac{1}{n}\mathbb{N}_{0}\cap[0,T]$
(4.33)

and 4.30 in $(H^{1}(\Omega))^{*}$.

We call
$c_{nm}:\frac{1}{n}\mathbb{N}_{0}\times\overline{\Omega}\rightarrow\mathbb{R}_{0}^{+}$
a global weak solution to 4.30 if it is a weak solution for all
$T\in\mathbb{N}$.

######  Lemma 4.8 (Weak solution to 4.30).

Let $\chi\in(0,1)$.

  1. (1)

(A priori bounds.) Let $m,T\in\mathbb{N}$,

$\displaystyle\Cr{fc1}<n\in\mathbb{N},$

and $(v,c_{0})$ satisfy 4.32. Then interpolations $\overline{\Pi}_{n}c_{nm}$
and $\Lambda_{n}c_{nm}$ of any corresponding weak solution
$c_{nm}:\left(\frac{1}{n}\mathbb{N}_{0}\cap[0,T]\right)\times\overline{\Omega}\rightarrow\mathbb{R}_{0}^{+}$
to 4.30 satisfy the estimate sets

$\displaystyle\|\overline{\Pi}_{n}{c_{nm}}\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\leq\Cl{CBNDdiscr}\left(\frac{1}{n},T,\Cr{fc1},\|c_{0}\|_{L^{\infty}(\Omega)}\right),$
(4.34a)
$\displaystyle\|\overline{\Pi}_{n}\nabla{c_{nm}}\|_{L^{1}((0,T);L^{1}(\Omega))}\leq\Cr{CD}\left(T,\Cr{CBNDdiscr}(\dots),\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\right),$
(4.34b)
$\displaystyle\frac{1}{m}\|\overline{\Pi}_{n}\nabla{c_{nm}}\|_{L^{2}((0,T);L^{2}(\Omega))}^{2}\leq\Cr{CD}\left(T,\Cr{CBNDdiscr}(\dots),\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\right),$
(4.34c)

and

$\displaystyle\|\Lambda_{n}{c_{nm}}\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\leq\Cr{CBNDdiscr}(\dots),$
(4.35a)
$\displaystyle\|\Lambda_{n}\nabla{c_{nm}}\|_{L^{1}((0,T);L^{1}(\Omega))}\leq\Cr{CD}\left(T,\Cr{CBNDdiscr}(\dots),\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\right)+\frac{1}{2n}\|\nabla
c_{0}\|_{L^{1}(\Omega)},$ (4.35b)
$\displaystyle\|\partial_{t}\Lambda_{n}{c}_{nm}\|_{L^{2}((0,T);(H^{1}(\Omega))^{*})}\leq\Cl{C22}\left(T,\Cr{CBNDdiscr}(\dots),\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\right),$
(4.35c)

respectively, where

$\displaystyle\Cr{CBNDdiscr}(\dots)=$
$\displaystyle\Cr{CBND_{1}}\left(1+T\left(1-\frac{\Cr{fc1}}{n}\right)^{-1}\right)e^{T\Cr{fc1}}\|c_{0}\|_{L^{\infty}(\Omega)}\underset{n\to\infty}{\to}\Cr{CBND}(T).$
(4.36)

  2. (2)

(Global existence.) Let

$\displaystyle(\Cr{fc1})_{+}+1<n\in\mathbb{N}$

and $(v,c_{0})$ satisfy

$\displaystyle 0\leq c_{0}\in L^{\infty}(\Omega)\cap H^{1}(\Omega),$
$\displaystyle 0\leq v\in
C(\mathbb{R}_{0}^{+};C^{1}_{\nu}(\overline{\Omega})).$

Then for all $m\in\mathbb{N}$ system 4.30 possesses a global weak solution.

  3. (3)

(Stability and uniqueness.) Let $m,n,T\in\mathbb{N}$. Set

$\displaystyle X(T,M):=$ $\displaystyle\left\\{0\leq c_{0}\in
L^{\infty}(\Omega)\cap H^{1}(\Omega):\ \|c_{0}\|_{L^{\infty}(\Omega)}\leq
M\right\\}$ $\displaystyle\times\left\\{0\leq v\in
C([0,T];C^{1}_{\nu}(\overline{\Omega}))\cap L^{1}((0,T);W^{2,1}(\Omega)):\
\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\leq M\right\\}.$ (4.37)

Let
$c_{nm}^{(1)},c_{nm}^{(2)}:\left(\frac{1}{n}\mathbb{N}_{0}\cap[0,T]\right)\times\overline{\Omega}\rightarrow\mathbb{R}_{0}^{+}$
be some weak solutions to 4.30 corresponding to respective pairs
$\left(c_{0}^{(1)},v^{(1)}\right),\left(c_{0}^{(2)},v^{(2)}\right)\in X(T,M)$.
Set

$\displaystyle M_{n}(T):=$
$\displaystyle\max\left\\{M,\Cr{CBNDdiscr}\left(\frac{1}{n},T,\Cr{fc1},M\right)\right\\}.$
(4.38)

If

$\displaystyle n>\underset{[0,M_{n}(T)]^{2}}{\max}\partial_{c}f,$ (4.39)

then

$\displaystyle\Lambda_{n}\left\|c_{nm}^{(1)}-c_{nm}^{(2)}\right\|_{C([0,T];L^{1}(\Omega))}$
$\displaystyle\leq$
$\displaystyle\operatorname{exp}_{n,\underset{[0,M_{n}(T)]^{2}}{\max}\partial_{c}f}(T)\left\|c_{0}^{(1)}-c_{0}^{(2)}\right\|_{L^{1}(\Omega)}$
$\displaystyle+\Cl{C361}(M_{n}(T))\left(\left\|v^{(1)}-v^{(2)}\right\|_{C([0,T];C^{1}(\overline{\Omega}))}+\left\|v^{(1)}-v^{(2)}\right\|_{L^{1}((0,T);W^{2,1}(\Omega))}\right),$
(4.40)

where $\operatorname{exp}_{n,\dots}$ is defined in Lemma A.4. In particular,
if $v^{(1)}\equiv v^{(2)}$ and $c_{0}^{(1)}\equiv c_{0}^{(2)}$, then
$c_{nm}^{(1)}\equiv c_{nm}^{(2)}$, so that in this case the solution is
unique.

Furthermore, there exists some $N\in\mathbb{N}_{0}$ such that if
$\left(c_{0}^{(1)},v^{(1)}\right),\left(c_{0}^{(2)},v^{(2)}\right)\in X(T,M)$
then 4.39 and, hence, 4.40 are valid for all $n>N$.

###### Proof.

  1. (1)

Let a weak solution $c_{nm}$ be given. Testing 4.31 with
$g(\overline{\Pi}_{n}{c_{nm}})$ for $g(u):=pu^{p-1}$, $p\geq 2$
($g(u):=\ln(\max\\{u,\omega\\})$) and arguing in a manner closely resembling
the approach taken in the proof of Lemma 4.1 (Lemma 4.2), thereby using the
tools provided in Appendix A, one obtains 4.34a and 4.36 (4.34b). Taking
$g(u):=u$ instead leads to 4.34c. Bounds 4.35a and 4.35b for
$\Lambda_{n}c_{nm}$ are a direct consequence of 4.34a and 4.34b, respectively,
combined with A.1c. Finally, using 4.34a and 4.34c and properties of
$a,\Phi,f$, one readily obtains 4.35c. Thus, any weak solution $c_{nm}$ of
4.30 satisfies 4.34 and 4.35. We omit further details.

  2. (2)

Observe that

$\displaystyle f_{c}(T_{0,L}(c),v)\leq$ $\displaystyle\Cr{fc1}T_{0,L}(c)$
$\displaystyle\leq$ $\displaystyle(\Cr{fc1})_{+}c\qquad\text{for all }L>0.$

Set

$\displaystyle
L_{k}:=\Cr{CBNDdiscr}\left(\frac{1}{n},\frac{k}{n},(\Cr{fc1})_{+},\|c_{0}\|_{L^{\infty}(\Omega)}\right)\qquad\text{for
}k\in\mathbb{N}.$

From Lemma 4.6 we know that $-{\cal B}_{n,L_{k},m}$ is a bounded coercive
pseudo-monotone operator between $H^{1}(\Omega)$ and its dual for all
$k\in\mathbb{N}$. Applying a standard existence result for such operators
(see, e.g. [37, Chapter 2 §2.4 Theorem 2.7]), we recursively deduce the
existence of weak solutions

$\displaystyle c_{nm}\left(\frac{k}{n},\cdot\right)\in
H^{1}(\Omega)\qquad\text{ for }k\in\mathbb{N},$

to the sequence of elliptic problems

$\displaystyle\frac{c_{nm}\left(\frac{k}{n},\cdot\right)-c_{nm}\left(\frac{k-1}{n},\cdot\right)}{\frac{1}{n}}={\cal
B}_{L_{k},m}\left[P_{n}v\left(\frac{k}{n},\cdot\right)\right]\left(c_{nm}\left(\frac{k}{n},\cdot\right)\right)\qquad\text{for
}k\in\mathbb{N}$ (4.41)

under zero Neumann boundary conditions. Well-known results on boundedness
(see, e.g. [35]) for such elliptic equations ensure 4.33 and the non-
negativity of $c_{nm}$.

For each $k\in\mathbb{N}$ we can apply the result of (1) for $T:=k/n$ and
$f_{c}$ replaced by $f_{c}(T_{0,L_{k}}(\cdot),\cdot)$, yielding

$\displaystyle c_{nm}\left(\frac{k}{n},\cdot\right)\leq L_{k}\qquad\text{for
all }k\in\mathbb{N}.$

Therefore 4.30b and 4.41 are equivalent, and so $c_{nm}$ is a global weak
solution to 4.30.

  3. (3)

Let
$c_{nm}^{(1)},c_{nm}^{(2)}:\left(\frac{1}{n}\mathbb{N}_{0}\cap[0,T]\right)\times\overline{\Omega}\rightarrow\mathbb{R}_{0}^{+}$
be some weak solutions to 4.30 corresponding to respective pairs
$\left(c_{0}^{(1)},v^{(1)}\right),\left(c_{0}^{(2)},v^{(2)}\right)\in X(T,M)$
and assume that 4.39 holds. Subtracting 4.30b for $c_{nm}^{(1)}$ and
$c_{nm}^{(2)}$, we test the resulting equation with

$g(u):=\operatorname{sign}_{\delta}(u)$

for $\delta>0$ and

$u:=c_{nm}^{(1)}-c_{nm}^{(2)}.$

Since both solutions satisfy 4.33 and $\operatorname{sign}_{\delta}$ is
Lipschitz, $g(u)$ is a valid test function. Making use of A.12, we estimate as
follows:

$\displaystyle\frac{d}{dt}\Lambda_{n}\int_{\Omega}\int_{0}^{c_{nm}^{(1)}-c_{nm}^{(2)}}\operatorname{sign}_{\delta}(s)\,ds\,dx$
$\displaystyle\leq$ $\displaystyle\overline{\Pi}_{n}\left<{\cal
A}[P_{n}v^{(1)}]\left(c_{nm}^{(1)}\right)-{\cal
A}[P_{n}v^{(2)}]\left(c_{nm}^{(2)}\right),\operatorname{sign}_{\delta}\left(c_{nm}^{(1)}-c_{nm}^{(2)}\right)\right>$
$\displaystyle-\overline{\Pi}_{n}\frac{1}{m}\int_{\Omega}\operatorname{sign}_{\delta}^{\prime}\left(c_{nm}^{(1)}-c_{nm}^{(2)}\right)\left|\nabla\left(c_{nm}^{(1)}-c_{nm}^{(2)}\right)\right|^{2}\,dx$
$\displaystyle+\overline{\Pi}_{n}\int_{\Omega}\left(f_{c}\left(c_{nm}^{(1)},P_{n}v^{(1)}\right)-f_{c}\left(c_{nm}^{(2)},P_{n}v^{(2)}\right)\right)\operatorname{sign}_{\delta}\left(c_{nm}^{(1)}-c_{nm}^{(2)}\right)\,dx$
$\displaystyle\leq$ $\displaystyle\overline{\Pi}_{n}\left<{\cal
A}[P_{n}v^{(1)}]\left(c_{nm}^{(1)}\right)-{\cal
A}[P_{n}v^{(2)}]\left(c_{nm}^{(2)}\right),\operatorname{sign}_{\delta}\left(c_{nm}^{(1)}-c_{nm}^{(2)}\right)\right>$
$\displaystyle+\overline{\Pi}_{n}\int_{\Omega}\left(f_{c}\left(c_{nm}^{(1)},P_{n}v^{(1)}\right)-f_{c}\left(c_{nm}^{(2)},P_{n}v^{(2)}\right)\right)\operatorname{sign}_{\delta}\left(c_{nm}^{(1)}-c_{nm}^{(2)}\right)\,dx.$
(4.42)

Notice that the left-hand side of 4.42 is time-independent in each time
interval $((k-1)/n,k/n)$, $k\in\mathbb{N}_{0}$. This allows to pass to the
limit superior as $\delta\to 0$ on both sides of 4.42 using 4.21 and, where
necessary, the dominated convergence theorem together with 4.34a and
properties of ${\cal A}$, $f$, $\operatorname{sign}_{\delta}$, and $v$. Thus
obtained inequality and estimates based on 4.34a, 4.38, and A.2 yield

$\displaystyle\frac{d}{dt}\Lambda_{n}\left\|c_{nm}^{(1)}-c_{nm}^{(2)}\right\|_{L^{1}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\Cr{C19}\overline{\Pi}_{n}\left(\left\|\nabla
c_{nm}^{(2)}\right\|_{L^{1}(\Omega)}\left\|P_{n}(v^{(1)}-v^{(2)})\right\|_{C^{1}(\overline{\Omega})}+\left\|c_{nm}^{(2)}\right\|_{L^{\infty}(\Omega)}\left\|P_{n}(v^{(1)}-v^{(2)})\right\|_{W^{2,1}(\Omega)}\right)$
$\displaystyle+\overline{\Pi}_{n}\int_{\Omega}\left(f_{c}\left(c_{nm}^{(1)},P_{n}v^{(1)}\right)-f_{c}\left(c_{nm}^{(2)},P_{n}v^{(2)}\right)\right)\operatorname{sign}\left(c_{nm}^{(1)}-c_{nm}^{(2)}\right)\,dx$
$\displaystyle\leq$
$\displaystyle\Cr{C19}\overline{\Pi}_{n}\left(\left\|\nabla
c_{nm}^{(2)}\right\|_{L^{1}(\Omega)}\left\|P_{n}(v^{(1)}-v^{(2)})\right\|_{C^{1}(\overline{\Omega})}+\left\|c_{nm}^{(2)}\right\|_{L^{\infty}(\Omega)}\left\|P_{n}(v^{(1)}-v^{(2)})\right\|_{W^{2,1}(\Omega)}\right)$
$\displaystyle+\underset{[0,M_{n}(T)]^{2}}{\max}|\partial_{v}f|\overline{\Pi}_{n}\left\|P_{n}(v^{(1)}-v^{(2)})\right\|_{L^{1}(\Omega)}+\underset{[0,M_{n}(T)]^{2}}{\max}\partial_{c}f\overline{\Pi}_{n}\left\|c_{nm}^{(1)}-c_{nm}^{(2)}\right\|_{L^{1}(\Omega)}$
$\displaystyle\leq$ $\displaystyle\Cr{C19}\overline{\Pi}_{n}\left\|\nabla
c_{nm}^{(2)}\right\|_{L^{1}(\Omega)}\left\|v^{(1)}-v^{(2)}\right\|_{C([0,T];C^{1}(\overline{\Omega}))}+\Cr{C19}(M_{n})\overline{\Pi}_{n}\left\|P_{n}(v^{(1)}-v^{(2)})\right\|_{W^{2,1}(\Omega)}$
$\displaystyle+\underset{[0,M_{n}(T)]^{2}}{\max}\partial_{c}f\overline{\Pi}_{n}\left\|c_{nm}^{(1)}-c_{nm}^{(2)}\right\|_{L^{1}(\Omega)}\qquad\text{in
}(0,T)\backslash\left(\frac{1}{n}\mathbb{N}_{0}\right).$ (4.43)

Due to 4.39 Lemma A.4 applies to 4.43. Together with 4.34b, A.1b, A.2, and A.7
it leads to 4.40.

Choosing

$\displaystyle\left(c_{0}^{(1)},v^{(1)}\right):=\left(c_{0}^{(2)},v^{(2)}\right):=(c_{0},v).$

in 4.40 yields $c_{mn}^{(1)}\equiv c_{mn}^{(2)}$, i.e. the solution is unique.

Finally, 4.36 and $f\in C^{1}$ imply that

$\displaystyle\left(\underset{\left[0,\max\left\\{M,\Cr{CBNDdiscr}\left(\frac{1}{n},T,\Cr{fc1},M\right)\right\\}\right]^{2}}{\max}\partial_{c}f\right)$

is convergent. Therefore, for sufficiently large $N=N(T,M)$ condition 4.39 is
met for all $n>N$ and
$\left(c_{0}^{(1)},v^{(1)}\right),\left(c_{0}^{(2)},v^{(2)}\right)\in X(T,M)$.

∎

###  4.4 Weak solutions to IBVP 3.1

Now we are ready to construct entropy solutions to the IBVP 3.1 for fixed $v$.
We formulate a result in terms of a solution mapping as this is needed in
Section 6.

######  Theorem 4.9 (Entropy solutions to 3.1).

Let $\chi\in(0,1)$ and assumptions 3.4, 3.6, and 3.8 hold. Then there exists a
mapping ${\cal S}_{1}$ that

  1. (1)

assigns any pair $(c_{0},v)$ that satisfies 3.10 and 3.13 to some $c$ such
that $(c,v)$ solves 3.1 in terms of Definition 3.2 and satisfies estimates 4.2
and, for all $T\in\mathbb{N}$,

$\displaystyle\|Dc\|_{L^{1}_{w}((0,T);{\cal
M}(\Omega))}\leq\Cr{CD}\left(T,\Cr{CBND}(T),\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\right);$
(4.44)

  2. (2)

is locally Lipschitz continuous in the following sense: if $c^{(1)}={\cal
S}_{1}\left(c^{(1)}_{0},v^{(1)}\right)$ and $c^{(2)}={\cal
S}_{1}\left(c^{(2)}_{0},v^{(2)}\right)$ for some

$\displaystyle 0\leq c^{(1)}_{0},c^{(2)}_{0}\in L^{\infty}(\Omega),$ (4.45a)
$\displaystyle 0\leq v^{(1)},v^{(2)}\in
C(\mathbb{R}_{0}^{+};C^{1}_{\nu}(\overline{\Omega}))\cap
L^{1}_{loc}(\mathbb{R}_{0}^{+};W^{2,1}(\Omega)),$ (4.45b)

then for all $T\in\mathbb{N}$ and $t\in[0,T]$

$\displaystyle\left\|c^{(1)}-c^{(2)}\right\|_{L^{1}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
e^{T\underset{(0,M(T))^{2}}{\max}\partial_{c}f}\left\|c_{0}^{(1)}-c_{0}^{(2)}\right\|_{L^{1}(\Omega)}$
$\displaystyle+\Cr{C361}(M(T))\left(\left\|v^{(1)}-v^{(2)}\right\|_{C([0,T];C^{1}(\overline{\Omega}))}+\left\|v^{(1)}-v^{(2)}\right\|_{L^{1}((0,T);W^{2,1}(\Omega))}\right),$
(4.46a) where $\displaystyle M(T):=$
$\displaystyle\max\left\\{\left\|(v^{(1)},v^{(2)})\right\|_{(L^{\infty}((0,T);L^{\infty}(\Omega)))^{2}},\Cr{CBND}(T)\right\\}.$
(4.46b)

######  Remark 4.10.

Theorem 4.9 provides existence of a solution class that depends on the data in
a stable fashion. This is sufficient for our needs as it allows to prove the
main Theorem 3.7. We do not address uniqueness of solutions to 3.1 in this
work.

######  Proof of Theorem 4.9.

To shorten the notation, we introduce

$\displaystyle J_{m}[c,v]:=\frac{1}{m}\nabla c+a(c,\nabla c)-\chi c\Phi(\nabla
v).$ (4.47)

Throughout the proof we assume

$\displaystyle T\in\mathbb{N},$ $\displaystyle 0\leq\rho\in D(\Omega),\qquad
0\leq\eta\in D((0,T)),$ (4.48) $\displaystyle
g:\mathbb{R}_{0}^{+}\rightarrow\mathbb{R}\text{ increasing and locally
Lipschtz},\qquad G(c):=\int_{0}^{c}g(u)\,du.$ (4.49)

######  Step 1 (Equation for the evolution of $c$).

Let $(c_{0},v)$ be such that 3.10 and 3.13 holds. Then $c_{0}$ can be
approximated by a sequence $(c_{n0})$ that satisfies

$\displaystyle 0\leq c_{n0}\in C^{1}(\overline{\Omega}),$ (4.50a)
$\displaystyle\|c_{n0}\|_{L^{\infty}(\Omega)}\leq\|c_{0}\|_{L^{\infty}(\Omega)},$
(4.50b)
$\displaystyle\|c_{n0}\|_{W^{1,1}(\Omega)}\leq\sqrt{n}\Cl{nabcn0}\|c_{0}\|_{L^{\infty}(\Omega)},$
(4.50c) $\displaystyle
c_{n0}\underset{n\rightarrow\infty}{\overset{*}{\rightharpoonup}}c_{0}\qquad\text{in
}L^{\infty}(\Omega).$ (4.50d)

Let $n>(\Cr{fc1})_{+}+1$ and $m\in\mathbb{N}$. Due to Lemma 4.8 (1) and (2)
and assumptions 4.50a and 3.13, the time-discrete approximation 4.30 possesses
a global weak solution (in terms of Definition 4.7) $c_{nm}$ that corresponds
to $(c_{n0},v)$ and satisfies for all $T\in\mathbb{N}$ estimates 4.34, 4.35c,
and 4.35a corresponding to $(c_{0},v)$ as well as

$\displaystyle\|\Lambda_{n}\nabla{c_{nm}}\|_{L^{1}((0,T);L^{1}(\Omega))}\leq$
$\displaystyle\Cr{CD}\left(T,\Cr{CBNDdiscr}(\dots),\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\right)+\frac{\Cr{nabcn0}}{2\sqrt{n}}\|c_{0}\|_{L^{\infty}(\Omega)}$
$\displaystyle\underset{n\rightarrow\infty}{\rightarrow}$
$\displaystyle\Cr{CD}\left(T,\Cr{CBND}(T),\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\right).$
(4.51)

In light of these facts, there exist functions
$(c,z):\mathbb{R}_{0}^{+}\times\overline{\Omega}\rightarrow\mathbb{R}_{0}^{+}\times(\mathbb{R}_{0}^{+})^{d}$,
with $c$ satisfying 3.12, and two sequences $(n_{l})$ and $(m_{l})$, such that

due to 4.35a, 4.35c, and 4.51, a version of the Lions-Aubin lemma [40, Section
8 Corollary 4], and the diagonal argument (which we use in what follows
without referring to it explicitly)

$\displaystyle\Lambda_{n_{l}}c_{n_{l}m_{l}}\underset{l\rightarrow\infty}{\rightarrow}c$
$\displaystyle\qquad\text{in }L^{1}_{loc}(\mathbb{R}_{0}^{+};L^{1}(\Omega)),$
(4.52) $\displaystyle\qquad\text{in }L^{1}(\Omega)\text{ a.e. in }(0,\infty),$
(4.53) $\displaystyle\lx@cref{creftype~refnum}{cbound}\text{ holds};$

due to 4.35a and 4.52 and Hölder’s inequality

$\displaystyle\Lambda_{n_{l}}c_{n_{l}m_{l}}\underset{l\rightarrow\infty}{\rightarrow}c\qquad\text{in
}L^{2}_{loc}(\mathbb{R}_{0}^{+};L^{2}(\Omega));$ (4.54)

due to 4.54 and A.6b

$\displaystyle\overline{\Pi}_{n_{l}}c_{n_{l}m_{l}}\underset{l\rightarrow\infty}{\rightarrow}c$
$\displaystyle\qquad\text{in }L^{2}_{loc}(\mathbb{R}_{0}^{+};L^{2}(\Omega)),$
(4.55) $\displaystyle\qquad\text{a.e. in }(0,\infty)\times\Omega;$ (4.56)

due to 4.34a, 4.56, and A.3, continuity of $\Phi$, $v$, and $\nabla v$, and
the dominated convergence theorem

$\displaystyle\overline{\Pi}_{n_{l}}F(c_{n_{l}m_{l}},P_{n_{l}}v)\underset{l\rightarrow\infty}{\rightarrow}F(c,v)$
$\displaystyle\qquad\text{in
}L^{2}_{loc}(\mathbb{R}_{0}^{+};L^{2}(\Omega))\qquad\text{for all }F\in
C(\mathbb{R}\times\overline{\Omega}),$ (4.57)
$\displaystyle\overline{\Pi}_{n_{l}}F(c_{n_{l}m_{l}},\Phi(P_{n_{l}}\nabla
v))\underset{l\rightarrow\infty}{\rightarrow}F(c,\Phi(\nabla v))$
$\displaystyle\qquad\text{in
}L^{2}_{loc}(\mathbb{R}_{0}^{+};L^{2}(\Omega))\qquad\text{for all }F\in
C(\mathbb{R}\times\overline{\Omega});$ (4.58)

due to 4.57

$\displaystyle\overline{\Pi}_{n_{l}}g(c_{n_{l}m_{l}})\underset{l\rightarrow\infty}{\rightarrow}g(c)$
$\displaystyle\qquad\text{in }L^{2}_{loc}(\mathbb{R}_{0}^{+};L^{2}(\Omega)),$
(4.59) $\displaystyle\qquad\text{in }L^{1}(\Omega)\text{ a.e. in }(0,\infty);$
(4.60)

due to 4.34a, 4.57, 4.58, and A.2, boundedness of $v$, and the Banach-Alaoglu
theorem

$\displaystyle\overline{\Pi}_{n_{l}}F(c_{n_{l}m_{l}},P_{n_{l}}v)\underset{l\rightarrow\infty}{\overset{*}{\rightharpoonup}}F(c,v)$
$\displaystyle\qquad\text{in
}L^{\infty}_{loc}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega))\qquad\text{for all
}F\in C(\mathbb{R}\times\overline{\Omega}),$ (4.61)
$\displaystyle\overline{\Pi}_{n_{l}}F(c_{n_{l}m_{l}},\Phi(P_{n_{l}}\nabla
v))\underset{l\rightarrow\infty}{\overset{*}{\rightharpoonup}}F(c,\Phi(\nabla
v))$ $\displaystyle\qquad\text{in
}L^{\infty}_{loc}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega))\qquad\text{for all
}F\in C(\mathbb{R}\times\overline{\Omega});$ (4.62)

due to 4.50b, 4.59, and A.6b

$\displaystyle\Lambda_{n_{l}}g(c_{n_{l}m_{l}})\underset{l\rightarrow\infty}{\rightarrow}g(c)$
$\displaystyle\qquad\text{in }L^{2}_{loc}(\mathbb{R}_{0}^{+};L^{2}(\Omega));$
(4.63)

due to 2.4, 4.34a, 4.34b, and 4.59, the chain rule, and the Banach-Alaoglu
theorem

$\displaystyle\nabla\overline{\Pi}_{n_{l}}g\left(c_{n_{l}m_{l}}\right)\underset{l\rightarrow\infty}{\overset{*}{\rightharpoonup}}Dg(c)\qquad\text{in
}{\cal M}_{loc}(\mathbb{R}_{0}^{+};{\cal M}(\Omega));$ (4.64)

due to 4.34a, 4.34b, and 4.60, the chain rule, and the lower semicontinuity of
the total variation with respect to the convergence in $L^{1}(\Omega)$

$\displaystyle g(c)(t,\cdot)\in BV(\Omega),$ (4.65)
$\displaystyle\left\|Dg(c)(t,\cdot)\right\|_{{\cal
M}(\Omega)}\leq\underset{l\rightarrow\infty}{\lim\inf}\left\|\nabla\overline{\Pi}_{n_{l}}g\left(c_{n_{l}m_{l}}\right)(t,\cdot)\right\|_{L^{1}(\Omega)}<\infty\
\ \ \text{a.e. in }(0,\infty);$

due to $g(c)\in L^{1}_{loc}(\mathbb{R}_{0}^{+};L^{1}(\Omega))$, 4.65, and [5,
Lemma 5]

$\displaystyle g(c):(0,\infty)\to BV(\Omega)\qquad\text{weakly measurable};$
(4.66)

due to $g(c)\in L^{1}_{loc}(\mathbb{R}_{0}^{+};L^{1}(\Omega))$, $Dg(c)\in{\cal
M}_{loc}(\mathbb{R}_{0}^{+};{\cal M}(\Omega))$, and 4.66

$\displaystyle g(c)\in L^{1}_{w,loc}(\mathbb{R}_{0}^{+};BV(\Omega)),$ (4.67)
$\displaystyle\lx@cref{creftype~refnum}{DCC_}\text{ holds};$

due to 4.67 and B.2

$\displaystyle\nabla g(c)\in L^{1}_{loc}(\mathbb{R}_{0}^{+};L^{1}(\Omega)),$
(4.68a) $\displaystyle D^{s}g(c)\in L^{1}_{w,loc}(\mathbb{R}_{0}^{+};{\cal
M}(\Omega)),$ (4.68b) $\displaystyle|D^{s}g(c)|\in
L^{1}_{w-*,loc}(\mathbb{R}_{0}^{+};{\cal M}(\Omega));$ (4.68c)

due to 4.2 and 4.65 and the chain rule for the superposition of a Lipschitz
and a BV functions

$\displaystyle\nabla g(c)(t,\cdot)=g^{\prime}(c)\nabla c(t,\cdot)$
$\displaystyle\qquad\text{a.e. in }\Omega\text{ for a.a. }t\in(0,\infty),$
(4.69a)
$\displaystyle|D^{s}g(c)(t,\cdot)|\leq\|g^{\prime}\|_{L^{\infty}((0,\Cr{CBND}(T)))}|D^{s}c(t,\cdot)|$
$\displaystyle\qquad\text{in }{\cal M}(\Omega)\text{ for a.a.
}t\in(0,\infty);$ (4.69b)

due to $\partial_{t}c\in
L^{\infty}_{loc}(\mathbb{R}_{0}^{+};(W^{1,1}(\Omega))^{*})$ and
$c_{0}\in(W^{1,1}(\Omega))^{*}$

$\displaystyle c\in C(\mathbb{R}_{0}^{+};(W^{1,1}(\Omega))^{*});$ (4.70)

due to 4.70, $c\in L^{\infty}_{loc}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega))$,
and a result on weak continuity [42, Chapter 3 §1.4 Lemma 1.4]

$\displaystyle c\in C_{w}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega));$

due to 3.5b and 4.34a and the Banach-Alaoglu theorem

$\displaystyle\overline{\Pi}_{n_{l}}a\left(c_{n_{l}m_{l}},\nabla
c_{n_{l}m_{l}}\right)\underset{l\rightarrow\infty}{\overset{*}{\rightharpoonup}}z$
$\displaystyle\qquad\text{in
}(L^{\infty}_{loc}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega)))^{d},$ (4.71)
$\displaystyle\|z\|_{(L^{\infty}((0,T);L^{\infty}(\Omega)))^{d}}\leq\Cr{CBND}(T)$
$\displaystyle\qquad\text{for all }T>0;$ (4.72)

due to 4.34c, 4.47, 4.62, and 4.71

$\displaystyle\overline{\Pi}_{n_{l}}J_{m_{l}}[c_{m_{l}n_{l}},P_{n_{l}}v]\underset{l\rightarrow\infty}{\overset{*}{\rightharpoonup}}z-\chi
c\Phi(\nabla v)=:$ $\displaystyle J\qquad\text{in
}(L^{\infty}_{loc}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega)))^{d};$ (4.73)

due to 3.7b, 4.2, and 4.72

$\displaystyle\|J\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\leq(1+\chi)\Cr{CBND}(T);$
(4.74)

due to 4.61 and continuity of $f_{c}$

$\displaystyle\overline{\Pi}_{n_{l}}f_{c}(c_{n_{l}m_{l}},P_{n_{l}}v)g(c_{n_{l}m_{l}})\underset{l\rightarrow\infty}{\overset{*}{\rightharpoonup}}f_{c}(c,v)g(c)$
$\displaystyle\qquad\text{in
}L^{\infty}_{loc}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega));$ (4.75)

due to 4.25a, 4.31, 4.73, 4.75, and 4.54

$\displaystyle\partial_{t}\Lambda_{n_{l}}c_{n_{l}}\underset{l\rightarrow\infty}{\overset{*}{\rightharpoonup}}\partial_{t}c\qquad\text{in
}L^{\infty}_{loc}(\mathbb{R}_{0}^{+};(W^{1,1}(\Omega))^{*})$ (4.76)

and for all $\varphi\in L^{1}_{loc}(\mathbb{R}_{0}^{+};W^{1,1}(\Omega))$

$\displaystyle\left<\partial_{t}c,\varphi\right>=\int_{\Omega}-J\cdot\nabla\varphi+f_{c}(c,v)\varphi\,dx\qquad\text{a.e.
in }(0,\infty);$ (4.77)

due to 4.77

$\displaystyle\partial_{t}c=\nabla\cdot J+f_{c}(c,v)\qquad\text{in
}W^{-1,\infty}(\Omega)\qquad\text{ a.e. in }(0,\infty);$ (4.78)

due to 4.50d, 4.54, and 4.76 and the trace theorem for Bochner integrals

$\displaystyle\Lambda_{n_{l}}c_{n_{l}m_{l}}(0,\cdot)\underset{l\rightarrow\infty}{\overset{*}{\rightharpoonup}}c(0,\cdot)=c_{0}\qquad\text{in
}L^{\infty}(\Omega).$

We also need certain inequalities that control evolution of $G(c)$ for $G$ as
in 4.49. Two sets of such inequalities are derived in the next Steps.

######  Step 2 (Inequalities for the evolution of $G(c)$: part I).

In this part of the proof we verify that

$\displaystyle\partial_{t}G(c)+J\cdot\nabla
g(c)-\nabla\cdot(g(c)J)-f_{c}(c,v)g(c)=:\mu_{g}$ (4.79a) with
$\displaystyle\mu_{g}\in L^{1}_{w-*,loc}(\mathbb{R}_{0}^{+};{\cal
M}(\Omega)),$ (4.79b)
$\displaystyle|\mu_{g}|\leq(1+\chi)\Cr{CBND}(T)|D^{s}g(c)|\qquad\text{in
}{\cal M}(\Omega)\text{ a.e. in }(0,T).$ (4.79c)

To begin with, we utilise 4.78. To avoid dealing with the distributional time
derivative of $c$ directly we again avail of Steklov averages. Let $t>0$ and
$\tau\in(0,1)$. Integrating both sides of equation 4.78 over $[t,t+\tau]$ in
Bochner’s sense, dividing by $\tau$, and using $c\in
C_{w}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega))\cap
L^{\infty}_{loc}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega))$, $J\in
L^{\infty}_{loc}(\mathbb{R}_{0}^{+};(L^{\infty}(\Omega))^{d})$, and the
continuity of $v$ and $f_{c}$, we obtain that

$\displaystyle\frac{1}{\tau}(c(t+\tau,\cdot)-c(t,\cdot))=\nabla\cdot\left(\frac{1}{\tau}\int_{t}^{t+\tau}J\,ds\right)+\frac{1}{\tau}\int_{t}^{t+\tau}f_{c}(c,v)\,ds\qquad\text{in
}L^{\infty}(\Omega)\text{ for all }t\in\mathbb{R}_{0}^{+}.$ (4.80)

Observe that $\frac{1}{\tau}\int_{\cdot}^{\cdot+\tau}J\,ds\in
L^{\infty}_{loc}(\mathbb{R}_{0}^{+};(L^{\infty}(\Omega))^{d})$ due to $J\in
L^{\infty}_{loc}(\mathbb{R}_{0}^{+};(L^{\infty}(\Omega))^{d})$, whereas, in
consequence of 4.80,
$\nabla\cdot\left(\frac{1}{\tau}\int_{\cdot}^{\cdot+\tau}J\,ds\right)\in
L^{\infty}_{loc}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega))$. Hence,

$\displaystyle\frac{1}{\tau}\int_{\cdot}^{\cdot+\tau}J\,ds\in
L^{\infty}_{loc}(\mathbb{R}_{0}^{+};X_{\infty}(\Omega)),$ (4.81)

where $X_{\infty}(\Omega)$ is the space defined in 2.2. Let us now consider
the time-dependent measures

$\displaystyle
M_{\theta\tau}(t,\cdot):=\left(\frac{1}{\tau}\int_{t-\theta\tau}^{t+(1-\theta)\tau}J\,ds,Dg(c)(t,\cdot)\right)\qquad\text{for
}\theta\in\\{0,1\\}.$

In the following bounds and convergences are understood to hold for both
choices of $\theta$. Set

$\displaystyle J:=0\qquad\text{in }(-\infty,0)\times\Omega.$ (4.82)

Combining 4.81 and 4.67 and the fact that pairing $(\cdot,\cdot)$ is a bounded
bilinear form (see (1) in Section 2.2) produces

$\displaystyle M_{\theta\tau}\in L^{1}_{w,loc}(\mathbb{R}_{0}^{+};{{\cal
M}(\Omega)}).$ (4.83a) Utilising 2.3a we compute $\displaystyle
M_{\theta\tau}^{ac}(t,\cdot)=$
$\displaystyle\frac{1}{\tau}\int_{t-\theta\tau}^{t+(1-\theta)\tau}J\,ds\cdot\nabla
g(c)(t,\cdot)\qquad\text{a.e. in }\Omega\text{ for a.a. }t\in(0,\infty).$
(4.83b) Further, due to 2.3b, 4.74, and 4.82
$\displaystyle\left|M_{\theta\tau}^{s}(t,\cdot)\right|\leq$
$\displaystyle\left\|\frac{1}{\tau}\int_{t-\theta\tau}^{t+(1-\theta)\tau}J\,ds\right\|_{L^{\infty}(\Omega)}|D^{s}g(c)(t,\cdot)|$
$\displaystyle\leq$
$\displaystyle(1+\chi)\Cr{CBND}(T)|D^{s}g(c)(t,\cdot)|\qquad\text{in }{\cal
M}(\Omega)\text{ for a.a. }t\in(0,T).$ (4.83c)

Due to 4.83c and 2.4 and the Banach-Alaoglu theorem there exist a
$\theta$-independent sequence

$\displaystyle(\tau_{l})\subset(0,1),\qquad\tau_{l}\rightarrow 0,$

and some $M_{\theta}^{s}$, such that

$\displaystyle
M_{\theta\tau_{l}}^{s}\underset{l\rightarrow\infty}{\overset{*}{\rightharpoonup}}M_{\theta}^{s}$
$\displaystyle\qquad\text{in }{\cal M}_{loc}(\mathbb{R}_{0}^{+};{\cal
M}(\Omega)),$ (4.84a)
$\displaystyle|M_{\theta}^{s}(E,\cdot)|\leq(1+\chi)\Cr{CBND}(T)|D^{s}g(c)(E,\cdot)|$
$\displaystyle\qquad\text{in }{\cal M}(\Omega)\qquad\text{for all Borel
}E\subset[0,T].$ (4.84b)

In particular, 4.84b and 4.68b imply that the assumptions of Lemma B.1(2) are
satisfied for $\mu:=M_{\theta}^{s}$ and
$g:=(1+\chi)\Cr{CBND}(T)\|D^{s}c\|_{{\cal M}(\Omega)}$. In view of this we
conclude that

$\displaystyle M_{\theta}^{s}\in L^{1}_{w-*,loc}(\mathbb{R}_{0}^{+};{\cal
M}(\Omega)),$ (4.85)

and 4.84b can then be replaced by

$\displaystyle|M_{\theta}^{s}(t,\cdot)|\leq(1+\chi)\Cr{CBND}(T)|D^{s}g(c)|(t,\cdot)\qquad\text{in
}{\cal M}(\Omega)\text{ for a.a. }t\in(0,T).$ (4.86)

Let

$\displaystyle 0\leq\varphi\in C^{1}([0,T]\times\overline{\Omega})\cap
C_{0}([0,T],C_{0}(\overline{\Omega}))).$ (4.87)

Combining 4.83c and 4.68c, we deduce that for $\varphi$ as in 4.87

$\displaystyle\left|\int_{0}^{\tau_{l}}\left<M_{1\tau_{l}}^{s}(t,\cdot),\varphi(t,\cdot)\right>dt\right|\leq$
$\displaystyle(1+\chi)\Cr{CBND}(T)\|\varphi\|_{L^{\infty}((0,{T});L^{\infty}(\Omega))}\int_{0}^{\tau_{l}}\|D^{s}c(t,\cdot)\|_{{\cal
M}(\Omega)}\,dt$ $\displaystyle\underset{l\to\infty}{\to}$ $\displaystyle 0,$
$\displaystyle\left|\int_{T-\tau_{l}}^{T}\left<M_{0\tau_{l}}^{s}(t,\cdot),\varphi(t,\cdot)\right>dt\right|\leq$
$\displaystyle(1+\chi)\Cr{CBND}(T)\|\varphi\|_{L^{\infty}((0,{T});L^{\infty}(\Omega))}\int_{T-\tau_{l}}^{T}\|D^{s}c(t,\cdot)\|_{{\cal
M}(\Omega)}\,dt$ $\displaystyle\underset{l\to\infty}{\to}$ $\displaystyle 0,$

which together with 4.84a yields

$\displaystyle\int_{\theta\tau_{l}}^{{T}-(1-\theta)\tau_{l}}\left<M_{\theta\tau_{l}}^{s}(t,\cdot),\varphi(t,\cdot)\right>dt\underset{l\to\infty}{\to}$
$\displaystyle\int_{0}^{T}\left<M^{s}_{\theta}(t,\cdot),\varphi(t,\cdot)\right>dt.$
(4.88)

Now we can proceed with 4.80. Multiplying this equality by $(-\varphi
g(c))(t+\theta\tau,\cdot)$ with $\varphi$ as in 4.87, applying 2.1,
integrating with respect to $t$ over $(0,{T}-\tau)$ (recall that ${T}\geq
1>\tau$), and using 4.83a and 4.83b, we conclude that

$\displaystyle-\int_{0}^{{T}-\tau}\int_{\Omega}\varphi(t+\theta\tau,\cdot)\frac{1}{\tau}(c(t+\tau,\cdot)-c(t,\cdot))g(c)(t+\theta\tau,\cdot)\,dxdt$
$\displaystyle=$
$\displaystyle\int_{0}^{{T}-\tau}\frac{1}{\tau}\int_{t}^{t+\tau}\int_{\Omega}\nabla\varphi(t+\theta\tau,\cdot)\cdot
J(s,\cdot)g(c)(t+\theta\tau,\cdot)-\varphi(t+\theta\tau,\cdot)f_{c}(c,v)(s,\cdot)\,g(c)(t+\theta\tau,\cdot)\,dxdsdt$
$\displaystyle+\int_{0}^{{T}-\tau}\left<\left(\frac{1}{\tau}\int_{t}^{t+\tau}J\,ds,Dg(c)(t+\theta\tau,\cdot)\right),\varphi(t+\theta\tau,\cdot)\right>dt$
$\displaystyle=$
$\displaystyle\int_{\theta\tau}^{{T}-(1-\theta)\tau}\frac{1}{\tau}\int_{t-\theta\tau}^{t+(1-\theta)\tau}\int_{\Omega}\left(\varphi(t,\cdot)J(s,\cdot)\cdot\nabla
g(c)(t,\cdot)+\nabla\varphi(t,\cdot)\cdot J(s,\cdot)g(c)(t,\cdot)\right.$
$\displaystyle\phantom{\int_{\theta\tau}^{{T}-(1-\theta)\tau}\frac{1}{\tau}\int_{t-\theta\tau}^{t+(1-\theta)\tau}\int_{\Omega}(}\left.-\varphi(t,\cdot)f_{c}(c,v)(s,\cdot)\,g(c)(t,\cdot)\right)\,dxdsdt$
$\displaystyle+\int_{\theta\tau}^{{T}-(1-\theta)\tau}\left<M_{\theta\tau}^{s}(t,\cdot),\varphi(t,\cdot)\right>dt.$
(4.89)

Since $c\in L^{\infty}_{loc}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega))$, $J\in
L^{\infty}_{loc}(\mathbb{R}_{0}^{+};(L^{\infty}(\Omega))^{d})$, $\nabla c\in
L^{1}_{loc}(\mathbb{R}_{0}^{+};L^{1}(\Omega))$, and
$v,f_{c},\varphi,\nabla\varphi$ are continuous, the Lebesgue differentiation
and the dominated convergence theorems can be used on the right-hand side of
4.89 along with 4.88 yielding

$\displaystyle\int_{\theta\tau_{l}}^{{T}-(1-\theta)\tau_{l}}\frac{1}{\tau_{l}}\int_{t-\theta\tau_{l}}^{t+(1-\theta)\tau_{l}}\int_{\Omega}\left(\varphi(t,\cdot)J(s,\cdot)\cdot\nabla
g(c)(t,\cdot)+\nabla\varphi(t,\cdot)\cdot J(s,\cdot)g(c)(t,\cdot)\right.$
$\displaystyle\phantom{\int_{\theta\tau_{l}}^{{T}-(1-\theta)\tau_{l}}\frac{1}{\tau_{l}}\int_{t-\theta\tau_{l}}^{t+(1-\theta)\tau_{l}}\int_{\Omega}(}\left.-\varphi(t,\cdot)f_{c}(c,v)(s,\cdot)\,g(c)(t,\cdot)\right)\,dxdsdt$
$\displaystyle+\int_{\theta\tau_{l}}^{{T}-(1-\theta)\tau_{l}}\left<M_{\theta\tau_{l}}^{s}(t,\cdot),\varphi(t,\cdot)\right>dt$
$\displaystyle\underset{l\to\infty}{\to}$
$\displaystyle\int_{0}^{T}\int_{\Omega}\varphi J\cdot\nabla
g(c)+\nabla\varphi\cdot Jg(c)-\varphi
f_{c}(c,v)\,g(c)\,dxdt+\int_{0}^{T}\left<M^{s}_{\theta}(t,\cdot),\varphi(t,\cdot)\right>dt.$
(4.90)

We turn to the left-hand side of 4.89. Thanks to 4.87 and A.13, $G(c)\in
L^{\infty}_{loc}(\mathbb{R}_{0}^{+};L^{\infty}(\Omega))$, and the dominated
convergence theorem we have

$\displaystyle-\int_{0}^{{T}-\tau_{l}}\int_{\Omega}\varphi(t,x)\frac{1}{\tau_{l}}(c(t+\tau_{l},x)-c(t,x))g(c)(t,x)\,dxdt$
$\displaystyle\geq$
$\displaystyle-\int_{0}^{{T}-\tau_{l}}\int_{\Omega}\varphi(t,x)\frac{1}{\tau_{l}}\left(G(c)(t+\tau_{l},x)-G(c)(t,x)\right)\,dxdt$
$\displaystyle=$
$\displaystyle\int_{\tau_{l}}^{T}\int_{\Omega}\frac{1}{-\tau_{l}}\left(\varphi(t-\tau_{l},x)-\varphi(t,x)\right)G(c)(t,x)\,dxdt$
$\displaystyle+\frac{1}{\tau_{l}}\int_{0}^{\tau_{l}}\int_{\Omega}\varphi(t,x)G(c)(t,x)\,dxdt-\frac{1}{\tau_{l}}\int_{{T}-\tau_{l}}^{T}\int_{\Omega}\varphi(t,x)G(c)(t,x)\,dxdt$
$\displaystyle\underset{l\to\infty}{\to}$
$\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{t}\varphi G(c)\,dxdt,$
(4.91a) and, similarly,
$\displaystyle-\int_{0}^{{T}-\tau_{l}}\int_{\Omega}\varphi(t+\tau_{l},x)\frac{1}{\tau_{l}}(c(t+\tau_{l},x)-c(t,x))g(c)(t+\tau_{l},x)\,dxdt$
$\displaystyle\leq$
$\displaystyle-\int_{0}^{{T}-\tau_{l}}\int_{\Omega}\varphi(t+\tau_{l},x)\frac{1}{\tau_{l}}\left(G(c)(t+\tau_{l},x)-G(c)(t,x)\right)\,dxdt$
$\displaystyle=$
$\displaystyle\int_{0}^{{T}-\tau_{l}}\int_{\Omega}\frac{1}{\tau_{l}}\left(\varphi(t+\tau_{l},x)-\varphi(t,x)\right)G(c)(t,x)\,dxdt$
$\displaystyle+\frac{1}{\tau_{l}}\int_{0}^{\tau_{l}}\int_{\Omega}\varphi(t,x)G(c)(t,x)\,dxdt-\frac{1}{\tau_{l}}\int_{{T}-\tau_{l}}^{T}\int_{\Omega}\varphi(t,x)G(c)(t,x)\,dxdt$
$\displaystyle\underset{l\to\infty}{\to}$
$\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{t}\varphi G(c)\,dxdt.$
(4.91b)

Altogether, 4.89, 4.90, and 4.91 lead to

$\displaystyle\int_{0}^{T}\left<M^{s}_{0}(t,\cdot),\varphi(t,\cdot)\right>dt$
$\displaystyle\leq$ $\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{t}\varphi
G(c)\,dxdt-\int_{0}^{T}\int_{\Omega}\varphi J\cdot\nabla
g(c)+\nabla\varphi\cdot Jg(c)-\varphi f_{c}(c,v)\,g(c)\,dxdt$
$\displaystyle\leq$
$\displaystyle\int_{0}^{T}\left<M^{s}_{1}(t,\cdot),\varphi(t,\cdot)\right>dt.$
(4.92)

Combining 4.92, 4.85, and 4.86, we obtain

$\displaystyle\left|\int_{0}^{T}\int_{\Omega}\partial_{t}\varphi
G(c)\,dxdt-\int_{0}^{T}\int_{\Omega}\varphi J\cdot\nabla
g(c)+\nabla\varphi\cdot Jg(c)-\varphi f_{c}(c,v)\,g(c)\,dxdt\right|$
$\displaystyle\leq$
$\displaystyle(1+\chi)\Cr{CBND}(T)\int_{0}^{T}\left<|D^{s}g(c(t,\cdot))|,\varphi(t,\cdot)\right>dt.$
(4.93)

Since $C^{1}([0,T]\times\overline{\Omega};\mathbb{R}_{0}^{+})\cap
C_{0}([0,T],C_{0}(\overline{\Omega};\mathbb{R}_{0}^{+})))$ is dense in
$C_{0}([0,T],C_{0}(\overline{\Omega};\mathbb{R}_{0}^{+}))$, inequality 4.93
implies for $\mu_{g}$ from 4.79a that

$\displaystyle\mu_{g}\in(C_{0}([0,T],C_{0}(\overline{\Omega})))^{*},$ (4.94a)
$\displaystyle|\left<\mu_{g},\varphi\right>|\leq(1+\chi)\Cr{CBND}(T)\int_{0}^{T}\left<|D^{s}g(c(t,\cdot))|,\varphi(t,\cdot)\right>dt\qquad\text{for
all }0\leq\varphi\in C_{0}([0,T];C_{0}(\overline{\Omega})).$ (4.94b)

In consequence of 4.94, for all $0\leq\zeta\in C_{0}([0,T])$ we have

$\displaystyle\zeta\mu\in(C([0,T],C_{0}(\overline{\Omega})))^{*},$ (4.95a)
$\displaystyle|\left<\zeta\mu_{g},\varphi\right>|\leq(1+\chi)\Cr{CBND}(T)\int_{0}^{T}\left<|D^{s}g(c(t,\cdot))|,\varphi(t,\cdot)\right>dt\qquad\text{for
all }\psi\in C([0,T];C_{0}(\overline{\Omega})).$ (4.95b)

Combining 2.4 and 4.95b, we conclude that

$\displaystyle\mu_{g}\in{\cal M}_{loc}((0,T);{\cal M}(\Omega)),$ (4.96a)
$\displaystyle|\mu_{g}(E,\cdot)|\leq(1+\chi)\Cr{CBND}(T)|D^{s}g(c)(E,\cdot)|\qquad\text{in
}{\cal M}(\Omega)\qquad\text{for all Borel }E\text{ such that
}\overline{E}\subset(0,T).$ (4.96b)

Lemma B.1(2) can now be applied to $\mu:=\mu_{g}$,
$g:=(1+\chi)\Cr{CBND}(T)\|D^{s}g(c)\|_{{\cal M}(\Omega)}$, and any closed
interval $I\subset(0,T)$, yielding

$\displaystyle\mu_{g}\in L^{1}_{w-*,loc}((0,\infty);{\cal M}(\Omega))$

and allowing to replace 4.96b by 4.79c. Finally, combining 4.79c and 4.68b, we
arrive at 4.79b.

######  Step 3 (Inequalities for the evolution of $G(c)$: part II).

Our next set of inequalities for the evolution of $G(c)$ is derived from the
approximating problems 4.31 and takes the form

$\displaystyle\int_{0}^{T}\eta(t)\left<\mu_{g}(t,\cdot),\rho\right>dt$
$\displaystyle\leq$
$\displaystyle-\underset{l\to\infty}{\lim\inf}\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho
a(c_{m_{l}n_{l}},\nabla c_{m_{l}n_{l}})\cdot\nabla
g(c_{m_{l}n_{l}})\,dxdt+\int_{0}^{T}\eta\int_{\Omega}\rho z\cdot\nabla
g(c)\,dxdt$
$\displaystyle+\chi\int_{0}^{T}\eta(t)\left<D^{s}\int_{0}^{c(t,\cdot)}ug^{\prime}(u)\,du\cdot\Phi(\nabla
v(t,\cdot)),\rho\right>dt,$ (4.97)

where $(m_{l})$ and $(n_{l})$ are as in Step 1.

Testing 4.31 with $\eta\rho\overline{\Pi}_{n_{l}}g(c_{m_{l}n_{l}})$ for
$l\in\mathbb{N}$, we estimate using A.12, 4.47, and 4.49, the continuity of
$\Phi$ and $\nabla v$, partial integration, and the chain rule:

$\displaystyle-\int_{0}^{T}\eta^{\prime}\int_{\Omega}\rho\Lambda_{n_{l}}G(c_{m_{l}n_{l}})\,dxdt$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\eta\int_{\Omega}\rho\partial_{t}\Lambda_{n_{l}}G(c_{m_{l}n_{l}})\,dxdt$
$\displaystyle\leq$
$\displaystyle\int_{0}^{T}\eta\int_{\Omega}\rho\partial_{t}\Lambda_{n_{l}}c_{m_{l}n_{l}}\overline{\Pi}_{n_{l}}g(c_{m_{l}n_{l}})\,dxdt$
$\displaystyle=$
$\displaystyle-\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho\left(\frac{1}{m_{l}}\nabla
c_{m_{l}n_{l}}+a(c_{m_{l}n_{l}},\nabla c_{m_{l}n_{l}})\right)\cdot\nabla
g(c_{m_{l}n_{l}})\,dxdt$
$\displaystyle+\chi\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho\nabla\int_{0}^{c_{m_{l}n_{l}}}ug^{\prime}(u)\,du\cdot\Phi(\nabla
P_{n_{l}}v)\,dxdt$
$\displaystyle-\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\nabla\rho\cdot
J_{m_{l}}[c_{m_{l}n_{l}},P_{n_{l}}v]g(c_{m_{l}n_{l}})-\rho
f_{c}(c_{m_{l}n_{l}},P_{n_{l}}v)g(c_{m_{l}n_{l}})\,dxdt$ $\displaystyle\leq$
$\displaystyle-\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho
a(c_{m_{l}n_{l}},\nabla c_{m_{l}n_{l}})\cdot\nabla g(c_{m_{l}n_{l}})\,dxdt$
$\displaystyle+\chi\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho\nabla\int_{0}^{c_{m_{l}n_{l}}}ug^{\prime}(u)\,du\cdot\Phi(\nabla
P_{n_{l}}v)\,dxdt$
$\displaystyle-\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\nabla\rho\cdot
J_{m_{l}}[c_{m_{l}n_{l}},P_{n_{l}}v]g(c_{m_{l}n_{l}})-\rho
f_{c}(c_{m_{l}n_{l}},P_{n_{l}}v)g(c_{m_{l}n_{l}})\,dxdt.$ (4.98)

Combining 4.63, 4.64, 4.73, 4.75, and A.3 and using the continuity of $\Phi$
and $\nabla v$ and the compensated compactness, we pass to the limit superior
on both sides of 4.98 and obtain

$\displaystyle-\int_{0}^{T}\eta^{\prime}\int_{\Omega}\rho G(c)\,dxdt$
$\displaystyle\leq$
$\displaystyle-\underset{l\to\infty}{\lim\inf}\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho
a(c_{m_{l}n_{l}},\nabla c_{m_{l}n_{l}})\cdot\nabla g(c_{m_{l}n_{l}})\,dxdt$
$\displaystyle+\chi\int_{0}^{T}\eta(t)\left<D\int_{0}^{c(t,\cdot)}ug^{\prime}(u)\,du\cdot\Phi(\nabla
v(t,\cdot)),\rho\right>dt-\int_{0}^{T}\eta\int_{\Omega}\nabla\rho\cdot
Jg(c)-\rho f_{c}(c,v)g(c)\,dxdt.$ (4.99)

Further, on the right-hand-side of 4.99 we make use of 4.68a, 4.68b, and 4.69
which yields

$\displaystyle-\int_{0}^{T}\eta^{\prime}\int_{\Omega}\rho G(c)\,dxdt$
$\displaystyle\leq$
$\displaystyle-\underset{l\to\infty}{\lim\inf}\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho
a(c_{m_{l}n_{l}},\nabla c_{m_{l}n_{l}})\cdot\nabla
g(c_{m_{l}n_{l}})\,dxdt+\int_{0}^{T}\eta\int_{\Omega}\rho z\cdot\nabla
g(c)\,dxdt$ $\displaystyle-\int_{0}^{T}\eta\int_{\Omega}\rho J\cdot\nabla
g(c)+\nabla\rho\cdot Jg(c)-\rho f_{c}(c,v)g(c)\,dxdt$
$\displaystyle+\chi\int_{0}^{T}\eta(t)\left<D^{s}\int_{0}^{c(t,\cdot)}ug^{\prime}(u)\,du\cdot\Phi(\nabla
v(t,\cdot)),\rho\right>dt.$ (4.100)

Combining 4.79a and 4.100, we arrive at 4.97.

######  Step 4 (Identifying $z$).

Our first application of the inequalities established in Steps 2 and 3 is to
the proof of

$\displaystyle z=a(c,\nabla c)\qquad\text{ a.e. in
}\mathbb{R}_{0}^{+}\times\Omega,$ (4.101)

so that

$\displaystyle J=a(c,\nabla c)-\chi c\Phi(\nabla v)\qquad\text{ a.e. in
}(0,\infty)\times\Omega$

in the limit equation 4.77. As is customary in such cases, we couple the
Minty-Browder method with compactness. Let

$\displaystyle y\in\mathbb{R}^{d}.$

Due to the monotonicity of $a$ with respect to its second argument, i.e.
property 3.5e, it holds that

$\displaystyle 0\leq$
$\displaystyle\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho\left(a(c_{m_{l}n_{l}},\nabla
c_{m_{l}n_{l}}\right)-a(c_{m_{l}n_{l}},y))\cdot(\nabla
c_{m_{l}n_{l}}-y)\,dxdt.$ (4.102)

Obviously, the right-hand side of 4.102 can be decomposed as follows:

$\displaystyle\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho\left(a(c_{m_{l}n_{l}},\nabla
c_{m_{l}n_{l}}\right)-a(c_{m_{l}n_{l}},y))\cdot(\nabla
c_{m_{l}n_{l}}-y)\,dxdt$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho
a(c_{m_{l}n_{l}},\nabla c_{m_{l}n_{l}})\cdot\nabla c_{m_{l}n_{l}}\,dxdt$
$\displaystyle-\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho\left(a(c_{m_{l}n_{l}},\nabla
c_{m_{l}n_{l}}\right)-a(c_{m_{l}n_{l}},y))\cdot y\,dxdt$
$\displaystyle-\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho
a(c_{m_{l}n_{l}},y)\cdot\nabla c_{m_{l}n_{l}}\,dxdt.$ (4.103)

We consider the limit behaviour of each of the terms in the decomposition in
4.103 separately. To handle the first term, we apply 4.97 for $g(c):=c$, so
that

$\displaystyle\underset{l\to\infty}{\lim\inf}\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho
a(c_{m_{l}n_{l}},\nabla c_{m_{l}n_{l}})\cdot\nabla c_{m_{l}n_{l}}\,dxdt$
$\displaystyle\leq$ $\displaystyle\int_{0}^{T}\eta\int_{\Omega}\rho
z\cdot\nabla
c\,dxdt+\chi\int_{0}^{T}\eta(t)\left<D^{s}c(t,\cdot)\cdot\Phi(\nabla
v(t,\cdot)),\rho\right>dt-\int_{0}^{T}\eta(t)\left<\mu_{id}(t,\cdot),\rho\right>dt.$
(4.104)

Combining 4.79c, 3.7b, and 4.68c, we estimate the singular measures on the
right-hand side of 4.104 and conclude that

$\displaystyle\underset{l\to\infty}{\lim\inf}\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho
a(c_{m_{l}n_{l}},\nabla c_{m_{l}n_{l}})\cdot\nabla c_{m_{l}n_{l}}\,dxdt$
$\displaystyle\leq$ $\displaystyle\int_{0}^{T}\eta\int_{\Omega}\rho
z\cdot\nabla
c\,dxdt+\C(T)\int_{0}^{T}\eta(t)\left<|D^{s}c(t,\cdot)|,\rho\right>dt.$
(4.105)

With the help of 4.59 and 4.71 we obtain

$\displaystyle\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho\left(a(c_{m_{l}n_{l}},\nabla
c_{m_{l}n_{l}}\right)-a(c_{m_{l}n_{l}},y))\cdot
y\,dxdt\underset{l\rightarrow\infty}{\rightarrow}$
$\displaystyle\int_{0}^{T}\eta\int_{\Omega}\rho(z-a(c,y))\cdot y\,dxdt.$
(4.106)

Due to the chain rule and 4.64 applied to $g(c):=\int_{0}^{c}a(u,y)\,du$ it
holds that

$\displaystyle\overline{\Pi}_{n_{l}}\left(a(c_{m_{l}n_{l}},y)\cdot\nabla
c_{m_{l}n_{l}}\right)=$
$\displaystyle\overline{\Pi}_{n_{l}}\nabla\cdot\int_{0}^{c_{m_{l}n_{l}}}a(u,y)\,du$
$\displaystyle\underset{l\rightarrow\infty}{\overset{*}{\rightharpoonup}}$
$\displaystyle\nabla\cdot\int_{0}^{c}a(u,y)\,du\qquad\text{in }{\cal
M}_{loc}(\mathbb{R}_{0}^{+};{\cal M}(\Omega)).$ (4.107)

Utilising 4.69 and the continuity of $a$, we find that

$\displaystyle\left(\nabla\cdot\int_{0}^{c(t,\cdot)}a(u,y)\,du\right)^{ac}=(a(c,y)\cdot\nabla
c)(t,\cdot)$ $\displaystyle\qquad\text{in }\Omega\text{ for a.a. }t\in(0,T),$
(4.108a)
$\displaystyle\left|\left(\nabla\cdot\int_{0}^{c(t,\cdot)}a(u,y)\,du\right)^{s}\right|\leq
T\Cr{CBND}(T)|D^{s}c(t,\cdot)|$ $\displaystyle\qquad\text{in }{\cal
M}(\Omega)\text{ for a.a. }t\in(0,\infty).$ (4.108b)

Combining 4.107 and 4.108, we obtain

$\displaystyle\underset{l\to\infty}{\lim}\int_{0}^{T}\eta\overline{\Pi}_{n_{l}}\int_{\Omega}\rho\left(a(c_{m_{l}n_{l}},y)\cdot\nabla
c_{m_{l}n_{l}}\right)\,dxdt$ $\displaystyle\leq$
$\displaystyle\int_{0}^{T}\eta\int_{\Omega}\rho a(c,y)\cdot\nabla
c\,dxdt+T\Cr{CBND}(T)\int_{0}^{T}\left<|D^{s}c(t,\cdot)|,\rho\right>dt.$
(4.109)

Collecting 4.102, 4.103, 4.106, 4.109, and 4.105, we finally arrive at the
inequality

$\displaystyle\int_{0}^{T}\eta\int_{\Omega}\rho(z-a(c,y))\cdot(\nabla
c-y)\,dxdt+T\Cl{Clast}\int_{0}^{T}\eta(t)\left<|D^{s}c(t,\cdot)|,\rho\right>dt\geq
0.$ (4.110)

Given that $\rho$ and $\eta$ are arbitrary, 4.110 implies

$\displaystyle((z-a(c,y))\cdot(\nabla
c-y))(t,\cdot)+T\Cr{Clast}|D^{s}c(t,\cdot)|\geq 0\qquad\text{in }{\cal
M}(\Omega)\text{ for a.a. }t\in(0,T).$ (4.111)

The ($d$-dimensional) Lebesgue measure and $|D^{s}c(t,\cdot)|$ are disjoint on
$\Omega$. Hence, 4.111 yields

$\displaystyle(z-a(c,y))\cdot(\nabla c-y)\geq 0\qquad\text{a.e. in
}(0,\infty)\times\Omega\text{ for all }y\in\mathbb{R}^{d}.$ (4.112)

Replacing $\mathbb{R}^{d}$ by the countable set $\mathbb{Q}^{d}$, we conclude
with 4.112 that

$\displaystyle(z-a(c,y))\cdot(\nabla c-y)\geq 0\qquad\text{for all
}y\in\mathbb{Q}^{d}\text{ a.e. in }(0,\infty)\times\Omega,$ (4.113)

Since $\mathbb{Q}^{d}$ is dense in $\mathbb{R}^{d}$ and $a$ is continuous in
the second variable, 4.113 implies

$\displaystyle(z-a(c,y))\cdot(\nabla c-y)\geq 0\qquad\text{for all
}y\in\mathbb{R}^{d}\text{ a.e. in }(0,\infty)\times\Omega.$ (4.114)

Combining 4.114 and a standard argument based on the continuity of $a$ with
respect to the second variable, we arrive at 4.101.

######  Step 5 (Entropy inequalities).

In order to establish the entropy inequalities 3.15, we choose $g$ of the form

$\displaystyle g:=T_{m,M}$ (4.115)

for $0\leq m<M<\infty$. This function is increasing, globally Lipschitz, and
satisfies

$\displaystyle g^{\prime}(u)=\begin{cases}1&\text{for }u\in(m,M),\\\
0&\text{for }u\in\mathbb{R}_{0}^{+}\backslash[m,M],\end{cases}$ (4.116a)
$\displaystyle ug^{\prime}(u)=\frac{1}{2}(g^{2}(u))^{\prime}\qquad\text{for
}u\in\mathbb{R}_{0}^{+}\backslash\\{m,M\\}.$ (4.116b)

We also make use of the auxiliary function

$\displaystyle L(u,\xi):=u\sqrt{u^{2}+|\xi|^{2}}-u^{2}\qquad\text{for }u\geq
0,\ \xi\in\mathbb{R}^{d}.$ (4.117)

One readily verifies that

$\displaystyle L\in C^{1}(\mathbb{R}_{0}^{+}\times\mathbb{R}^{d})\cap
C^{2}((0,\infty)\times\mathbb{R}^{d}),$ (4.118a) $\displaystyle L(u,0)=0$
$\displaystyle\qquad\text{for }\xi\in\mathbb{R}^{d},$ (4.118b) $\displaystyle
0<L(u,\xi)<u|\xi|$ $\displaystyle\qquad\text{for }u>0,\
\xi\in\mathbb{R}^{d}\backslash\\{0\\},$ (4.118c) $\displaystyle
L(u,\cdot)\text{ is convex in }\mathbb{R}^{d}$ $\displaystyle\qquad\text{for
}u\geq 0,$ (4.118d) $\displaystyle\nabla_{\xi}L(u,\xi)=a(u,\xi)\ ,$
$\displaystyle\qquad\text{for }u\geq 0,\ \xi\in\mathbb{R}^{d},$ (4.118e)
$\displaystyle a(u,\xi)\cdot\xi\geq L(u,\xi)$ $\displaystyle\qquad\text{for
}u\geq 0,\ \xi\in\mathbb{R}^{d},$ (4.118f) $\displaystyle
L^{\infty}(u,\xi)=u|\xi|=(a(u,\cdot)\cdot(\cdot))^{\infty}(\xi)$
$\displaystyle\qquad\text{for }u\geq 0,\ \xi\in\mathbb{R}^{d}.$ (4.118g)

Since $a(\cdot,0)\equiv 0$, we have in consequence of 4.116a and 4.118f and
the chain rule that

$\displaystyle a(u,\nabla u)\cdot\nabla g(u)=$ $\displaystyle a(u,\nabla
g(u))\cdot\nabla g(u)$ $\displaystyle\geq$ $\displaystyle L(g(u),\nabla
g(u))\qquad\text{a.e. in }\Omega\text{ for all }u\in W^{1,1}(\Omega).$ (4.119)

Consider now

$\displaystyle f(x,w,\xi):=\rho(x)L\left(w_{+},\xi\right)\qquad\text{for
}(x,w,\xi)\in\Omega\times\mathbb{R}\times\mathbb{R}^{d}$

and the corresponding relaxation functional [26] for $0\leq w\in
BV(\Omega)\cap L^{\infty}(\Omega)$:

$\displaystyle{\cal R}_{f}(w):=$ $\displaystyle\int_{\Omega}f(x,w(x),\nabla
w(x))\,dx+\int_{\Omega\backslash
S_{w}}f^{\infty}(x,\widetilde{w}(x),\nu_{w}(x))\,d|D^{c}w|(x)$
$\displaystyle+\int_{J_{u}}\frac{\int_{w^{-}(x)}^{w^{+}(x)}f^{\infty}(x,z,\nu_{w}(x))\,dz}{w^{+}(x)-w^{-}(x)}\,d|D^{j}w|(x)$
$\displaystyle=$ $\displaystyle\int_{\Omega}\rho L\left(w,\nabla
w\right)\,dx+\frac{1}{2}\int_{\Omega}\rho\,d|D^{s}w^{2}|.$ (4.120)

Here we used 4.118g and the chain rule for BV functions to simplify the
expression for ${\cal R}_{f}$. By the assumptions on $\rho$ in 4.48 and
properties 4.118a, 4.118d, and 4.118c of $L$ the requirements of [27, Theorem
1.1] are satisfied for $f$. Therefore, ${\cal R}_{f}$ is lower semi-continuous
in $BV(\Omega)$ equipped with the $L^{1}(\Omega)$ norm. Together with 4.60
this implies

$\displaystyle\underset{l\to\infty}{\lim\inf}\int_{\Omega}\rho
L(\overline{\Pi}_{n_{l}}g(c_{m_{l}n_{l}}),\nabla\overline{\Pi}_{n_{l}}g(c_{m_{l}n_{l}}))\,dx(t,\cdot)$
$\displaystyle\geq$ $\displaystyle\int_{\Omega}\rho L\left(g(c),\nabla
g(c)\right)\,dx(t,\cdot)+\left<\frac{1}{2}|D^{s}g^{2}(c(t,\cdot))|,\rho\right>\qquad\text{
for a.a. }t\in(0,T).$ (4.121)

Consequently, Fatou’s lemma combined with 4.68c and 4.121 allows to conclude
that

$\displaystyle\underset{l\to\infty}{\lim\inf}\int_{0}^{T}\eta\int_{\Omega}\rho
L(\overline{\Pi}_{n_{l}}g(c_{m_{l}n_{l}}),\nabla\overline{\Pi}_{n_{l}}g(c_{m_{l}n_{l}}))\,dxdt$
$\displaystyle\geq$
$\displaystyle\int_{0}^{T}\eta\,\underset{l\to\infty}{\lim\inf}\int_{\Omega}\rho
L(\overline{\Pi}_{n_{l}}g(c_{m_{l}n_{l}}),\nabla\overline{\Pi}_{n_{l}}g(c_{m_{l}n_{l}}))\,dxdt$
$\displaystyle\geq$ $\displaystyle\int_{0}^{T}\eta\int_{\Omega}\rho
L\left(g(c),\nabla
g(c)\right)\,dxdt+\int_{0}^{T}\eta(t)\left<\frac{1}{2}|D^{s}g^{2}(c(t,\cdot))|,\rho\right>\,dt.$
(4.122)

We use 4.119 and 4.122 to estimate the first term on right-hand side of 4.97
and then 4.116b, yielding

$\displaystyle\int_{0}^{T}\eta(t)\left<\mu_{g}(t,\cdot),\rho\right>dt\leq$
$\displaystyle\int_{0}^{T}\eta\int_{\Omega}\rho\left(-L\left(g(c),\nabla
g(c)\right)+z\cdot\nabla g(c)\right)\,dx$
$\displaystyle+\int_{0}^{T}\eta(t)\left<-\frac{1}{2}|D^{s}g^{2}(c(t,\cdot))|+\frac{\chi}{2}D^{s}g^{2}(c(t,\cdot))\cdot\Phi(\nabla
v(t,\cdot)),\rho\right>dt.$ (4.123)

Since $\rho$ and $\eta$ are arbitrary and $\mu_{g}(t,\cdot)$ is singular for
a.a. $t$, 4.123 implies

$\displaystyle\mu_{g}(t,\cdot)\leq-\frac{1}{2}|D^{s}g^{2}(c(t,\cdot))|+\frac{\chi}{2}D^{s}g^{2}(c(t,\cdot))\cdot\Phi(\nabla
v(t,\cdot))\qquad\text{in }{\cal M}(\Omega)\text{ a.e. in }(0,\infty).$
(4.124)

Finally, combining 4.79a and 4.124 and recalling 4.115, we obtain the entropy
inequality 3.15b. As to 3.15a, it is a special case of the lower bound for
$\mu_{g}$ provided in 4.79.

######  Step 6 (Stability).

In order to obtain 4.46, we aim to pass to the limit along suitable
subsequences in 4.40. We need some preparation.

As is well-known, each $0\leq c_{0}\in L^{\infty}(\Omega)$ can be approximated
in the $L^{1}(\Omega)$-norm by an increasing sequence of nonnegative simple
functions from a countable set ${\cal C}_{0}$. Further, by exploiting standard
separability results, one readily confirms that each $v$ that belongs to

$\displaystyle C(\mathbb{R}_{0}^{+};C^{1}_{\nu}(\overline{\Omega}))\cap
L^{1}_{loc}(\mathbb{R}_{0}^{+};W^{2,1}(\Omega))$

can be approximated by a sequence $(v_{k})$ from a countable subset ${\cal
C}_{v}$ of this space in the following way: for all $T\in\mathbb{N}$

$\displaystyle\|v_{k}-v\|_{C([T,T+1];C^{1}_{\nu}(\overline{\Omega}))}\underset{k\to\infty}{\to}0,$
$\displaystyle\|v_{k}-v\|_{L^{1}_{loc}((T,T+1);W^{2,1}(\Omega))}\underset{k\to\infty}{\to}0,$
$\displaystyle\|v_{k}\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\leq\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\qquad\text{for
all }k\in\mathbb{N}.$

For $(c_{0},v)$ that satisfies 4.45 let $(m_{l}(c_{0},v))$ and
$(n_{l}(c_{0},v))$ denote two corresponding sequences as constructed in Step 1
of the present proof. Since ${\cal C}_{0}\times{\cal C}_{v}$ is countable, the
diagonal argument allows to choose these sequences in the following way:

$\displaystyle m_{l}(c_{0},v)=m_{l}^{(*)}$ $\displaystyle\qquad\text{for all
}l\in\mathbb{N},\ (c_{0},v)\in{\cal C}_{0}\times{\cal C}_{v},$ $\displaystyle
n_{l}(c_{0},v)=n_{l}^{(*)}$ $\displaystyle\qquad\text{for all
}l\in\mathbb{N},\ (c_{0},v)\in{\cal C}_{0}\times{\cal C}_{v},$

and

$\displaystyle(m_{l}(c_{0},v))\text{ is a subsequence of
}\left(m_{l}^{(*)}\right)$ $\displaystyle\qquad\text{for all }(c_{0},v)\text{
as in }\lx@cref{creftype~refnum}{c0vreg},$
$\displaystyle(n_{l}(c_{0},v))\text{ is a subsequence of
}\left(n_{l}^{(*)}\right)$ $\displaystyle\qquad\text{for all }(c_{0},v)\text{
as in }\lx@cref{creftype~refnum}{c0vreg},$

where $\left(m_{l}^{(*)}\right)$ and $\left(n_{l}^{(*)}\right)$ are some fixed
sequences.

Now we are ready to prove 4.46 for the case when
$\left(c_{0}^{(1)},v^{(1)}\right)$ satisfies 4.45 and
$\left(c_{0}^{(2)},v^{(2)}\right)\in{\cal C}_{0}\times{\cal C}_{v}$. Choosing

$m:=m_{l}\left(c_{0}^{(1)},v^{(1)}\right)\qquad\text{and}\qquad
n:=n_{l}\left(c_{0}^{(1)},v^{(1)}\right)$

in 4.40 for sufficiently large $l\in\mathbb{N}$, so that
$n_{l}\left(c_{0}^{(1)},v^{(1)}\right)>N$ for $N$ as in Lemma 4.8(3), we can
pass to the limit in 4.40 using 4.36, 4.37, 4.38, 4.46b, and 4.53 and the
elementary fact that $\operatorname{exp}_{n,C}(t)\to e^{tC}$ as $n\to\infty$.
This yields 4.46 for almost all $t\in(0,T)$. Taking into account
$c^{(1)}-c^{(2)}\in C_{w}([0,T];L^{1}(\Omega))$ and the lower weak
semicontinuity of norms, we conclude that 4.46 is valid for all $t\in[0,T]$.
With this knowledge at hand we turn to the general data
$\left(c_{0}^{(i)},v^{(i)}\right)$ satisfying 4.45 for $i\in\\{1,2\\}$.
Approximating each component in the way as described at the beginning of this
Step of the proof by the respective sequences
$\left(c_{k0}^{(i)},v_{k}^{(i)}\right)$, we obtain for the corresponding
solutions, $c^{(1)}$ and $c^{(2)}$,

$\displaystyle\left\|c^{(1)}-c^{(2)}\right\|_{L^{1}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\underset{k\to\infty}{\lim\inf}\left(\left\|c_{k}^{(1)}-c_{k}^{(2)}\right\|_{L^{1}(\Omega)}+\left\|c_{k}^{(1)}-c^{(1)}\right\|_{L^{1}(\Omega)}+\left\|c_{k}^{(2)}-c^{(2)}\right\|_{L^{1}(\Omega)}\right)$
$\displaystyle\leq$ $\displaystyle
e^{T\underset{(0,M(T))^{2}}{\max}\partial_{c}f}\underset{k\to\infty}{\lim}\left(\left\|c_{k0}^{(1)}-c_{k0}^{(2)}\right\|_{L^{1}(\Omega)}+\left\|c_{k0}^{(1)}-c_{0}^{(1)}\right\|_{L^{1}(\Omega)}+\left\|c_{k0}^{(2)}-c_{0}^{(2)}\right\|_{L^{1}(\Omega)}\right)$
$\displaystyle+\Cr{C361}(M(T))$ $\displaystyle\ \ \
\cdot\underset{k\to\infty}{\lim}\left(\left\|v_{k}^{(1)}-v_{k}^{(2)}\right\|_{C([0,T];C^{1}(\overline{\Omega}))}+\left\|v_{k}^{(1)}-v^{(1)}\right\|_{C([0,T];C^{1}(\overline{\Omega}))}+\left\|v_{k}^{(2)}-v^{(2)}\right\|_{C([0,T];C^{1}(\overline{\Omega}))}\right)$
$\displaystyle+\Cr{C361}(M(T))$ $\displaystyle\ \ \
\cdot\underset{k\to\infty}{\lim}\left(\left\|v_{k}^{(1)}-v_{k}^{(2)}\right\|_{L^{1}((0,T);W^{2,1}(\Omega))}+\left\|v_{k}^{(1)}-v^{(1)}\right\|_{L^{1}((0,T);W^{2,1}(\Omega))}+\left\|v_{k}^{(2)}-v^{(2)}\right\|_{L^{1}((0,T);W^{2,1}(\Omega))}\right)$
$\displaystyle=$ $\displaystyle
e^{T\underset{(0,M(T))^{2}}{\max}\partial_{c}f}\left\|c_{0}^{(1)}-c_{0}^{(2)}\right\|_{L^{1}(\Omega)}$
$\displaystyle+\Cr{C361}(M(T))\left(\left\|v^{(1)}-v^{(2)}\right\|_{C([0,T];C^{1}(\overline{\Omega}))}+\left\|v^{(1)}-v^{(2)}\right\|_{L^{1}((0,T);W^{2,1}(\Omega))}\right),$

as required. This completes the proof of Theorem 4.9.

∎

Several remarks are in order.

######  Remark 4.11.

Since Step 2 of the above proof relies on the weak formulation 4.78 alone, the
lower bound 3.15a holds for every weak solution.

######  Remark 4.12.

Let us compare the way we have handled the existence of solutions to the FL
reaction-diffusion-taxis equation 3.1a with the treatment of a FL
(reaction-)diffusion equation such as presented, e.g. in [9, 8, 6]. These are
the main differences.

  1. 1.

Due to the presence of the taxis and a general reaction term, bounds for
$\|c\|_{L^{\infty}(\Omega)}$ had to be derived using a method that is common
in analysis of taxis systems. In the present case, $c$ is, in general, only
bounded over finite time intervals.

  2. 2.

We chose not to study the non-regularised, i.e. without the inclusion of
operator $1/m\Delta$ in 4.30 (recall 4.25b), disrectisations of 3.1. As a
result, we cannot avail of the mild solution typically obtained in the limit
as $n\to\infty$ by the theory developed in [25]. Instead, our solution, as
constructed in Step 1 of the above proof, is a limit of a subsequence
$(n_{l})$ that may depend on $c_{0}$ and $v$. This mostly affected the proof
of the stability property 4.46, making it more involved than it would have
been otherwise.

On the whole, we made more use of compactness. It was necessary due to the
presence of terms other than diffusion.

  3. 3.

We made an extensive use of interpolation operators and their properties
(presented in Appendix A). It allowed us to embed the elliptic approximations
into the framework of continuous-time equations.

  4. 4.

We used several lemmas about functions and measures with values in Banach
spaces (presented in Appendix B) that further streamlined the analysis.

##  5 Strong well-posedness of 3.2 for fixed $c$

Unlike the non-trivial flux-limited equation 3.1a, the semilinear parabolic
PDE 3.2a can be handled using a classical theory. Similarly to Theorem 4.9, we
provide a solvability result that is formulated in terms of a solution
mapping. This time, however, we have chosen not to track the dependence upon
the initial value since the equation is a standard one.

######  Lemma 5.1 (Strong solutions to 3.2).

Let $D_{v}>0$ and assumptions 3.9 and 3.11 be satisfied. Then there exists a
mapping ${\cal S}_{2}$ that maps any $c$ that satisfies 3.16 to the unique $v$
such that $(c,v)$ solves 3.2 in terms of Definition 3.4 and for all $T>0$

$\displaystyle\|v\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\leq\C\left(T,\|c\|_{L^{\infty}((0,T);L^{\infty}(\Omega))}\right).$
(5.1)

Mapping ${\cal S}_{2}$ is locally Lipschitz continuous in the following sense:
if $v^{(i)}:={\cal S}_{2}\left(c^{(i)}\right)$, then for all $T>0$

$\displaystyle\left\|v^{(1)}-v^{(2)}\right\|_{W^{1,p_{0}}((0,T);L^{p_{0}}(\Omega))}+\left\|v^{(1)}-v^{(2)}\right\|_{L^{p_{0}}((0,T);W^{2,p_{0}}(\Omega))}$
$\displaystyle\leq$
$\displaystyle\C\left(T,M\left[c^{(1)},c^{(2)}\right](T)\right)\left\|c^{(1)}-c^{(2)}\right\|_{L^{p_{0}}((0,T);L^{p_{0}}(\Omega))},$
(5.2a) where $\displaystyle
M\left[c^{(1)},c^{(2)}\right](T):=\left\|\left(c^{(1)},c^{(2)}\right)\right\|_{(L^{\infty}((0,T);L^{\infty}(\Omega)))^{2}}.$
(5.2b)

###### Proof of Lemma 5.1.

######  Step 1 (Existence, uniqueness, and bounds).

Thanks to the assumptions on $c$ and $f_{v}$, estimate 5.1 as well as the non-
negativity hold a priori by the maximum principle. Standard theory of
semilinear parabolic equations provides the existence and uniqueness of a
global weak solution $v$ to 3.2 that satisfies 5.1. Exploiting the maximal
Sobolev regularity for linear parabolic equations in $L^{p}$ (e.g. as stated
in [1, Chapter 4 Remark 4.10.9$(c)$]) yields 3.17.

######  Step 2 (Stability).

Consider solutions $v^{(1)}$ and $v^{(2)}$ that correspond to some $c^{(1)}$
and $c^{(2)}$, respectively, and have the same initial value. Using [1,
Chapter 4 Remark 4.10.9$(c)$], 5.1, and $f_{v}\in C^{1}$, we obtain for
$0<t<T<\infty$

$\displaystyle\left\|\left(v^{(1)}-v^{(2)}\right)(t,\cdot)\right\|_{L^{p_{0}}(\Omega)}^{p_{0}}+\left\|v^{(1)}-v^{(2)}\right\|_{W^{1,p_{0}}((0,t);L^{p_{0}}(\Omega))}^{p_{0}}+\left\|v^{(1)}-v^{(2)}\right\|_{L^{p_{0}}((0,t);W^{2,p_{0}}(\Omega))}^{p_{0}}$
$\displaystyle\leq$
$\displaystyle\C(T)\int_{0}^{t}\left\|f_{v}\left(c^{(1)},v^{(1)}\right)-f_{v}\left(c^{(2)},v^{(2)}\right)\right\|_{L^{p_{0}}(\Omega)}^{p_{0}}\,ds$
$\displaystyle\leq$
$\displaystyle\Cl{C21}\left(T,M\left[c^{(1)},c^{(2)}\right](T)\right)\left(\int_{0}^{t}\left\|v^{(1)}-v^{(2)}\right\|_{L^{p_{0}}(\Omega)}^{p_{0}}\,ds+\int_{0}^{t}\left\|c^{(1)}-c^{(2)}\right\|_{L^{p_{0}}(\Omega)}^{p_{0}}\,ds\right).$
(5.3)

Omitting the second and third summands on the left-hand side of 5.3 and using
Gronwall’s lemma, we arrive at

$\displaystyle\left\|v^{(1)}-v^{(2)}\right\|_{L^{\infty}((0,T),L^{p_{0}}(\Omega))}^{p_{0}}\leq\C\left(T,M\left[c^{(1)},c^{(2)}\right](T)\right)\left\|c^{(1)}-c^{(2)}\right\|_{L^{p_{0}}((0,T);L^{p_{0}}(\Omega))}^{p_{0}}.$
(5.4)

Altogether, 5.3 and 5.4 yield 5.2.

∎

##  6 Existence of solutions to system 3.1-3.2

Theorems 4.9 and 5.1 provide solvability of 3.1 for given $v$ and 3.2 for
given $c$, respectively. With these results at hand, we can now prove our main
Theorem 3.7 on the solvability of the complete system 3.1-3.2.

###### Proof of Theorem 3.7.

######  Step 1 (Reduction to local existence).

Observe that it suffices to establish the existence of a joint solution to
3.1-3.2 on a finite time interval of a fixed length, e.g. on $[0,1]$. This is
because the system is autonomous and we are looking for solutions that due to
3.12a and 3.19 satisfy

$\displaystyle(c,v)(1,\cdot)\in\left(L^{\infty}(\Omega),W^{2\left(1-\frac{1}{p_{0}}\right),p_{0}}_{\nu}\right),$

i.e. have the value at $t=1$ with the same regularity as prescribed at $t=0$,
so that this value can serve as an initial value for the time interval $[1,2]$
and so on. Thus, a solution for all times
$t\in\mathbb{R}_{0}^{+}=\underset{N\in\mathbb{N}_{0}}{\cup}[N,N+1]$ can be
build recursively. It is evident that it necessarily satisfies 3.12 and 3.17.

######  Step 2 (Fixed point argument).

We aim at applying Schauder’s fixed point theorem in order to solve 3.1-3.2
for $t\in[0,1]$. Let

$\displaystyle{\cal V}:=L^{1}((0,1);L^{1}(\Omega)),$ $\displaystyle{\cal
X}:=\left\\{u\in L^{\infty}((0,1);L^{\infty}(\Omega)):\
\|u\|_{L^{\infty}((0,1);L^{\infty}(\Omega))}\leq\Cr{CBND}(1)\right\\},$
$\displaystyle{\cal S}:{\cal X}\to{\cal X},\qquad{\cal S}(c):={\cal
S}_{1}(c_{0},{\cal S}_{2}(c))\text{ for }c\in{\cal X}.$

By Theorems 4.9 and 5.1, mapping ${\cal S}$ is well-defined. Its fixed points
are local solutions of the kind we need. To complete the proof, we verify the
assumptions of Schauder’s theorem for ${\cal S}$ in ${\cal X}\subset{\cal V}$.

Set ${\cal X}$ is a closed bounded convex subset of ${\cal V}$. By
construction, ${\cal S}({\cal X})$ consists of entropy solutions to the IBVP
3.1 that belong to the closure of

$\displaystyle\left\\{c\in{\cal X}:\ c\in L^{1}((0,1);W^{1,1}(\Omega))\cap
W^{1,1}((0,1);W^{-1,\infty}(\Omega))\right.$
$\displaystyle\left.\phantom{c\in{\cal X}:\ \ }\text{ and
}\|c\|_{L^{1}((0,1);W^{1,1}(\Omega))}+\|c\|_{W^{1,1}((0,1);W^{-1,\infty}(\Omega))}\leq\C\right\\}.$

The set is precompact in ${\cal V}$ and, hence, in its closed subset ${\cal
X}$. This is a consequence of a version of the Lions-Aubin lemma [40, Section
8 Corollary 4]. Consequently, ${\cal S}({\cal X})$ is precompact in ${\cal
V}$. It remains to verify the continuity of ${\cal S}$. Let $c^{(i)}:={\cal
S}\left(\bar{c}^{(i)}\right)$ and $v^{(i)}:={\cal
S}_{2}\left(\bar{c}^{(i)}\right)$ for $i\in\\{1,2\\}$. Combining 4.46, 5.2,
4.2, and 5.1, embeddings 3.18, and the interpolation inequality for $L^{p}$
spaces, we estimate as follows:

$\displaystyle\left\|c^{(1)}-c^{(2)}\right\|_{L^{1}((0,1);L^{1}(\Omega))}\leq$
$\displaystyle\C\left(\left\|v^{(1)}-v^{(2)}\right\|_{C([0,1];C^{1}(\overline{\Omega}))}+\left\|v^{(1)}-v^{(2)}\right\|_{L^{1}((0,1);W^{2,1}(\Omega))}\right)$
$\displaystyle\leq$
$\displaystyle\C\left(\left\|v^{(1)}-v^{(2)}\right\|_{W^{1,p_{0}}((0,T);L^{p_{0}}(\Omega))}+\left\|v^{(1)}-v^{(2)}\right\|_{L^{p_{0}}((0,T);W^{2,p_{0}}(\Omega))}\right)$
$\displaystyle\leq$
$\displaystyle\C\left\|\bar{c}^{(1)}-\bar{c}^{(2)}\right\|_{L^{p_{0}}((0,1);L^{p_{0}}(\Omega))}$
$\displaystyle\leq$
$\displaystyle\C\left\|\bar{c}^{(1)}-\bar{c}^{(2)}\right\|_{L^{\infty}((0,1);L^{\infty}(\Omega))}^{1-\frac{1}{p_{0}}}\left\|\bar{c}^{(1)}-\bar{c}^{(2)}\right\|_{L^{1}((0,1);L^{1}(\Omega))}^{\frac{1}{p_{0}}}$
$\displaystyle\leq$
$\displaystyle\C\left\|\bar{c}^{(1)}-\bar{c}^{(2)}\right\|_{L^{1}((0,1);L^{1}(\Omega))}^{\frac{1}{p_{0}}}.$

This means that ${\cal S}$ is a (Hölder) continuous operator in ${\cal X}$.

∎

## Acknowledgement

  * •

The author was supported by the Engineering and Physical Sciences Research
Council [grant number EP/T03131X/1].

  * •

For the purpose of open access, the author has applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising.

  * •

No new data were generated or analysed during this study.

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##  Appendix A Interpolation and discretisation in Bochner spaces

In this Section, we collect some useful properties of the disrectisation and
interpolation operators introduced in Section 2.4. These properties are quite
well-known. Since we could not find the complete statements and proofs in a
single source, we include them here for the sake of completeness.

######  Lemma A.1.

Let $n\in\mathbb{N}$, $T\in\frac{1}{n}\mathbb{N}$, $X$ be some Banach space,
$u,u_{n}:\frac{1}{n}\mathbb{N}_{0}\cap[0,T]\to X$, and $\widetilde{u}\in
L^{p}((0,T);X)$ for some $p\in[1,\infty]$. Then

$\displaystyle\|\underline{\Pi}_{n}u\|_{L^{p}((0,T);X)}=\begin{cases}\left(\frac{1}{n}\sum_{k=0}^{nT-1}\left\|u\left(\frac{k}{n}\right)\right\|_{X}^{p}\right)^{\frac{1}{p}}&\text{for
}p\in[1,\infty),\\\ \max\left\\{\left\|u\left(\frac{k}{n}\right)\right\|_{X}:\
k\in\\{0,\dots,nT-1\\}\right\\}&\text{for }p=\infty,\end{cases}$ (A.1a)
$\displaystyle\|\overline{\Pi}_{n}u\|_{L^{p}((0,T);X)}=\begin{cases}\left(\frac{1}{n}\sum_{k=1}^{nT}\left\|u\left(\frac{k}{n}\right)\right\|_{X}^{p}\right)^{\frac{1}{p}}&\text{for
}p\in[1,\infty),\\\ \max\left\\{\left\|u\left(\frac{k}{n}\right)\right\|_{X}:\
k\in\\{1,2,\dots,nT\\}\right\\}&\text{for }p=\infty,\end{cases}$ (A.1b)
$\displaystyle\|\Lambda_{n}u\|_{L^{p}((0,T);X)}\leq\begin{cases}\left(\|\overline{\Pi}_{n}u\|_{L^{p}((0,T);X)}^{p}+\frac{1}{2n}\|u_{n}(0)\|_{X}^{p}\right)^{\frac{1}{p}}&\text{for
}p\in[1,\infty),\\\ \max\left\\{\left\|u\left(\frac{k}{n}\right)\right\|_{X}:\
k\in\\{0,1,\dots,nT\\}\right\\}&\text{for }p=\infty,\end{cases}$ (A.1c)
$\displaystyle\|\Lambda_{n}u-\underline{\Pi}_{n}u\|_{L^{p}((0,T);X)}=\|\overline{\Pi}_{n}u-\Lambda_{n}u\|_{L^{p}((0,T);X)}=C_{p}\|\overline{\Pi}_{n}u-\underline{\Pi}_{n}u\|_{L^{p}((0,T);X)},$
$\displaystyle\text{where
}C_{p}:=\begin{cases}\frac{1}{(p+1)^{\frac{1}{p}}}&\text{for
}p\in[1,\infty),\\\ 1&\text{for }p=\infty;\end{cases}$ (A.1d)
$\displaystyle\|\widetilde{u}\|_{L^{p}((0,T);X)}\geq\begin{cases}\left(\frac{1}{n}\sum_{k=1}^{nT}\left\|P_{n}\widetilde{u}\left(\frac{k}{n}\right)\right\|_{X}^{p}\right)^{\frac{1}{p}}&\text{for
}p\in[1,\infty),\\\
\max\left\\{\left\|P_{n}\widetilde{u}\left(\frac{k}{n}\right)\right\|_{X}:\
k\in\\{1,2,\dots,nT\\}\right\\}&\text{for }p=\infty;\end{cases}$ (A.2)
$\displaystyle\|{\cal
T}_{n}P_{n}\widetilde{u}-\widetilde{u}\|_{L^{p}((0,T);X)}\underset{n\rightarrow\infty}{\rightarrow}0\qquad\text{for
}\begin{cases}p\in[1,\infty),\\\ p=\infty\text{ and }\widetilde{u}\in
C([0,T];X)\end{cases}\text{ and }{\cal
T}\in\\{\underline{\Pi},\overline{\Pi},\Lambda\\}.$ (A.3)

If $p\in[1,\infty)$, then

$\displaystyle\quad\|\underline{\Pi}_{n}u_{n}-\widetilde{u}\|_{L^{p}((0,T);X)}\underset{n\rightarrow\infty}{\rightarrow}0\quad\text{and}\quad\frac{1}{n^{\frac{1}{p}}}\|u_{n}(T)\|_{X}\underset{n\rightarrow\infty}{\rightarrow}0$
$\displaystyle\Leftrightarrow$
$\displaystyle\quad\|\overline{\Pi}_{n}u_{n}-\widetilde{u}\|_{L^{p}((0,T);X)}\underset{n\rightarrow\infty}{\rightarrow}0\quad\text{and}\quad\frac{1}{n^{\frac{1}{p}}}\|u_{n}(0)\|_{X}\underset{n\rightarrow\infty}{\rightarrow}0$
(A.4a) $\displaystyle\Rightarrow$
$\displaystyle\quad\|\Lambda_{n}u_{n}-\widetilde{u}\|_{L^{p}((0,T);X)}\underset{n\rightarrow\infty}{\rightarrow}0.$
(A.4b)

If $X$ is a Hilbert space and $p=2$, then

$\displaystyle\|\Lambda_{n}u\|_{L^{2}((0,T);X)}\geq\frac{1}{\sqrt{6}}\left(\|\underline{\Pi}_{n}u\|_{L^{2}((0,T);X)}^{2}+\|\overline{\Pi}_{n}u\|_{L^{2}((0,T);X)}^{2}\right)^{\frac{1}{2}}$
(A.5)

and

$\displaystyle\quad\|\underline{\Pi}_{n}u_{n}-\widetilde{u}\|_{L^{2}((0,T);X)}\underset{n\rightarrow\infty}{\rightarrow}0\quad\text{and}\quad\frac{1}{n}\|u_{n}(T)\|_{X}^{2}\underset{n\rightarrow\infty}{\rightarrow}0$
$\displaystyle\Leftrightarrow$
$\displaystyle\quad\|\overline{\Pi}_{n}u_{n}-\widetilde{u}\|_{L^{2}((0,T);X)}\underset{n\rightarrow\infty}{\rightarrow}0\quad\text{and}\quad\frac{1}{n}\|u_{n}(0)\|_{X}^{2}\underset{n\rightarrow\infty}{\rightarrow}0$
(A.6a) $\displaystyle\Leftrightarrow$
$\displaystyle\quad\|\Lambda_{n}u_{n}-\widetilde{u}\|_{L^{2}((0,T);X)}\underset{n\rightarrow\infty}{\rightarrow}0.$
(A.6b)

Finally, if $u:\frac{1}{n}\mathbb{N}_{0}\cap[0,T]\to[0,\infty)$, then

$\displaystyle\|\Lambda_{n}u\|_{L^{\infty}((0,T))}=\max\left\\{\left\|u\left(\frac{k}{n}\right)\right\|_{X}:\
k\in\\{0,1,\dots,nT\\}\right\\}.$ (A.7)

###### Proof.

######  Step 1 (A.1, A.2, A.5, and A.7).

Identities A.1a, A.1b, and A.7 are evident.

Let $p\in[1,\infty)$. We compute using the convexity of $s\mapsto s^{p}$ and
A.1b that

$\displaystyle\int_{0}^{T}\|\Lambda_{n}u\|_{X}^{p}\,dt=$
$\displaystyle\sum_{k=1}^{nT}\int_{\frac{k-1}{n}}^{\frac{k}{n}}\left\|\frac{\frac{k}{n}-t}{\frac{1}{n}}u\left(\frac{k-1}{n}\right)+\frac{t-\frac{k-1}{n}}{\frac{1}{n}}u\left(\frac{k}{n}\right)\right\|_{X}^{p}\,dt$
$\displaystyle\leq$
$\displaystyle\sum_{k=1}^{nT}\int_{\frac{k-1}{n}}^{\frac{k}{n}}\left(\frac{\frac{k}{n}-t}{\frac{1}{n}}\left\|u\left(\frac{k-1}{n}\right)\right\|_{X}+\frac{t-\frac{k-1}{n}}{\frac{1}{n}}\left\|u\left(\frac{k}{n}\right)\right\|_{X}\right)^{p}\,dt$
$\displaystyle\leq$
$\displaystyle\sum_{k=1}^{nT}\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\frac{k}{n}-t}{\frac{1}{n}}\left\|u\left(\frac{k-1}{n}\right)\right\|_{X}^{p}+\frac{t-\frac{k-1}{n}}{\frac{1}{n}}\left\|u\left(\frac{k}{n}\right)\right\|_{X}^{p}\,dt$
$\displaystyle=$
$\displaystyle\frac{1}{2n}\sum_{k=1}^{nT}\left(\left\|u\left(\frac{k-1}{n}\right)\right\|_{X}^{p}+\left\|u\left(\frac{k}{n}\right)\right\|_{X}^{p}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{k=1}^{nT}\left\|u\left(\frac{k}{n}\right)\right\|_{X}^{p}+\frac{1}{2n}\|u_{n}(0)\|_{X}^{p}-\frac{1}{2n}\|u_{n}(T)\|_{X}^{p}$
$\displaystyle\leq$
$\displaystyle\|\overline{\Pi}_{n}u\|_{L^{p}((0,T);X)}^{p}+\frac{1}{2n}\|u_{n}(0)\|_{X}^{p},$

yielding A.1c. Identity A.1d follows with

$\displaystyle\int_{0}^{T}\|\underline{\Pi}_{n}u-\Lambda_{n}u\|_{X}^{p}\,dt=$
$\displaystyle\sum_{k=1}^{nT}\int_{\frac{k-1}{n}}^{\frac{k}{n}}\left\|-\frac{u\left(\frac{k}{n}\right)-u\left(\frac{k-1}{n}\right)}{\frac{1}{n}}\left(t-\frac{k}{n}\right)\right\|_{X}^{p}\,dt$
$\displaystyle=$
$\displaystyle\frac{1}{p+1}\frac{1}{n}\sum_{k=1}^{nT}\left\|u\left(\frac{k}{n}\right)-u\left(\frac{k-1}{n}\right)\right\|_{X}^{p}$
$\displaystyle=$