# Weak-strong uniqueness for
energy-reaction-diffusion systems

Katharina Hopf  Katharina Hopf, Weierstrass Institute for Applied Analysis and
Stochastics (WIAS), Mohrenstrasse 39, 10117 Berlin, Germany, E-Mail:
<EMAIL_ADDRESS>
###### Abstract.

We establish weak-strong uniqueness and stability properties of renormalised
solutions to a class of energy-reaction-diffusion systems. The systems
considered are motivated by thermodynamically consistent models, and their
formal entropy structure allows us to use as a key tool a suitably adjusted
relative entropy method. The weak-strong uniqueness principle holds for
dissipative renormalised solutions, which in addition to the renormalised
formulation obey suitable dissipation inequalities consistent with previous
existence results. We treat general entropy-dissipating reactions without
growth restrictions, and certain models with a non-integrable diffusive flux.

The results also apply to a class of (isoenergetic) reaction-cross-diffusion
systems.

###### Key words and phrases:

Energy-reaction-diffusion systems, weak-strong uniqueness, entropy method,
convexity method, dissipative renormalized solutions, cross diffusion.

###### 2020 Mathematics Subject Classification:

35A02, 35K51, 35K57, 35Q79

##  1\. Introduction

It is well acknowledged that the evolution of a system of diffusing and
reacting chemicals is influenced by the thermal state of the system. _Energy-
reaction-diffusion systems_ (ERDS) take into account this thermal dependency
by consistently coupling the evolution of the chemical concentrations
$c=(c_{1},\dots,c_{n})$ to a heat-type equation for the _internal energy
density_ $u$. Choosing the internal energy density as the thermal variable (as
opposed to temperature for instance) has the advantage that the underlying
physical entropy is jointly concave in the state variables $z=(u,c)$ [38, 39].

Recently, global existence of weak and renormalised solutions has been
obtained for a class of thermodynamically consistent ERDS [28] taking the form
(with $z:=(u,c)$)

$\left\\{\begin{array}[]{rlll}\partial_{t}u\\!\\!&=\nabla\cdot\big{(}A_{0j}(z)\nabla
z_{j}\big{)},&t>0,x\in\Omega,&\vspace{.1cm}\\\
\partial_{t}c_{i}\\!\\!&=\nabla\cdot\big{(}A_{ij}(z)\nabla
z_{j}\big{)}+R_{i}(z),&t>0,x\in\Omega,&i\in\\{1,\dots,n\\},\vspace{.1cm}\\\
0\\!\\!&=A_{ij}(z)\nabla
z_{j}\cdot\nu,&t>0,x\in\partial\Omega,&i\in\\{0,\dots,n\\},\end{array}\right.$
(1.1)

see also the more explicit system (2.9). Eq. (1.1) is supplemented with an
initial condition $(u,c)_{|t=0}=(u^{\mathrm{in}},c^{\mathrm{in}})$ for
$x\in\Omega$, where $\Omega\subset\mathbb{R}^{d}$ is a bounded Lipschitz
domain with outer unit normal $\nu$. Note that we use the summation convention
omitting the summation symbol in repeatedly occurring indices (here
$\sum_{j=0}^{n}$). The diffusion matrix $A(z)=(A_{ij}(z))_{i,j=0,\dots,n}$ and
the reactions $(R_{i}(z))_{i=1,\dots,n}$ are obtained from an underlying
formal gradient structure based on entropy functionals
$H(z)=\int_{\Omega}h(z)\,\differential x$ with convex densities $h(z)=h(u,c)$
taking the form

$\displaystyle
h(u,c_{1},\dots,c_{n})=-\sigma(u)+\sum_{i=1}^{n}b(c_{i},w_{i}(u)).$ (1.2)

Here, $\sigma$ denotes the thermal part when the concentrations $c_{i}$ are in
their thermodynamic equilibrium $w_{i}=w_{i}(u)$, and $b(s,e):=e\lambda(s/e)$
with $\lambda$ denoting the Boltzmann function, see Section 1.1 for details.
The absence of reactions in the $u$-component of (1.1) reflects the property
of conservation of the total (internal) energy $\int_{\Omega}u$.

ERDS genuinely feature cross-diffusion effects, such as concentration flux
driven by gradients of the internal energy density and energy flux due to
concentration gradients, which are one of the main sources of difficulties in
their analysis. These phenomena are closely linked to the thermodynamic origin
of ERDS, and are related to the _Soret effect_ and the _Dufour effect_ , well-
known in physics, which describe concentration flux due to temperature
gradients resp. heat flux driven by concentration gradients.

In the present manuscript, we aim to derive stability properties including a
weak-strong uniqueness result for ERDS (1.1) based on their thermodynamic
structure. This can be seen as an attempt to justify the weak solution concept
in [28]. The main contribution of [28] was to identify non-trivial classes of
thermodynamically consistent models that allow for an existence theory of
generalised solutions. Interestingly, even in a cross-diffusion dominant
regime and without physically restrictive growth conditions on the reactions,
existence has been obtained in [28] based on the notion of renormalised
solutions [26]. Our weak-strong uniqueness principle covers models involving
various cross-diffusion phenomena (Soret effect, Dufour effect, cross
diffusion between species), and applies in particular to a class of
isoenergetic reaction-cross-diffusion systems, thus generalising [15]. Weak-
strong uniqueness is obtained from a weak stability estimate for a generalised
distance involving as in [27] an adjusted relative entropy. By suitably
exploiting the thermodynamic structure of the system some of the technical
issues arising in the proof of [15] will be avoided. We also obtain an
asymptotic stability result.

###  1.1. Thermodynamic modelling

Let us now briefly specify the thermodynamic structure considered in the
present manuscript. For more background on the modelling, we refer to [28, 43,
32, 40]. Models compatible with thermodynamics can be derived using the
Onsager formalism in [40]. Here, the main ingredient is a triple
$(Z,H,\mathbb{K})$ consisting of a state space $Z$, a driving functional $H$,
and a so-called Onsager operator $\mathbb{K}$. Typically, $Z\subset X$ is a
convex subset of a Banach space $X$, $H:Z\to\mathbb{R}\cup\\{\infty\\}$ a
differentiable and convex functional on $Z$ (below usually referred to as
entropy due to its correspondence to the negative of the physical entropy),
while $\mathbb{K}$ can be seen as a generalised inverse Riemannian metric
tensor on $Z$. More specifically, for every $z\in Z$, $\mathbb{K}(z)$ defines
a symmetric and positive semi-definite (unbounded) linear operator from
$T_{z}^{*}Z$ to $T_{z}Z$. If $\mathbb{G}=\mathbb{K}^{-1}$ exists, the triple
$(Z,H,\mathbb{G})$ forms a gradient system. Then, motivated by the classical
gradient flow equation $\mathbb{G}(z)\dot{z}=-DH(z)$, with $\dot{z}$ denoting
the time derivative of $z=z(t)$, one considers the evolution law

$\dot{z}=-\mathbb{K}(z)DH(z),$

where here $DH$ denotes the Fréchet derivative of the functional $H$. An
advantage of this Onsager form is that it facilitates the consistent coupling
of different physical phenomena, which can be realised by an additive
decomposition of $\mathbb{K}$ [40]. Observe that, formally, the above
structure encodes the following core entropy dissipation property

$\displaystyle\tfrac{\differential}{\differential t}H(z)=-\langle
DH(z),\mathbb{K}(z)DH(z)\rangle\leq 0$ (1.3)

along any solution curve $z=z(t)$ of the above law. Conservation of the total
energy $E(z),$ with $E$ denoting the energy functional on $Z$, can be
guaranteed by imposing the condition
$\mathbb{K}DE=\mathbb{K}^{*}DE\overset{!}{=}0$, which implies that
$\tfrac{\differential}{\differential t}E(z)=-\langle
DE(z),\mathbb{K}(z)DH(z)\rangle=0.$

In the context of ERDS, we consider, as introduced above, $z=(u,c)$ with $u$
the internal energy density and $c=(c_{1},\dots,c_{n})$ the vector of
concentrations. We focus on entropies of the form

$H(z)=\int_{\Omega}h(z)\,\differential x$

with densities

$h(z)=h(u,c)=-\sigma(u)+\sum_{i=1}^{n}b(c_{i},w_{i}(u)),$

composed of a thermal part $\sigma(u)$ and a relative Boltzmann entropy
$b(s,e)=e\lambda(s/e)$, where

$\displaystyle\lambda(r):=r\log(r)-r+1,$ (1.4)

and with $w_{i}=w_{i}(u)$, $i=1,\dots,n$, denoting the thermodynamic
equilibria of the concentrations $c_{i}$. The dependence of $w_{i}$ on the
internal energy density $u$ results in a strong coupling of the system and is
one of the main sources of difficulties in the analysis. It will be convenient
to introduce the function $\hat{\sigma}(u)=\sigma(u)-\sum_{i=1}^{n}w_{i}(u)+n$
and rewrite $h(u,c)$ in the following more explicit form

$\displaystyle
h(u,c)=-\hat{\sigma}(u)+\sum_{i=1}^{n}\Big{(}\lambda(c_{i})-c_{i}\log
w_{i}(u)\Big{)}.$ (h1)

To simplify the exposition, we will impose the following concrete conditions
on the coefficient functions (cf. [28]):

$\left.\begin{tabular}[]{@{}l@{}}$\hat{\sigma}\in C^{2}((0,\infty))$ strictly
concave \& non-decreasing.\\\ $w_{i}\in C([0,\infty))\cap C^{2}((0,\infty))$
concave \& non-decr.\ with $w_{i}(0)>0$ for all $i$.\\\ $\lim_{u\downarrow
0}\hat{\sigma}^{\prime}(u)=+\infty$,
$\lim_{u\uparrow\infty}\hat{\sigma}^{\prime}(u)=0$;\quad$\sup_{u\in(0,1]}\hat{\sigma}^{\prime\prime}(u)<0$.\\\
$\exists\beta\in(0,1)$ such that $w_{i}(u)\lesssim 1+u^{\beta}$ for all
$i\in\\{1,\dots,n\\}$.\end{tabular}\right\\}$ (h2)

Typical choices are $\hat{\sigma}(u)=a\log(u)$ or $\hat{\sigma}(u)=au^{\nu}$
for some $\nu\in(0,1)$, $a>0$, and $w_{i}(u)=(b_{i,1}u+b_{i,0})^{\beta_{i}}$
or $w_{i}(u)=b_{i,1}u^{\beta_{i}}+b_{i,0}$ for $\beta_{i}\in(0,1),$
$b_{i,0}>0,b_{i,1}\geq 0$.

Our weak-strong uniqueness principle does not rely, in an essential way, on
this specific form of the entropy density.111See Sec. 2.3.2 for an example of
a different entropy density that our technique can be adapted to. In fact, in
the proof of our main theorem (Thm 2.9), identity (h1) is only used to
guarantee the coercivity properties in Proposition 3.2. The crucial point in
the assumptions (h2) on the coefficient functions is that they ensure good
convexity properties, and more specifically the locally uniform positive
definiteness of the Hessian $D^{2}h$, which is essential for estimate (3.13)
in Prop. 3.2. The monotonicity assumptions on $\hat{\sigma}$ and $w_{i}$ are
relevant from the modelling point of view, since they ensure that $u\mapsto
h(u,c)$ is non-increasing, so that temperature, which is given by
$-\tfrac{1}{\partial_{u}h}$, is non-negative.

As in [43, 28] we are primarily interested in Onsager operators $\mathbb{K}$
of the form

$\mathbb{K}(z)\zeta=\mathbb{K}_{\mathrm{diff}}(z)\zeta+\mathbb{K}_{\mathrm{react}}(z)\zeta=-\nabla\cdot(\mathbb{M}(z)\nabla\zeta)+\mathbb{L}(z)\zeta,$

where $\mathbb{M}(z),\mathbb{L}(z)\in\mathbb{R}^{(1+n)\times(1+n)}$ are
positive semi-definite symmetric matrices and where $\nabla=\nabla_{x}$ is the
gradient with respect to $x\in\Omega$. We will complement $\mathbb{K}$ with
the no-flux boundary conditions $\mathbb{M}\nabla\zeta\cdot\nu=0$ on
$\partial\Omega$, where $\nu$ denotes the outer unit normal vector to
$\partial\Omega$. Observing that $E(u,c)=\int_{\Omega}u$ describes the total
(internal) energy, the condition $\mathbb{K}DE\equiv 0$, ensuring energy
conservation, means that
$\ker\mathbb{L}(z)\supseteq\mathrm{span}\\{(1,0)^{T}\\}$. Thus, by the
symmetry of $\mathbb{L}$, the zeroth component $R_{0}$ of
$R(z):=-\mathbb{L}(z)Dh(z)$ vanishes. Moreover, positive semi-definiteness of
$\mathbb{L}$ implies the inequality

$\displaystyle D_{i}h(z)R_{i}(z)\leq 0.$ (1.5)

In this paper, the specific form of $\mathbb{L}(z)$ will not be relevant.
Instead, we directly work with reactions $R(z)$ of the form

$R(z):=(0,R_{1}(z),\dots,R_{n}(z))$

satisfying (1.5).

With $\mathbb{K}$ as above, the equation $\dot{z}=-\mathbb{K}(z)DH(z)$ can be
written in the form (1.1) by choosing

$A(z):=\mathbb{M}(z)D^{2}h(z).$

In short,

$\begin{array}[]{rlcl}\partial_{t}z&\\!\\!=\,\nabla\cdot\big{(}A(z)\nabla
z\big{)}+R(z),&&\quad t>0,x\in\Omega,\vspace{1mm}\\\ 0&\\!\\!=\,A(z)\nabla
z\cdot\nu,&&\quad t>0,x\in\partial\Omega,\end{array}$ (erds)

subject to an initial condition $z_{|t=0}=z^{\mathrm{in}}$. In the above
setting, the entropy dissipation property (1.3) takes the form

$\displaystyle\frac{\differential}{\differential
t}H(z)+\int_{\Omega}\mathscr{P}(z)\,\differential
x=\int_{\Omega}D_{i}h(z)R_{i}(z)\,\differential x\leq 0,$ (1.6)

where $\mathscr{P}(z):=\nabla D_{i}h(z)\cdot\mathbb{M}_{il}(z)\nabla
D_{l}h(z)\geq 0$, by the positive semi-definiteness of the mobility matrix
$\mathbb{M}$, which will be assumed throughout. Supposing, for instance, that
$\mathscr{P}(z)\gtrsim\sum_{i=1}^{n}|\nabla\sqrt{c_{i}}|^{2}$ (as it can be
proved for many of the models considered in [28], see Section 2.3 and Lemma
6.1), and using conservation of $\int_{\Omega}u$ together with suitable bounds
on $H(z)$ (cf. Lemma 6.3), the entropy dissipation property provides a priori
control of $\sum_{i=1}^{n}\|\nabla\sqrt{c_{i}}\|^{2}_{L^{2}_{t,x}}$. Let us
further note that the fact that $R_{0}(z)$ vanishes gives to some extent a
scalar-like structure to the $u$-component of (erds), and if, for instance,
$\mathbb{M}$ is chosen such that $A_{0j}=a\,\delta_{0j}$ for some function
$a=a(z)\geq 0$, any $L^{p}$-type energy $\tfrac{1}{p}\|u(t)\|_{L^{p}}^{p}$,
$p\in(1,\infty)$, is formally non-increasing in time.

###  1.2. Motivation and strategy

Being able, for a given PDE, to identify concepts of solutions for which both
existence and uniqueness can be established is a fundamental concern in
modelling and analysis. For _scalar_ equations, there are various tools to
identify frameworks allowing for the existence of a unique solution, even in
regimes of low regularity and with strong nonlinearities. One approach is
based on the ‘doubling variables’ technique first employed by Kružkov [36] to
entropy solutions of first-order equations, and extended by Carrillo [9] to
hyperbolic-parabolic-elliptic equations. The concept was adapted to situations
where $L^{\infty}$ bounds are not available to give uniqueness in a class of
renormalised solutions [5, 10]. See also [3, 44, 1] for more recent
developments. Let us also mention the Young measure approach to conservation
laws going back to Tartar [47] and DiPerna [23] who obtained, in the scalar
case, uniqueness of solutions obeying an entropy inequality. For second order
parabolic and elliptic equations the viscosity solution technique and
associated comparison principles [34, 17] are powerful tools and the key to a
variety of wellposedness results in geometric, highly nonlinear or degenerate
settings, see e.g. [24, 16, 8, 11]. Some extensions of the viscosity solution
approach to systems are available for weakly coupled problems with a
monotonicity condition [33]. Further uniqueness results applying to specific
systems and typically in more regular situations include [29, 35, 13, 45, 30,
6].

In general, the case of _strongly coupled_ (parabolic) _systems_ tends to be
much more difficult. While under a parabolicity condition the existence of
suitable generalised solutions (here referred to as ‘weak’ solutions) can
often be established, positive uniqueness results in such general settings are
rare. It is therefore common, to relax the quest for uniqueness to the problem
of whether weak solutions are uniquely determined in situations where a
sufficiently regular solution (a ‘strong’ solution) happens to exist. In other
words, one is interested in the question of whether such strong solutions are
unique in a potentially much larger class of weak solutions. The question of
weak-strong uniqueness is classical in fluid dynamics problems and goes back
to Leray’s fundamental work [37], where it was established for the
incompressible Navier–Stokes equations. We refer to the survey by Wiedemann
[49] for more details and further references. For recent advances on
conditional uniqueness results for dissipative measure-valued solutions to
conservation laws, see [31] and references therein. Rather significant in the
thermodynamics context is moreover the relative entropy technique employed by
DiPerna [22] and Dafermos [19] for hyperbolic conservation laws.

Relative entropy methods are nowadays a standard tool to study weak stability
properties of nonlinear systems endowed with a (convex) entropy structure.
Generally speaking (using the notation introduced in Sec. 1.1), a relative
entropy of the form

$\displaystyle
H_{\mathrm{rel}}(z,\tilde{z})=H(z)-\int_{\Omega}D_{i}h(\tilde{z})(z_{i}-\tilde{z}_{i})\,\differential
x-H(\tilde{z})$ (1.7)

is used to measure the distance between a weak solution $z$ and a strong
solution $\tilde{z}$. Observe that for convex entropies $H(z)$, the map
$z\mapsto H_{\mathrm{rel}}(z,\tilde{z})$ is a non-negative, convex functional
vanishing in $z=\tilde{z}$. The thermodynamic structure ensures that regular
solutions automatically satisfy an entropy dissipation balance (cf. eq.
(1.3)). Physically relevant processes may, however, in general possess less
regularity and here an entropy inequality is often added as an admissibility
criterion for weak solutions. The goal is then to obtain an upper bound on the
time evolution of the relative entropy that implies stability of a regular
flow on a finite time horizon among generalised solutions. This means that
given a regular flow, any weak solution that is initially close (in relative
entropy) will remain close for some time.

In the present paper, we pursue such a strategy in the context of ERDS. Weak-
strong uniqueness has recently been obtained for entropy-dissipating reaction-
diffusion systems with a uniformly elliptic and bounded diagonal diffusion
matrix [27], where the main difficulty consists in a lack of control of the
reaction rates. Extensions to a cross-diffusion system from population
dynamics with weak cross diffusion can be found in [15]. Both references are
based on the relative entropy method, but their arguments rely on the specific
structure of the diffusion matrix of their systems. Here, we would like to
present a more general strategy to deduce stability from an underlying
thermodynamic structure.

Given the strong coupling and lack of a priori bounds in $L^{\infty}$, there
are several difficulties in our ERDS that require an adaptation of the
classical relative entropy approach to weak-strong stability. First, due to
the lack of growth restrictions on the reactions and in some cases even the
flux term (see the models in [28]), the evolution of the classical relative
entropy used to measure the distance between a renormalised solution $z$ and a
strong solution $\tilde{z}$, cannot be properly controlled. This is due to the
term

$\displaystyle-\int_{\Omega}D_{i}h(\tilde{z})\tfrac{\differential}{\differential
t}z_{i}\,\differential x=\int_{\Omega}\nabla
D_{i}h(\tilde{z})\cdot(A_{ij}(z)\nabla z_{j})\,\differential
x-\int_{\Omega}D_{i}h(\tilde{z})R_{i}(z)\,\differential x\quad$ (1.8)

arising in the formal computation of the time derivative of
$H_{\mathrm{rel}}(z,\tilde{z})$. In fact, the available a priori estimates do
not ensure that $A_{ij}(z)\nabla z_{j}\in L^{1}(\Omega)$ and $R_{i}(z)\in
L^{1}(\Omega)$ for a.e. time. At the same time, the corresponding integrands
in (1.8) do not have a sign, and there is no hope for the uncontrolled parts
to cancel with some of the remaining terms appearing in
$\tfrac{\differential}{\differential t}H_{\mathrm{rel}}(z,\tilde{z})$. It is
therefore necessary to adjust the relative entropy
$H_{\mathrm{rel}}(z,\tilde{z})$. This issue has already been encountered in
[27]; it can be resolved by introducing a suitable smooth and compactly
supported truncation function $\xi^{*}=\xi^{*}(z)$ with $\xi^{*}(z)=1$ if
$\sum_{i=0}^{n}z_{i}\leq E$ for some $E\gg\sum_{i=0}^{n}\tilde{z}_{i}$ (see
Section 3.1 for details) in the formula for the relative entropy via

$\displaystyle
H_{\mathrm{rel}}^{*}(z,\tilde{z}):=H(z)-\int_{\Omega}D_{i}h(\tilde{z})(\xi^{*}(z)z_{i}-\tilde{z}_{i})\,\differential
x-H(\tilde{z}).$ (1.9)

The relative entropy density adjusted in this fashion allows to remove the
issue pointed out above. (Strictly speaking, in the term
$D_{0}h(\tilde{z})(z_{0}-\tilde{z}_{0})$ the truncation function is not needed
in the models considered in this paper, and for other applications it may be
helpful to use a different choice such as
$D_{i}h(\tilde{z})(\xi^{*}_{i}(z)z_{i}-\tilde{z}_{i})$ with
$\xi^{*}_{i}(z)\equiv 1$ for $i=0$, or versions thereof.)

A second difficulty arising in the case of ERDS is the inherent coupling
between concentrations and energy density, which manifests itself in the
circumstance that the entropy density cannot be additively decomposed into
terms depending only on an individual component $z_{i}$. This in turn leads to
a non-diagonal diffusion matrix $A(z)$ and renders estimating the evolution of
$H^{*}_{\mathrm{rel}}(z,\tilde{z})$ substantially more delicate than in the
diagonal case. One of the main contributions of this manuscript is to show
that such estimates can be achieved, with relatively little technical effort,
by carefully exploiting the entropy structure.

The energy component $u$ plays a distinguished role in ERDS that has to be
taken advantage of when interested in a general analysis. At a technical
level, the physical constraint of the convex function $h(u,c)$ being non-
increasing in $u$ (to ensure a non-negative temperature) restricts the range
of relevant functions $\hat{\sigma}$ to sublinearities such as
$\hat{\sigma}(u)=u^{\nu}$ for some $\nu\in[0,1)$ (with $\nu=0$ corresponding
to $\log$). Unless $\hat{\sigma}(u)$ has close to linear growth for large
values of $u$, even the possibility of an existence theory solely based on the
entropy estimate is questionable in general dimensions. We are interested in
covering more degenerate choices of $\hat{\sigma}$, and therefore cannot
purely rely on the (adjusted) relative entropy to measure the distance of a
weak to the strong solution. Instead, we exploit the absence of source terms
in the $u$-component of the evolution system, which allows to give an extra,
scalar-like structure to the evolution law for $u$. Here, we content ourselves
with the arguably simplest choice of an $L^{2}$-structure, meaning that we
consider weighted generalised distances of the form

$\displaystyle\operatorname{Dist}^{*}_{\alpha}(z,\tilde{z})=H_{\mathrm{rel}}^{*}(z,\tilde{z})+\tfrac{\alpha}{2}\|u-\tilde{u}\|_{L^{2}(\Omega)}^{2},\quad\alpha\in(0,\infty).$
(1.10)

This is consistent with the approach in [28] and allows us in particular to
show the weak-strong uniqueness property for the potentially pathological
solutions constructed in [28, Theorem 1.8], where a cross-diffusion dominant
regime was considered with gradients of the internal energy density inducing a
(possibly) non-integrable concentration flux. Furthermore, by exploiting the
existence of such an additional quantity that up to some error term is
dissipated along the flow, we can relax the conditions on the entropy
functional in [43] required for proving exponential convergence to
equilibrium.

###  1.3. Technique

Here, we briefly outline, at a formal level, the main points of our argument
showing a weak-strong stability estimate of the form

$\displaystyle\tfrac{\differential}{\differential
t}\operatorname{Dist}^{*}_{\alpha}(z,\tilde{z})\lesssim_{T,\alpha,\xi^{*}}\operatorname{Dist}^{*}_{\alpha}(z,\tilde{z}),$
(1.11)

on any finite time horizon $(0,T)$, $T<T^{*}$, where $z$ is assumed to be a
‘weak’ (renormalised) solution and $\tilde{z}$ a ‘strong’ solution of (erds)
in $(0,T^{*})\times\Omega$ for some $T^{*}\in(0,\infty]$.

First, letting
$\operatorname{dist}^{*}_{\alpha}(z,\tilde{z})=h_{\mathrm{rel}}^{*}(z,\tilde{z})+\tfrac{\alpha}{2}|u-\tilde{u}|^{2},$
where

$\displaystyle
h_{\mathrm{rel}}^{*}(z,\tilde{z})=h(z)-D_{i}h(\tilde{z})(\xi^{*}(z)z_{i}-\tilde{z}_{i})-h(\tilde{z}),$
(1.12)

we can write
$\operatorname{Dist}^{*}_{\alpha}(z,\tilde{z})=\int_{\Omega}\operatorname{dist}^{*}_{\alpha}(z,\tilde{z})\,\differential
x$. We further recall that the function $\xi^{*}=\xi^{*}(z)$ will be chosen
such that $\xi^{*}(z)=1$ if $\sum_{i=0}^{n}z_{i}\leq E$ for an auxiliary
parameter $E\gg\sum_{i=0}^{n}\tilde{z}_{i}$. Then, if
$E=E(\tilde{z},\min\\{\alpha,1\\})$ is chosen large enough,
$\operatorname{dist}^{*}_{\alpha}(z,\tilde{z})\geq 0$ for all
$z\in[0,\infty)^{1+n}$ with equality if and only if $z=\tilde{z}$.

To sketch the argument leading to (1.11), let us for simplicity only consider
the case where $A_{0j}(z)=a(z)\delta_{0j}$ with $a\gtrsim 1$. In this case, it
will suffice to take $\alpha\geq 1$. We now assume that $z$ and $\tilde{z}$
are sufficiently regular solutions of (erds) (with $A=\mathbb{M}D^{2}h$,
$\mathbb{M}\geq 0$, $D_{i}hR_{i}\leq 0$), where the strong solution
$\tilde{z}$ be such that
$\|\tilde{z}\|_{C^{0,1}([0,T]\times\bar{\Omega})}<\infty$ and
$\inf_{(0,T)\times\Omega}\tilde{z}_{i}>0$ for all $i\in\\{0,\dots,n\\}$ and
all $T<T^{*}$. To estimate the time evolution of
$\operatorname{Dist}^{*}_{\alpha}(z,\tilde{z})$, one formally computes

$\displaystyle\tfrac{\differential}{\differential
t}\operatorname{Dist}^{*}_{\alpha}(z,\tilde{z})=\int_{\Omega}\;\rho^{(h)}\;\differential
x+\alpha\int_{\Omega}\;\rho^{(g)}\;\differential x,$ (1.13)

where (see Lemma 3.4)

$\displaystyle\rho^{(h)}$ $\displaystyle:=-\nabla
D_{i}h(z)\cdot\mathbb{M}_{il}(z)\nabla D_{l}h(z)$ (1.14)
$\displaystyle\qquad+\nabla(D_{i}(\xi^{*}(z)z_{j})D_{j}h(\tilde{z}))\cdot\mathbb{M}_{il}(z)\nabla
D_{l}h(z)$ (1.15)
$\displaystyle\qquad+\nabla\Big{(}D_{ij}h(\tilde{z})(\xi^{*}(z)z_{j}-\tilde{z}_{j})\Big{)}\cdot\mathbb{M}_{il}(\tilde{z})\nabla
D_{l}h(\tilde{z})$ (1.16) $\displaystyle\qquad-
D_{ij}h(\tilde{z})(\xi^{*}(z)z_{j}-{\tilde{z}}_{j})R_{i}(\tilde{z})$ (1.17)
$\displaystyle\qquad+(D_{i}h(z)-D_{j}h(\tilde{z})D_{i}(\xi^{*}(z)z_{j}))R_{i}(z),$
(1.18)

and

$\displaystyle\rho^{(g)}$ $\displaystyle:=-a(z)|\nabla
u|^{2}-a(\tilde{z})|\nabla\tilde{u}|^{2}+a(z)\nabla
u\cdot\nabla\tilde{u}+a(\tilde{z})\nabla u\cdot\nabla\tilde{u}$ (1.19)
$\displaystyle=-a(z)|\nabla u-\nabla\tilde{u}|^{2}-(a(z)-a(\tilde{z}))(\nabla
u-\nabla\tilde{u})\cdot\nabla\tilde{u}.$ (1.20)

Thus, to show (1.11) it suffices to obtain a pointwise upper bound of the form

$\displaystyle\rho_{\alpha}:=\rho^{(h)}+\alpha\rho^{(g)}\lesssim\operatorname{dist}^{*}_{\alpha}(z,\tilde{z}).$
(1.21)

This pointwise estimate will be proved by distinguishing four cases determined
by the value of the weak solution $z=z(t,x)\in[0,\infty)^{1+n}$ at any given
point $(t,x)$. This case distinction is motivated by the following
observations:

First, if $z\in[0,\infty)^{1+n}$ with $\sum_{i=0}^{n}z_{i}\leq E$ for
$E=E(\tilde{z})$ large enough, we want $h_{\mathrm{rel}}^{*}(z,\tilde{z})$ to
coincide with the classical relative entropy density
$h_{\mathrm{rel}}(z,\tilde{z})=h(z)-D_{i}h(\tilde{z})(z_{i}-\tilde{z}_{i})-h(\tilde{z})$
to be able to use its distance-like properties. This will be ensured by
choosing $\xi^{*}(z)=1$ with $D^{k}\xi^{*}(z)=0$ for all $k\in\mathbb{N}^{+}$
whenever $\sum_{i=0}^{n}z_{i}\leq E$ (cf. the definition of $\xi^{*}$ in Sec.
3.1). Thus, if $|z-\tilde{z}|$ is close to zero, the strict convexity, non-
negativity and vanishing in $z=\tilde{z}$ of
$\operatorname{dist}^{*}_{\alpha}(\cdot,\tilde{z})$ imply that
$\operatorname{dist}^{*}_{\alpha}(z,\tilde{z})\sim_{\|\tilde{z}\|_{L^{\infty}},E}|z-\tilde{z}|^{2}$
for $|z|\leq E$. In this case, to show that $\rho_{\alpha}$ is quadratically
small in $|z-\tilde{z}|$, we write (using $\sum_{i}z_{i}\leq E$)

$\displaystyle\rho^{(h)}$
$\displaystyle=-\nabla(D_{i}h(z)-D_{i}h(\tilde{z}))\cdot\mathbb{M}_{il}(z)\nabla(D_{l}h(z)-D_{l}h(\tilde{z}))$
(1.22)
$\displaystyle\qquad-\nabla(D_{i}h(z)-D_{i}h(\tilde{z}))\cdot(\mathbb{M}_{il}(z)-\mathbb{M}_{il}(\tilde{z}))\nabla
D_{l}h(\tilde{z})$ (1.23)
$\displaystyle\qquad-\nabla\big{(}D_{i}h(z)-D_{i}h(\tilde{z})-D_{ij}h(\tilde{z})(z_{j}-\tilde{z}_{j})\big{)}\cdot\mathbb{M}_{il}(\tilde{z})\nabla
D_{l}h(\tilde{z})$ (1.24)
$\displaystyle\qquad+\big{(}D_{i}h(z)-D_{i}h(\tilde{z})-D_{ij}h(\tilde{z})(z_{j}-{\tilde{z}}_{j})\big{)}R_{i}(\tilde{z})$
(1.25)
$\displaystyle\qquad+(D_{i}h(z)-D_{i}h(\tilde{z}))(R_{i}(z)-R_{i}(\tilde{z})),$
(1.26)

see case $\mathcal{A}_{+}$ in the proof of Theorem 2.9 for details. In order
to deal with the terms involving a gradient of $z$ that appear in the second
and the third term on the RHS, one would like to exploit the non-positive
first term on the RHS. Typically (such as in the ERDS models considered in
[28]), the submatrix $(\mathbb{M}_{il}(z))_{i,l=1,\dots,n}$ will, however,
degenerate as soon as $c_{i}\searrow 0$ for some $i\in\\{1,\dots,n\\}$. Yet if
$\min\\{z_{1},\dots,z_{n}\\}\geq\iota$ for some $\iota>0$, then it is possible
to assume that
$(\mathbb{M}_{il}(z))_{i,l=1,\dots,n}\gtrsim_{\iota}\mathbb{I}_{n}$. This,
combined with the second line in (1.20) and suitable smoothness assumptions on
$\mathbb{M}$ and $h$, will allow us to infer that

$\displaystyle\rho_{\alpha}\lesssim_{E,\iota,\tilde{z}}|z-\tilde{z}|^{2}$
(1.27)

whenever
$z(t,x)\in\mathcal{A}_{+}:=\\{z^{\prime}\in[0,\infty)^{1+n}:\min\\{z_{0}^{\prime},\dots,z_{n}^{\prime}\\}\geq\iota,\;\sum_{i=0}^{n}z_{i}^{\prime}\leq
E\\}$ for some $\iota>0$ and sufficiently large $E\geq 1$.

To deal with the case
$z(t,x)\in\mathcal{A}_{0}:=\\{z^{\prime}:\min\\{z_{0}^{\prime},\dots,z_{n}^{\prime}\\}<\iota,\;\sum_{i=0}^{n}z_{i}^{\prime}\leq
E\\}$, we fix $\iota=\iota(\tilde{z})>0$ small enough such that
$\inf\tilde{z}_{i}\geq 2\iota$ for all $i=0,\dots,n$. This implies that
$|z-\tilde{z}|\geq\iota$ whenever $z\in\mathcal{A}_{0}$. Thus, since
$|z-\tilde{z}|\gtrsim 1$ is bounded away from zero, so is
$\operatorname{dist}^{*}_{\alpha}(z,\tilde{z})$ (see Prop. 3.2). It then
suffices to have suitable coercivity estimates on $\mathscr{P}(z)$ that allow
to absorb those terms on the RHS of (1.14) that involve gradients of $z$ and
do not have a sign by the first term on the RHS, which equals
$-\mathscr{P}(z)$. (Such coercivity estimates are typically already needed in
the construction of solutions.)

It remains to consider the case $\sum_{i}z_{i}>E$, where $E$ will be chosen
large enough, in particular such that
$E\geq\|\sum_{i}\tilde{z}_{i}\|_{L^{\infty}((0,T)\times\Omega)}+1$ and $E\geq
E_{0}$ with $E_{0}$ being such that
$\operatorname{dist}^{*}_{\alpha}(z,\tilde{z})\gtrsim\sum_{i}{c_{i}}\log_{+}(c_{i})+u^{2}+1$
for all $\tilde{E}\geq E_{0}$. If $z\not\in\operatorname{supp}\xi^{*}$,
$\rho^{(h)}$ takes a simple form. The entropy dissipating property of
diffusion and reactions, and the Lipschitz regularity of the strong solution
$\tilde{z}$ are sufficient to deduce (1.21) in this case (referred to as
$z\in\mathcal{C}$).

The intermediate case (below referred to as case $z\in\mathcal{B}$), where
$0<\xi^{*}<1$, is more delicate as can be seen in formula (1.14), where the
second term on the RHS involves summands that are quadratic in the gradient of
the renormalised solution. Here, we take advantage of an idea by Fischer [27].
In order to be able to absorb this bad term by the first term on the RHS of
(1.14), another scale $E^{\prime}\gg E$ is introduced (for convenience we
choose $E^{\prime}=E^{N}$, as in [27]), and $\xi^{*}$ will be taken such that
$\xi^{*}(z)=0$ if and only if $\sum_{i=0}^{n}z_{i}\geq E^{\prime}$,
$\xi^{*}(z)=1$ if and only if $\sum_{i=0}^{n}z_{i}\leq E$. On these scales,
$\xi^{*}$ can be chosen in such a way that derivatives of $\xi^{*}$ have an
additional decay property enabling the desired absorption if
$\tfrac{E^{\prime}}{E}$ is large enough. Finding $\xi^{*}$ such that
absorption is possible is non-trivial and relies on a logarithmic gain.

###  1.4. Outline

The rest of the article is structured as follows. In Section 2 we introduce
relevant definitions and hypotheses, and formulate our main results: a weak-
strong uniqueness principle for dissipative renormalised solutions to (erds)
(see Thm 2.9), a strong entropy dissipation inequality as used in the proof of
Theorem 2.9 (see Prop. 2.10), and a result on the exponential convergence to
equilibrium (see Prop. 2.11). In Section 2.3 we present selected examples that
our main results apply to, including the class of ERDS considered in [28] as
well as a class of models with cross diffusion between species.

The weak-strong uniqueness principle is proved in Section 3, starting with
several auxiliary results with the actual proof of Theorem 2.9 being given in
Section 3.4. In Sections 4 and 5 respectively, we establish the entropy
dissipation inequality (ed) and the exponential convergence to equilibrium for
a specific ERDS, below referred to as Model (M0). Some auxiliary results are
gathered in Appendix 6.1. In Appendix 6.2 we explain how to derive inequality
(ed) for the renormalised solutions constructed in [28] for a model with non-
integrable diffusive flux.

###  1.5. Notations

  * •

_Summation convention_ : any unspecified summations of the form $\sum_{i}$ are
to be understood as $\sum_{i=0}^{n}$. For brevity, we use a summation
convention for summing over the system’s components $i=0,\dots,n$ in case of
repeatedly occurring indices while omitting the summation symbol. In ambiguous
situations the summation symbol will be used. Summations restricted to
$i=1,\dots,n$ (excluding the $u$-component) will always be made explicit. In
our convention, summation over repeated indices has priority over other
mathematical operations such as integration or taking the absolute value. For
instance, by default we let $|A_{ik}(z)\nabla
z_{k}|=|\sum_{k=0}^{n}A_{ik}(z)\nabla z_{k}|$.

  * •

For technical concerns regarding the notation $\mathbb{M}_{il}(z)\nabla
D_{l}h(z)$, we refer to Remark 2.3.

  * •

We denote by $R=(0,R_{1},\dots,R_{n})^{T}$ the vectors of reaction rates.

  * •

Given $T^{*}\in(0,\infty]$, we let $I=[0,T^{*})$ denote the time horizon of
interest. For $T>0$, we abbreviate $\Omega_{T}:=(0,T)\times\Omega$.

  * •

For functions $f=f(z_{0},\dots,z_{n})$ we let $D_{i}f=\tfrac{\partial
f}{\partial z_{i}}$ and $D_{ij}f=\tfrac{\partial^{2}f}{\partial z_{i}\partial
z_{j}}$ for $i,j\in\\{0,\dots,n\\}$.

  * •

In estimates, $C<\infty$ typically denotes a finite (sufficiently large)
constant that may change from line to line, while we often use $\epsilon>0$ to
denote a (sufficiently small) positive constant.

  * •

For quantities $A,B\geq 0$ we write $A\lesssim B$ if there exists a fixed
constant $C<\infty$ such that $A\leq CB$. The notation $A\gtrsim B$ means
$B\lesssim A$, while $A\sim B$ is to be understood as both $A\lesssim B$ and
$A\gtrsim B$ being satisfied. In order to indicate dependencies of the
constant $C=C(p_{1},\dots,p_{k})$ on certain parameters $p_{1},\dots,p_{k}$,
we write $A\lesssim_{p_{1},\dots,p_{k}}B$, and analogously for $\gtrsim$ and
$\sim$.

  * •

Any dependence of constants and estimates on the regular solution
$\tilde{z}\in C^{0,1}$ will usually not be explicitly indicated.

  * •

We let $\min(z):=\min\\{z_{0},z_{1},\dots,z_{n}\\}$ for
$z=(z_{0},\dots,z_{n})\in[0,\infty)^{1+n}$.

  * •

By default, $|\cdot|$ denotes the Euclidean norm, e.g.
$|z|=(\sum_{i}|z_{i}|^{2})^{\frac{1}{2}}$

  * •

$|z|_{1}=\sum_{i=0}^{n}z_{i}$ and $|c|_{1}=\sum_{i=1}^{n}c_{i}$.

  * •

For time-dependent integral functionals
$\int_{\Omega}f((t,x),z(t,x))\,\differential x$, where $z=z(t,x)$ denotes a
‘weak’ solution of (erds) taking in a suitable sense the data
$z^{\mathrm{in}}$, we use the convention

$\evaluated{\int_{\Omega}f((t,x),z(t,x))\,\differential
x}_{t=0}^{t=T}:=\int_{\Omega}f((T,x),z(T,x))\,\differential
x-\int_{\Omega}f((0,x),z^{\mathrm{in}}(x))\,\differential x$

provided the terms on the RHS are well-defined.

  * •

For an open set $U\subset\mathbb{R}^{N}$, $C^{k}(U)$ denotes the space of
continuous functions on $U$ that are continuously differentiable up to order
$k\in\mathbb{N}$. By $C^{k,\nu}(U)=C^{k,\nu}_{\mathrm{loc}}(U)$, we denote the
space of functions in $C^{k}(U)$, whose $k$-th derivative is $\nu$-Hölder
continuous for some $\nu\in(0,1]$ on compact subsets $K\subseteq U$. (We use
the symbol $C^{k,\nu}_{\mathrm{loc}}(U)$ for clarity’s sake.)

  * •

For $\Omega$ bounded, we let $L\log L(\Omega):=\\{f\in L^{1}(\Omega):f\geq
0\text{ a.e.\ and }\int_{\Omega}f\log f\,\differential x<\infty\\}$.

  * •

The abbreviation ‘hp.’ stands for hypothesis.

##  2\. Main results

###  2.1. Assumptions

Throughout these notes, we let $d\geq 1$ and $\Omega\subset\mathbb{R}^{d}$ be
a bounded Lipschitz domain with $|\Omega|=1$. We further let
$T^{*}\in(0,\infty]$ and $I=[0,T^{*})$.

To prepare for stating our main result, the weak-strong uniqueness principle
(Theorem 2.9), we gather the following conditions.

  1. (A$1$)

Entropy: $h\in C^{4}((0,\infty)^{1+n})$ is of the form (h1), where
$\hat{\sigma}$ and $w_{i}$ are supposed to satisfy (h2).

  1. (A$2$)

Reactions: $R=(0,R_{1},\dots,R_{n})\in C([0,\infty)^{1+n})^{1+n}$

    1. (i)

satisfy $\sum_{i=1}^{n}D_{i}h(z)R_{i}(z)\leq 0$ in $(0,\infty)^{1+n}$

    2. (ii)

are locally Lipschitz continuous in $(0,\infty)^{1+n}$.

  1. (A$3$)

Mobility matrix:
$\mathbb{M}\in\big{(}C^{0,1}_{\mathrm{loc}}((0,\infty)^{1+n})\cap
C([0,\infty)^{1+n})\big{)}^{(1+n)\times(1+n)}$ and there exist non-negative
functions $m,a\in C^{0,1}_{\mathrm{loc}}((0,\infty)^{1+n})$ and
$\varpi\in\\{0,1\\}$ with $0\leq m\lesssim\varpi$ and $a\gtrsim 1$ such that

$\displaystyle\mathbb{M}_{0l}=\mathbb{\widetilde{M}}_{0l}+\delta_{0l}m\quad\text{
for }l=0,\dots,n,$ (A3.a)

for suitable $\mathbb{\widetilde{M}}_{0l}$ satisfying
$\sum_{l=0}^{n}\mathbb{\widetilde{M}}_{0l}D_{lj}h=\delta_{0j}a.$

Moreover, for all $z\in[0,\infty)^{1+n}$ with $\min_{i}z_{i}\geq\iota$ for
some $\iota>0$ there exists $\epsilon(\iota)>0$ such that

$\displaystyle\mathbb{M}(z)\geq\operatorname{diag}(m(z),0,\dots,0)+\epsilon(\iota)\operatorname{diag}(0,1,\dots,1).$
(A3.b)

By continuity, when $\iota=0$, (A3.b) holds true with $\epsilon(\iota)=0$.

Using our standard notation $A(z)=\mathbb{M}(z)D^{2}h(z)$, hp. (A3.a) implies
that, formally,

$\displaystyle\sum_{j=0}^{n}A_{0j}(z)\nabla
z_{j}=\sum_{l=0}^{n}\mathbb{M}_{0l}(z)\nabla D_{l}h(z)=a(z)\nabla u+m(z)\nabla
D_{0}h(z).$ (A3.c)

We further need certain bounds on the flux and the concentration gradients in
terms of the entropy dissipation. For this purpose we define for non-negative
functions $z_{j}\in L^{1}_{\mathrm{loc}}(I,L^{1}(\Omega))$ such that
$\nabla(z_{j}^{s})\in L^{2}_{\mathrm{loc}}(I;L^{2}(\Omega))$ for some
$s\in\\{\tfrac{1}{2},1\\}$ for each $j\in\\{0,\dots,n\\}$, the quantity

$\displaystyle\mathscr{P}(z):$ $\displaystyle=\nabla z:(D^{2}h(z)A(z)\nabla
z)$ (2.1) $\displaystyle=\nabla D_{i}h(z)\cdot(\mathbb{M}_{il}(z)\nabla
D_{l}h(z)),$ (2.2)

where the second equality is to be understood in a formal sense, see Remark
2.3. By the positive semi-definiteness of $\mathbb{M}$ imposed by hp. (A3.b),
we have $\mathscr{P}(z)\geq 0$ for any such $z$, and more specifically,
$\mathscr{P}(z)\geq m(z)|\nabla D_{0}h(z)|^{2}$.

  1. (A$4$)

For all $K\geq 1$

$\displaystyle\chi_{\\{|z|\leq K\\}}|\nabla c|$
$\displaystyle\lesssim_{K}\sqrt{\mathscr{P}(z)},$ (A4.a)
$\displaystyle\chi_{\\{|z|\leq K\\}}|\sum_{j=0}^{n}A_{ij}(z)\nabla z_{j}|$
$\displaystyle\lesssim_{K}\sqrt{\mathscr{P}(z)}$ (A4.b)

for all $i=0,\dots,n$.

  2. (A$5$)

For $a(z)$ as in (A$3$),

$\displaystyle|a(z)\nabla u|\lesssim(1+u)\sqrt{\mathscr{P}(z)}.$ (2.3)

Additionally, we often impose the following bound:

  1. (A$6$)

For all $0\leq i\leq n$

$\chi_{\\{|z|_{1}\geq 1\\}}|\nabla z||\sum_{j=0}^{n}A_{ij}(z)\nabla
z_{j}|\lesssim|z|\mathscr{P}(z)+|z||\nabla u|^{2}.$

If (A$6$) is not satisfied, we have to assume that $\varpi=0$ in (A$3$)
together with the condition:

  1. (A$6$’)

For all $0\leq i\leq n$ and all $\underline{u}\in(0,1]$

$\chi_{\\{u\geq\underline{u}\\}}\chi_{\\{|z|_{1}\geq 1\\}}|\nabla
z||\sum_{j=0}^{n}A_{ij}(z)\nabla
z_{j}|\lesssim|z|\mathscr{P}(z)+C(|z|,\underline{u})|\nabla u|^{2}.$

Let us observe that (A$6$)$\implies$(A$6$’).

A selection of relevant examples fulfilling the above hypotheses is provided
in Section 2.3.

###  2.2. Definitions and Results

Throughout this text, we write $A(z):=\mathbb{M}(z)D^{2}h(z)$, where $h$ takes
the form (h1), (h2). We further recall our summation convention (see Notations
1.5).

######  Definition 2.1 (Renormalised solution).

Let $I=[0,T^{*})$ and suppose that the vector-valued function
$z=(u,c_{1},\dots,c_{n})$ has non-negative components $z_{i}\geq 0$ satisfying
$\sqrt{z_{i}}\in L^{2}_{\mathrm{loc}}(I;H^{1}(\Omega))$ or $z_{i}\in
L^{2}_{\mathrm{loc}}(I;H^{1}(\Omega))$ for all $i=0,\dots,n$. Further suppose
that for all $E\geq 1$

$\chi_{\\{|z|\leq E\\}}A_{ik}(z)\nabla z_{k}\in
L^{2}_{\mathrm{loc}}(I;L^{2}(\Omega)),$ (2.4)

for every $i\in\\{0,\dots,n\\}$.

We call such $z$ a _renormalised solution_ of the energy-reaction-diffusion
system (erds) in $\Omega_{T^{*}}:=(0,T^{*})\times\Omega$ with initial data
$z^{\mathrm{in}}$ if for all $\xi\in C^{\infty}(\mathbb{R}_{\geq 0}^{1+n})$
with compactly supported derivative $D\xi$, all $\psi\in
C^{\infty}(I\times\bar{\Omega})$ and almost all $T\in(0,T^{*})$

$\begin{split}&\int_{\Omega}\xi(z(T,\cdot))\psi(T,\cdot)\,\differential
x-\int_{\Omega}\xi(z^{\mathrm{in}})\psi(0,\cdot)\,\differential
x-\int_{0}^{T}\\!\\!\int_{\Omega}\xi(z)\partial_{t}\psi\,\differential
x\differential t\\\
&\qquad=-\int_{0}^{T}\\!\\!\int_{\Omega}D_{ij}\xi(z)A_{ik}(z)\nabla
z_{k}\cdot\nabla z_{j}\psi\,\differential x\differential t\\\
&\qquad\quad-\int_{0}^{T}\\!\\!\int_{\Omega}D_{i}\xi(z)A_{ik}(z)\nabla
z_{k}\cdot\nabla\psi\,\differential x\differential
t+\int_{0}^{T}\\!\\!\int_{\Omega}D_{i}\xi(z)R_{i}(z)\psi\,\differential
x\differential t.\end{split}$ (2.5)

The renormalised formulation (2.5) and the required functional setting alone
are in general too weak to deduce the weak-strong uniqueness principle. In
Definition 2.6 below, we introduce a more satisfactory generalised solution
concept that strengthens Definition 2.1, but is still general enough to be
consistent with the existence results in [28].

######  Remark 2.2.

By approximation, given a renormalised solution $z$, the equality (2.5) can be
seen to hold true for a larger set of test functions $\psi\in
C(I\times\bar{\Omega})$ with $\partial_{t}\psi\in
L^{1}_{\mathrm{loc}}(I;L^{1}(\Omega))$, $\nabla\psi\in
L^{2}_{\mathrm{loc}}(I;L^{2}(\Omega))$, and for truncation functions $\xi\in
C^{2}(\mathbb{R}_{\geq 0}^{1+n})$ with $\operatorname{supp}D\xi$ compact.

######  Remark 2.3 (Notation).

Let $z$ denote a renormalised solution of (erds) in the sense of Def. 2.1. To
keep notation simple and better emphasise the entropy structure of the
diffusive part, we will often use a ‘symbolic’ notation writing
$\mathbb{M}_{il}(z)\nabla D_{l}h(z)$ instead of $A_{ik}(z)\nabla z_{k}$, where
as before the summation convention is used. Likewise, we write $\nabla
D_{i}h(z)$ instead of the more precise notation $D_{ij}h(z)\nabla z_{j}$. The
point here is that while, by hypothesis, the weak derivatives $\nabla
z_{j},j=0,\dots,n,$ are well-defined (in the standard Sobolev/distributional
sense), the function $D_{i}h(z)$ may not be weakly differentiable.

######  Remark 2.4.

The following equation, equivalent to (2.5), can be obtained by ‘reversing the
product rule’ for $\nabla$

$\begin{split}&\int_{\Omega}\xi(z(T,\cdot))\psi(T,\cdot)\,\differential
x-\int_{\Omega}\xi(z^{\mathrm{in}})\psi(0,\cdot)\,\differential
x-\int_{0}^{T}\\!\\!\int_{\Omega}\xi(z)\partial_{t}\psi\,\differential
x\differential t\\\
&\quad=-\int_{0}^{T}\\!\\!\int_{\Omega}\nabla\big{(}D_{i}\xi(z)\psi\big{)}\cdot\mathbb{M}_{il}(z)\nabla
D_{l}(z)\,\differential x\differential
t+\int_{0}^{T}\\!\\!\int_{\Omega}D_{i}\xi(z)R_{i}(z)\psi\,\differential
x\differential t,\end{split}$ (2.6)

where we recall our convention $\mathbb{M}_{il}(z)\nabla
D_{l}h(z):=A_{ik}(z)\nabla z_{k}$, see Remark 2.3.

Notice that thanks to the hypothesis of $D\xi$ being compactly supported, no
growth restrictions have to be imposed on the reaction term $R(z)$ in (2.5),
and none of the flux terms $A_{ik}(z)\nabla z_{k}$ is necessarily required to
be integrable in order for the integrals in (2.5) to converge. At the same
time, this restrictive condition on the set of admissible truncation functions
$\xi$ means that recovering a separate (e.g. weak) formulation of a single
component $i_{0}\in\\{0,\dots,n\\}$ of the system (assuming integrability of
$A_{i_{0}k}(z)\nabla z_{k}$ and $R_{i_{0}}(z)$) is not immediate unless all
components of the flux and the reactions are known to be integrable. Thus,
consistent with the existence result for (erds) in [28], the present analysis
additionally assumes a weak formulation for the energy component.

######  Definition 2.5 (Weak solution of energy equation).

Let $z=(u,c)$ be as in Def. 2.1. We say that $u$ is a _weak solution_ of the
energy component $\partial_{t}u=\operatorname{div}(A_{0j}(z)\nabla z_{j})$ in
$\Omega_{T^{*}}$ with no-flux boundary conditions and initial condition
$u^{\mathrm{in}}$ if $A_{0j}(z)\nabla z_{j}\in
L^{1}_{\mathrm{loc}}(I;L^{1}(\Omega))$ and if for all $\varphi\in
C^{1}(I\times\bar{\Omega})$ and almost all $T<T^{*}$

$\begin{split}\int_{\Omega}u(T,\cdot)\varphi(T,\cdot)\,\differential
x-\int_{\Omega}u^{\mathrm{in}}\varphi(0,\cdot)\,\differential
x&-\int_{0}^{T}\\!\\!\int_{\Omega}u\partial_{t}\varphi\,\differential
x\differential t\\\ =&-\int_{0}^{T}\\!\\!\int_{\Omega}A_{0j}(z)\nabla
z_{j}\cdot\nabla\varphi\,\differential x\differential t.\end{split}$ (2.7)

By carefully using lower semicontinuity-type properties of the entropy and
entropy dissipation, the existence proof of global-in-time weak and
renormalised solutions to ERDS as provided in [28] allows to show that for
almost all $T\in(0,\infty)$ the constructed solutions satisfy the entropy
dissipation inequality

$H(z(T))-H(z^{\mathrm{in}})\leq-\int_{0}^{T}\\!\\!\int_{\Omega}\mathscr{P}(z)\,\differential
x\differential
t+\int_{0}^{T}\\!\\!\int_{\Omega}R_{i}(z)D_{i}h(z)\,\differential
x\differential t,$ (ed)

cf. Section 6.2, where we recall the notation $\mathscr{P}(z):=\nabla
D_{i}h(z)\cdot\mathbb{M}_{il}(z)\nabla D_{l}h(z)\geq 0$. Observe that thanks
to the non-negativity of $\mathscr{P}(z)$ and $-R_{i}(z)D_{i}h(z)$ (to be
assumed throughout), any function $z=(u,c)$ with $u\in
L^{\infty}_{t}L^{1}_{x}$ and well-defined, measurable $\mathscr{P}(z)$ that
satisfies (ed) with $h(z^{\mathrm{in}})\in L^{1}(\Omega)$ necessarily has the
regularity $\mathscr{P}(z)\in L^{1}(\Omega_{T})$ and $R_{i}(z)D_{i}h(z)\in
L^{1}(\Omega_{T})$. (For the estimate on $H(z(T))$ required in this argument,
see (6.7) in the appendix.)

We further note that the energy equation
$\partial_{t}u=\operatorname{div}(A_{0j}(z)\nabla z_{j})$ is satisfied in the
weak sense by the solutions constructed in ref. [28], that $\nabla u\in
L^{2}(\Omega_{T})$, and that the quantity
$G(u)=\tfrac{1}{2}\|u\|_{L^{2}(\Omega)}^{2}$ satisfies (with an equality)

$G(u(T))-G(u^{\mathrm{in}})\leq-\int_{0}^{T}\\!\\!\int_{\Omega}a(z)|\nabla
u|^{2}\,\differential x\differential
t-\int_{0}^{T}\\!\\!\int_{\Omega}m(z)\nabla D_{0}h(z)\cdot\nabla
u\,\differential x\differential t$ (ene)

for almost all $T\in(0,\infty)$. (In [28], the case $m\equiv 0$ was
considered.)

The weak-strong uniqueness principle we will establish in our main theorem
(Theorem 2.9 below) is valid for renormalised solutions that are _dissipative_
in the sense that they conform to (ed) and (ene) (and satisfy some extra
hypotheses related to the energy component).

######  Definition 2.6 (Dissipative renormalised solution).

We call $z=(u,c)$ a _dissipative renormalised solution_ of system (erds) in
$\Omega_{T^{*}}$ with initial data $z^{\mathrm{in}}$ if it is a renormalised
solution with data $z^{\mathrm{in}}$ in the sense of Definition 2.1 that
fulfils the energy equation in the weak sense of Definition 2.5 with
$\int_{0}^{T}\\!\\!\int_{\Omega}a(z)|\nabla u|^{2}\,\differential
x\differential t<\infty$ for all $T\in(0,T^{*})$ and further obeys the
inequalities (ed) and (ene) for almost all $T\in(0,T^{*})$.

Let us stress that the renormalised solutions in [28, Theorem 1.8] are in fact
dissipative renormalised solutions in the sense of the above definition.222In
Section 6.2 we explain how to deduce (ed) along the construction in [28,
Theorem 1.8]. Thus, dissipative renormalised solutions provide a natural
framework for studying (erds). It is interesting to note that there are
parallels of the present setting to classical literature such as [4, 7] on
elliptic/parabolic equations, where a satisfactory solution concept is
obtained by additionally imposing certain entropy or regularity conditions
that are indispensable for the uniqueness proof. See also [20, 10] and
references therein as well as the classical literature on conservation laws
mentioned in the introduction.

Before stating our main theorem, we need to specify what we understand by a
‘strong’ solution.

######  Definition 2.7 (Weak solution).

Let $I=[0,T^{*})$. We call a function $z\in
L^{1}_{\mathrm{loc}}(I;W^{1,1}(\Omega))$, $z=(u,c_{1},\dots,c_{n})$ with
$z_{i}\geq 0$ for all $i$, a _weak solution_ of (erds) in $\Omega_{T^{*}}$
with initial data $z^{\mathrm{in}}$ if $A_{ij}(z)\nabla z_{j},R_{i}(z)\in
L^{1}_{\mathrm{loc}}(I;L^{1}(\Omega))$ for all $i$, and if for all $\psi\in
C^{\infty}(I\times\bar{\Omega})^{1+n}$ and almost all $T\in(0,T^{*})$

$\begin{split}\int_{\Omega}z_{i}(T,\cdot)\psi_{i}(T,\cdot)\,\differential
x&-\int_{\Omega}z^{\mathrm{in}}_{i}\psi_{i}(0,\cdot)\,\differential
x-\int_{0}^{T}\\!\\!\int_{\Omega}z_{i}\partial_{t}\psi_{i}\,\differential
x\differential t\\\ &=-\int_{0}^{T}\\!\\!\int_{\Omega}A_{ij}(z)\nabla
z_{j}\cdot\nabla\psi_{i}\,\differential x\differential
t+\int_{0}^{T}\\!\\!\int_{\Omega}R_{i}(z)\psi_{i}\,\differential
x\differential t.\end{split}$ (2.8)

######  Definition 2.8 (‘Strong’ solution).

Let $I=[0,T^{*})$. We call $z=z(t,x)$ a _strong solution_ of system (erds) in
$\Omega_{T^{*}}$ with data $z^{\mathrm{in}}$ if it is a weak solution in the
sense of Definition 2.7, has the regularity $z\in
C^{0,1}_{\mathrm{loc}}(I\times\bar{\Omega})$ and satisfies
$\inf_{\Omega_{T}}z_{i}>0$ for $i=0,\dots,n$ and every $T\in(0,T^{*})$.

For strictly positive, Lipschitz continuous initial data, Amann’s work [2]
guarantees the local-in-time existence of a strong solution for the
applications considered in the present manuscript (see Section 2.3).

######  Theorem 2.9 (Weak-strong uniqueness).

Let $T^{*}\in(0,\infty]$. Assume hp. (A$1$)–(A$5$). Further suppose that
(A$6$) holds true, or alternatively that $\varpi=0$ and that (A$6$’) is
fulfilled. Let $z^{\mathrm{in}}=(u^{\mathrm{in}},c^{\mathrm{in}})\in
L^{1}(\Omega)^{1+n}$, $z^{\mathrm{in}}_{i}\geq 0$ for all $i$, and
$h(z^{\mathrm{in}})\in L^{1}(\Omega),\;u^{\mathrm{in}}\in L^{2}(\Omega)$. Let
$z=(u,c)$ be a dissipative renormalised solution of system (erds) in
$\Omega_{T^{*}}$ in the sense of Definition 2.6 taking the initial data
$z^{\mathrm{in}}$. If $\tilde{z}$ is a strong solution in $\Omega_{T^{*}}$ in
the sense of Definition 2.8 taking the same initial data $z^{\mathrm{in}}$,
then $z=\tilde{z}$ a.e. in $\Omega_{T^{*}}$.

The proof of Theorem 2.9 will be completed in Section 3.4. In Section 2.3 we
provide a list of examples covered by this theorem, including a class of
(isoenergetic) cross-diffusion systems.

As mentioned in the introduction, the proof of Theorem 2.9 is based on a weak-
strong stability type estimate with a generalised distance involving a
modified relative entropy and an $L^{2}$-part for the energy component (cf.
Sec. 1.3). The entropy inequality (ed) that dissipative renormalised solutions
conform to is a fundamental ingredient in the proof. As we will see in Section
6.2, under suitable hypotheses on the data, the solutions constructed in ref.
[28] enjoy this estimate. Note that similar situations are encountered in the
context of weak-strong uniqueness in fluid dynamics problems, see e.g. [49,
46] and references therein. We would, however, like to point out that for many
of the models we are interested in, it is possible to derive the entropy
dissipation inequality (ed) from the renormalised formulation if suitable
regularity conditions are met (the integrability of $\mathscr{P}(z)$ and
$a(z)|\nabla u|^{2}$ are sufficient). In order to illustrate the ideas, we
provide a proof of inequality (ed) for one of the models in the ref. [28]
considering ERDS of the form

$\displaystyle\dot{u}$
$\displaystyle=\operatorname{div}\\!\big{(}\,a(u,c)\nabla u+m(z)\nabla
D_{0}h(z)\big{)},$ (2.9a) $\displaystyle\dot{c}_{i}$
$\displaystyle=\operatorname{div}\Big{(}m_{i}(u,c)\nabla\log(\tfrac{c_{i}}{w_{i}(u)})+a(u,c)c_{i}\tfrac{w^{\prime}_{i}(u)}{w_{i}(u)}\nabla
u\bigg{)}+R_{i}(u,c),$ (2.9b)

where, as before, $h(u,c)$ satisfies (h1), (h2), and where
$a(z):=\pi_{1}(z)\gamma(z)$, with

$\displaystyle\gamma(u,c):=\big{(}\partial^{2}_{u}h-\sum_{i=1}^{n}c_{i}\big{(}\tfrac{w^{\prime}_{i}}{w_{i}}\big{)}^{2}\big{)}=-\hat{\sigma}^{\prime\prime}(u)-\sum_{i=1}^{n}c_{i}\tfrac{w_{i}^{\prime\prime}(u)}{w_{i}(u)}>0.$
(2.10)

System (2.9) is obtained as a special case of (erds) by choosing

$\mathbb{M}(z):=\operatorname{diag}\big{(}m,m_{1},\dots,m_{n}\big{)}+\pi_{1}\mu\otimes\mu$
(2.11)

with non-negative functions $m,m_{i},\pi_{1}\in C([0,\infty)^{1+n})$ to be
specified below, and $\mu=(1,\mu_{1},\dots,\mu_{n})$ determined by

$\displaystyle\mu_{i}(u,c):=\frac{w_{i}^{\prime}(u)}{w_{i}(u)}c_{i}\qquad\qquad\text{
for }i\in\\{1,\dots,n\\}.$ (2.12)

Throughout, the bound

$\displaystyle 0\leq\sqrt{\pi_{1}(u,c)}\lesssim 1+u$ (2.13)

and the mild regularity condition $\sqrt{\pi_{1}(u,c)}w_{i}^{\prime}(u)\in
C([0,\infty)^{1+n}),i=1,\dots,n,$ will be imposed, the latter ensuring the
continuity of $\mathbb{M}$ in $[0,\infty)^{1+n}$.

The functions $m_{i}(u,c)$, $i\in\\{1,\dots,n\\}$, are assumed to take the
following form for certain $a_{i}\in C([0,\infty)^{1+n})$ and
$\kappa_{0,i},\kappa_{1,i}\geq 0:$

$\begin{split}&m_{i}(u,c)=c_{i}a_{i}(u,c),\qquad\quad\text{where}\quad\;a_{i}(u,c)\sim\kappa_{0,i}+\kappa_{1,i}c_{i}.\end{split}$
(2.14)

Hypotheses of Model (M0). Model (M0) consists of the following conditions:

  * •

Entropy density $h$ is given by (h1), (h2) with coefficient functions
satisfying

$\displaystyle(w_{i}^{\prime})^{2}\lesssim-w_{i}^{\prime\prime}w_{i}$ (2.15)

for all $i\in\\{1,\dots,n\\}$.

  * •

Reactions $R_{i}\in C([0,\infty)^{1+n})$, $i=1,\dots,n$, satisfy
A$2$(it:react.i), where $R_{0}\equiv 0$.

  * •

$\mathbb{M}$ is given by (2.11)–(2.14) with

    * $\circ$

rank-one part: $\pi_{1}(z)\sim\tfrac{1}{\gamma(z)}$, where $\gamma$ is given
by (2.10).

    * $\circ$

diagonal part: $0\leq m(z)\lesssim\varpi$ for some $\varpi\in\\{0,1\\}$, and
$m_{i}(u,c)$ given by (2.14) with $\kappa_{0,i}=1,\kappa_{1,i}=0$ for all
$i\in\\{1,\dots,n\\}$.

We recall that the evolution law (erds) associated with Model (M0) takes the
form (2.9a), (2.9b). (Cf. Lemma 6.1 and [28] for details.) It is further easy
to see that condition (2.13) is compatible with the choice
$\pi_{1}(z)\sim\tfrac{1}{\gamma(z)}$ for any power law
$\hat{\sigma}(u)=u^{\nu}$, $\nu\in(0,1)$ and $\hat{\sigma}(u)=\log(u)$.

Model (M0) generalises the special case
$\mathbb{M}(u,c)=\text{const}\cdot(D^{2}h(u,c))^{-1}$ considered in [32]. It
allows for species-dependent diffusivities, and genuinely contains cross terms
in this case. More precisely, for models with species-dependent diffusion
coefficients, thermodynamical consistency always leads to cross-diffusion
effects, since for a diagonal diffusion matrix
$A=\operatorname{diag}(\dots)\in\mathbb{R}^{(1+n)\times(1+n)}$ that is not a
multiple of the identity matrix the product $\mathbb{M}=A(D^{2}h)^{-1}$ is not
symmetric.

In the derivation of the entropy dissipation inequality (ed), we need to
assume that

$\displaystyle m(u,c)\sum_{l=1}^{n}c_{l}\lesssim(1+u)^{2},$ (2.16)

and have to impose the following conditions mainly serving to avoid issues for
small values of $u$ close to zero:

$\displaystyle\begin{cases}m(u,c)|\hat{\sigma}^{\prime\prime}(u)|\lesssim
1,\\\
-\hat{\sigma}^{\prime\prime}(v)\lesssim-\hat{\sigma}^{\prime\prime}(u)+1\quad\text{
for all }v\geq u.\end{cases}$ (2.17)

######  Proposition 2.10 (Strong entropy dissipation inequality).

Let $T^{*}\in(0,\infty]$. Let the hypotheses of Model (M0) hold, and assume
locally $\epsilon_{0}$-Hölder continuous reactions $R\in
C^{\epsilon_{0}}_{\mathrm{loc}}([0,\infty)^{1+n})$ for some
$\epsilon_{0}\in(0,1)$. In addition, assume the bounds (2.16), (2.17), and
suppose that $w_{i}\in C^{2}([0,\infty))$ for $i=1,\dots,n$. Let
$z^{\mathrm{in}}=(u^{\mathrm{in}},c^{\mathrm{in}})$, $z^{\mathrm{in}}_{i}\geq
0$, and $u^{\mathrm{in}},\hat{\sigma}_{-}(u^{\mathrm{in}})\in L^{1}(\Omega)$
and $c^{\mathrm{in}}_{i}\in L\log L(\Omega)$ for all $i$. Let $z=(u,c)$ have
non-negative components and suppose that

$\displaystyle u\in L^{\infty}_{\mathrm{loc}}([0,T^{*});L^{1}(\Omega)),$
(2.18) $\displaystyle\mathscr{P}(z)\in L^{1}(\Omega_{T}),\quad a(z)|\nabla
u|^{2}\in L^{1}(\Omega_{T})\quad\text{ for all }T<T^{*}.$ (2.19)

If $z$ is a renormalised solution (in the sense of Def. 2.1) of system (erds)
in $\Omega_{T^{*}}$ with initial data $z^{\mathrm{in}}$, then the strong
entropy dissipation inequality is satisfied, i.e.

$\displaystyle
H(z(t))-H(z(s))\leq-\int_{s}^{t}\\!\int_{\Omega}\mathscr{P}(z)\,\differential
x\differential\tau+\int_{s}^{t}\\!\int_{\Omega}R_{i}(z)D_{i}h(z)\,\differential
x\differential\tau$ (ed.s)

for a.e. $0\leq s<t<T^{*}$, and for $s=0$333With the understanding that
$H(z(0))=H(z^{\mathrm{in}})$. and a.e. $t\in(0,T^{*})$. In particular, ineq.
(ed) holds true for a.e. $T\in(0,T^{*})$.

For the proof of Proposition 2.10, see Section 4. Some comments on
generalisations of Proposition 2.10 to other models are provided in Section
2.3.

In our final main result, we illustrate that a version of the generalised
distance can further be used to prove exponential convergence to equilibrium.
Exponential convergence in relative entropy has been studied at a formal level
in [43] by means of log-Sobolev type inequalities leading to entropy-entropy
dissipation estimates (see also [32]). In contrast to the present approach,
[43] solely relies on the relative entropy, which restricts the results to
thermal parts $\sigma$ close to linear; for instance, the choice
$\sigma(u)\sim\log u$ related to gas dynamics is only admitted in dimensions
$d\leq 2$. For simplicity, in the following result we disregard the reaction
term and refer to [43] for applications on mass action-type reactions. We will
further assume the strong energy inequality i.e.

$\displaystyle G(u(t))-G(u(s))\leq-\int_{s}^{t}\\!\int_{\Omega}a(z)|\nabla
u|^{2}\,\differential
x\differential\tau-\int_{s}^{t}\\!\int_{\Omega}m(z)\nabla D_{0}h(z)\cdot\nabla
u\,\differential x\differential\tau\;\quad\qquad$ (ene.s)

for a.e. $0\leq s<t<T^{*}$, and for $s=0$ and a.e. $t\in(0,T^{*})$.

######  Proposition 2.11 (Exponential convergence to equilibrium).

Let $\Omega$ be smooth, recall that $|\Omega|=1$, and let the hypotheses of
Model (M0) hold. Let $z^{\mathrm{in}}=(u^{\mathrm{in}},c^{\mathrm{in}})$ have
non-negative components with $\hat{\sigma}_{-}(u^{\mathrm{in}})\in
L^{1}(\Omega)$, $u^{\mathrm{in}}\in L^{2}(\Omega)$, $c^{\mathrm{in}}_{i}\in
L\log L$. Let $z=(u,c)$ with $u\in
L^{\infty}_{\mathrm{loc}}([0,\infty),L^{1}(\Omega))$ and $\nabla u\in
L^{2}_{\mathrm{loc}}([0,\infty),L^{2}(\Omega))$ be a global-in-time
renormalised solution of (erds) with $R\equiv 0$, and suppose that (ed.s) and
(ene.s) are satisfied (with $T^{*}=\infty$).

Then $\int_{\Omega}z_{i}(t,x)\,\differential
x=\int_{\Omega}z^{\mathrm{in}}_{i}(x)\,\differential x=:\bar{z}_{i}$ for all
$i\in\\{0,\dots,n\\}$ and a.a. $t>0$, and there exist constants
$\alpha\in(0,1]$ and $\lambda=\lambda(\bar{z},\alpha,\Omega)>0$ such that for
a.e. $t>0$

$\displaystyle\operatorname{Dist}_{\alpha}(z(t),\bar{z})\leq\mathrm{e}^{-\lambda
t}\operatorname{Dist}_{\alpha}(z^{\mathrm{in}},\bar{z}),$ (2.20)

where
$\operatorname{Dist}_{\alpha}(z,\bar{z})=H_{\mathrm{rel}}(z,\bar{z})+\alpha
G_{\mathrm{rel}}(u,\bar{u})$ (cf. eq. (1.7)).

See Section 5 for the proof of Prop. 2.11. For non-trivial reactions obeying
mass action kinetics, the steady state $\bar{z}$ associated with (erds) is
determined by solving a constrained minimisation problem for the convex
entropy functional $H(u,c)$ imposing energy conservation
$E(u,c)=\int_{\Omega}u\overset{!}{=}E_{0}$ and further linear constraints
taking into account possible conservation laws for the concentrations (see
[32, 43]). Prop. 2.11 considers the simplest case, where all species have a
conserved mass. Extension to more general reactions is usually achieved by
means of suitable coercivity estimates for the dissipation term
$-D_{i}h(z)R_{i}(z)\geq 0$ associated with the reactions. See [32, 43] for
applications in a non-isothermal setting; for previous works in the isothermal
case, we refer to [42, 21, 41] and references therein. Let us observe that
since (M0) allows for $m\not=0$, leading to energy flux induced by temperature
gradients, a maximum principle for the internal energy density (as is
crucially used in [32]) is not available here. We further note that, with a
standard Csiszár–Kullback–Pinsker inequality [18, 48], the bound (2.20) can be
used to deduce exponential convergence to equilibrium in
$L^{2}(\Omega)\times(L^{1}(\Omega))^{n}$.

######  Remark 2.12.

Observe that the condition (ed.s) in Prop. 2.11 is satisfied under the
additional hypotheses on the coefficient functions imposed in Prop. 2.10.
Under suitable regularity hypotheses, the strong energy inequality (ene.s)
(with an equality) can be proved similarly as in [28, Lemma 6.1].

###  2.3. Examples

Below, we provide relevant applications of the weak-strong uniqueness result.

####  2.3.1. Energy-reaction-diffusion systems

The hypotheses of Theorem 2.9 are compatible with the class of ERDS introduced
in [28]. In that work, the existence of generalised solutions (weak or
renormalised) has been established for two classes of models, both taking the
form (2.9) with $m\equiv 0$.

One of the models in [28] is a special case of (M0) (see page 2.15) with
$m\equiv 0$. A brief verification of the model hypotheses of Theorem 2.9 for
(M0) is provided in the appendix, see Lemma 6.2. The existence analysis for
(M0) focuses on reactions obeying suitable growth conditions, in which case
there are global-in-time weak solutions. However, renormalised solutions can
be constructed by adapting the proof of [28, Theorem 1.8], and in this case no
growth restrictions on $|R(z)|$ are required. Conceptually, the construction
of renormalised solutions for (M0) is simpler than in [28, Theorem 1.8], since
the diffusive flux is integrable and the strong convergence of $\nabla u$ is
not required in case of (M0).

The second class of models considered in [28] will here be referred to as
(M1). It again takes the form (2.9) with $m\equiv 0$, and assumes the
following conditions:

  * •

Entropy density $h$ is given by (h1), (h2).

  * •

Reactions $R_{i}\in C([0,\infty)^{1+n})$, $i=1,\dots,n$, satisfy
A$2$(it:react.i), where $R_{0}\equiv 0$.

  * •

$\mathbb{M}$ is given by (2.11)–(2.14) with $m(z)\equiv 0$, and $m_{i}(u,c)$
given by (2.14), where $\kappa_{0,i},\kappa_{1,i}$ satisfy one of the
following:

    1. (i)

$\kappa_{0,i}=1$ and $\kappa_{1,i}=0$ for all $i$

    2. (ii)

$\kappa_{0,i}\geq 0$ and $\kappa_{1,i}=1$ for all $i$

  * •

Moreover,

    * $\circ$

$\pi_{1}\gamma\gtrsim 1$ ($\gamma$ as in (2.10))

    * $\circ$

$w_{i}^{\prime}(u)\lesssim-w_{i}^{\prime\prime}(u)\sqrt{\pi_{1}}$

    * $\circ$

$\sqrt{\pi_{1}}\tfrac{w_{i}^{\prime}}{w_{i}}\lesssim 1$.

Global weak solutions in case (ii) have been constructed in [28, Theorem 1.2]
for reactions obeying suitable growth hypotheses. More interesting is case
(i), in which the existence of global renormalised solutions has been
established in [28, Theorem 1.8] for general continuous reactions satisfying
A$2$(it:react.i). A pecularity of this model lies in the circumstance that the
renormalised formulation is needed not only to give a meaning to the reaction
rates, but also to the diffusion flux $A(z)\nabla z$, which may be non-
integrable. Model (M1.i) satisfies conditions (A3.a) (with $\varpi=0$),
(A3.b), (A$4$), (A$5$) and (A$6$’) of Theorem 2.9. We refer to [28, Section
2.2] (notably the proof of Lemma 2.3 therein), where the necessary estimates
can be found. For verifying (A$6$’), one should also use the fact that the
coefficient function $a(z)=\pi_{1}(z)\gamma(z)$ satisfies the bound
$\chi_{\\{u\geq\underline{u}\\}}|a(u,c)|\lesssim C(|z|,\underline{u})$ for any
$\underline{u}>0$. Thus, under the extra smoothness assumptions
$\hat{\sigma},w_{i}\in C^{4}((0,\infty))$ and $R_{i},m,m_{i},\pi_{1}\in
C^{0,1}_{\mathrm{loc}}((0,\infty)^{1+n})$, Theorem 2.9 is applicable. We
caution that verifying (ed) as it was done in Prop. 2.10 for (M0) is much more
delicate for Model (M1.i) due to the possibility of strong cross diffusion
caused by a non-integrable diffusion flux. Whether or not (M1.i) admits an
analogue of Prop. 2.10 is an open question. Recall, however, that for the
solutions constructed in [28, Theorem 1.8] inequality (ed) can be deduced from
the construction using lower semicontinuity arguments (cf. Section 6.2).

####  2.3.2. Reaction-cross-diffusion systems

Our results further apply to (isoenergetic) population models generalising the
two-species system for pattern formation by Shigesada, Kawasaki, and Teramoto
(SKT). Reduction to the isoenergetic case is achieved by choosing $\varpi=0$
(see (A$3$)) and $u\equiv u^{\mathrm{in}}$ to be spatially constant, which is
consistent with the evolution law for the energy density $u$ if $\varpi=0$.
Then, the given energy density $u=u^{\mathrm{in}}\in\mathbb{R}^{+}$ can be
regarded as a fixed system’s parameter (in particular $\nabla u=0$) and one
can write $h=h(c),A=A(c),R=R(c),$ and $\mathscr{P}=\mathscr{P}(c)$ etc.

The generalised SKT system as considered in [12] states

$\begin{array}[]{rlcl}\partial_{t}c_{i}&\\!\\!=\,\nabla\cdot(A_{ij}(c)\nabla
c_{j})+R_{i}(c),&&\quad t>0,x\in\Omega,\vspace{1mm}\\\
0&\\!\\!=\,A_{ij}(c)\nabla c_{j}\cdot\nu,&&\quad
t>0,x\in\partial\Omega,\end{array}$ (skt)

where $A_{ij}(c)=\delta_{ij}p_{i}(c)+c_{i}\tfrac{\partial p_{i}}{\partial
c_{j}}(c)$ for $i,j=1,\dots,n$ with
$p_{i}(c)=a_{i0}+\sum_{k=1}^{n}a_{ik}c_{k}^{s}$ for suitable $a_{il}\geq 0$
and some $s>0$. Under certain hypotheses, this system has a generalised
gradient structure with entropy density given by
$h(c)=\sum_{i=1}^{n}\pi_{i}\lambda_{s}(c_{i})$ for constants $\pi_{i}>0$ and
$\lambda_{1}$ given by (1.4), $\lambda_{s}(r)=\tfrac{r^{s}-sr}{s-1}+1$ for
$s\not=1$. Under a weak cross-diffusion (wc) condition (see eq. (12) in [12])
or the detailed balance (db) condition $\pi_{i}a_{ij}=\pi_{j}a_{ji}$ for all
$i,j\in\\{1,\dots,n\\}$ together with $a_{i0}>0,a_{ii}>0$ for all $i$, the
matrix $\mathbb{M}(z)=A(z)(D^{2}h(z))^{-1}$ satisfies the non-degeneracy
condition (A3.b), i.e. $\mathbb{M}(c)\gtrsim_{\iota}I_{n}$ whenever
$\min\\{c_{1},\dots,c_{n}\\}\geq\iota$, see the explicit estimate in [15,
Lemma 2.1] for $s=1$, and [12, Section 2] for the general case under certain
extra hypotheses. Observe that the detailed balance condition is equivalent to
the symmetry of the mobility matrix $\mathbb{M}=A(D^{2}h)^{-1}$, and thus
ensures the symmetry of the diffusive part $\mathbb{K}_{\mathrm{diff}}$ of the
Onsager operator.

  1. a)

Linear transition rates $s=1$. In this case, assuming (wc) or (db) with
$a_{i0}>0,a_{ii}>0$, one has
$\mathscr{P}(c)\gtrsim\sum_{i=1}^{n}\big{(}|\nabla
c_{i}|^{2}+|\nabla\sqrt{c_{i}}|^{2}\big{)}$ and $|A(c)\nabla
c|\lesssim|c||\nabla c|$. Thus, assumptions (A$1$), (A$3$)–(A$6$) on the
entropy density and the mobility matrix are satisfied. (The more general
weights $\pi_{i}>0$ in $h(c)$ as opposed to the unit weights in (A$1$) do not
impact the analysis.) We leave it to reader to verify that, by adapting the
proof of Prop. 2.10 (see also [27, Prop. 5]), the entropy dissipation
inequality (ed) can be proved for renormalised solutions to this system, when
assuming the regularity

$\displaystyle c_{i},\sqrt{c_{i}}\in
L^{2}_{\mathrm{loc}}(I;H^{1}(\Omega))\quad\text{for all }i.$ (2.21)

Observe that this regularity is essentially equivalent to the assumption in
Prop. 2.10 that $\mathscr{P}(z)\in L^{1}(\Omega_{T})$ for all $T<T^{*}$. Thus,
for (skt) with $s=1$ and reactions satisfying (A$2$), Theorem 2.9 yields the
weak-strong uniqueness of renormalised solutions of the regularity (2.21). A
similar result has been obtained previously in [15, Theorem 1]. Here, we
should caution that the regularity assumption $c_{i}\in
L^{2}_{\mathrm{loc}}(I;H^{1}(\Omega))$ _and_ $\sqrt{c_{i}}\in
L^{2}_{\mathrm{loc}}(I;H^{1}(\Omega))$ is also needed in the proof of [15,
Theorem 1], is, however, incompletely stated in this theorem. Moreover, our
result shows that hypothesis ‘(H2.iii)’ in [15] on the reactions can be
dropped.

  2. b)

Nonlinear transition rates. An adjustment of our hypotheses further allows to
treat system (skt) with superlinear transition rates $s\in(1,2]$. In this
case, condition (A$6$) is no longer fulfilled, but with a suitable adjustment
of the truncation function $\xi^{*}(c)$ in the definition of the modified
relative entropy density $h^{*}_{\mathrm{rel}}(z,\tilde{z})$ (given by (1.12))
our technique can still be applied. See Remark 3.1 for technical details.

  3. For sublinear transition rates, $s<1$, the problem becomes more delicate since a direct analogue of condition (A$6$) is not available. When relying exclusively on entropy estimates for a priori bounds, even the construction of renormalised solutions, which to the author’s knowledge is currently only available in the case of linear transition rates [14] (but likely to be extendable to the case $s\in[1,2]$), is open for small $s\in(0,1)$.

##  3\. Weak-strong uniqueness principle

In this section we will establish a stability estimate implying Theorem 2.9.
Throughout this section, we therefore assume the hypotheses of Theorem 2.9.
Before turning to the actual proof in Subsection 3.4, we gather some technical
auxiliary results.

###  3.1. Truncation function $\xi^{*}(z)$

Let $\iota,B$ be fixed constants, $0<\iota<1<B<\infty$, that are such that for
all $i\in\\{0,\dots,n\\}$

$2\iota\leq\tilde{z}_{i}\leq B.$ (3.1)

##### Different regimes/case distinction

We henceforth abbreviate $|z|_{1}=\sum_{i=0}^{n}z_{i}$. For sufficiently large
but fixed parameters $E,N\geq 2$ (to be chosen later) we decompose
$[0,\infty)^{1+n}$ into the following three sets:

$\begin{split}\mathcal{A}&=\\{z\in[0,\infty)^{1+n}:|z|_{1}\leq E\\},\\\
\mathcal{B}&=\\{z\in[0,\infty)^{1+n}:E<|z|_{1}<E^{N}\\},\\\
\mathcal{C}&=\\{z\in[0,\infty)^{1+n}:|z|_{1}\geq E^{N}\\}.\end{split}$ (3.2)

A motivation for this decomposition is given in the introduction (see Section
1.3). Let us also mention that in the proof of Theorem 2.9 the set
$\mathcal{A}$ will be further decomposed into $\mathcal{A}_{+}$ and
$\mathcal{A}_{0}$ (defined in (3.45)). The parameter $E$ will always be
supposed to satisfy $E\geq 2B$. A finite number of further lower bounds on $E$
will be imposed later on.

##### Adjusted relative entropy

We now define

$h_{\mathrm{rel}}^{*}(z,\tilde{z})=h(z)-D_{i}h(\tilde{z})(\xi^{*}(z)z_{i}-\tilde{z}_{i})-h(\tilde{z}),$
(3.3)

where $\xi^{*}=\xi^{(E,N)}\in C^{\infty}([0,\infty)^{1+n})$, $0\leq\xi^{*}\leq
1$, is a truncation function subordinate to the above case distinction
enjoying the following properties:

  1. (t$1$)

$\xi^{*}(z)=1$ and $D^{k}\xi^{*}(z)=0$ for all $k\in\mathbb{N}_{+}$ if
$z\in\mathcal{A}$,

  2. (t$2$)

$\xi^{*}(z)=0$ and $D^{k}\xi^{*}(z)=0$ for all $k\in\mathbb{N}_{+}$ if
$z\in\mathcal{C}$,

and

  1. (t$3$)

$|D\xi^{*}(z)|\lesssim\tfrac{1}{N|z|_{1}},\quad|D^{2}\xi^{*}(z)|\lesssim\tfrac{1}{N(|z|_{1})^{2}}$
for all $z\in[0,\infty)^{1+n}.$

A function $\xi^{*}$ that has these properties can be obtained as follows: let
$\vartheta\in C^{\infty}_{b}(\mathbb{R})$ be non-increasing with
$\vartheta(r)=1$ for $r\leq 0$ and $\vartheta(r)=0$ for $r\geq 1$ and define
(cf. [27])

$\xi^{*}(z)=\xi^{(E,N)}(z)=\vartheta\left(\frac{\log(|z|_{1})-\log(E)}{\log(E^{N})-\log(E)}\right).$
(3.4)

It is elementary to check that this choice satisfies the above properties. For
instance, note that

$|D\xi^{(E,N)}(z)|\lesssim\tfrac{1}{N|z|_{1}}\cdot\tfrac{1}{\log(E)}$

if $N\geq 2$.

######  Remark 3.1 (Superlinear transition rates).

When dealing with (isoenergetic) reaction-cross-diffusion population systems
with _superlinear_ transition rates $s\in(1,2]$ for concentrations
$c_{1},\dots,c_{n}$ (see Example b) in Sec. 2.3.2), the decay property (t$3$)
is no longer sufficient, and the above choice of $\xi^{*}$ should be replaced
by

$\xi^{*}(c):=\xi^{(E,N,s)}(c):=\vartheta\left(\frac{\log(\rho_{s}^{s}(c))-\log(E^{s})}{\log(E^{sN})-\log(E^{s})}\right)=\vartheta\left(\frac{\log(\rho_{s}(c))-\log(E)}{\log(E^{N})-\log(E)}\right),$
(3.5)

where $\vartheta$ is as before, and

$\displaystyle\rho_{s}(c):=\Big{(}\sum_{i=1}^{n}(c_{i}+\delta)^{s}\Big{)}^{\frac{1}{s}}$
(3.6)

with $\delta=\delta(s)=1$ for $s\in(1,2)$ and $\delta(s)=0$ for $s=2$. The
parameter $\delta\in\\{0,1\\}$ ensures the smoothness of $\rho_{s}^{s}$ and
thus of $\xi^{*}$ .

Introducing the regimes $\mathcal{A}^{(s)}=\\{\rho_{s}(c)\leq E\\},$
$\mathcal{B}^{(s)}=\\{E<\rho_{s}(c)<E^{N}\\}$,
$\mathcal{C}^{(s)}=\\{\rho_{s}(c)\geq E^{N}\\}$, properties (t$1$) and (t$2$)
remain valid with $\mathcal{A},$ $\mathcal{C}$ replaced by $\mathcal{A}^{(s)}$
and $\mathcal{C}^{(s)}$, respectively. Derivatives of $\xi^{*}$ given by (3.5)
enjoy the following decay properties, specifically adapted to the problem at
hand,

$|D_{i}\xi^{*}(c)|\lesssim\tfrac{(c_{i}+\delta)^{s-1}}{N|c|_{s}^{s}},\qquad|D_{ik}\xi^{*}(c)|\lesssim\tfrac{(c_{i}+\delta)^{s-1}(c_{k}+\delta)^{s-1}}{N|c|_{s}^{2s}}+\tfrac{(c_{i}+\delta)^{s-2}}{N|c|_{s}^{s}}\delta_{ik},$
(3.7)

where $\delta_{ik}$ denotes the Kronecker delta. As will become clear in
Section 3.4, if $c\in\mathcal{A}^{(s)}$, one can simply follow the reasoning
in the proof of Theorem 2.9 using the fact that (skt) satisfies (under
reasonable hypotheses) a non-degeneracy condition analogous to (A3.b).

If $c\in\mathcal{B}^{(s)}$, we need an analogue of assumption (A$6$) to be
able to absorb terms without a good sign involving products of gradients of
the renormalised solution by the entropy dissipation. Typical systems (skt)
satisfy the coercivity bound (cf. [12])

$\displaystyle\mathscr{P}(c)\gtrsim\sum_{i=1}^{n}|\nabla
c_{i}^{s}|^{2}+\sum_{i=1}^{n}|\nabla c_{i}^{s/2}|^{2}.$ (3.8)

Confining to systems (skt) enjoying this bound, a suitable generalisation of
hp. (A$6$) (in the isoenergetic case) that is satisfied by such systems is

$\displaystyle\chi_{\\{|c|_{s}\geq
1\\}}\cdot\big{(}c_{i}^{s-1}+c_{i}^{s/2-1}\big{)}|A_{ij}(c)\nabla
c_{j}|\lesssim|c|^{s}_{s}\sqrt{\mathscr{P}(c)}.$ (3.9)

Observe that since $s\in(1,2]$, the factor
$\big{(}c_{i}^{s-1}+c_{i}^{s/2-1}\big{)}$ in this condition can be
equivalently replaced by $\big{(}(c_{i}+\delta)^{s-1}+c_{i}^{s/2-1}\big{)}$.
Using the above model bounds and decay properties of derivatives of $\xi^{*}$,
one can verify the estimate

$\begin{split}\chi_{\\{|c|_{s}\geq 1\\}}|\nabla
D_{i}(\xi^{*}(c)c_{l})||A_{ij}(c)\nabla
c_{j}|\lesssim\tfrac{1}{N}\mathscr{P}(c),\end{split}$ (3.10)

which allows to deal with the case $c\in\mathcal{B}^{(s)}$ (cf. ineq. (3.76)
in the proof of Thm 2.9). When verifying this bound, one uses the fact that
$|\nabla c_{l}|\lesssim|\nabla c_{l}^{s}|+|\nabla c_{l}^{s/2}|$, which holds
true since $s\in(1,2]$.

If $c\in\mathcal{C}^{(s)}$, one can follow the reasoning in the proof of Thm
2.9.

Thus, for models with superlinear transition rates $s\in(1,2]$ that satisfy
(3.8), weak-strong uniqueness is obtained by adapting the proof of Theorem 2.9
as sketched above. Here, we also use the fact that thanks to the locally
uniform convexity of the entropy density associated with (skt) (see Sec.
2.3.2) an analogue of Prop. 3.2 is immediate.

###  3.2. Coercivity properties of the generalised distance

We henceforth let $g(s)=\tfrac{1}{2}s^{2}$,

$g_{\mathrm{rel}}(u,\tilde{u}):=g(u)-g^{\prime}(\tilde{u})(u-\tilde{u})-g(\tilde{u})=\tfrac{1}{2}|u-\tilde{u}|^{2},$
(3.11)

and define for $\alpha\in(0,\infty)$

$\operatorname{dist}^{*}_{\alpha}(z,\tilde{z})=h_{\mathrm{rel}}^{*}(z,\tilde{z})+\alpha
g_{\mathrm{rel}}(u,\tilde{u}).$ (3.12)

For the following assertion we recall that $z\in\mathcal{A}$ if and only if
$\sum_{i=0}^{n}z_{i}\leq E$.

######  Proposition 3.2 (Coercivity properties).

Recall that $h(z)=h(u,c)$ is given by (h1), (h2), and that $\tilde{z}$
satisfies (3.1). For any $E\in[1,\infty)$, we have444We recall that any
dependence of estimates on $\iota$ and $B$ (see (3.1)) will usually not be
indicated explicitly.

$\begin{split}|z-\tilde{z}|^{2}\lesssim_{E}h_{\mathrm{rel}}^{*}(z,\tilde{z})\quad\text{if
}z\in\mathcal{A}.\end{split}$ (3.13)

There exists $\underline{E}=\underline{E}(\tilde{z})<\infty$ such that for any
$E\geq\underline{E}$ and any $\alpha\geq 1$

$\begin{split}h_{\mathrm{rel}}^{*}(z,\tilde{z})+\alpha
g_{\mathrm{rel}}(u,\tilde{u})\geq\epsilon\bigg{(}\sum_{i=1}^{n}c_{i}\log_{+}c_{i}+u^{2}\bigg{)}+1\quad\text{if
}z\in\mathcal{A}^{c},\end{split}$ (3.14)

where $\epsilon>0$ is a positive constant only depending on model parameters.

For any $\alpha\in(0,1]$ there exists
$\underline{E}=\underline{E}(\alpha,\tilde{z})<\infty$ such that for any
$E\geq\underline{E}$

$\begin{split}h_{\mathrm{rel}}^{*}(z,\tilde{z})+\alpha
g_{\mathrm{rel}}(u,\tilde{u})\geq\epsilon\bigg{(}\sum_{i=1}^{n}c_{i}\log_{+}c_{i}+\alpha\,u^{2}\bigg{)}+1\quad\text{if
}z\in\mathcal{A}^{c},\end{split}$ (3.15)

where $\epsilon>0$ is a positive constant only depending on model parameters.

######  Remark 3.3.

Note that as long as $\alpha\in[1,\infty)$, the lower bound $\underline{E}$
can be chosen independently of $\alpha$. This property is essential for
obtaining the stability estimate (1.11) in the case where (A$6$) is not
satisfied (including model (M1) in case (i), where $\kappa_{1,i}=0$; cf.
Section 2.3).

###### Proof of Prop. 3.2.

For the first assertion, we note that, as can be seen from the proof of [43,
Prop. 2.1], the entropy density $h$ is locally uniformly convex on
$[0,\infty)^{1+n}$ with $D_{ij}h(z)\geq\epsilon_{0}(E)\delta_{ij}$ if
$|z|_{1}\leq E$. Since, by construction, $\xi^{*}(z)=1$ whenever
$z\in\mathcal{A}$, we thus infer

$h_{\mathrm{rel}}^{*}(z,\tilde{z})=h(z)-D_{i}h(\tilde{z})(z-\tilde{z})_{i}-h(\tilde{z})\gtrsim_{E}|z-\tilde{z}|^{2}.$

Let us now turn to assertions (3.14) and (3.15). Since $\tilde{u}\leq B$, we
have the bound $|u-\tilde{u}|^{2}\geq\tfrac{1}{4}u^{2}\chi_{\\{u\geq 2B\\}}$.
By Lemma 6.3, we further have for some $\nu\in[0,1)$ and some positive
constant $\epsilon>0$

$h(z)\geq\epsilon\sum_{i=1}^{n}c_{i}\log_{+}c_{i}-C(\hat{\sigma}_{+}(u)+u^{\nu})-C.$

Hence,

$\displaystyle h_{\mathrm{rel}}^{*}(z,\tilde{z})+\alpha
g_{\mathrm{rel}}(u,\tilde{u})$
$\displaystyle\geq\epsilon\sum_{i=1}^{n}c_{i}\log_{+}c_{i}+\alpha
u^{2}\chi_{\\{u\geq 2B\\}}-C\sum_{i=0}^{n}z_{i}-C.$ (3.16)

Inequality (3.14) is now immediate since
$\sum_{i=1}^{n}c_{i}\log(c_{i})+u^{2}$ dominates $\sum_{0=1}^{n}z_{i}$ for
$|z|_{1}\geq\underline{E}$ whenever $\underline{E}$ is large enough. Observe
that the lower bound on $\underline{E}$ can be chosen independently of
$\alpha\in[1,\infty)$.

Let now $\alpha\in(0,1]$ be given. Inequality (3.16) shows that if
$\underline{E}=\underline{E}(\alpha)$ is large enough, we obtain (3.15) if
$E\geq\underline{E}$. ∎

###  3.3. Evolution inequality for the generalised distance

We will abbreviate

$\displaystyle
H_{\mathrm{rel}}^{*}(z,\tilde{z}):=\int_{\Omega}\,h_{\mathrm{rel}}^{*}(z,\tilde{z})\,\differential
x,\qquad
G_{\mathrm{rel}}(u,\tilde{u}):=\int_{\Omega}g_{\mathrm{rel}}(u,\tilde{u})\,\differential
x.$ (3.17)

In this subsection, we exploit the evolution laws satisfied by the dissipative
renormalised solution $z$ and the strong solution $\tilde{z}$ of Theorem 2.9
to derive an evolution inequality for the generalised distance

$\operatorname{Dist}^{*}_{\alpha}(z,\tilde{z}):=\int_{\Omega}\operatorname{dist}^{*}_{\alpha}(z,\tilde{z})\,\differential
x=H_{\mathrm{rel}}^{*}(z,\tilde{z})+\alpha\,G_{\mathrm{rel}}(u,\tilde{u}),$
(3.18)

where $\alpha\in(0,\infty)$ is a suitably chosen weight (to be specified in
Section 3.4).

We recall that $I:=[0,T^{*})$. Furthermore, we note that since
$\partial_{t}\tilde{z}\in L^{\infty}(\Omega_{T})$ for any $T<T^{*}$, we can
integrate by parts with respect to time in the weak formulation (2.8)
satisfied by the strong solution $\tilde{z}$ to find

$\begin{split}\int_{0}^{T}\\!\\!\int_{\Omega}\partial_{t}\tilde{z}_{i}\psi_{i}\,\differential
x\differential
t=-\int_{0}^{T}\\!\\!\int_{\Omega}A_{ij}(\tilde{z})\nabla\tilde{z}_{j}\cdot\nabla\psi_{i}\,\differential
x\differential
t+\int_{0}^{T}\\!\\!\int_{\Omega}R_{i}(\tilde{z})\psi_{i}\,\differential
x\differential t.\end{split}$ (3.19)

By a density argument, one can see that eq. (3.19) holds true for all $\psi\in
L^{1}_{\mathrm{loc}}(I;W^{1,1}(\Omega))$.

We consider separately the two quantities $H_{\mathrm{rel}}^{*}(z,\tilde{z})$
and $G_{\mathrm{rel}}(u,\tilde{u})$ appearing in (3.18), beginning with the
former.

######  Lemma 3.4 (Evolution of the entropic part).

For a.e. $T<T^{*}$ one has

$\begin{split}\evaluated{H_{\mathrm{rel}}^{*}(z,\tilde{z})}_{t=0}^{t=T}&\leq\int_{0}^{T}\\!\\!\int_{\Omega}\;\rho^{(h)}\;\differential
x\differential t,\end{split}$ (3.20)

where

$\begin{split}\rho^{(h)}&:=-\nabla D_{i}h(z)\cdot\mathbb{M}_{il}(z)\nabla
D_{l}h(z)\\\
&\qquad+\nabla(D_{i}(\xi^{*}(z)z_{j})D_{j}h(\tilde{z}))\cdot\mathbb{M}_{il}(z)\nabla
D_{l}h(z)\\\
&\qquad+\nabla\Big{(}D_{ij}h(\tilde{z})(\xi^{*}(z)z_{j}-\tilde{z}_{j})\Big{)}\cdot\mathbb{M}_{il}(\tilde{z})\nabla
D_{l}h(\tilde{z})\\\ &\qquad-
D_{ij}h(\tilde{z})(\xi^{*}(z)z_{j}-{\tilde{z}}_{j})R_{i}(\tilde{z})\\\
&\qquad+(D_{i}h(z)-D_{j}h(\tilde{z})D_{i}(\xi^{*}(z)z_{j}))R_{i}(z).\end{split}$
(3.21)

###### Proof of Lemma 3.4.

The subsequent observations apply to a.e. $T<T^{*}$.

We write

$H_{\mathrm{rel}}^{*}(z,\tilde{z})=H(z)-\int_{\Omega}D_{i}h(\tilde{z})\xi^{*}(z)z_{i}\,\differential
x+\int_{\Omega}(D_{i}h(\tilde{z})\tilde{z}_{i}-h(\tilde{z}))\,\differential
x.$

For the first term on the RHS we use the fact that, by hypothesis, $z$
satisfies (ed), i.e.

$\begin{split}\evaluated{H(z)}_{t=0}^{t=T}&\leq-\int_{0}^{T}\\!\\!\int_{\Omega}\nabla
D_{i}h(z)\cdot\mathbb{M}_{il}(z)\nabla D_{l}h(z)\,\differential x\differential
t+\int_{0}^{T}\\!\\!\int_{\Omega}D_{i}h(z)R_{i}(z)\,\differential
x\differential t.\end{split}$

For the second term, we want to use the fact that $z$ satisfies the
renormalised formulations (2.5) and (2.6) with the truncation function
$\xi(z)=\xi^{*}(z)z_{j}$ and the test function $\psi=D_{j}h(\tilde{z})\in
W^{1,\infty}(\Omega_{T})$. (For the admissibility of this choice, see Remark
2.2.) Inserting these choices in eq. (2.6), we obtain

$\begin{split}-\evaluated{\int_{\Omega}D_{j}h(\tilde{z})\xi^{*}(z)z_{j}\,\differential
x}_{t=0}^{t=T}&+\int_{0}^{T}\\!\\!\int_{\Omega}\xi^{*}(z)z_{j}\tfrac{\differential}{\differential
t}D_{j}h(\tilde{z})\,\differential x\differential t\\\
&=\int_{0}^{T}\\!\\!\int_{\Omega}\nabla(D_{i}(\xi^{*}(z)z_{j})D_{j}h(\tilde{z}))\cdot\mathbb{M}_{il}(z)\nabla
D_{l}h(z)\,\differential x\differential t\\\
&\quad-\int_{0}^{T}\\!\\!\int_{\Omega}D_{i}(\xi^{*}(z)z_{j})R_{i}(z)D_{j}h(\tilde{z})\,\differential
x\differential t.\end{split}$ (3.22)

We next rewrite the second term on the LHS choosing in the weak form (3.19)
for $\tilde{z}$ the test function $\psi:=D_{ij}h(\tilde{z})\xi^{*}(z)z_{j}$.
This yields

$\begin{split}\int_{0}^{T}\\!\\!\int_{\Omega}D_{ij}h(\tilde{z})\xi^{*}(z)z_{j}\partial_{t}{\tilde{z}}_{i}\,\differential
x\differential
t&=-\int_{0}^{T}\\!\\!\int_{\Omega}\nabla\big{(}D_{ij}h(\tilde{z})\xi^{*}(z)z_{j})\cdot\mathbb{M}_{il}(\tilde{z})\nabla
D_{l}h(\tilde{z})\,\differential x\differential t\\\
&\qquad+\int_{0}^{T}\\!\\!\int_{\Omega}D_{ij}h(\tilde{z})\xi^{*}(z)z_{j}R_{i}(\tilde{z})\,\differential
x\differential t.\end{split}$ (3.23)

Observe that since $\chi_{\\{|z|\leq E\\}}\nabla z\in
L^{2}_{\mathrm{loc}}(I;L^{2}(\Omega))$ for any $E<\infty$, the function
$\psi:=D_{ij}h(\tilde{z})\xi^{*}(z)z_{j}\in
L^{2}_{\mathrm{loc}}(I;H^{1}(\Omega))$ is indeed admissible in the weak
equation (3.19) for $\tilde{z}$.

We finally need to determine the evolution of the term

$\begin{split}\int_{\Omega}(D_{i}h(\tilde{z})\tilde{z}_{i}-h(\tilde{z}))\,\differential
x.\end{split}$ (3.24)

To this end, note that thanks to the regularity of $\tilde{z}$,

$\begin{split}\evaluated{\int_{\Omega}h(\tilde{z})\,\differential
x}_{t=0}^{t=T}&=\int_{0}^{T}\\!\\!\int_{\Omega}D_{i}h(\tilde{z})\partial_{t}{\tilde{z}}_{i}\,\differential
x\differential t,\\\
\evaluated{\int_{\Omega}D_{i}h(\tilde{z})\tilde{z}_{i}\,\differential
x}_{t=0}^{t=T}&=\int_{0}^{T}\\!\\!\int_{\Omega}D_{i}h(\tilde{z})\partial_{t}{\tilde{z}}_{i}\,\differential
x\differential
t+\int_{0}^{T}\\!\\!\int_{\Omega}\big{(}\tfrac{\differential}{\differential
t}D_{i}h(\tilde{z})\big{)}{\tilde{z}}_{i}\,\differential x\differential
t.\end{split}$ (3.25)

Subtracting the first from the second equality then yields

$\displaystyle\evaluated{\int_{\Omega}\big{(}D_{i}h(\tilde{z})\tilde{z}_{i}-h(\tilde{z})\big{)}\,\differential
x}_{t=0}^{t=T}$
$\displaystyle=\int_{0}^{T}\\!\\!\int_{\Omega}D_{ij}h(\tilde{z}){\tilde{z}}_{i}\partial_{t}{\tilde{z}}_{j}\,\differential
x\differential t$ (3.26)
$\displaystyle=\int_{0}^{T}\\!\\!\int_{\Omega}D_{ij}h(\tilde{z}){\tilde{z}}_{j}\partial_{t}{\tilde{z}}_{i}\,\differential
x\differential t$ (3.27)
$\displaystyle=-\int_{0}^{T}\\!\\!\int_{\Omega}\nabla\big{(}D_{ij}h(\tilde{z}){\tilde{z}}_{j}\big{)}\cdot\mathbb{M}_{il}(\tilde{z})\nabla
D_{l}h(\tilde{z})\,\differential x\differential t$ (3.28)
$\displaystyle\qquad+\int_{0}^{T}\\!\\!\int_{\Omega}D_{ij}h(\tilde{z}){\tilde{z}}_{j}R_{i}(\tilde{z})\,\differential
x\differential t,$ (3.29)

where in the second step we have used the symmetry of the Hessian of $h$.

In combination, the above estimates yield the bound

$\displaystyle\evaluated{H_{\mathrm{rel}}^{*}(z,\tilde{z})}_{t=0}^{t=T}$
$\displaystyle\leq-\int_{0}^{T}\\!\\!\int_{\Omega}\nabla
D_{i}h(z)\cdot\mathbb{M}_{il}(z)\nabla D_{l}h(z)\,\differential x\differential
t$ (3.30)
$\displaystyle\qquad+\int_{0}^{T}\\!\\!\int_{\Omega}D_{i}h(z)R_{i}(z)\,\differential
x\differential t$ (3.31)
$\displaystyle\qquad+\int_{0}^{T}\\!\\!\int_{\Omega}\nabla(D_{i}(\xi^{*}(z)z_{j})D_{j}h(\tilde{z}))\cdot
M_{il}(z)\nabla D_{l}h(z)\,\differential x\differential t$ (3.32)
$\displaystyle\qquad-\int_{0}^{T}\\!\\!\int_{\Omega}D_{i}(\xi^{*}(z)z_{j})R_{i}(z)D_{j}h(\tilde{z})\,\differential
x\differential t$ (3.33)
$\displaystyle\qquad+\int_{0}^{T}\\!\\!\int_{\Omega}\nabla\big{(}D_{ij}h(\tilde{z})\xi^{*}(z)z_{j})\cdot\mathbb{M}_{il}(\tilde{z})\nabla
D_{l}h(\tilde{z})\,\differential x\differential t$ (3.34)
$\displaystyle\qquad-\int_{0}^{T}\\!\\!\int_{\Omega}D_{ij}h(\tilde{z})\xi^{*}(z)z_{j}R_{i}(\tilde{z})\,\differential
x\differential t$ (3.35)
$\displaystyle\qquad-\int_{0}^{T}\\!\\!\int_{\Omega}\nabla\big{(}D_{ij}h(\tilde{z}){\tilde{z}}_{j}\big{)}\cdot\mathbb{M}_{il}(\tilde{z})\nabla
D_{l}h(\tilde{z})\,\differential x\differential t$ (3.36)
$\displaystyle\qquad+\int_{0}^{T}\\!\\!\int_{\Omega}D_{ij}h(\tilde{z}){\tilde{z}}_{j}R_{i}(\tilde{z})\,\differential
x\differential t.$ (3.37)

The asserted inequality is now obtained upon rearranging the integrals on the
RHS. ∎

We next turn to the energetic part. We first note that equation (3.19) and the
fact that $R_{0}\equiv 0$ imply that

$\begin{split}\int_{0}^{T}\\!\\!\int_{\Omega}\partial_{t}\tilde{u}\varphi\,\differential
x\differential
t=-\int_{0}^{T}\\!\\!\int_{\Omega}A_{0j}(\tilde{z})\nabla\tilde{z}_{j}\cdot\nabla\varphi\,\differential
x\differential t\end{split}$ (3.38)

for all $\varphi\in L^{1}_{\mathrm{loc}}(I;W^{1,1}(\Omega))$.

######  Lemma 3.5 (Evolution of the energetic part).

Recall the definition of $g_{\mathrm{rel}}$ in (3.11) and the notation
$G_{\mathrm{rel}}(u,\tilde{u}):=\int_{\Omega}g_{\mathrm{rel}}(u,\tilde{u})\,\differential
x$. For almost every $T>0$, we have

$\begin{split}\evaluated{G_{\mathrm{rel}}(u,\tilde{u})}_{t=0}^{t=T}=\int_{0}^{T}\\!\\!\int_{\Omega}\;\rho^{g}\;\differential
x\differential t,\end{split}$ (3.39)

where

$\begin{split}\rho^{(g)}:=-a(z)&|\nabla u-\nabla\tilde{u}|^{2}\\\
&-(a(z)-a(\tilde{z}))(\nabla u-\nabla\tilde{u})\cdot\nabla\tilde{u}\\\
&-m(z)(\nabla u-\nabla\tilde{u})\cdot(\nabla D_{0}h(z)-\nabla
D_{0}h(\tilde{z}))\\\ &-(m(z)-m(\tilde{z}))(\nabla
u-\nabla\tilde{u})\cdot\nabla D_{0}h(\tilde{z}).\end{split}$ (3.40)

###### Proof.

We expand
$g_{\mathrm{rel}}(u,\tilde{u})=\tfrac{1}{2}u^{2}-u\tilde{u}+\tfrac{1}{2}\tilde{u}^{2}$.

To deal with the first term on the RHS, we use the energy inequality (ene),
i.e. the property that for a.e. $T<T^{*}$

$\evaluated{G(u)}_{t=0}^{t=T}\leq-\int_{0}^{T}\\!\\!\int_{\Omega}a(z)|\nabla
u|^{2}\,\differential x\differential
t-\int_{0}^{T}\\!\\!\int_{\Omega}m(z)\nabla D_{0}h(z)\cdot\nabla
u\,\differential x\differential t.$

To determine the time evolution of the term
$\int_{\Omega}u\tilde{u}\,\differential x$, we assert that the Lipschitz
function $\tilde{u}$ is admissible in the weak formulation (2.7) of the
equation for $u$, thus yielding

$\begin{split}-\evaluated{\int_{\Omega}u\tilde{u}\,\differential
x}_{t=0}^{t=T}&+\int_{0}^{T}\\!\\!\int_{\Omega}u\partial_{t}\tilde{u}\,\differential
x\differential t\\\ &=\int_{0}^{T}\\!\\!\int_{\Omega}\big{(}a(z)\nabla
u\cdot\nabla\tilde{u}+m(z)\nabla
D_{0}h(z)\cdot\nabla\tilde{u}\big{)}\,\differential x\differential
t,\end{split}$

where we used (A3.c). The admissibility of $\tilde{u}$ can be shown as
follows: first exploit the regularity properties of $\nabla u\in
L^{2}_{\mathrm{loc}}(I;L^{2}(\Omega))$, $u\in
L^{\infty}_{\mathrm{loc}}(I;L^{2}(\Omega))$, which hold true by hypothesis
resp. follow from (ene) and the fact that $\mathscr{P}(z)\in
L^{1}_{\mathrm{loc}}(I;L^{1}(\Omega))$. Assumption (A$5$) and the
Gagliardo–Nirenberg interpolation applied to $u$ as well as the estimate

$m|\nabla
D_{0}h(z)|\leq\sqrt{m}\sqrt{\mathscr{P}(z)}\lesssim\sqrt{\mathscr{P}(z)}$

then imply improved integrability of the flux term, namely for some $s=s(d)>1$

$A_{0j}(z)\nabla z_{j}=a(z)\nabla u+m(z)\nabla D_{0}h(z)\in
L^{s}_{\mathrm{loc}}(I;L^{s}(\Omega)).$

With these bounds one can now use an approximation argument to show that,
under the current hypotheses, eq. (2.7) can be extended in particular to
Lipschitz functions $\varphi\in C^{0,1}(I\times\bar{\Omega})$.

Finally, using the test function $\varphi=u-\tilde{u}\in
L^{1}_{\mathrm{loc}}(I;W^{1,1}(\Omega))$ in the weak equation (3.38) for
$\tilde{u}$ gives

$-\int_{0}^{T}\\!\\!\int_{\Omega}\partial_{t}\tilde{u}(u-\tilde{u})\,\differential
x\differential
t=\int_{0}^{T}\\!\\!\int_{\Omega}a(\tilde{z})\nabla\tilde{u}\cdot\nabla(u-\tilde{u})\,\differential
x\differential t.$

The asserted identity (3.39) is now obtained by adding up the above equations
and rearranging appropriately the terms on the RHS. ∎

The evolution inequality for our generalised distance is an immediate
consequence of the previous two propositions.

######  Corollary 3.6.

Let $\alpha\in(0,\infty)$. We have

$\begin{split}\evaluated{\operatorname{Dist}^{*}_{\alpha}(z,\tilde{z})}_{t=0}^{t=T}\leq\int_{0}^{T}\\!\\!\int_{\Omega}\;\rho_{\alpha}\;\differential
x\differential t,\end{split}$ (3.41)

where $\rho_{\alpha}:=\rho^{(h)}+\alpha\rho^{(g)}$ with $\rho^{(h)}$,
$\rho^{(g)}$ given by (3.21) resp. (3.40).

###  3.4. Stability estimate

###### Proof of Theorem 2.9.

Since (A$6$) implies (A$6$’), it suffices to prove the assertion for the case
‘$\varpi=1$ (and thus, by hp., (A$6$))’ and the case ‘$\varpi=0$ and (A$6$’)’,
henceforth referred to as Case $\varpi=1$ resp. Case $\varpi=0$.

We will show the following.

  * •

Case $\varpi=1:$ if $\alpha\in(0,1]$ is sufficiently small, and if
$E=E(\alpha)$ and $N=N(E)$ are large enough, then for almost all
$T\in(0,T^{*})$

$\evaluated{\operatorname{Dist}^{*}_{\alpha}(z(t,\cdot),\tilde{z}(t,\cdot))}_{t=0}^{t=T}\lesssim_{E,N,\alpha}\int_{0}^{T}\operatorname{Dist}^{*}_{\alpha}(z,\tilde{z})\,\differential
t.$ (3.42)

  * •

Case $\varpi=0:$ if $E$, $N=N(E)$ and $\alpha\in[1,\infty)$ are chosen large
enough ($\alpha$ possibly depending on $E,N$), then for a.a. $T\in(0,T^{*})$
ineq. (3.42) holds true.

Once ineq. (3.42) has been established, we can invoke Gronwall’s inequality to
infer that for a.e. $T\in(0,T^{*})$

$\displaystyle\operatorname{Dist}^{*}_{\alpha}(z(T,\cdot),\tilde{z}(T,\cdot))\leq\operatorname{Dist}^{*}_{\alpha}(z^{\mathrm{in}},\widetilde{z^{\mathrm{in}}})\exp(kT),$
(3.43)

where $k=k(E,N,\alpha)>0.$ The estimates will also depend on the fixed
constant $\iota>0$, i.e. on the pointwise lower bound for $\min(\tilde{z})$.
This dependency will only be indicated occasionally and for the sake of
clarity.

In view of inequality (3.41) it suffices to show the pointwise bound

$\displaystyle\rho_{\alpha}\;\lesssim_{E,N,\alpha}\;\operatorname{dist}^{*}_{\alpha}(z,\tilde{z}).$
(3.44)

An elementary ingredient in the proof of this bound will be the coercivity
properties of $\operatorname{dist}^{*}_{\alpha}$ (see Prop. 3.2). We
anticipate that referring to Prop. 3.2 will be the only instance, where the
present proof makes use of the more specific form of the entropy density
$h(u,c)$ assumed in (A$1$). Loosely speaking, besides the locally strict
convexity ensuring (3.13), we will rely on a lower bound on the generalised
distance of the form $\operatorname{dist}^{*}_{\alpha}(z,\tilde{z})\gtrsim
1+u^{2}$ for $|z|\gg 1$.

We will distinguish the cases $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$
introduced on page 3.2, where, owing to the degeneracies of $\mathbb{M}(z)$
occurring when one of the concentrations vanishes, we further decompose the
set $\mathcal{A}$ into

$\mathcal{A}_{+}:=\\{z^{\prime}:\min(z^{\prime})\geq\iota\\}\cap\mathcal{A}\qquad\text{and}\qquad\mathcal{A}_{0}:=\\{z^{\prime}:\min(z^{\prime})<\iota\\}\cap\mathcal{A},$
(3.45)

where $\min(z^{\prime}):=\min\\{z_{0}^{\prime},\dots,z_{n}^{\prime}\\}$ for
$z^{\prime}=(z_{0}^{\prime},\dots,z_{n}^{\prime})\in[0,\infty)^{1+n}$. This
decomposition further serves to avoid regularity issues of $h$ as
$z_{i}\searrow 0$ for some $i\in\\{0,\dots,n\\}$.

If $z\in(\mathcal{A}_{+})^{c}$, we will make use of the following equivalent
formula for $\rho^{(g)}$

$\begin{split}\rho^{(g)}=-a(z)|\nabla
u|^{2}&-a(\tilde{z})|\nabla\tilde{u}|^{2}+a(z)\nabla
u\cdot\nabla\tilde{u}+a(\tilde{z})\nabla u\cdot\nabla\tilde{u}\\\
&-m(z)(\nabla u-\nabla\tilde{u})\cdot\nabla D_{0}h(z)\\\ &+m(\tilde{z})(\nabla
u-\nabla\tilde{u})\cdot\nabla D_{0}h(\tilde{z}).\end{split}$ (3.46)

Since, by hypothesis, $0\leq m(z)\lesssim\varpi$ and $a(z)\gtrsim 1$, this
implies that

$\displaystyle\rho^{(g)}$ $\displaystyle\leq-\tfrac{a(z)}{2}|\nabla
u|^{2}+C|a(z)\nabla u|+Cm(z)\varpi|\nabla D_{0}h(z)|^{2}+C.$ (3.47)

Using this form in the case when $z\in(\mathcal{A}_{+})^{c}$, we can avoid for
instance issues due to $a(z)$ becoming singular as $u\to 0$ by using the bound
(A$5$) on the energy flux.

Finally, note that $z\in\mathcal{A}$ implies $\xi^{*}(z)=1$ and
$D^{k}\xi^{*}(z)=0$ $\forall k\in\mathbb{N}_{+}$, so that, if
$z\in\mathcal{A}$, one has by formula (3.21)

$\begin{split}\rho^{(h)}&=-\nabla D_{i}h(z)\cdot\mathbb{M}_{il}(z)\nabla
D_{l}h(z)\\\ &\qquad+\nabla D_{i}h(\tilde{z})\cdot M_{il}(z)\nabla
D_{l}h(z)\\\
&\qquad+\nabla\Big{(}D_{ij}h(\tilde{z})(z_{j}-\tilde{z}_{j})\Big{)}\cdot\mathbb{M}_{il}(\tilde{z})\nabla
D_{l}h(\tilde{z})\,\differential x\differential t\\\ &\qquad-
D_{ij}h(\tilde{z})(z_{j}-{\tilde{z}}_{j})R_{i}(\tilde{z})\\\
&\qquad+(D_{i}h(z)-D_{i}h(\tilde{z}))R_{i}(z).\end{split}$ (3.48)

We are now ready to tackle the four cases.

Case $z\in\mathcal{A}_{+}:$ in this case we have the control $\iota\leq
z_{i}\leq E$ for all $i\in\\{0,\dots,n\\}$, and we need to show that
$\rho_{\alpha}\lesssim|z-\tilde{z}|^{2}$. We therefore rewrite formula (3.48)
as

$\begin{split}\rho^{(h)}&=-\nabla(D_{i}h(z)-D_{i}h(\tilde{z}))\cdot\mathbb{M}_{il}(z)\nabla
D_{l}h(z)\\\
&\qquad+\nabla\big{(}D_{ij}h(\tilde{z})(z_{j}-\tilde{z}_{j})\big{)}\cdot\mathbb{M}_{il}(\tilde{z})\nabla
D_{l}h(\tilde{z})\\\
&\qquad+\big{(}D_{i}h(z)-D_{i}h(\tilde{z})-D_{ij}h(\tilde{z})(z_{j}-{\tilde{z}}_{j})\big{)}R_{i}(\tilde{z})\\\
&\qquad+(D_{i}h(z)-D_{i}h(\tilde{z}))(R_{i}(z)-R_{i}(\tilde{z}))\\\
&=-\nabla(D_{i}h(z)-D_{i}h(\tilde{z}))\cdot\mathbb{M}_{il}(z)\nabla(D_{l}h(z)-D_{l}h(\tilde{z}))\\\
&\qquad-\nabla(D_{i}h(z)-D_{i}h(\tilde{z}))\cdot(\mathbb{M}_{il}(z)-\mathbb{M}_{il}(\tilde{z}))\nabla
D_{l}h(\tilde{z})\\\
&\qquad-\nabla\big{(}D_{i}h(z)-D_{i}h(\tilde{z})-D_{ij}h(\tilde{z})(z_{j}-\tilde{z}_{j})\big{)}\cdot\mathbb{M}_{il}(\tilde{z})\nabla
D_{l}h(\tilde{z})\\\
&\qquad+\big{(}D_{i}h(z)-D_{i}h(\tilde{z})-D_{ij}h(\tilde{z})(z_{j}-{\tilde{z}}_{j})\big{)}R_{i}(\tilde{z})\\\
&\qquad+(D_{i}h(z)-D_{i}h(\tilde{z}))(R_{i}(z)-R_{i}(\tilde{z})).\end{split}$
(3.49)

Since, by hp. (A3.b),
$\mathbb{M}(z)\geq\operatorname{diag}(m(z)\varpi,0,\dots,0)+\epsilon(\iota)\operatorname{diag}(0,1,\dots,1)$
for a suitable constant $\epsilon(\iota)>0$, we have

$\nabla(D_{i}h(z)-D_{i}h(\tilde{z}))\cdot\mathbb{M}_{il}(z)\nabla(D_{l}h(z)-D_{l}h(\tilde{z}))\\\
\geq m(z)\varpi|\nabla D_{0}h(z)-\nabla
D_{0}h(\tilde{z})|^{2}+\epsilon(\iota)\delta|\nabla D_{c}h(z)-\nabla
D_{c}h(\tilde{z})|^{2}$ (3.50)

for any $\delta\in(0,1]$. The auxiliary parameter $\delta=\delta(\alpha)$ will
eventually be chosen small enough to be specified below. For
$i\in\\{1,\dots,n\\}$ we observe that since $\iota\leq z_{j}\leq E$ for all
$j\in\\{0,\dots,n\\}$, the triangle inequality yields

$\begin{split}|\nabla D_{i}h(z)-\nabla
D_{i}h(\tilde{z})|&\geq|\nabla\log(c_{i})-\nabla\log(\tilde{c}_{i})|-|\nabla\log(w_{i}(u))-\nabla\log(w_{i}(\tilde{u}))|\\\
&\geq\tfrac{1}{c_{i}}|\nabla
c_{i}-\nabla\tilde{c}_{i}|-\big{|}\tfrac{1}{c_{i}}-\tfrac{1}{\tilde{c}_{i}}\big{|}|\nabla\tilde{c}_{i}|-|\nabla
u-\nabla\tilde{u}|\tfrac{w_{i}^{\prime}(u)}{w_{i}(u)}\\\
&\qquad-\big{|}\tfrac{w_{i}^{\prime}(u)}{w_{i}(u)}-\tfrac{w_{i}^{\prime}(\tilde{u})}{w_{i}(\tilde{u})}\big{|}|\nabla\tilde{u}|\\\
&\geq\epsilon(E,\iota)|\nabla c-\nabla\tilde{c}|-C(E,\iota)|\nabla
u-\nabla\tilde{u}|-C(E,\iota)|z-\tilde{z}|,\end{split}$

where $\epsilon(E,\iota)>0$ is some sufficiently small constant.

We next estimate, using the local Lipschitz continuity of $\mathbb{M}$,

$\begin{split}|\nabla(D_{i}h(z)-D_{i}h(\tilde{z}))\cdot(\mathbb{M}_{il}(z)-\mathbb{M}_{il}(\tilde{z}))\nabla
D_{l}h(\tilde{z})|&\lesssim_{E,\iota}|\nabla Dh(z)-\nabla
Dh(\tilde{z})||z-\tilde{z}|\\\ &\lesssim_{E,\iota}|z-\tilde{z}||\nabla
z-\nabla\tilde{z}|+|z-\tilde{z}|^{2}.\end{split}$ (3.51)

Before estimating the remaining terms, we compute for $f\in
C^{3}((0,\infty)^{1+n})$

$\begin{split}\nabla\big{(}f(z)-f(\tilde{z})-D_{j}f(\tilde{z})(z_{j}-\tilde{z}_{j})\big{)}&=D_{j}f(z)\nabla
z_{j}-D_{j}f(\tilde{z})\nabla\tilde{z}_{j}-D_{j}f(\tilde{z})\nabla(z_{j}-\tilde{z}_{j})\\\
&\qquad-D_{jk}f(\tilde{z})(z_{j}-\tilde{z}_{j})\nabla\tilde{z}_{k}\\\
&=(D_{j}f(z)-D_{j}f(\tilde{z}))\nabla
z_{j}-D_{jk}f(\tilde{z})(z_{j}-\tilde{z}_{j})\nabla\tilde{z}_{k}\\\
&=(D_{j}f(z)-D_{j}f(\tilde{z}))\nabla(z_{j}-\tilde{z}_{j})\\\
&\qquad+[D_{k}f(z)-D_{k}f(\tilde{z})-D_{jk}f(\tilde{z})(z_{j}-\tilde{z}_{j})]\nabla\tilde{z}_{k}\end{split}$
(3.52)

and, using Taylor’s theorem, for $k=0,\dots,n$

$|D_{k}f(z)-D_{k}f(\tilde{z})-D_{jk}f(\tilde{z})(z_{j}-\tilde{z}_{j})|\lesssim_{E,\iota,f}|z-\tilde{z}|^{2}.$

Letting $f(z)=D_{i}h(z)$, we infer since $h\in C^{4}((0,\infty)^{1+n})$ (see
hp. (A$1$)) that

$\begin{split}|\nabla\big{(}D_{i}h(z)-D_{i}h(\tilde{z})-D_{ij}h(\tilde{z})(z_{j}-\tilde{z}_{j})\big{)}|&\lesssim_{\iota,E}|z-\tilde{z}||\nabla
z-\nabla\tilde{z}|+|z-\tilde{z}|^{2}.\end{split}$ (3.53)

Using the previous bounds to estimate the RHS of (3.49), recalling also hp.
A$2$(it:react.ii), and applying Young’s inequality and an absorption argument,
we thus infer for suitable $\epsilon(\iota,E)>0$

$\begin{split}\rho^{(h)}\leq-\epsilon(\iota,E)\delta|\nabla
c-\nabla\tilde{c}|^{2}-m(z)&\varpi|\nabla D_{0}h(z)-\nabla
D_{0}h(\tilde{z})|^{2}\\\ &+C_{1}(E,\iota)\delta|\nabla
u-\nabla\tilde{u}|^{2}+C(\delta,E,\iota)|z-\tilde{z}|^{2}.\end{split}$ (3.54)

On the other hand, using the fact that $z\mapsto a(z)$ is locally Lipschitz
continuous in $(0,\infty)^{1+n}$, we deduce from eq. (3.40) for suitable
$\epsilon_{1}>0$ and $C_{2}<\infty$ (independent of $E,\iota$)

$\rho^{(g)}\leq-\epsilon_{1}|\nabla
u-\nabla\tilde{u}|^{2}+C_{2}m(z)\varpi|\nabla D_{0}h(z)-\nabla
D_{0}h(\tilde{z})|^{2}+C(E,\iota)|z-\tilde{z}|^{2},$ (3.55)

where we used the fact that $0\leq m(z)\lesssim\varpi$.

If $\varpi=1$, we choose $\alpha\in(0,1]$ small enough such that $\alpha
C_{2}\leq 1$ and subsequently $\delta=\delta(\alpha,E,\iota)$ sufficiently
small such that $\delta C_{1}(E,\iota)\leq\alpha\epsilon_{1}$. We may then
conclude that

$\displaystyle\rho_{\alpha}=\rho^{(h)}+\alpha\rho^{(g)}\leq
C(\alpha,E,\iota)|z-\tilde{z}|^{2}.$ (3.56)

Let us emphasise that the smallness condition of $\alpha$ is independent of
$E$.

If instead $\varpi=0$, we choose for given555In the case $\varpi=0$, it
suffices to restrict $\alpha$ to the range $1\leq\alpha<\infty$.
$\alpha\in[1,\infty)$ the parameter $\delta=\delta(\alpha,E,\iota)$ small
enough such that $\delta\,C_{1}(E,\iota)\leq\epsilon_{1}\alpha$, and obtain as
before

$\displaystyle\rho_{\alpha}\leq C(\alpha,E,\iota)|z-\tilde{z}|^{2}.$ (3.57)

Case $z\in\mathcal{A}_{0}:$ in this case, we have no lower bound on $z_{i}$
away from zero, but since $\min(z)\leq\iota$ and $\min(\tilde{z})\geq 2\iota$,
we know that $|z-\tilde{z}|\geq\iota$. By Prop. 3.2 (cf. (3.13)) it thus
suffices to prove that $\rho_{\alpha}\lesssim_{E}1$.

Recalling (3.48) and A$2$(it:react.i), we estimate

$\begin{split}\rho^{(h)}&\leq-\nabla D_{i}h(z)\cdot\mathbb{M}_{il}(z)\nabla
D_{l}h(z)\\\ &\qquad+\nabla D_{i}h(\tilde{z})\cdot\mathbb{M}_{il}(z)\nabla
D_{l}h(z)\\\
&\qquad+\nabla\Big{(}D_{ij}h(\tilde{z})(z_{j}-\tilde{z}_{j})\Big{)}\cdot\mathbb{M}_{il}(\tilde{z})\nabla
D_{l}h(\tilde{z})\\\ &\qquad+C(E)\\\
&\leq-\mathscr{P}(z)+C(E)\sqrt{\mathscr{P}(z)}+C|\nabla u|+C(E),\end{split}$
(3.58)

where the last step uses hp. (A4.a) and hp. (A4.b). Hence,

$\rho^{(h)}\leq-\tfrac{1}{2}\mathscr{P}(z)+C(E)|\nabla u|+C(E).$

Next, by ineq. (3.47) and hp. (A4.b),

$\displaystyle\rho^{(g)}$ $\displaystyle\leq-\tfrac{a(z)}{2}|\nabla
u|^{2}+C(\delta_{1},E)+\tfrac{\delta_{1}}{4}\mathscr{P}(z)+Cm(z)\varpi|\nabla
D_{0}h(z)|^{2}$ (3.59)

for any $\delta_{1}>0$.

If $\varpi=1$, we let $\delta_{1}=1$ and use the estimate $m(z)\varpi|\nabla
D_{0}h(z)|^{2}\leq\mathscr{P}(z)$, which follows from hp. (A$3$), to see that
after possibly decreasing $\alpha\in(0,1]$ we have
$\rho_{\alpha}\leq-\alpha\tfrac{a(z)}{2}|\nabla u|^{2}+C(E)|\nabla
u|+C(E)\leq-\alpha\tfrac{a(z)}{4}|\nabla u|^{2}+C(\alpha,E)$.

If $\varpi=0$, we choose $\delta_{1}=\tfrac{1}{\alpha}\leq 1$. Then

$\begin{split}\rho_{\alpha}&\leq-\tfrac{1}{2}\mathscr{P}(z)-\alpha\tfrac{a(z)}{2}|\nabla
u|^{2}+\tfrac{a(z)}{4}|\nabla u|^{2}+C(\alpha,E)+\tfrac{1}{4}\mathscr{P}(z)\\\
&\lesssim_{\alpha,E}1.\end{split}$

Case $z\in\mathcal{B}:$ in this case derivatives of $\xi^{*}$ do in general
not vanish, but we know that $1\ll E<|z|_{1}<E^{N}$. We estimate

$\displaystyle\rho^{(h)}$ $\displaystyle=-\nabla
D_{i}h(z)\cdot\mathbb{M}_{il}(z)\nabla D_{l}h(z)$ (3.60)
$\displaystyle\qquad+\nabla(D_{i}(\xi^{*}(z)z_{j})D_{j}h(\tilde{z}))\cdot\mathbb{M}_{il}(z)\nabla
D_{l}h(z)$ (3.61)
$\displaystyle\qquad+\nabla\Big{(}D_{ij}h(\tilde{z})(\xi^{*}(z)z_{j}-\tilde{z}_{j})\Big{)}\cdot\mathbb{M}_{il}(\tilde{z})\nabla
D_{l}h(\tilde{z})$ (3.62) $\displaystyle\qquad-
D_{ij}h(\tilde{z})(\xi^{*}(z)z_{j}-{\tilde{z}}_{j})R_{i}(\tilde{z})$ (3.63)
$\displaystyle\qquad+D_{i}h(z)R_{i}(z)-D_{j}h(\tilde{z})D_{i}(\xi^{*}(z)z_{j})R_{i}(z)$
(3.64) $\displaystyle\leq-\mathscr{P}(z)$ (3.65) $\displaystyle\qquad+C|\nabla
D_{i}(\xi^{*}(z)z_{j})\cdot\mathbb{M}_{il}(z)\nabla D_{l}h(z)|$ (3.66)
$\displaystyle\qquad+C(E,N)\sqrt{\mathscr{P}(z)}+C(E,N)|\nabla u|\hskip
56.9055pt(\text{by hp.}\leavevmode\nobreak\
\eqref{eq:hp.fluxtrunc.P},\leavevmode\nobreak\ \eqref{eq:hp.dztrunc.P})$
(3.67) $\displaystyle\qquad+C(E,N)\hskip 56.9055pt(\text{using
hp.}\leavevmode\nobreak\ \ref{eq:hp.Rdiss})$ (3.68)
$\displaystyle\leq-\tfrac{1}{2}\mathscr{P}(z)$ (3.69)
$\displaystyle\qquad+C|\nabla
D_{i}(\xi^{*}(z)z_{j})\cdot\mathbb{M}_{il}(z)\nabla D_{l}h(z)|$ (3.70)
$\displaystyle\qquad+C(E,N)|\nabla u|+C(E,N).$ (3.71)

In order to estimate the term $|\nabla
D_{i}(\xi^{*}(z)z_{j})\cdot\mathbb{M}_{il}(z)\nabla D_{l}h(z)|$, we observe
that

$\begin{split}|\nabla D_{i}(\xi^{*}(z)z_{j})|&\lesssim|D\xi^{*}(z)||\nabla
z|+|D^{2}\xi^{*}(z)z||\nabla z|\\\ &\lesssim\tfrac{1}{N|z|_{1}}|\nabla
z|\hskip 71.13188pt(\text{by }\ref{eq:Dxi*}).\end{split}$ (3.72)

We first consider the case $\varpi=1$. Then (A$6$) is at our disposal, which
yields using (3.72)

$\begin{split}|\nabla D_{i}(\xi^{*}(z)z_{j})||\mathbb{M}_{il}(z)\nabla
D_{l}h(z)|&\lesssim\tfrac{1}{N|z|_{1}}|\nabla z||\sum_{j}A_{ij}(z)\nabla
z_{j}|\lesssim\tfrac{1}{N}\mathscr{P}(z)+\tfrac{1}{N}|\nabla
u|^{2}.\end{split}$

Thus, choosing $N=N(\min\\{1,\alpha\\})$ sufficiently large, we infer

$\displaystyle\rho^{(h)}$
$\displaystyle\leq-\tfrac{1}{4}\mathscr{P}(z)+\min\\{1,\alpha\\}\tfrac{a(z)}{8}|\nabla
u|^{2}+C(E,N,\alpha).$ (3.73)

Next, simiarly as in (3.59), we estimate

$\displaystyle\rho^{(g)}$ $\displaystyle\leq-\tfrac{a(z)}{2}|\nabla
u|^{2}+C(E,N)+\tfrac{1}{8}\mathscr{P}(z)+C_{4}m(z)\varpi|\nabla
D_{0}h(z)|^{2}.$ (3.74)

Decreasing $\alpha\in(0,1]$, if necessary, to ensure that $\alpha
C_{4}m(z)\varpi|\nabla D_{0}h(z)|^{2}\leq\tfrac{1}{8}\mathscr{P}(z)$, we
obtain $\rho_{\alpha}\lesssim_{N,E}1.$

It remains to consider the case where $\varpi=0$ and (A$6$’) are fulfilled.
Since $\underline{u}:=\inf u^{\mathrm{in}}=\inf\tilde{u}(0,\cdot)\geq 2\iota$,
Lemma 6.4 yields $\inf u\geq\underline{u}>0$. By hp. (A$6$’), we infer for all
$0\leq i\leq n$

$\displaystyle|\nabla z||\mathbb{M}_{il}(z)\nabla D_{l}h(z)|\leq
C|z|_{1}\mathscr{P}(z)+C(E,N,\underline{u})|\nabla u|^{2}.$ (3.75)

Thus, recalling ineq. (3.72), we can estimate for $N$ large enough

$\displaystyle|\nabla D_{i}(\xi^{*}(z)z_{j})\cdot\mathbb{M}_{il}(z)\nabla
D_{l}h(z)|$
$\displaystyle\leq\tfrac{1}{4}\mathscr{P}(z)+C(E,N,\underline{u})|\nabla
u|^{2}$ (3.76)

to infer

$\rho^{(h)}\leq-\tfrac{1}{4}\mathscr{P}(z)+C_{1}(E,N,\underline{u})|\nabla
u|^{2}+C(E,N).$

Next, since $\varpi=0$, ineq. (3.47) yields

$\displaystyle\rho^{(g)}$ $\displaystyle\leq-\tfrac{a(z)}{2}|\nabla
u|^{2}+C(E,N)\sqrt{\mathscr{P}(z)}+C$ (3.77)
$\displaystyle\leq-\tfrac{a(z)}{2}|\nabla
u|^{2}+\tfrac{1}{8\alpha}\mathscr{P}(z)+C(\alpha,E,N).$ (3.78)

Increasing $\alpha=\alpha(E,N,\underline{u})$, if necessary, to ensure that
$\alpha\tfrac{a(z)}{2}\geq C_{1}(E,N,\underline{u})$, we conclude

$\displaystyle\rho_{\alpha}\leq-\tfrac{1}{8}\mathscr{P}(z)+C(\alpha,E,N,\underline{u}).$
(3.79)

Case $z\in\mathcal{C}:$ in this case, $\xi^{*}(z)=0$ and $D^{k}\xi^{*}(z)=0$
for all $k\in\mathbb{N}_{+}$. Thus

$\begin{split}\rho^{(h)}&=-\nabla D_{i}h(z)\cdot\mathbb{M}_{il}(z)\nabla
D_{l}h(z)\\\
&\qquad-\nabla(D_{ij}h(\tilde{z})\tilde{z}_{j})\cdot\mathbb{M}_{il}(\tilde{z})\nabla
D_{l}h(\tilde{z})\\\
&\qquad+D_{ij}h(\tilde{z}){\tilde{z}}_{j}R_{i}(\tilde{z})\\\
&\qquad+D_{i}h(z)R_{i}(z)\\\ &\leq-\mathscr{P}(z)+C,\end{split}$ (3.80)

where we used hp. A$2$(it:react.i).

If $\varpi=1$, we have thanks to (3.47) and hp. (A$3$), (A$5$)

$\rho^{(g)}\leq-\tfrac{a(z)}{2}|\nabla u|^{2}+C_{4}\mathscr{P}(z)+u^{2}+C,$

where $C_{4}$ is independent of $E,N$. Hence, after possibly decreasing
$\alpha\in(0,1]$ to ensure that $\alpha C_{4}\leq 1$, we find

$\displaystyle\rho_{\alpha}$ $\displaystyle\leq C+\alpha
u^{2}\lesssim_{\alpha}\operatorname{dist}^{*}_{\alpha}(z,\tilde{z}),$ (3.81)

where the second step follows from (3.15) (after choosing $E=E(\alpha)$ large
enough).

If $\varpi=0$, we estimate using again (3.47) and hp. (A$5$)

$\displaystyle\rho^{(g)}$ $\displaystyle\leq-\tfrac{a(z)}{2}|\nabla
u|^{2}+C(1+u)\sqrt{\mathscr{P}(z)}+C$ (3.82)
$\displaystyle\leq-\tfrac{a(z)}{2}|\nabla
u|^{2}+\tfrac{1}{2\alpha}\mathscr{P}(z)+C(\alpha)u^{2}+C(\alpha),$ (3.83)

and infer

$\displaystyle\rho_{\alpha}$
$\displaystyle\leq-\tfrac{1}{2}\mathscr{P}(z)-\alpha\tfrac{a(z)}{2}|\nabla
u|^{2}+C(\alpha)u^{2}+C(\alpha)$ (3.84)
$\displaystyle\lesssim_{\alpha}\operatorname{dist}^{*}_{\alpha}(z,\tilde{z}).$
(3.85)

The second step follows from the coercivity property (3.14) and the fact that
$\alpha\geq 1$.

This proves the bound (3.44) and thus completes the proof of Theorem 2.9. ∎

##  4\. Strong entropy dissipation property

###### Proof of Proposition 2.10.

We first establish (ed.s) for $s=0$ and a.e. $t=T\in(0,T^{*})$, that is, we
first prove (ed). In a second step (see page 4.51), we point out how to extend
the result to a.e. $0<s<t<T^{*}$.

Case 1: $s=0,t=T\in(0,T^{*})$.

We consider for a small parameter $\delta>0$ the regularised entropy density

$h_{\delta}(u,c)=\delta u+h(u,c).$

The additive term $\delta u$ serves to ensure coercivity, since the original
density $h(z)$ may in general allow for cancellations at infinity reflecting
the coupling between concentrations and energy component. In particular, for
every $L\in\mathbb{N}$ the sublevel set $\\{z\in\mathbb{R}_{\geq
0}^{1+n}:h_{\delta}(z)\leq L\\}$ is bounded. This coercivity property easily
follows from the lower bound

$h(z)\geq-\hat{\sigma}(u)+\epsilon_{*}\sum_{i=1}^{n}c_{i}\log
c_{i}-Cu^{\nu}-C,$ (4.1)

valid for suitable $\nu\in[0,1)$, $\epsilon_{*}>0$ (see Lemma 6.3), together
with the sublinearity of the increasing function $\hat{\sigma}(u)$ as
$u\to\infty$ (see (h2)).

In order to define an admissible truncation function, we consider as in [27,
Proof of Prop. 5] for $L\geq 2$ an auxiliary function $\theta_{L}\in
C^{\infty}(\mathbb{R})$ satisfying $\theta_{L}(s)=s$ for $|s|\leq L$,
$0\leq\theta_{L}^{\prime}\leq 1$,

$\displaystyle|\theta^{\prime\prime}_{L}(s)|\lesssim\tfrac{C}{1+|s|\log(|s|+\mathrm{e})}$
(4.2)

for all $s\in\mathbb{R}$ and $\theta^{\prime}_{L}(s)=0$ for $|s|\geq L^{C}$
for some sufficiently large constant $C\geq 2$, which is kept fixed throughout
the proof.

To derive the entropy dissipation inequality, we would like to choose the
truncation function $\theta_{L}(h_{\delta}(\cdot))$ and the test function
$\psi\equiv 1$ in the renormalised formulation satisfied by $z$, and then let
$L\to\infty$ and subsequently $\delta\to 0$. Since derivatives of $h_{\delta}$
are in general not bounded as $u\searrow 0$ or $c_{i}\searrow 0$, further
regularisation is required. We let

$h_{\delta,\varepsilon}(u,c)=h_{\delta}(z)+\hat{\sigma}(u)-\hat{\sigma}(u+\varepsilon),$

abbreviate for $z=(u,c_{1},\dots,c_{n})$

$z^{\tilde{\varepsilon}}:=(u,c_{1}+\tilde{\varepsilon},\dots,c_{n}+\tilde{\varepsilon}),\quad\tilde{\varepsilon}\in(0,1],$

and then consider the function $z\mapsto
h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\in C^{2}([0,\infty)^{1+n})$.
Thanks to (4.1), it is easy to see that for fixed $\delta\in(0,1]$ sublevel
sets of $z\mapsto h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})$ are
bounded; in fact

$\displaystyle
h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\geq\tfrac{\delta}{2}u+\epsilon_{*}\sum_{i=1}^{n}c_{i}\log_{+}c_{i}-C_{\delta}$
(4.3)

for $\delta,\varepsilon,\tilde{\varepsilon}\in(0,1]$. Hence, the
$C^{2}$-function
$\xi(z):=\theta_{L}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))$ has
compactly supported derivative $D\xi$, and is thus an admissible truncation in
eq. (2.5) (cf. Remark 2.2). This, combined with the choice $\psi\equiv 1$ in
eq. (2.5), yields for a.e. $T<T^{*}$

$\begin{split}LHS:=&\evaluated{\int_{\Omega}\theta_{L}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))\,\differential
x}_{t=0}^{t=T}\\\
=&-\int_{0}^{T}\\!\\!\int_{\Omega}\theta_{L}^{\prime}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))D_{ij}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\nabla
z_{j}\cdot A_{ik}(z)\nabla z_{k}\,\differential x\differential t\\\
&-\int_{0}^{T}\\!\\!\int_{\Omega}\theta_{L}^{\prime\prime}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))D_{j}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})D_{i}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\nabla
z_{j}\cdot A_{ik}(z)\nabla z_{k}\,\differential x\differential t\\\
&+\int_{0}^{T}\\!\\!\int_{\Omega}\theta_{L}^{\prime}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))D_{i}h(z^{\tilde{\varepsilon}})R_{i}(z)\,\differential
x\differential t\\\ =&:I+II+III,\end{split}$ (4.4)

where we recall the summation convention (see Notations 1.5). In the reaction
term we have used the fact that $D_{i}h_{\delta,\varepsilon}(z)=D_{i}h(z)$
whenever $i\not=0$ together with $R_{0}\equiv 0$.

We will establish the asserted inequality (ed) by taking the $\liminf$ of the
LHS and the $\limsup$ of the RHS of the above equation (4.4) as
$\tilde{\varepsilon}\to 0$, $L\to\infty$ and $\delta,\varepsilon\to 0$, in the
stated order. We perform the corresponding limits separately in $LHS$ and in
each of the three terms $I,II,III$. Below we use, without explicit reference,
the following basic properties satisfied under the hypotheses of Model (M0),
see Lemma 6.1:

$\displaystyle a(z)|\nabla u|^{2}\sim|\nabla u|^{2},$ (4.5)
$\displaystyle\mathscr{P}(z)\gtrsim\sum_{i=1}^{n}|\nabla\sqrt{c_{i}}|^{2}+|\sqrt{\gamma}\,\nabla
u|^{2}+|\sqrt{m}\nabla D_{0}h(z)|^{2},$ (4.6)

where
$\gamma(u,c)=-\hat{\sigma}^{\prime\prime}(u)-\sum_{l=1}^{n}\tfrac{w_{l}^{\prime\prime}(u)}{w_{l}(u)}c_{l}$,

$\displaystyle|A_{ik}(z)\nabla z_{k}|$
$\displaystyle\lesssim\sqrt{c_{i}}\sqrt{\mathscr{P}(z)}\qquad\text{for }i\geq
1,$ (4.7) $\displaystyle|A_{0k}(z)\nabla z_{k}|$ $\displaystyle\lesssim|\nabla
u|+\sqrt{m}\sqrt{\mathscr{P}(z)}.$ (4.8)

_LHS:_
The limit $\tilde{\varepsilon}\to 0$ of the $LHS$ is immediate due to the
boundedness of $\theta_{L}$, and yields

$\displaystyle\evaluated{\int_{\Omega}\theta_{L}(h_{\delta,\varepsilon}(z))\,\differential
x}_{t=0}^{t=T}.$ (4.9)

We next take

$\liminf_{\varepsilon,\delta\to\infty}\liminf_{L\to\infty}$

of the last expression using a combination of the dominated convergence
theorem and Fatou’s lemma. At initial time, we estimate using the lower and
upper bounds on $h$ in Lemma 6.3

$|\theta_{L}(h_{\delta,\varepsilon}(z^{\mathrm{in}}))|\leq|h_{\delta,\varepsilon}(z^{\mathrm{in}})|\lesssim
u^{\mathrm{in}}+|\hat{\sigma}_{-}(u^{\mathrm{in}})|+\sum_{i=0}^{n}c^{\mathrm{in}}_{i}\log_{+}c^{\mathrm{in}}_{i}+1.$

We can hence use dominated convergence to deduce that, as $L\to\infty$ and
$\delta,\varepsilon\to 0$,

$\displaystyle\int_{\Omega}\theta_{L}(h_{\delta,\varepsilon}(z^{\mathrm{in}}))\,\differential
x\to\int_{\Omega}h(z^{\mathrm{in}})\,\differential x.$ (4.10)

To deal with the integral at time $t=T$, we first observe that, thanks to the
regularity $u\in L^{\infty}_{\mathrm{loc}}([0,T^{*}),L^{1}(\Omega))$ and
(4.1), the negative part of $\theta_{L}(h_{\delta,\varepsilon}(z(T,\cdot)))$
is controlled pointwise in $x$, for a.e. $T\in(0,T^{*}]$, by an integrable
function, uniformly in $L,{\delta,\varepsilon}$, and its integral can thus be
shown to converge in the same way as the term at initial time. For the
positive part of $\theta_{L}(h_{\delta,\varepsilon}(z(T,\cdot)))$ we use
Fatou’s lemma:

$\displaystyle\int_{\Omega}\max\\{h_{\delta,\varepsilon}(z(T,\cdot)),0\\}\,\differential
x\leq\liminf_{L\to\infty}\int_{\Omega}\max\\{\theta_{L}(h_{\delta,\varepsilon}(z(T,\cdot))),0\\}\,\differential
x,$ (4.11) $\displaystyle\int_{\Omega}\max\\{h(z(T,\cdot)),0\\}\,\differential
x\leq\liminf_{\delta,\varepsilon\to
0}\int_{\Omega}\max\\{h_{\delta,\varepsilon}(z(T,\cdot)),0\\}\,\differential
x.$ (4.12)

In combination, we infer

$\displaystyle\evaluated{\int_{\Omega}h(z)\,\differential
x}_{t=0}^{t=T}\leq\liminf_{\delta,\varepsilon\to
0}\liminf_{L\to\infty}\lim_{\tilde{\varepsilon}\to
0}\evaluated{\int_{\Omega}\theta_{L}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))\,\differential
x}_{t=0}^{t=T}.$ (4.13)

_Diffusive dissipation term I:_
We assert that

$\limsup_{\delta,\varepsilon\to
0}\limsup_{L\to\infty}\limsup_{\tilde{\varepsilon}\to
0}\Big{(}-\int_{0}^{T}\\!\\!\int_{\Omega}\theta_{L}^{\prime}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))\nabla
z_{j}\cdot
D_{ji}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})A_{ik}(z)\nabla
z_{k}\,\differential x\differential t\Big{)}$ (4.14)

can be bounded above by the non-positive term

$\displaystyle-\int_{0}^{T}\\!\\!\int_{\Omega}\nabla z_{j}\cdot
D_{ji}h(z)A_{ik}(z)\nabla z_{k}\,\differential x\differential t.$ (4.15)

To show this, we will mainly rely on the dominated convergence theorem. We
therefore start by listing several uniform pointwise estimates on the terms
involved.

We first estimate for $i,j$ fixed the term

$\displaystyle
p_{ij}:=\theta_{L}^{\prime}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))D_{ji}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\nabla
z_{j}\cdot\sum_{k}A_{ik}(z)\nabla z_{k},$ (4.16)

where we recall that $0\leq\theta^{\prime}_{L}\leq 1.$

Case $i,j\geq 1:$ in this case
$D_{ij}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})=\tfrac{1}{c_{i}+\tilde{\varepsilon}}\delta_{ij}$
and thus

$\displaystyle|p_{ij}|\leq|\tfrac{1}{c_{i}+\tilde{\varepsilon}}\nabla
c_{i}\cdot\sum_{k}A_{ik}(z)\nabla z_{k}|\lesssim\tfrac{1}{c_{i}}|\nabla
c_{i}|\sqrt{c_{i}}\sqrt{\mathscr{P}(z)}\lesssim\mathscr{P}(z).$ (4.17)

Case $i\geq 1,j=0:$ observing that
$-D_{i0}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})=\tfrac{w_{i}^{\prime}}{w_{i}}\lesssim\sqrt{\tfrac{-w_{i}^{\prime\prime}}{w_{i}}}$
(cf. (2.15)), we find

$\displaystyle|p_{ij}|\leq|\tfrac{w_{i}^{\prime}}{w_{i}}\nabla
u\cdot\sum_{k}A_{ik}(z)\nabla
z_{k}|\lesssim|\sqrt{\tfrac{-w_{i}^{\prime\prime}}{w_{i}}c_{i}}\nabla
u|\sqrt{\mathscr{P}(z)}\lesssim\mathscr{P}(z).$ (4.18)

Case $i=0,j\geq 0:$ we split the sum over $k$ in the definition of $p_{0j}$
into two parts:

$\sum_{k}A_{0k}(z)\nabla z_{k}=a(z)\nabla u+m(z)\nabla D_{0}h(z)$

and split $p_{0j}$ accordingly into $p_{0j}=p_{0j}^{(0)}+p_{0j}^{(1)}.$

For $j\geq 1$ we estimate as above

$\displaystyle|p_{0j}^{(0)}|$
$\displaystyle\leq|\tfrac{w_{j}^{\prime}}{w_{j}}\nabla c_{j}\cdot a(z)\nabla
u|\lesssim|\sqrt{\tfrac{-w_{j}^{\prime\prime}}{w_{j}}c_{j}}\nabla
u||\nabla\sqrt{c_{j}}|\lesssim\mathscr{P}(z).$ (4.19)

For $j=0$, we have
$p_{00}^{(0)}=\theta_{L}^{\prime}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))D_{00}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})a(z)|\nabla
u|^{2}\geq 0$, and since in the equation it comes with a minus sign, its
integral can easily be handled using Fatou’s lemma.

It remains to estimate the part $p_{0j}^{(1)}$. To deal with the limit
$\tilde{\varepsilon}\to 0$ (for finite $L$), we estimate

$\displaystyle|\nabla
D_{0}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\cdot m(z)\nabla
D_{0}h(z)|$ $\displaystyle\lesssim\sqrt{m}|\nabla
D_{0}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})|\sqrt{\mathscr{P}(z)}$
(4.20)

and for $|z^{\tilde{\varepsilon}}|\leq C(L,\delta)$

$\displaystyle\sqrt{m}|\nabla
D_{0}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})|$
$\displaystyle\leq|D_{00}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\nabla
u|+\sum_{j=1}^{n}|D_{0j}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\nabla
c_{j}|$ (4.21) $\displaystyle\lesssim_{L,\delta,\varepsilon}|\nabla
u|+\sqrt{\mathscr{P}(z)}.$ (4.22)

Thus the limit $\tilde{\varepsilon}\to 0$ can be handled using dominated
convergence.

We can now let $\tilde{\varepsilon}=0$ and compute

$\displaystyle\nabla$ $\displaystyle D_{0}h_{\delta,\varepsilon}(z)\cdot
m(z)\nabla D_{0}h(z)$ (4.23) $\displaystyle=m(z)|\nabla
D_{0}h(z)|^{2}-(\hat{\sigma}^{\prime\prime}(u+\varepsilon)-\hat{\sigma}^{\prime\prime}(u))\nabla
u\cdot m(z)\nabla D_{0}h(z).$ (4.24)

Note that the first term on the RHS is bounded above by $\mathscr{P}(z)\in
L^{1}(\Omega_{T})$. Concerning the second term, we estimate

$\displaystyle|(\hat{\sigma}^{\prime\prime}(u+\varepsilon)-\hat{\sigma}^{\prime\prime}(u))$
$\displaystyle\nabla u\cdot m(z)\nabla D_{0}h(z)|$ (4.25)
$\displaystyle\lesssim\sqrt{m}|(\hat{\sigma}^{\prime\prime}(u+\varepsilon)-\hat{\sigma}^{\prime\prime}(u))\nabla
u|\sqrt{\mathscr{P}(z)}$ (4.26)
$\displaystyle\lesssim|(1+\sqrt{-\hat{\sigma}^{\prime\prime}(u)})\nabla
u|\sqrt{\mathscr{P}(z)}$ (4.27) $\displaystyle\lesssim\mathscr{P}(z)+|\nabla
u|^{2},$ (4.28)

where in the penultimate step we have used hypothesis (2.17).

Combining the above estimates allows to take the successive limits

$\limsup_{\varepsilon,\delta\to
0}\limsup_{L\to\infty}\limsup_{\tilde{\varepsilon}\to 0}\dots$

as above, thus yielding the asserted inequality for term $I$.

_Remainder gradient term II:_
We will show that

$\limsup_{L\to\infty}\limsup_{\tilde{\varepsilon}\to
0}\Big{(}-\int_{0}^{T}\\!\\!\int_{\Omega}\theta_{L}^{\prime\prime}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))D_{j}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\nabla
z_{j}\cdot D_{i}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})A_{ik}(z)\nabla
z_{k}\,\differential x\differential t\Big{)}\leq 0.$ (4.29)

As in the previous paragraph, the main task is to obtain uniform pointwise
estimates on the terms involved, where here we can afford a dependence of our
estimates on $\delta$ and $\varepsilon$. We introduce for $i,j$ fixed the term

$\displaystyle
q_{ij}=-\theta_{L}^{\prime\prime}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))D_{j}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\nabla
z_{j}\cdot
D_{i}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\sum_{k}A_{ik}(z)\nabla
z_{k}$ (4.30)

and observe that, by (4.2) and (4.3), for $L\geq L_{0}(\delta)$ (henceforth to
be assumed)

$\displaystyle|\theta^{\prime\prime}_{L}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))|\lesssim_{\delta}\tfrac{1}{1+(u+\sum_{i}c_{i}\log_{+}(c_{i}))\log(|z|_{1}+\mathrm{e})}.$
(4.31)

Case $i,j\geq 1:$ in this case,

$\displaystyle|q_{ij}|\lesssim|\theta^{\prime\prime}_{L}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))||\log(\tfrac{c_{i}+\tilde{\varepsilon}}{w_{i}(u)})||\log(\tfrac{c_{j}+\tilde{\varepsilon}}{w_{j}(u)})|\sqrt{c_{i}}\sqrt{c_{j}}|\nabla\sqrt{c_{j}}|\sqrt{\mathscr{P}(z)}.$
(4.32)

Estimating

$\displaystyle\sqrt{c_{i}}|\log(\tfrac{c_{i}+\tilde{\varepsilon}}{w_{i}(u)})|\lesssim\sqrt{c_{i}}\log_{+}(c_{i})+\sqrt{w_{i}(u)}\log_{+}(w_{i}(u))+1,$
(4.33)

we find, using (h2) and (4.31),

$|\theta^{\prime\prime}_{L}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))|\,\sqrt{c_{i}}|\log(\tfrac{c_{i}+\tilde{\varepsilon}}{w_{i}(u)})|\,\sqrt{c_{j}}|\log(\tfrac{c_{j}+\tilde{\varepsilon}}{w_{j}(u)})|\\\
\lesssim|\theta^{\prime\prime}_{L}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))|(\sqrt{c_{i}}\log_{+}(c_{i})+\sqrt{u}+1)(\sqrt{c_{j}}\log_{+}(c_{j})+\sqrt{u}+1)\;\lesssim_{\delta}\;1.\quad$
(4.34)

Thus, $|q_{ij}|\lesssim_{\delta}\mathscr{P}(z)$.

Case $i\geq 1,j=0:$ here,

$\displaystyle|q_{ij}|$
$\displaystyle\lesssim\sqrt{c_{i}}|\log(\tfrac{c_{i}+\tilde{\varepsilon}}{w_{i}(u)})||D_{0}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\nabla
u|\sqrt{\mathscr{P}(z)}\,|\theta^{\prime\prime}_{L}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))|.$
(4.35)

In view of the factor $\sqrt{c_{i}},$ this shows that $|q_{ij}|\leq
C(L,\delta,\varepsilon)|\nabla u|\sqrt{\mathscr{P}(z)}$ uniformly in
$\tilde{\varepsilon}$, allowing us to infer by dominated convergence

$\displaystyle\limsup_{\tilde{\varepsilon}\to
0}\int_{0}^{T}\\!\\!\int_{\Omega}q_{ij}\,\differential x\differential
t\leq-\int_{0}^{T}\\!\\!\int_{\Omega}\theta_{L}^{\prime\prime}(h_{\delta,\varepsilon}(z))D_{0}h_{\delta,\varepsilon}(z)\nabla
u\cdot\log(\tfrac{c_{i}}{w_{i}(u)})A_{ik}(z)\nabla z_{k}\,\differential
x\differential t.$ (4.36)

Since $\varepsilon>0$ and
$\tfrac{w^{\prime}_{l}(u)}{w_{l}(u)}\lesssim\sqrt{\tfrac{-w^{\prime\prime}_{l}(u)}{w_{l}(u)}}$,
we have the rough bound

$\begin{split}|D_{0}h_{\delta,\varepsilon}(z)\nabla
u|&\lesssim_{\varepsilon}|\nabla u|+\sum_{l}\sqrt{c_{l}}\sqrt{\gamma}|\nabla
u|\\\ &\lesssim|\nabla
u|+\sum_{l}\sqrt{c_{l}}\sqrt{\mathscr{P}(z)}.\end{split}$ (4.37)

Moreover,

$\displaystyle|\log(\tfrac{c_{i}}{w_{i}(u)})A_{ik}(z)\nabla
z_{k}|\lesssim(\log_{+}(c_{i})+1+\log_{+}(w_{i}(u)))\sqrt{c_{i}}\sqrt{\mathscr{P}(z)},$
(4.38)

and hence

$\displaystyle|\theta_{L}^{\prime\prime}(h_{\delta,\varepsilon}(z))$
$\displaystyle D_{0}h_{\delta,\varepsilon}(z)\nabla u\cdot
D_{i}h_{\delta,\varepsilon}(z)A_{ik}(z)\nabla
z_{k}|\lesssim_{\varepsilon}\mathscr{P}(z)+|\nabla u|^{2}.$ (4.39)

Case $i=0,j\geq 0:$ using the fact that for $|z|_{1}\leq C(L,\delta)$

$\displaystyle|D_{j}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\nabla
z_{j}||D_{0}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})||a(z)\nabla
u+m(z)\nabla D_{0}h(z)|$
$\displaystyle\lesssim_{L,\delta,\varepsilon}\mathscr{P}(z)+|\nabla u|^{2},$
(4.40)

one can take the limit $\tilde{\varepsilon}\to 0$

$\displaystyle\limsup_{\tilde{\varepsilon}\to 0}\big{(}$
$\displaystyle-\int_{0}^{T}\\!\\!\int_{\Omega}\theta_{L}^{\prime\prime}(h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}}))D_{j}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\nabla
z_{j}\cdot
D_{0}h_{\delta,\varepsilon}(z^{\tilde{\varepsilon}})\sum_{k}A_{0k}(z)\nabla
z_{k}\,\differential x\differential t\big{)}$ (4.41)
$\displaystyle\leq-\int_{0}^{T}\\!\\!\int_{\Omega}\theta_{L}^{\prime\prime}(h_{\delta,\varepsilon}(z))D_{j}h_{\delta,\varepsilon}(z)\nabla
z_{j}\cdot D_{0}h_{\delta,\varepsilon}(z)\sum_{k}A_{0k}(z)\nabla
z_{k}\,\differential x\differential t.$ (4.42)

To obtain $L$-uniform bounds of the integrand on the RHS, we split

$\sum_{k}A_{0k}(z)\nabla z_{k}=a(z)\nabla u+m(z)\nabla D_{0}h(z).$

Using (4.37) we have for $j\geq 1$

$|\nabla
z_{j}D_{j}h_{\delta,\varepsilon}(z)||D_{0}h_{\delta,\varepsilon}(z)a(z)\nabla
u|\\\
\lesssim_{\varepsilon}\sqrt{c_{j}}(\log_{+}(c_{j})+1+\log_{+}(w_{j}(u)))(|\nabla
u|\sqrt{\mathscr{P}(z)}+\sum_{l}\sqrt{c_{l}}\mathscr{P}(z)),$ (4.43)

and for $j=0$

$\displaystyle|\nabla
uD_{0}h_{\delta,\varepsilon}(z)||D_{0}h_{\delta,\varepsilon}(z)a(z)\nabla
u|\lesssim_{\epsilon}\big{(}|\nabla
u|+\sqrt{|c|_{1}}\sqrt{\mathscr{P}(z)}\big{)}^{2}\lesssim|\nabla
u|^{2}+|c|_{1}\mathscr{P}(z).$ (4.44)

It remains to consider the term involving $m(z)$. We estimate for $j\geq 1$

$|D_{j}h_{\delta,\varepsilon}(z)\nabla z_{j}\cdot
D_{0}h_{\delta,\varepsilon}(z)||m(z)\nabla D_{0}h(z)|\\\
\lesssim_{\varepsilon}\sqrt{c_{j}}(\log_{+}(c_{j})+1+\log_{+}(w_{j}(u)))(1+\tfrac{\sum_{l}c_{l}}{1+u})\sqrt{m}\mathscr{P}(z),$
(4.45)

where we used the fact that
$\tfrac{w_{l}^{\prime}(u)}{w_{l}(u)}\lesssim\tfrac{1}{1+u}$ for all $l$
allowing us to estimate

$\displaystyle|D_{0}h_{\delta,\varepsilon}(z)|\lesssim_{\varepsilon}1+\tfrac{\sum_{l=1}^{n}c_{l}}{1+u}.$
(4.46)

Similarly, for $j=0$ we estimate, using also the bound (4.37),

$|D_{0}h_{\delta,\varepsilon}(z)\nabla u\cdot
D_{0}h_{\delta,\varepsilon}(z)||m(z)\nabla D_{0}h(z)|\\\
\lesssim_{\varepsilon}(1+\tfrac{\sum_{l^{\prime}=1}^{n}c_{l^{\prime}}}{1+u})\big{(}|\nabla
u|+\sum_{l=1}^{n}\sqrt{c_{l}}\sqrt{\mathscr{P}(z)}\big{)}\sqrt{m}\sqrt{\mathscr{P}(z)}.$
(4.47)

Thus, recalling the conditions (2.16), (2.17) on $m(z)$, we infer the
$L$-uniform bound

$\displaystyle|\theta_{L}^{\prime\prime}(h_{\delta,\varepsilon}(z))D_{j}h_{\delta,\varepsilon}(z)\nabla
z_{j}\cdot D_{0}h_{\delta,\varepsilon}(z)\sum_{k}A_{0k}(z)\nabla
z_{k}|\lesssim_{\delta,\varepsilon}\mathscr{P}(z)+|\nabla u|^{2}.$ (4.48)

Combining the above estimates, we can take the limits

$\limsup_{L\to\infty}\limsup_{\tilde{\varepsilon}\to 0}\dots$

of term $II$ and obtain ineq. (4.29) by the pointwise convergence
$\theta_{L}^{\prime\prime}(s)\to 0$ as $L\to\infty$.

_Reactions III:_
Concerning the reaction term $III$, we first need to take care of the fact