# Characterisations for the depletion of reactant in a one-dimensional dynamic
combustion model

Siran Li  Siran Li: School of Mathematical Sciences $\&$ CMA-Shanghai,
Shanghai Jiao Tong University, No. 6 Science Buildings, 800 Dongchuan Road,
Minhang District, Shanghai, China (200240<EMAIL_ADDRESS>and  Jianing
Yang  Jianing Yang: School of Mathematical Sciences, Shanghai Jiao Tong
University, No. 6 Science Buildings, 800 Dongchuan Road, Minhang District,
Shanghai, China (200240<EMAIL_ADDRESS>
###### Abstract.

In this paper, a novel observation is made on a one-dimensional compressible
Navier–Stokes model for the dynamic combustion of a reacting mixture of
$\gamma$-law gases ($\gamma>1$) with discontinuous Arrhenius reaction rate
function, on both bounded and unbounded domains. We show that the mass
fraction of the reactant (denoted as $Z$) satisfies a weighted gradient
estimate $Z_{y}/\sqrt{Z}\in L^{\infty}_{t}L^{2}_{y}$, provided that at time
zero the density is Lipschitz continuous and bounded strictly away from zero
and infinity. Consequently, the graph of $Z$ cannot form cusps or corners near
the points where the reactant in the combustion process is completely depleted
at any instant, and the entropy of $Z$ is bounded from above. The key
ingredient of the proof is a new estimate based on the Fisher information,
first exploited by [7, 2] with applications to PDEs in chemorepulsion and
thermoelasticity. Along the way, we also establish a Lipschitz estimate for
the density.

###### Key words and phrases:

Compressible fluid; Navier–Stokes equations; dynamic combustion; reacting
mixture; Fisher information.

##  0\. Introduction

###  0.1. The model

This paper is devoted to the analysis of a one-dimensional (1D) compressible
Navier–Stokes model for a reacting mixture in dynamic combustion with
discontinuous reacting rate function. This system of partial differential
equations (PDEs), written in the following form, was first studied by Chen [3]
in 1992 and has been an active problem of research ever since (_cf_. [6, 5, 9,
10, 11, 22, 31, 32, 41, 38, 28, 33, 37, 13, 12, 30, 42, 40, 24, 35]):

$\begin{cases}\rho_{t}+(\rho u)_{x}=0,\\\ (\rho u)_{t}+(\rho u^{2}+p)_{x}=(\mu
u_{x})_{x},\\\ (\rho E)_{t}+[u(\rho E+p)]_{x}=(\mu
uu_{x})_{x}+(\nu\theta_{x})_{x}+(qD\rho Z_{x})_{x},\\\ (\rho Z)_{t}+(\rho
uZ)_{x}=-K\phi(\theta)\rho Z+(D\rho Z_{x})_{x}.\end{cases}$ (0.1)

See also Williams [34], Majda [25], Glimm [14], Collela–Majda–Roytburd [8],
and many subsequent developments for mathematical models of dynamic
combustion.

In Eq. (0.1): $\rho$, $u$, $\theta$, $E$, and $Z$ denote respectively the
density, velocity, temperature, total specific energy, and mass fraction of
the reactant. The _positive_ constants $\mu$, $\nu$, $D$, and $K$ are the
coefficients of bulk viscosity, heat conduction, diffusion, and rate of
reactant, respectively. The total specific energy $E$ is given by

$\displaystyle E=e+\frac{u^{2}}{2}+qZ,$ (0.2)

where $e$ is the specific internal energy and $q>0$ is the difference between
heat of formation of the reactant and the product. For an ideal gas mixture
with the same $\gamma$-law ($\gamma>1$), one has the constitutive relations:

$\displaystyle e=\frac{p}{\rho(\gamma-1)},\qquad\theta=\frac{p}{\rho RM},$

where $R$ is the Boltzmann constant and $M$ is the molecular weight. We
abbreviate

$a=RM$

in the sequel, which is a positive constant.

The reaction rate function $\phi$ in Eq. (0.1) obeys the _Arrhenius’ law_ :

$\phi(\theta)=\begin{cases}\theta^{\alpha}e^{-A/\theta}\qquad\text{ for
}\theta>\theta_{\rm ignite}>0,\\\ 0\qquad\text{ for }\theta<\theta_{\rm
ignite}.\end{cases}$ (0.3)

Here, $\theta_{\rm ignite}$ is the ignition temperature at which the
combustion reaction is initiated; $A>0$ is known as the activation energy.
Note that $\phi\in L^{\infty}(]0,\infty[)$ with a jump discontinuity at
$\theta_{\rm ignite}$.

Consider as in [3, Eqs. (1.5)–(1.8)] the initial-boundary value problem (IBVP)
of Eq. (0.1) in the space-time domain $[0,T]\times\Omega$ with impermeably
insulated boundaries:

$\begin{cases}(\rho,u,T,Z)\big{|}_{t=0}=\big{(}\rho_{0}(x),u_{0}(x),T_{0}(x),Z_{0}(x)\big{)}\qquad\text{
for all }x\in\Omega,\\\
\left(u,\theta_{x},Z_{x}\right)\big{|}_{\partial\Omega}=0\qquad\text{ for all
}t\in[0,T],\\\ 0<m_{0}\leq\rho_{0}(x),\,\theta_{0}(x)\leq
M_{0}<\infty\quad\text{and}\quad 0\leq Z_{0}(x)\leq 1\qquad\text{ for all
}x\in\Omega,\end{cases}$ (0.4)

where $m_{0}$ and $M_{0}$ are positive constants.

The initial-boundary data in Eq. (0.4) are subject to the natural
compatibility conditions which, in the case of $\gamma$-law ideal gases, read
as follows:

$\displaystyle\Big{(}u_{0},\,(\theta_{0})_{x},\,(Z_{0})_{x},\,a\rho_{0}\theta_{0}-\big{[}\mu(u_{0})_{x}\big{]}_{x}\Big{)}\Big{|}_{\partial\Omega}={\bf
0}.$ (0.5)

######  Remark 0.1.

The arguments in this paper rely crucially on the _Neumann_ boundary condition
for $Z$. It is also important to assume that the diffusion constant $D$ is
strictly positive.

We investigate PDEs (0.1) and (0.4) (subject to Eq. (0.5)) on both bounded
domain (without loss of generality, $\Omega=[0,1]$) and unbounded domain
$\Omega=\mathbb{R}$. In the latter case, the boundary conditions — _i.e._ ,
the second line in Eq. (0.4) — are understood as the far-field conditions:

$\displaystyle\lim_{|x|\to\infty}\left(u,\theta_{x},Z_{x}\right)=0\qquad\text{
for all }t\in[0,T].$ (0.6)

###  0.2. Lagrangian coordinates

As is often the case for 1D compressible fluid dynamics, the Lagrangian
formulation of Eq. (0.1) proves to be convenient for PDE analysis. We shall
reserve the symbol $(t,y)$ for the Lagrangian coordinates, while $(t,x)$
always designates the Eulerian coordinates as in §0.1 above. They are related
by

$\displaystyle y=\int_{x(t)}^{x}\rho(t,s)\,{\rm d}s\qquad$ where $x(t)$ is the
particle $\displaystyle\text{trajectory obeying the ODE
}x^{\prime}(t)=u\big{(}x(t),t\big{)}.$

Here and hereafter, denote by

$\displaystyle v:=\frac{1}{\rho}$

the specific volume of the gas mixture.

Eq. (0.1) in Lagrangian coordinates $(t,y)$ can be expressed as follows:

$\left\\{\begin{aligned} &v_{t}-u_{y}=0,\\\
&u_{t}+\left(\frac{a\theta}{v}\right)_{y}=\left(\frac{\mu
u_{y}}{v}\right)_{y},\\\
&\left(\theta+\frac{u^{2}}{2}\right)_{t}+\left(\frac{au\theta}{v}\right)_{y}=\left(\frac{\mu
uu_{y}}{v}\right)_{y}+\left(\frac{\nu\theta_{y}}{v}\right)_{y}+qK\Phi(\theta)Z,\\\
&Z_{t}+K\Phi(\theta)Z=\left(\frac{D}{v^{2}}Z_{y}\right)_{y}.\end{aligned}\right.$
(0.7)

This is a PDE system for $(v,u,\theta,Z)$ in $[0,T]\times\Omega$.

In [3, §2, Theorem 1.1], Chen established the equivalence of generalised
solutions to Eqs. (0.1) and (0.7), adapting the arguments in [36] by Wagner
for 1D compressible Navier–Stokes equations. More precisely, it is proved that
the transform from Eulerian to Lagrangian coordinates induces a one-to-one
correspondence between the generalised solutions to Eqs. (0.1) and (0.7), for
which $\rho$ (or, equivalently, $v$) is essentially bounded away from zero and
infinity. Therefore, throughout this paper, we shall work with Eq. (0.7) in
the Lagrangian coordinates, equipped with the initial-boundary conditions in
Eq. (0.4).

The notion of generalised solution in this work is defined as in [3, p.611].

######  Definition 0.2.

Let $\Omega=[0,1]$ or $]-\infty,\infty[$. For any given $T>0$, the quadruplet

$(v,u,\theta,Z):[0,T]\times\Omega\to\mathbb{R}^{4}$

is a generalised solution to the initial-boundary value problem (0.7) $\&$
(0.4) if

  * •

Eq. (0.7) holds almost everywhere on $[0,T]\times\Omega$;

  * •

one has the following estimates:

$\left\\{\begin{aligned} &(u,\theta-1,Z)\in
L^{\infty}\big{(}0,T;H^{1}(\Omega)\big{)}\cap
L^{2}\big{(}0,T;H^{2}(\Omega)\big{)},\\\ &(v_{t},u_{t},\theta_{t},Z_{t})\in
L^{2}\big{(}0,T;L^{2}(\Omega)\big{)},\\\ &v^{-1}-1\in
L^{\infty}\big{(}0,T;H^{1}(\Omega)\big{)};\end{aligned}\right.$

and

  * •

initial conditions in Eq. (0.4) hold in the sense of trace in the above
function spaces.

Throughout, for our PDE system in consideration, the initial-boundary
conditions (0.4) are always assumed to satisfy the compatibility condition
(0.5).

###  0.3. Known results

Various results concerning the well-posedness, stability, and asymptotic
behaviour of the 1D dynamic combustion model, namely Eqs. (0.7) $\&$ (0.4),
have been established in the literature.

In [3], Chen proved the existence of generalised solutions on bounded domain
$\Omega=[0,1]$, with quantitative bounds in Eq. (3.33) therein. Moreover,
large-time behaviour as follows was proved in [3, Theorem 5(A)]:

$\displaystyle\lim_{t\to\infty}\left\|\left(v-\int_{0}^{1}v_{0}(y)\,{\rm
d}y,\,u,\,\theta-\theta^{\infty},\,Z-Z^{\infty}\right)\right\|_{H^{1}(\Omega)}=0,$

with the constant asymptotic states $\theta^{\infty}$, $Z^{\infty}$ determined
via

$\theta^{\infty}+qZ^{\infty}=\int_{0}^{1}\left\\{\theta_{0}(y)+qZ_{0}(y)+\frac{u_{0}^{2}(y)}{2}\right\\}\,{\rm
d}y$

and

$\phi\left(\theta^{\infty}\right)Z^{\infty}=0.$

More refined decay estimates have been obtained by Wang–Wen [37] and Wang–Wu
[38], among other works. The first named author in [22] generalised the
aforementioned existence and asymptotic results in [3] to the unbounded domain
case $\Omega=\mathbb{R}$. Other types of boundary conditions have also been
considered in [3, 22, 37, 38].

For closely related works, we refer to Feng–Zhang–Zhu [12], Feng–Hong–Zhu
[13], Meng [28], Peng [30], Li–Peng [31, 32, 33], Xu–Feng [40], and Yin–Zhu
[41], etc. for asymptotic and stability analysis on the combustion model with
different initial and/or boundary data, to Chen–Kratka [4], Liao–Wang–Zhao
[24], and Zhu [42], etc. for studies on spherically symmetric reacting flows
in multi-dimensions, and to Chen–Hoff–Trivisa [5] for analysis of a two-
species model.

The above list of references is by no means exhaustive.

###  0.4. Main theorem and consequences

This note aims at providing a novel estimate for the mass fraction of
reactant, _i.e._ , the function $Z$ in Eq. (0.1) or Eq. (0.7). As a
consequence, we show that $Z$ cannot be merely Hölder or Lipschitz near its
nodal set (_a.k.a._ zero locus) $Z^{-1}{\\{0\\}}$ at any instant, so the
reactant cannot be depleted abruptly. This, to the best of the authors’
knowledge, is among the very first refined characterisations of solutions to
the 1D dynamic combustion model — Eqs. (0.7) $\&$ (0.4) — apart from those
obtained via more conventional $L^{2}$-based energy estimates.

Our main theorem is as follows.

######  Theorem 0.3.

Let $Z$ be the mass fraction of the reactant in a generalised solution to the
1D dynamic combustion model Eqs. (0.7) $\&$ (0.4) in $[0,T]\times\Omega$,
where $T>0$ and $\Omega=[0,1]$ or $\mathbb{R}$. Suppose the initial data
satisfy, in addition to the conditions in Eq. (0.4), that

$(v_{0}(y),u_{0}(y),\theta_{0}(y),Z_{0}(y))\in\left[H^{1}(\Omega)\right]^{4}\quad\text{and}\quad
v_{0}\in W^{1,\infty}(\Omega).$

Then we have the estimate

$\displaystyle\sup_{t\in[0,T]}\left\\{\int_{\Omega}\frac{\left|Z_{y}\right|^{2}}{Z}\,{\rm
d}y\right\\}\leq\Lambda<\infty,$

where $\Lambda$ is a uniform constant as in Notation 0.4 below.

######  Notation 0.4.

We reserve the symbol $\Lambda$ for positive constants depending only on the
physical parameters $a$, $q$, $m_{0}$, $M_{0}$, $D$, $K$, $\mu$, and $\nu$;
the lifespan $T$; the $L^{\infty}$-norm of the discontinuous reaction rate
function $\phi$; as well as the natural norm of the initial data

$\left\|\big{(}v_{0},u_{0},\theta_{0},Z_{0}\big{)}\right\|_{[H^{1}(\Omega)\cap
W^{1,\infty}(\Omega)]\times[H^{1}(\Omega)]^{3}}.$

It may change from line to line.

By Definition 0.2, $Z\in L^{\infty}\big{(}0,T;H^{1}(\Omega)\big{)}$ and
$Z_{t}\in L^{2}\big{(}0,T;L^{2}(\Omega)\big{)}$ whenever $(v,u,\theta,Z)$ is a
generalised solution. Theorem 0.3 does not provide new information on
$\\{Z\geq b_{0}\\}$ for positive definite $b_{0}>0$. It instead gives refined
characterisations for $Z$ near its nodal set $Z^{-1}\\{0\\}$; _i.e._ , where
the reactant of the dynamic combustion is depleted. This explains the title of
the paper.

The study of $Z^{-1}\\{0\\}$ is an interesting topic and appears by now
elusive in the literature. In contrast, the other _a priori_ non-negative
variables $v$ and $\theta$ are shown to be strictly positive. More precisely,
it is proved that

$c\leq\theta,\,v\leq C\text{ for strictly positive uniform constants }0<c\leq
C<\infty.$

See [3, Lemmata 3, 4, and 6] and [22, Theorem 3.1, Lemma 5.1, and Eq. (5.25)].
In physical terms, the gas mixture in dynamic combustion cannot develop vacua
or concentrations, and the temperature remains finite and cannot approach
absolute zero. For $Z$, it has been shown by maximum principle arguments that
$0\leq Z\leq 1$ (_cf_. [3, Lemma 2] and [22, Lemma 2.2]. Strictly speaking,
these inequalities only hold almost everywhere; see however Remark 0.5 below),
but we have had no knowledge about the set of degeneracy $Z^{-1}\\{0\\}$.

######  Remark 0.5.

By virtue of the Aubin–Lions lemma, $Z$ equals almost everywhere to a function

$\tilde{Z}\in\bigcap_{\gamma\in[0,1[}C^{0}\big{(}[0,T];H^{\gamma}(\Omega)\big{)}.$

In the sequel, we say that $\tilde{Z}$ is a _continuous (in-time)
representative_ of $Z$ and identify $\tilde{Z}\equiv Z$ without relabelling.

Under the above provision, we readily deduce the following from Theorem 0.3.

######  Corollary 0.6.

Near any $y_{\star}\in Z^{-1}{\\{0\\}}$ and at any time $t\in[0,T]$, the
function $Z$ cannot be locally of the form

$\displaystyle Z(t,y)=C\left|y-y_{\star}\right|^{\beta}+{\rm
l.o.t.}\qquad\text{for any }C\in\mathbb{R},\,\beta\in[0,1].$

Here ${\rm l.o.t.}$ denotes a (smooth) perturbation of magnitude
$\sim\mathfrak{o}\left(\left|y-y_{\star}\right|^{\beta}\right)$ as $y\to
y_{\star}$. The constants $C$ and $\beta$ and the term ${\rm l.o.t.}$ may
depend on time.

In view of Definition 0.2 of the generalised solution, one has $Z\in
L^{2}\big{(}0,T;H^{2}(\Omega)\big{)}$, which implies by Sobolev–Morrey
embedding that $Z\in L^{2}\big{(}0,T;C^{\frac{3}{2}}(\Omega)\big{)}$. This of
course shows that $Z(t,\bullet)\in C^{\frac{3}{2}}(\Omega)$ for almost every
$t\in[0,T]$. Nonetheless, it leaves open the possibility that
$Z|_{\mathcal{N}\times\Omega}$ exhibits rather singular behaviours on some
Lebesgue null set $\mathcal{N}\subset[0,T]$. ($\mathcal{N}$ does not even need
to be discrete here; for example, it may be some fat Cantor sets.)

In this context, Corollary 0.6 rules out, at any instant, the local geometries
of the graph of $Z$ near its nodal set as shown in Figure 0.1, where
$y_{0},y_{1},y_{2}\in Z^{-1}\\{0\\}$. In other words, the graph of $Z$ —
understood as its continuous representative as before — cannot instantaneously
develop cusps or corners at any time on $[0,T]$.

Figure 0.1. Inadmissible shapes of $Z$, the mass fraction of reactant

###### Proof of Corollary 0.6.

The arguments below are local — _i.e._ , only pertaining to arbitrarily small
neighbourhoods of $Z^{-1}{\\{0\\}}$. So we may treat the cases $\Omega=[0,1]$
and $\mathbb{R}$ at one stroke.

If $\beta\in[-0,1[$, then $Z=\left|y-y_{\star}\right|^{\beta}$ (without loss
generality, assuming $C=1$ and ${\rm l.o.t.}=0$ here) leads to
${Z_{y}^{2}}/{Z}=\beta^{2}\left|y-y_{\star}\right|^{\beta-2}$, which is not in
$L^{1}$ locally about $y_{\star}$.

If $\beta=1$ and $Z=\left|y-y_{\star}\right|$, then $Z_{y}^{2}=1$ on
$\Omega\setminus\\{y_{\star}\\}$, so almost everywhere
$Z_{y}^{2}/Z=\left|y-y_{\star}\right|^{-1}$, which again is not in $L^{1}_{\rm
loc}(\Omega)$. ∎

Another consequence of Theorem 0.3 is the boundedness of _entropy_ :

######  Corollary 0.7.

Let $Z$ be the mass fraction of the reactant in a generalised solution to the
1D dynamic combustion model Eqs. (0.7) $\&$ (0.4) in $[0,T]\times\Omega$,
where $T>0$ and $\Omega=[0,1]$. Then, the entropy of $Z$ with respect to the
Lebesgue measure on $[0,1]$, defined by

$\displaystyle{\rm Ent}_{\mathcal{L}^{1}\mathbin{\vrule
height=4.82224pt,depth=0.0pt,width=0.39182pt\vrule
height=0.39182pt,depth=0.0pt,width=3.91806pt}[0,1]}(Z):=\int_{0}^{1}Z\log
Z\,{\rm d}y-\left(\int_{0}^{1}Z\,{\rm d}y\right)\log\left(\int_{0}^{1}Z\,{\rm
d}y\right),$

is bounded from above on $[0,T]$ by a uniform constant, depending on the same
parameters as $\Lambda_{0}$ in Theorem 0.3.

Note that ${\rm Ent}_{\mathcal{L}^{1}\mathbin{\vrule
height=4.82224pt,depth=0.0pt,width=0.39182pt\vrule
height=0.39182pt,depth=0.0pt,width=3.91806pt}[0,1]}(Z)\geq 0$ by convexity;
here we understand $0\log 0=0$. For the unbounded domain case
$\Omega=\mathbb{R}$, one may replace the entropy by the following variant:

$\displaystyle{\rm Ent}_{\gamma}(Z):=\int_{0}^{1}Z\log Z\,{\rm
d}\gamma(y)-\left(\int_{0}^{1}Z\,{\rm
d}\gamma(y)\right)\log\left(\int_{0}^{1}Z\,{\rm d}\gamma(y)\right),$

where $\gamma$ is the Gaussian probability measure ${\rm
d}\gamma(y)=(2\pi)^{-1/2}\exp\\{-y^{2}/2\\}\,{\rm d}y$ on $\mathbb{R}$.

###### Proof of Corollary 0.7.

It follows immediately from Theorem 0.3 and the (tight) logarithmic Sobolev
inequality

${\rm Ent}_{\mathcal{L}^{1}\mathbin{\vrule
height=4.82224pt,depth=0.0pt,width=0.39182pt\vrule
height=0.39182pt,depth=0.0pt,width=3.91806pt}[0,1]}(Z)\leq
c_{0}\int_{0}^{1}\frac{Z_{y}^{2}}{Z}\,{\rm d}y.$

Here $c_{0}$ is a universal constant determined, for example, from the
curvature-dimension condition of $\mathcal{L}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}[0,1]$ on $\mathbb{R}$. See
Bakry–Gentil–Ledoux [1, §5.1.1]. ∎

The key ingredient of the proof for our main Theorem 0.3 is an inequality
based on Fisher information. See Lemma 2.1. This inequality was proved in the
recent preprint [7] on a 3D chemorepulsion system by
Cieślak–Fuest–Hajduk–Sierżȩga, and later applied to 1D thermoelasticity PDE
models by Bies–Cieślak in [2]. In addition, careful uniform _a priori_
estimates on the generalised solution $(v,u,\theta,Z)$ are performed to deal
with the discontinuity of the reaction rate function $\phi$.

###  0.5. Organisation

The remaining sections of the paper are organised as follows.

In §1, we collect several uniform _a priori_ bounds for the generalised
solution, which are crucial for later developments. Then, in §2 we introduce
the Fisher information-based inequality. This is the key tool for this paper.
§§3 $\&$ 4 are devoted to the proof of the main Theorem 0.3, in the cases of
$\Omega=[0,1]$ and $\Omega=\mathbb{R}$, respectively.

##  1\. Preliminary estimates

In this section, we collect from the existing literature (especially Chen [3];
Jiang [15]) several uniform estimates for the dynamic combustion model Eqs.
(0.7) $\&$ (0.4), which are used to establish existence results — summarised
as Proposition 1.1 below — for generalised solutions. We also fix some
notations that will be used throughout this work.

######  Proposition 1.1.

Suppose that the initial data

$(v_{0}(y),u_{0}(y),\theta_{0}(y),Z_{0}(y))\in\left[H^{1}(\Omega)\right]^{4}$

satisfy Eqs. (0.4) $\&$ (0.5). There exists a generalised solution
$(v,u,\theta,Z)$ to Eqs. (0.7) $\&$ (0.4).

######  Remark 1.2.

For the existence of generalised solutions, one does not need to assume the
stronger condition $v_{0}\in W^{1,\infty}(\Omega)$ as in the main Theorem 0.3.
$H^{1}$-regularity will suffice.

###  1.1. Regularised system

The proof of Proposition 1.1 in [3, 22] relies on the $C^{1}$-regularisation
of the discontinuous reaction rate function $\phi(\bullet)$ defined in Eq.
(0.3), which shall be described as follows for convenience of the readers.

Define the _nondecreasing_ function

$\displaystyle\Phi_{\epsilon}(\theta)=\left\\{\begin{aligned}
&\theta^{\alpha}e^{-A/\theta},\quad\theta\geq\theta_{\rm ignite}+\epsilon,\\\
&0,\qquad\theta\leq\theta_{\rm
ignite}-\epsilon\end{aligned}\right.\qquad\text{and}\qquad\|\Phi_{\epsilon}\|_{C^{1}}\leq\frac{1}{\epsilon}.$
(1.1)

One obtains uniform estimates independently of the regularisation parameter
$\varepsilon>0$, and then passes to limits in suitable Sobolev spaces by
sending $\varepsilon\searrow 0$.

The aforementioned uniform estimates are naturally derived in the (parabolic)
Hölder framework. We adopt the following notations from [3]. Let
$Q_{T}=[0,T]\times\Omega$ and set

$\left\\{\begin{aligned} &B^{\sigma/2,\sigma}(Q_{T})=\left\\{u\in
C^{0}(Q_{T}):\sup_{Q_{T}}|u|+\sup_{t\neq s,x\neq
y}\frac{|u(t,x)-u(s,y)|}{|t-s|^{\sigma/2}+|x-y|^{\sigma}}<\infty\right\\},\\\
&B^{1+\sigma}(Q_{T})=\left\\{u\in B^{\sigma/2,\sigma}(Q_{T}):u_{t},u_{y}\in
B^{\sigma/2,\sigma}(Q_{T})\right\\},\\\ &B^{2+\sigma}(Q_{T})=\left\\{u\in
B^{\sigma/2,\sigma}(Q_{T}):u_{t},u_{y},u_{yy}\in
B^{\sigma/2,\sigma}(Q_{T})\right\\}.\end{aligned}\right.$

We denote the norms $\|\bullet\|_{1+\sigma}:=\|\bullet\|_{B^{1+\sigma}}$ and
$\|\bullet\|_{2+\sigma}:=\|\bullet\|_{B^{2+\sigma}}$, where $T>0$ is
suppressed.

In the case $\Omega=[0,1]$, it is established in [3] by linearisation and
classical Schauder estimates that, for regular initial data, there exists a
unique classical solution in parabolic Hölder spaces. The analogue on
$\Omega=\mathbb{R}$ is obtained in [22]. The crucial ingredients of the
arguments therein are adapted from Jiang [15, 16] (which, in turn, rely on the
localisation lemma by Kazhikhov [20] and Kazhikhov–Shelukhin [18, 19]) and
Li–Liang [23].

######  Proposition 1.3 (Theorem 3 in [3] and Theorem 1.2 in [22]).

Let $\Omega=[0,1]$ or $\mathbb{R}$. Suppose that for some $\alpha\in]0,1]$ the
initial data $(v_{0},u_{0},\theta_{0},Z_{0})$ satisfy the conditions in Eq.
(0.4) and

$v_{0}\in C^{1,\alpha}(\Omega),\quad(u_{0},\theta_{0},Z_{0})\in
C^{2,\alpha}(\Omega),\quad\|\left(u_{0},\theta_{0},Z_{0}\right)\|_{H^{1}(\Omega)}\leq
M_{1}.$ (1.2)

Consider the system (0.7) $\&$ (0.4), in which the discontinuous reaction rate
function $\phi$ is replaced by its $C^{1}$-regularisation $\Phi$ defined in
Eq. (1.1). Then for aribitrary $T>0$, there exists a unique classical solution

$\left(v^{\varepsilon},u^{\varepsilon},\theta^{\varepsilon},Z^{\varepsilon}\right)\in
B^{1+\sigma}(Q_{T})\times\big{[}B^{2+\sigma}(Q_{T})\big{]}^{3}$

for some exponent $\sigma\in]0,1]$ depending on $\alpha$ and $M_{1}$.

In addition, the following estimates hold for all $(t,y)\in Q_{T}$:

$\left\\{\begin{aligned} &C_{0}^{-1}\leq v^{\varepsilon}\leq C_{0},\quad
0<C^{-1}\leq\theta^{\varepsilon}\leq C_{0},\quad|u^{\varepsilon}|\leq
C_{0},\quad 0\leq Z^{\varepsilon}\leq 1,\\\
&\left\|\left(v^{\varepsilon}_{t},v^{\varepsilon}_{y}\right)\right\|^{2}(t)+\int_{0}^{t}\left\|\left(v^{\varepsilon}_{t},v^{\varepsilon}_{y}\right)\right\|^{2}(\tau)\,\mathrm{d}\tau\leq
C_{0},\\\
&\left\|\left(u^{\varepsilon}_{y},\theta^{\varepsilon}_{y},Z^{\varepsilon}_{y}\right)\right\|^{2}(t)+\int_{0}^{t}\left\|\left(u^{\varepsilon}_{yy},\theta^{\varepsilon}_{yy},Z^{\varepsilon}_{yy},u^{\varepsilon}_{y},\theta^{\varepsilon}_{y},Z^{\varepsilon}_{y},u^{\varepsilon}_{t},\theta^{\varepsilon}_{t},Z^{\varepsilon}_{t}\right)\right\|^{2}(\tau)\,\mathrm{d}\tau\leq
C_{0},\\\
&\|v^{\varepsilon}\|_{1+\sigma}+\left\|\left(u^{\varepsilon},\theta^{\varepsilon},Z^{\varepsilon}\right)\right\|_{2+\sigma}\leq
C.\end{aligned}\right.$

Here $C_{0}$ depend only on $M_{1}$, while $C$ depend only on $M_{1}$,
$\epsilon$, and $T$.

In particular, note that Proposition 1.3 together with the monotonicity of
$\Phi_{\epsilon}$ yields that $\Phi_{\epsilon}(\theta)$ is uniformly bounded
from above independently of $\epsilon$.

Next, let us document several conserved quantities of the system (0.7) $\&$
(0.4), which both play a crucial role in the proof of Proposition 1.3 and have
enormous physical significance. See [3, Lemma 1] and [22, Propositions 2.3
$\&$ 2.4] for proofs.

######  Lemma 1.4.

Let $\Omega=[0,1]$ or $\mathbb{R}$; $T>0$. For any $t\in[0,T]$, the classical
solution
$\left(v^{\varepsilon},u^{\varepsilon},\theta^{\varepsilon},Z^{\varepsilon}\right)$
on $Q_{T}$ (whose existence is guaranteed by Proposition 1.3) satisfies the
identities:

$\displaystyle\int_{\Omega}v(t,y)\,\mathrm{d}y=\int_{\Omega}v_{0}(y)\,\mathrm{d}y,$
$\displaystyle\int_{\Omega}Z(t,y)\,\mathrm{d}y+\int_{0}^{t}\int_{\Omega}K\Phi(\theta)Z\,\mathrm{d}y\,\mathrm{d}\tau=\int_{0}^{1}Z_{0}(y)\,\mathrm{d}y,$
$\displaystyle\int_{\Omega}\left[a(v-1-\log
v)+\frac{u^{2}}{2}+(\theta-1-\log\theta)\right]\,\mathrm{d}y+\int_{0}^{t}\int_{\Omega}\left[\frac{\mu
u_{y}^{2}}{v\theta}+\frac{\nu\theta_{y}^{2}}{v\theta^{2}}-q\frac{\theta-1}{\theta}K\Phi(\theta)Z\right]\,\mathrm{d}y\,\mathrm{d}\tau$
$\displaystyle\qquad=\int_{\Omega}\left[a(v_{0}-1-\log
v_{0})+\frac{u_{0}^{2}}{2}+(\theta_{0}-1-\log\theta_{0})\right]\,\mathrm{d}y$
$\displaystyle\qquad=:{\mathcal{E}}_{0}<\infty.$

###  1.2. Uniform bounds on $v_{y}$

For the proof of Theorem 0.3, one crucial estimate we need is the spacetime
$L^{\infty}$-bound for $v_{y}$, the first spatial derivative of $v$. Of
course, such estimate is impossible unless it holds at $t=0$. This motivates
the assumption

$\displaystyle v_{0}\in H^{1}(\Omega)\cap W^{1,\infty}(\Omega)$

in Theorem 0.3. As commented in Remark 1.2, the Lipschitz assumption is in
fact not needed to establish the existence of generalised solutions (see
Definition 0.2).

The following result shows that the Lipschitzness of $v$ is propagated through
time.

######  Theorem 1.5.

Let $T>0$ and $\Omega=[0,1]$. Suppose that the initial data
$(v_{0},u_{0},\theta_{0},Z_{0})$ satisfy the conditions in Theorem 0.3. There
is a finite constant $\Lambda$ as in Notation 0.4 such that

$\displaystyle\left\|v_{y}\right\|_{L^{\infty}(Q_{T})}\leq\Lambda.$

Proposition 1.1 and Theorem 1.5 together imply that

$v\in L^{\infty}\big{(}0,T;H^{1}(\Omega)\cap W^{1,\infty}(\Omega)\big{)}.$

The essential tool needed for the proof, which has already appeared in [3], is
an integral representation formula for $v$ (Eq. (1.2) below) discovered by
Kazhikhov [18]. We shall only sketch the arguments here, with emphasis on the
dependency of constants. Special care is taken to show that the Lipschitz
estimate of $v$ requires only the Lipschitz bounds for the initial specific
volume $v_{0}$, but _not_ for $u_{0}$, $Z_{0}$, $\theta_{0}$; the
$H^{1}$-bounds for the later three variables would suffice, as in Definition
0.2 of the generalised solutions.

Meanwhile, to avoid undesired complications that may blur the main ideas of
the proof, we focus on the case of bounded domain $\Omega=[0,1]$. Theorem 1.5
for $\Omega=\mathbb{R}$ is also valid, but we postpone its proof to §4 towards
the end of the paper.

###### Proof of Theorem 1.5.

We first introduce some notations, most of which are taken from [3]. Denote by
$\alpha_{0}$, $\beta_{0}$ the roots $y_{0}$ of the transcendental equation

$\displaystyle y_{0}-1-\log y_{0}={\mathcal{E}}_{1},$

where

$\displaystyle{\mathcal{E}}_{1}:=\frac{1}{\min\left\\{1,a\right\\}}\left\\{{\mathcal{E}}_{0}+q\int_{0}^{1}Z_{0}(y)\,\mathrm{d}y\right\\}.$

Clearly $\alpha_{0}$, $\beta_{0}$ are positive numbers (as $q>0$ for
combustion reaction).

Throughout this proof, we consider the _regularised_ system as in §1.1, with
$\Phi_{\varepsilon}$ (see Eq. (1.1)) in lieu of $\phi$. For notational
convenience, we shall drop the superscipt ε in the remaining parts of the
proof, namely that

$(v_{0},u_{0},\theta_{0},Z_{0})\equiv(v^{\varepsilon}_{0},u^{\varepsilon}_{0},\theta^{\varepsilon}_{0},Z^{\varepsilon}_{0})\quad\text{
and
}\quad(v,u,\theta,Z)\equiv(v^{\varepsilon},u^{\varepsilon},\theta^{\varepsilon},Z^{\varepsilon}).$

The process for passage of limits is the same as in Chen [3], so all we need
to complete the proof is to obtain an $L^{\infty}$-bound for $v_{y}$ _uniform
in $\varepsilon$ and independent of higher-order norms_. See the following
paragraph for details.

Under this provision, we may infer from Notation 0.4 that the constants
$\Lambda$ are allowed to depend only on the physical parameters $a$, $q$, $D$,
$K$, $\mu$, and $\nu$; the constants $m_{0}$, $M_{0}$, $\alpha_{0}$,
$\beta_{0}$, ${\mathcal{E}}_{0}$, $\|\phi\|_{L^{\infty}}$, $T$; as well as the
natural norms
$\left\|\big{(}v_{0},u_{0},\theta_{0},Z_{0}\big{)}\right\|_{[H^{1}(\Omega)\cap
W^{1,\infty}(\Omega)]\times[H^{1}(\Omega)]^{3}}$ of the initial data. It, in
particular, is independent of the regularisation parameter $\varepsilon$ and
the Hölder norms $\|v_{0}\|_{1+\sigma}$,
$\|(u_{0},\theta_{0},Z_{0})\|_{2+\sigma}$. As a consequence, $\Lambda$ is
persistent under the limit $\varepsilon\searrow 0$ passing from the classical
solution to the regularised system to the generalised solution to the original
system (0.4) $\&$ (0.7). Especially, the remark ensuing Proposition 1.3
explains why the dependence on $\|\phi\|_{L^{\infty}}$ works here.

The following representation formula for $v=v(t,y)$ was obtained by Kazhikhov
[18], using the first two equations in (0.7) and the boundary conditions for
$u$ and $\theta$: For any $t\in]0,T]$, there exists a point
$y_{1}(t)\in[0,1]=\Omega$ such that

$\displaystyle v(t,y)$
$\displaystyle=B(t,y)\,\exp\left\\{-\frac{1}{\mu}\int_{0}^{t}\int_{0}^{1}\Big{(}u(\tau,\xi)^{2}+a\theta(\tau,\xi)\Big{)}\,\mathrm{d}\xi\,\mathrm{d}\tau\right\\}\times$
$\displaystyle\qquad\times\left\\{1+\frac{a}{\mu}\int_{0}^{t}\frac{\theta(\tau,y)}{B(\tau,y)}\exp\left\\{\frac{1}{\mu}\int_{0}^{\theta(\tau,y)}au(\tau,y)\int_{0}^{1}\Big{(}u(\tau,\xi)^{2}+a\theta(\tau,\xi)\Big{)}\,\mathrm{d}\xi\,\mathrm{d}s\right\\}\mathrm{d}\tau\right\\},$
(1.3)

where

$B(t,y)=v_{0}(y)\exp\left\\{\frac{1}{\mu}\left[\int_{y_{1}(t)}^{y}u(t,\xi)\,\mathrm{d}\xi-\int_{0}^{y}u_{0}(\xi)\,\mathrm{d}\xi+\int_{0}^{1}v_{0}(y)\left(\int_{0}^{\eta}u_{0}(\xi)\,\mathrm{d}\xi\right)\,\mathrm{d}\eta\right]\right\\}.$

Note that for any $t$, the Cauchy–Schwarz inequality gives us

$\left|\int_{y_{1}(t)}^{y}u(t,\xi)\,\mathrm{d}\xi\right|\leq\|u\|_{L^{2}(\Omega)}\leq\sqrt{2{\mathcal{E}}_{0}}.$

This together with Eq. (1.2) implies that there exists $\Lambda>0$ satisfying

$0<\Lambda^{-1}\leq B(t,y)\leq\Lambda<\infty\qquad\text{for all }(t,y)\in
Q_{T}.$ (1.4)

Now, one may deduce from the representation formula (1.2) an explicit
expression for $v_{y}(t,y)$. Indeed, by chain rule and the fundamental theorem
of calculus, we compute that

$\displaystyle v_{y}(t,y)=$
$\displaystyle\,B_{y}(t,y)\exp\left\\{-\frac{1}{\mu}\int_{0}^{t}\int_{0}^{1}\Big{(}u(\tau,\xi)^{2}+a\theta(\tau,\xi)\Big{)}\,\mathrm{d}\xi\,\mathrm{d}\tau\right\\}\times$
(1.5)
$\displaystyle\quad\times\left\\{1+\frac{a}{\mu}\int_{0}^{t}\frac{\theta(\tau,y)}{B(\tau,y)}\exp\left\\{\frac{1}{\mu}\int_{0}^{\theta(\tau,y)}au(\tau,y)\int_{0}^{1}\Big{(}u(\tau,\xi)^{2}+a\theta(\tau,\xi)\Big{)}\,\mathrm{d}\xi\,\mathrm{d}s\right\\}\,\mathrm{d}\tau\right\\}$
$\displaystyle+B(t,y)\exp\left\\{-\frac{1}{\mu}\int_{0}^{t}\int_{0}^{1}(u^{2}+a\theta)\,\mathrm{d}\xi\,\mathrm{d}\tau\right\\}\times$
$\displaystyle\quad\times\Bigg{\\{}\frac{a}{\mu}\int_{0}^{t}\frac{\theta_{y}(\tau,y)B(\tau,y)-\theta(\tau,y)B_{y}(\tau,y)}{B^{2}(\tau,y)}\times$
$\displaystyle\qquad\qquad\qquad\times\exp\left\\{\frac{1}{\mu}\int_{0}^{\theta(\tau,y)}au(\tau,y)\int_{0}^{1}\Big{(}u(\tau,\xi)^{2}+a\theta(\tau,\xi)\Big{)}\,\mathrm{d}\xi\,\mathrm{d}s\right\\}\,\mathrm{d}\tau\Bigg{\\}}.$

Here one has

$\displaystyle
B_{y}(t,y)=\left\\{v_{0y}(y)+v_{0}(y)\frac{1}{\mu}\big{[}u(t,y)-u_{0}(y)\big{]}\right\\}\times\frac{B(t,y)}{v_{0}(y)}.$

Thus, by the $L^{\infty}$-bound for $u$ (see Proposition 1.3), the assumptions
on the initial data in the statement of Theorem 1.5 (in particular, the
strictly positive lower bound for $v_{0}$ and its Lipschitz upper bound), and
estimate (1.4) above, we deduce that

$|B_{y}(t,y)|\leq\Lambda\qquad\text{for all }(t,y)\in Q_{T}.$ (1.6)

It is crucial that the above expression for $B_{y}$ does not involve any
derivative of $u$ or $u_{0}$.

Next, let us estimate the term $\int_{0}^{1}(u^{2}+a\theta)\,\mathrm{d}y$ in
Eq. (1.5). Since $s\mapsto s-1-\log s$ is a convex function on $[0,\infty[$,
we see by Jensen’s inequality that

$\displaystyle\int_{0}^{1}\theta(t,y)\,\mathrm{d}y-1-\log\left(\int_{0}^{1}\theta(t,y)\,\mathrm{d}y\right)$
$\displaystyle\qquad\leq\int_{0}^{1}\Big{\\{}\theta(t,y)-1-\log\theta(t,y)\Big{\\}}\,\mathrm{d}y\leq{\mathcal{E}}_{0},$

where ${\mathcal{E}}_{0}$ is given in Lemma 1.4. In view of the definition of
$\alpha_{0}$, $\beta_{0}$ and the third identity in Lemma 1.4, we thus have

$\displaystyle 0<\alpha_{0}a$
$\displaystyle\leq\int_{0}^{1}(u^{2}+a\theta)\,\mathrm{d}y$
$\displaystyle\leq\beta_{0}a+2\left\\{{\mathcal{E}}_{0}+q\int_{0}^{1}Z_{0}(y)\,\mathrm{d}y\right\\}.$
(1.7)

Also observe the following simple bound: for any $t\in[0,T]$ and $y\in\Omega$,

$\displaystyle\left|\int_{0}^{t}\theta_{y}(\tau,y)\mathrm{d}\tau\right|$
$\displaystyle=\left|\int_{0}^{t}\int_{0}^{y}\theta_{yy}(\tau,\xi)\,\mathrm{d}\xi\,\mathrm{d}\tau\right|$
$\displaystyle\leq\int_{0}^{t}\left\|\theta_{yy}(\tau,\bullet)\right\|_{L^{2}(\Omega)}\,\mathrm{d}\tau\leq\Lambda.$
(1.8)

We used here the fundamental theorem of calculus, the boundary condition for
$\theta$ at $y=0$, the Cauchy–Schwarz inequality, as well as Proposition 1.3.

Therefore, putting together the bounds in Eqs. (1.4), (1.6), (1.2), $\&$ (1.2)
and substituting them into Eq. (1.5), we readily conclude that

$|v_{y}(t,y)|\leq\Lambda\qquad\text{for all }(t,y)\in Q_{T}.$

This completes the proof. ∎

Note that the Lipschitzness of $v$ is equivalent to that of $\rho=v^{-1}$.
This is because

$\displaystyle\frac{1}{v(t,y_{1})}-\frac{1}{v(t,y_{2})}=\frac{v(t,y_{2})-v(t,y_{1})}{v(t,y_{1})v(t,y_{2})}\qquad\text{for
any $t\in[0,T]$ and $y_{1},y_{2}\in\Omega$},$

and $v$ is strictly bounded below by a positive constant (see Proposition
1.3).

##  2\. An inequality based on Fisher information

Now let us introduce the key inequality in this work. It was first discovered
in the recent preprint [7] by Cieślak–Fuest–Hajduk–Sierżȩga, which relates the
Hessian of the square-root of a positive function to the dissipation of Fisher
information along heat flow.

######  Lemma 2.1.

Let $-\infty\leq a<b\leq\infty$, and let $\Psi$ be a positive $C^{2}$-function
on $]a,b[$ such that $\Psi_{x}(a)=\Psi_{x}(b)=0$. Then it holds that

$\int_{a}^{b}\left[(\Psi^{1/2})_{xx}\right]^{2}\,\mathrm{d}x\leq\frac{13}{8}\int_{a}^{b}\Psi\left[(\log\Psi)_{xx}\right]^{2}\,\mathrm{d}x.$

When $a=-\infty$, the condition $\Psi_{x}(a)=0$ is understood as
$\lim_{y\searrow-\infty}\Psi_{x}(y)=0$; similarly for the case $b=\infty$. For
convenience of the readers, we shall sketch a proof below. We essentially
follow [7] and do not claim any originality for our arguments. In [7] the
inequality is established only for finite $a$ and $b$, but the same proof
carries over to the general case without much alterations. Of course, if
either $a$ or $b$ is infinite, the inequality is effective only when
$\int_{a}^{b}\Psi\left[(\log\Psi)_{xx}\right]^{2}\,\mathrm{d}x<\infty$.

###### Proof of Lemma 2.1.

For $0<\Psi\in C^{2}(]a,b[)$, one has

$\displaystyle\left[(\Psi^{1/2})_{xx}\right]^{2}=\left(\frac{\Psi_{xx}}{2\Psi^{1/2}}-\frac{\Psi_{x}^{2}}{4\Psi^{3/2}}\right)^{2}=\frac{1}{4}\left(\frac{\Psi_{xx}}{\Psi^{1/2}}-\frac{1}{2}\frac{\Psi_{x}^{2}}{\Psi^{3/2}}\right)^{2},$
$\displaystyle\Psi\left[(\log\Psi)_{xx}\right]^{2}=\Psi\left(\frac{\Psi_{xx}}{\Psi}-\frac{\Psi_{x}^{2}}{\Psi^{2}}\right)^{2}=\left(\frac{\Psi_{xx}}{\Psi^{1/2}}-\frac{\Psi_{x}^{2}}{\Psi^{3/2}}\right)^{2}.$

Completing the square and using the inequality $(a+b)^{2}\leq 2(a^{2}+b^{2})$,
we obtain that

$\displaystyle\left[(\Psi^{1/2})_{xx}\right]^{2}$
$\displaystyle=\frac{1}{4}\left(\frac{\Psi_{xx}}{\Psi^{1/2}}-\frac{\Psi_{x}^{2}}{\Psi^{3/2}}+\frac{1}{2}\frac{\Psi_{x}^{2}}{\Psi^{3/2}}\right)^{2}$
$\displaystyle\leq\frac{1}{4}\left[2\left(\frac{\Psi_{xx}}{\Psi^{1/2}}-\frac{\Psi_{x}^{2}}{\Psi^{3/2}}\right)^{2}+\frac{1}{2}\left(\frac{\Psi_{x}^{2}}{\Psi^{3/2}}\right)^{2}\right]$
$\displaystyle=\frac{1}{2}\Psi\left[(\log\Psi)_{xx}\right]^{2}+\frac{1}{8}\left(\frac{\Psi_{x}^{2}}{\Psi^{3/2}}\right)^{2}.$

To conclude, one invokes the following inequality attribute in [7] to Winkler
[39]. In full generality, it reads:

######  Lemma (Bernis-type inequality).

Let $h\in C^{1}(\mathbb{R}_{+})$ be a positive function, and let
$\Omega\subset\mathbb{R}^{N}$ be a smooth Euclidean domain. Set
$\Theta(s):=\int_{1}^{s}\frac{\mathrm{d}\sigma}{h(\sigma)}$ for $s>0$. Then,
for any positive function $\varphi\in C^{2}\big{(}\overline{\Omega}\big{)}$
with $\frac{\partial\varphi}{\partial\nu}=0$ on $\partial\Omega$, it holds
that

$\int_{\Omega}\frac{h^{\prime}(\varphi)}{h^{3}(\varphi)}\left|\nabla\varphi\right|^{4}\,{\rm
d}x\leq(2+\sqrt{N})^{2}\int_{\Omega}\frac{h(\varphi)}{h^{\prime}(\varphi)}\left|D^{2}\Theta(\varphi)\right|^{2}\,{\rm
d}x.$

Applying the above inequality to $\Psi=\varphi$, $h={\rm Id}$, $N=1$, and
integrating the previous pointwise inequality over space, we arrive at

$\displaystyle\int_{0}^{1}\left[(\Psi^{1/2})_{xx}\right]^{2}\,\mathrm{d}x\leq\frac{13}{8}\int_{0}^{1}\Psi\left[(\log\Psi)_{xx}\right]^{2}\,\mathrm{d}x,$

which is what we want to prove. ∎

In passing, we note that the reverse inequality is immediate:

$\displaystyle\int_{a}^{b}\Psi\left[(\log\Psi)_{xx}\right]^{2}\,\mathrm{d}x\leq
4\int_{a}^{b}\left[(\Psi^{1/2})_{xx}\right]^{2}\,\mathrm{d}x.$

In fact, one has the pointwise identity

$\displaystyle\left(\Psi^{1/2}\right)_{yy}-\frac{1}{2}\Psi^{1/2}\left(\log\Psi\right)_{yy}=\frac{1}{4}\frac{\left(\Psi_{y}\right)^{2}}{\Psi^{3/2}}$

for $\Psi$ as in Lemma 2.1.

##  3\. Weighted gradient estimate for reactant fraction on bounded domain

In this section, we prove Theorem 0.3 for $\Omega=[0,1]$.

###### Proof of Theorem 0.3 for bounded domain.

As in the proof for Theorem 1.5, let us consider the regularised system —
_i.e._ , with $\Phi\equiv\Phi_{\varepsilon}$ in place of $\phi$ — and we
systematically drop the superscripts ε in our variables.

It suffices to seek a uniform-in-$\varepsilon$ bound for
$\frac{Z_{y}}{\sqrt{Z}}$ in $L^{\infty}_{t}L^{2}_{y}$ by $\Lambda$ (see
Notation 0.4). Then the theorem follows from a standard passage of limit
process, which can be directly adapted from [3]. To avoid potential dangers of
dividing by zero, we introduce the new variable

$X:=Z+\delta\equiv Z^{\epsilon}+\delta\qquad\text{ for }\varepsilon,\delta>0,$
(3.1)

and look for uniform estimates independent of both the regularisation
parameter $\varepsilon$ and the “shift” parameter $\delta$. Say
$\delta<100^{-1}$.

Note that $X$ satisfies the PDE

$\left\\{\begin{aligned}
&X_{t}+K\Phi(\theta)X-K\Phi(\theta)\delta=\left(\frac{D}{v^{2}}X_{y}\right)_{y}\qquad\text{in
}Q_{T},\\\ &X_{0}(y)=Z_{0}(y)+\delta\qquad\text{at }t=0,\\\
&X_{y}=0\qquad\text{on }\partial\Omega=\\{0,1\\}.\end{aligned}\right.$ (3.2)

Here $X_{0}$ is the initial data $X|_{t=0}$. Dividing $X^{1/2}$ on both sides
of Eq. (3.2), we obtain the evolution equation for $X^{1/2}$:

$2\frac{\partial}{\partial
t}X^{1/2}+K\Phi(\theta)X^{1/2}-\frac{K\Phi(\theta)\delta}{X^{1/2}}=\frac{(\frac{D}{v^{2}}X_{y})_{y}}{X^{1/2}}.$

As $X$ is strictly positive and has the same regularity as $Z$, the above PDE
is understood in the classical sense.

We deduce by integrating over $\Omega=[0,1]$ that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{0}^{1}\frac{X_{y}^{2}}{X}\,\mathrm{d}y$
$\displaystyle\quad=4\frac{\mathrm{d}}{\mathrm{d}t}\int_{0}^{1}(X^{1/2})_{y}^{2}\,\mathrm{d}y$
$\displaystyle\quad=8\int_{0}^{1}(X^{1/2})_{y}(X^{1/2})_{ty}\,\mathrm{d}y$
$\displaystyle\quad=4\int_{0}^{1}(X^{1/2})_{y}\left[\left(\frac{D}{v^{2}}X_{y}\right)_{y}\frac{1}{X^{1/2}}-K\Phi(\theta)X^{1/2}\right]_{y}\,\mathrm{d}y$
$\displaystyle\quad=4\int_{0}^{1}(X^{1/2})_{y}\left[2\frac{D}{v^{2}}(X^{1/2})_{yy}+\frac{1}{2}\frac{D}{v^{2}}\frac{X_{y}^{2}}{X^{3/2}}+\left(\frac{D}{v^{2}}\right)_{y}\frac{X_{y}}{X^{1/2}}+K\Phi(\theta)\left(\frac{\delta}{X^{1/2}}-X^{1/2}\right)\right]_{y}\,\mathrm{d}y.$

The penultimate equality follows from the PDE for $X^{1/2}$.

Let us write

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{0}^{1}\frac{X_{y}^{2}}{X}\,\mathrm{d}y=I_{1}+I_{2}+I_{3}$
(3.3)

where

$\displaystyle
I_{1}:=4\int_{0}^{1}(X^{1/2})_{y}\left[2\frac{D}{v^{2}}(X^{1/2})_{yy}+\frac{1}{2}\frac{D}{v^{2}}\frac{X_{y}^{2}}{X^{3/2}}\right]_{y}\,\mathrm{d}y,$
$\displaystyle
I_{2}:=4\int_{0}^{1}(X^{1/2})_{y}\left[\left(\frac{D}{v^{2}}\right)_{y}\frac{X_{y}}{X^{1/2}}\right]_{y}\,\mathrm{d}y,$
$\displaystyle
I_{3}:=4\int_{0}^{1}(X^{1/2})_{y}\left[K\Phi(\theta)\left(\frac{\delta}{X^{1/2}}-X^{1/2}\right)\right]_{y}\,\mathrm{d}y.$

In what follows, we shall repetitively argue by integration by parts,
employing the boundary conditions $X_{y}(t,0)=X_{y}(t,1)=0$ as in Eq. (0.4).

For $I_{1}$, we compute that

$\displaystyle I_{1}$
$\displaystyle=-8\int_{0}^{1}\left[(X^{1/2})_{yy}\right]^{2}\frac{D}{v^{2}}\,\mathrm{d}y+8\int_{0}^{1}(X^{1/2})_{y}\left(\frac{D}{v^{2}}\right)_{y}\frac{(X^{1/2})_{y}^{2}}{X^{1/2}}\,\mathrm{d}y$
$\displaystyle\qquad+8\int_{0}^{1}(X^{1/2})_{y}\frac{D}{v^{2}}\left[\frac{2(X^{1/2})_{y}(X^{1/2})_{yy}}{X^{1/2}}-\frac{(X^{1/2})_{y}^{3}}{X}\right]\,\mathrm{d}y$
$\displaystyle=-8\int_{0}^{1}X\frac{D}{v^{2}}\left[(\log
X^{1/2})_{yy}\right]^{2}\,\mathrm{d}y+8\int_{0}^{1}(X^{1/2})_{y}\left(\frac{D}{v^{2}}\right)_{y}\frac{(X^{1/2})_{y}^{2}}{X^{1/2}}\,\mathrm{d}y$
$\displaystyle=-2\int_{0}^{1}X\frac{D}{v^{2}}\left[(\log
X)_{yy}\right]^{2}\,\mathrm{d}y+\int_{0}^{1}\left(\frac{D}{v^{2}}\right)_{y}\frac{\left(X_{y}\right)^{3}}{X^{2}}\,{\rm
d}y.$

Hence,

$\displaystyle I_{1}+I_{2}$
$\displaystyle=\underbrace{-2\int_{0}^{1}\frac{D}{v^{2}}X\left[(\log
X)_{yy}\right]^{2}\,\mathrm{d}y}_{=:\,[{\rm
GOOD}]}\underbrace{-2\int_{0}^{1}\left(\frac{D}{v^{2}}\right)_{y}X_{y}(\log
X)_{yy}\,\mathrm{d}y}_{=:\,[{\rm BAD}]}.$ (3.4)

The $[{\rm GOOD}]$ term is of favourable sign. For the $[{\rm BAD}]$ term, one
may resort to Young’s inequality to deduce, for arbitrarily small $\eta>0$,
that

$\displaystyle\big{|}[{\rm BAD}]\big{|}$
$\displaystyle=\left|\int_{0}^{1}\frac{D^{1/2}X^{1/2}}{v}\left[\log(X^{1/2})\right]_{yy}\cdot\frac{X_{y}}{X^{1/2}}\frac{(D/v^{2})_{y}}{D^{1/2}/v}\,{\rm
d}y\right|$ $\displaystyle\leq\eta\int_{0}^{1}\frac{D}{v^{2}}X\left[\left(\log
X\right)_{yy}\right]^{2}\,\mathrm{d}y+C_{\eta}D\int_{0}^{1}\frac{(X_{y})^{2}}{X}\frac{(v_{y})^{2}}{v^{4}}\,{\rm
d}y.$ (3.5)

Here $C_{\eta}\sim\eta^{-1}$ is a positive constant. For the second term on
the right-hand side of Eq. (3), we have $v\geq m_{0}>0$ by Proposition 1.3 and
$|v_{y}|\leq\Lambda$ by Theorem 1.5, both uniformly in spacetime. Thus,
recalling Notation 0.4, one may continue Eq. (3) by

$\big{|}[{\rm BAD}]\big{|}\leq\eta\int_{0}^{1}\frac{D}{v^{2}}X\left[\left(\log
X\right)_{yy}\right]^{2}\,\mathrm{d}y+C_{\eta}\Lambda\int_{0}^{1}\frac{(X_{y})^{2}}{X}\,{\rm
d}y.$ (3.6)

Now let us turn to $I_{3}$. Integration by parts yields that

$\displaystyle
I_{3}=4\int_{0}^{1}(X^{1/2})_{yy}K\Phi(\theta)\left[X^{1/2}-\frac{\delta}{X^{1/2}}\right]\,\mathrm{d}y.$

Thus, for any $\eta^{\prime}>0$ we have that

$\displaystyle\left|I_{3}\right|$
$\displaystyle\leq\eta^{\prime}\int_{0}^{1}\left[(X^{1/2})_{yy}\right]^{2}\,{\rm
d}y+C_{\eta^{\prime}}K^{2}\int_{0}^{1}\Phi^{2}(\theta)\left[X^{1/2}-\frac{\delta}{X^{1/2}}\right]^{2}\,\mathrm{d}y$
$\displaystyle\leq\frac{13\eta^{\prime}}{8}\int_{0}^{1}X\left[\left(\log
X\right)_{yy}\right]^{2}\,{\rm
d}y+C_{\eta^{\prime}}\Lambda\int_{0}^{1}\left[X^{1/2}-\frac{\delta}{X^{1/2}}\right]^{2}\,\mathrm{d}y$
$\displaystyle\leq\frac{13\eta^{\prime}(M_{0})^{2}}{8D}\int_{0}^{1}\frac{D}{v^{2}}X\left[\left(\log
X\right)_{yy}\right]^{2}\,\mathrm{d}y+C_{\eta^{\prime}}\Lambda\int_{0}^{1}\left[X^{1/2}-\frac{\delta}{X^{1/2}}\right]^{2}\,\mathrm{d}y,$

where $C_{\eta^{\prime}}\sim(\eta^{\prime})^{-1}$. In the above, we used the
Young’s inequality, Lemma 2.1, and the upper bounds for $v$ and $\theta$ in
Proposition 1.3. Recall Notation 0.4 for the constant $\Lambda$.

To proceed, simply notice that since $0\leq Z\leq 1$ (see Proposition 1.3;
$Z\equiv Z^{\varepsilon}$ is the $C^{1}$-regularised variable as before), one
has $\delta\leq X\leq 1+\delta$. This implies that $0\leq
1-\frac{\delta}{X}\leq\frac{1}{1+\delta}<1$, and hence we have a bound uniform
in $\delta$:

$\displaystyle
0\leq\left[X^{1/2}-\frac{\delta}{X^{1/2}}\right]^{2}\leq{1+\delta}\leq 1.01$

whenever $\delta\in\left[0,100^{-1}\right]$. Therefore, choosing
$\eta^{\prime}$ via

$\displaystyle\eta=\frac{13\eta^{\prime}(M_{0})^{2}}{8D}$

where $\eta$ is as in Eq. (3.6), we arrive at the bound

$\displaystyle|I_{3}|\leq\eta\int_{0}^{1}\frac{D}{v^{2}}X\left[\left(\log
X\right)_{yy}\right]^{2}\,\mathrm{d}y+C_{\eta}\Lambda.$ (3.7)

Here $\eta>0$ is arbitrary; the constants $C_{\eta}\sim\eta^{-1}$ and
$\Lambda$ are both independent of $\delta$ and $\varepsilon$.

Now, substituting the estimates (3.4), (3.6), and (3.7) into Eq. (3.3), we
deduce that

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\int_{0}^{1}\frac{X_{y}^{2}}{X}\,\mathrm{d}y+2(1-\eta)\int_{0}^{1}\frac{D}{v^{2}}X\left[(\log
X)_{yy}\right]^{2}\,\mathrm{d}y\leq
C_{\eta}\Lambda\left\\{1+\int_{0}^{1}\frac{X_{y}^{2}}{X}\,\mathrm{d}y\right\\}.$

Fix any $0<\eta<1$ and hence $C_{\eta}$. Since $\Lambda$ is independent of
both the “shift” and regularisation parameters $\delta$ and $\varepsilon$, we
can now conclude the proof — _i.e._ , recover the
$L^{\infty}_{t}L^{2}_{y}$-bound for $\frac{Z_{y}}{\sqrt{Z}}$ with $Z$ non-
regularised — from Grönwall’s lemma. ∎

##  4\. Gradient estimate for reactant fraction on unbounded domains

In this section we prove Theorem 0.3 for the unbounded domain case. For this
purpose, we need the analogue of Theorem 1.5, namely the uniform Lipschitz
bound for $v$, on $\Omega=\mathbb{R}$. To avoid unnecessary repetitions with
the bounded domain case, the parallel arguments are safely omitted. We shall
only highlight the essential differences from the developments in §3.

######  Theorem 4.1.

Theorem 1.5 remains valid when $\Omega=\mathbb{R}$. That is, for any given
$T>0$, we have $\left\|v_{y}\right\|_{L^{\infty}(Q_{T})}\leq\Lambda$. The
constant $\Lambda$ is as in Notation 0.4.

The proof is based on arguments by Jiang in [15], which relies on the
localisation lemma _à la_ Kazhikhov [19]. See also Li [22].

###### Proof of Theorem 4.1.

Throughout the proof we denote $I_{k}:=[k,k+1]$ for each $k\in\mathbb{Z}$.

Consider the cutoff function $\chi=\chi_{k}\in W^{1,\infty}(\mathbb{R})$
defined by

$\eta(y)=\left\\{\begin{aligned} &1\qquad\text{for }y\leq k,\\\
&k+1-y\qquad\text{for }k\leq y\leq k+1,\\\ &0\qquad\text{for }y\geq
k+1.\end{aligned}\right.$

Multiply the equation

$u_{t}+\left(\frac{a\theta}{v}\right)_{y}=\left(\frac{\mu
u_{y}}{v}\right)_{y}$

by $\chi$ to obtain that

$\chi(y)u_{t}(t,y)=\left[\left(\frac{\mu
u_{y}}{v}-\frac{a\theta}{v}\right)(t,y)\chi(y)\right]_{y}-\chi^{\prime}(y)\left(\frac{\mu
u_{y}}{v}-\frac{a\theta}{v}\right)(t,y).$

Then, integrating the above equality over $[0,t]\times]y,\infty[$ and taking
exponential, we deduce that

$\displaystyle
v(t,y)\exp\left\\{-\frac{a}{\mu}\int_{0}^{t}\frac{\theta}{v}(\tau,y)\mathrm{d}\tau\right\\}=v_{0}(y)P(t)Q(t,y),$

where

$P(t):=\exp\left\\{\frac{1}{\mu}\int_{0}^{t}\int_{k}^{k+1}\left(\frac{\mu
u_{y}}{v}-\frac{a\theta}{v}\right)(\tau,\xi)\mathrm{d}\tau\mathrm{d}\xi\right\\},$
$Q(t,y):=v_{0}(y)\exp\left\\{\frac{1}{\mu}\int_{y}^{\infty}[u_{0}(\xi)-u(t,\xi)]\eta(\xi)\mathrm{d}\xi\right\\}.$

One may infer the following from the above identity:

$\displaystyle
v(t,y)=Q(t,y)P(t)+\frac{a}{\mu}\int_{0}^{t}\frac{Q(t,y)P(t)}{Q(\tau,y)P(\tau)}\theta(\tau,y)\,\mathrm{d}\tau\qquad\text{for
any }y\in I_{k},\,t\geq 0.$

Moreover, the bounds below are established by Jiang [15]:

$0\leq P(t)\leq\Lambda\exp\left\\{-t/\Lambda\right\\}\quad\text{ and
}\quad\frac{P(t)}{P(\tau)}\leq\Lambda\exp\left\\{-(t-\tau)/\Lambda\right\\}.$
(4.1)

Now, straightforward computation yields that

$\displaystyle v_{y}(t,y)$
$\displaystyle=Q_{y}(t,y)P(t)+\frac{a}{\mu}\int_{0}^{t}\frac{P(t)}{P(\tau)}\frac{Q_{y}(t,y)\theta(\tau,y)+Q(t,y)\theta_{y}(\tau,y)}{Q(\tau,y)}\,\mathrm{d}\tau$
$\displaystyle\qquad-\frac{a}{\mu}\int_{0}^{t}\frac{P(t)}{P(\tau)}\frac{Q(t,y)\theta(\tau,y)Q_{y}(\tau,y)}{Q^{2}(\tau,y)}\,\mathrm{d}\tau,$
(4.2)

where

$\displaystyle Q_{y}(t,y)$
$\displaystyle=v_{0y}(y)\exp\left\\{\frac{1}{\mu}\int_{y}^{\infty}[u_{0}(\xi)-u(t,\xi)]\eta(\xi)\mathrm{d}\xi\right\\}$
$\displaystyle\qquad+v_{0}(y)\left[\frac{1}{\mu}[u(t,y)-u_{0}(y)](k+1-y)\right]\exp\left\\{\frac{1}{\mu}\int_{y}^{\infty}[u_{0}(\xi)-u(t,\xi)]\eta(\xi)\mathrm{d}\xi\right\\}.$

The above expressions are well defined, for it is assumed that $v_{0}\in
H^{1}(\mathbb{R})\cap W^{1,\infty}(\mathbb{R})$, as well as that
$u(t,\bullet)\in H^{1}(\mathbb{R})$ for every $t\in[0,T]$.

We are ready to conclude. Indeed, Eq. (4.1) gives uniform bounds on $P(t)$ and
$P(t)/P(\tau)$. Also, in view of the expressions for $Q$ and $Q_{y}$ above,
the $L^{\infty}_{t,x}$-bound for $(u,\theta)$ and the
$L^{2}_{t}H^{2}_{x}$-bound (hence the $L^{2}_{t}C^{0,1/2}_{x}$-bound by
Sobolev–Morrey embedding) for $\theta$ — proved in Proposition 1.3 — as well
as the assumptions on $v_{0}$, also imply that $|Q_{y}|\leq\Lambda$ and
$0<\Lambda^{-1}\leq Q\leq\Lambda<\infty$ uniformly in spacetime. The constants
$\Lambda$ follows the same rule of Notation 0.4. Most importantly, all these
above bounds are uniform in $k$. Substituting these bounds into the
representation formula (4) for $v_{y}$, we complete the proof. ∎

###### Sketch of proof of Theorem 0.3 for unbounded domain.

The arguments in §3 almost carry over verbatim, with the understanding of the
boundary condition for $Z$ as
$\lim\limits_{|y|\rightarrow\infty}Z_{y}(t,y)=0$. Indeed, consider
$X=Z^{\varepsilon}+\delta$ as in §3. The following holds for the case
$\Omega=\mathbb{R}$ in place of Eq. (3.3):

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{-\infty}^{\infty}\frac{X_{y}^{2}}{X}\,\mathrm{d}y$
$\displaystyle=8\left(\sqrt{X}\right)_{yy}\left(\sqrt{X}\right)_{y}\frac{D}{v^{2}}\bigg{|}_{-\infty}^{\infty}-4\left(\sqrt{X}\right)_{y}K\Phi(\theta)\sqrt{X}\bigg{|}_{-\infty}^{\infty}+8\left(\sqrt{X}\right)_{y}^{2}\left(\frac{D}{v^{2}}\right)_{y}\bigg{|}_{-\infty}^{\infty}$
$\displaystyle-2\int_{-\infty}^{\infty}X\frac{D}{v^{2}}\left[\left(\log
X\right)_{yy}\right]^{2}\mathrm{d}y-2\int_{-\infty}^{\infty}\left(\frac{D}{v^{2}}\right)_{y}X_{y}\left(\log
X\right)_{yy}\,\mathrm{d}y$
$\displaystyle+4\int_{-\infty}^{\infty}\left(\sqrt{X}\right)_{yy}K\Phi(\theta)\sqrt{X}\,\mathrm{d}y.$

This identity holds pointwise in $t$. By virtue of the boundary conditions,
the first three terms on the right-hand side are equal to zero. The control
for the remaining terms is exactly the same as for the case $\Omega=[0,1]$,
which relies essentially on Theorems 4.1 and Lemma 1.3. ∎

Acknowledgement. The research of SL is supported by NSFC (National Natural
Science Foundation of China) Project $\\#$12201399, the SJTU-UCL joint seed
fund WH610160507/067, and the Shanghai Frontier Research Institute for Modern
Analysis. This paper was written during SL’s visiting scholarship at New York
University-Shanghai; SL is grateful to NYUSH for providing very nice working
atmosphere.

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