# Hyperbolic–parabolic normal form and local classical solutions for cross-
diffusion systems with incomplete diffusion

Pierre-Étienne Druet  Pierre-Étienne Druet, Weierstrass Institute for Applied
Analysis and Stochastics (WIAS), Mohrenstrasse 39, 10117 Berlin, Germany
<EMAIL_ADDRESS>,  Katharina Hopf  Katharina Hopf,
Weierstrass Institute for Applied Analysis and Stochastics (WIAS),
Mohrenstrasse 39, 10117 Berlin, Germany<EMAIL_ADDRESS>and
Ansgar Jüngel  Institute of Analysis and Scientific Computing, Technische
Universität Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria
<EMAIL_ADDRESS>
###### Abstract.

We investigate degenerate cross-diffusion equations with a rank-deficient
diffusion matrix that are considered to model populations which move as to
avoid spatial crowding and have recently been found to arise in a mean-field
limit of interacting stochastic particle systems. To date, their analysis in
multiple space dimensions has been confined to the purely convective case with
equal mobility coefficients. In this article, we introduce a normal form for
an entropic class of such equations which reveals their structure of a
symmetric hyperbolic–parabolic system. Due to the state-dependence of the
range and kernel of the singular diffusive matrix, our way of rewriting the
equations is different from that classically used for symmetric second-order
systems with a nullspace invariance property. By means of this change of
variables, we solve the Cauchy problem for short times and positive initial
data in $H^{s}(\mathbb{T}^{d})$ for $s>d/2+1$.

###### Key words and phrases:

Hyperbolic–parabolic systems, cross diffusion, quasilinear second-order
symmetric systems, initial-value problem, entropy structure.

###### 2020 Mathematics Subject Classification:

35M11, 35L45, 35K40, 35A09.

##  1\. Introduction

In the present paper, we are interested in the Cauchy problem for a class of
cross-diffusion systems with a rank-deficient diffusion matrix and a porous
medium-type degeneracy. For $n$ species with partial densities
$u=(u_{1},\dots,u_{n})$ and partial velocity fields ${\bf v}_{1},\ldots,{\bf
v}_{n}$, we consider the conservation laws

$\displaystyle\partial_{t}u_{i}+\operatorname{div}(u_{i}\textbf{v}_{i})=0,\quad
t>0,\ x\in\mathbb{T}^{d},\ i=1,\dots,n.$ (1.1)

Here, $\mathbb{T}^{d}$ denotes the periodic box and $n$, $d\in\mathbb{N}$ are
arbitrary integers. From a simple phenomenological viewpoint, the type of
diffusion system we are interested in arises when the motion is driven by
density gradients according to

$\displaystyle{\bf v}_{i}=-\sum_{j=1}^{n}b_{ij}\nabla u_{j},$ (1.2)

where $b_{ij}$ are phenomenological coefficients. For a derivation of such
equations from a weakly interacting stochastic many-particle system, we refer
to [6].

In the special case where $b_{ij}=k\partial_{u_{j}}\widehat{p}$ for some $k>0$
and a suitable equation of state for the thermodynamic pressure
$\widehat{p}=\widehat{p}(u_{1},\ldots,u_{n})$, we obtain a purely convective
motion with Darcy law ${\bf v}_{i}={\bf v}=-k{\nabla}\widehat{p}$. Problems of
this type have been studied in several works concerning the well-posedness of
classical and weak solutions [8, 14] as well as with regard to their
segregation property, see for instance [4, 15], where the model is enhanced by
competition-type reactions arising in tissue growth modelling.

A prototypical example motivating the present work is obtained for the choice

$\displaystyle b_{ij}=k_{i}a_{j}\quad\text{with}\quad k_{1},\dots,k_{n}>0,\
a_{1},\dots,a_{n}>0,$ (1.3)

leading to the equations

$\displaystyle\partial_{t}u_{i}-\operatorname{div}(k_{i}u_{i}{\nabla}\mathfrak{p}(u))=0\quad\text{with}\quad\mathfrak{p}(u)=\sum_{i=1}^{n}a_{i}u_{i}.$
(Rk1)

This is the model considered by Bertsch et al. [2, 3] for $d=1$, $n=2$ and
segregated solutions. It was first proposed in [13] to describe populations
that disperse in order to avoid spatial crowding. Few further rigorous studies
exist for this or similar models when the $k_{i}$ are not all equal. The study
in [21] on a two-species model illustrates numerically and through travelling
wave analysis that unequal coefficients $k_{1}\not=k_{2}$ can lead to
instabilities in the presence of a reaction term.

There are several energy functionals
$F(u)=\int_{\mathbb{T}^{d}}f(u)\,\mathrm{d}x$ available for the simple model
(Rk1), allowing to exhibit an underlying gradient-flow structure,

$\displaystyle\partial_{t}u-\operatorname{div}(\mathbb{M}(u)\nabla\mu)=0,$
(1.4)

where $\mu=\partial_{u}F(u)$ is the vector of chemical potentials and
$\mathbb{M}(u)$ denotes the mobility tensor. First, we note the so-called
Shannon entropy [27] generated by the density

$\displaystyle f_{1}(u):=\sum_{i=1}^{n}\pi_{i}(u_{i}\log
u_{i}-u_{i})\quad\text{with}\quad\pi_{i}:=\frac{a_{i}}{k_{i}}.$

In this case, the $i^{\rm th}$ chemical potential is $\mu_{i}=\pi_{i}\log
u_{i}$, and (1.4) is true for $\mathbb{M}(u)=(ku)\otimes(ku)$ with
$ku=(k_{1}u_{1},\ldots,k_{n}u_{n})$. Every density function $f$ of the form
$f(u)=(\varphi\circ\mathfrak{p})(u)$ with a strictly convex function
$\varphi\in C^{2}(0,\infty)$ generates another Lyapunov functional for system
(Rk1). If we choose, for instance,

$\displaystyle
f_{2}(u)=\frac{1}{2}\bigg{(}\sum_{i=1}^{n}a_{i}u_{i}\bigg{)}^{2},$ (1.5)

then the chemical potential is given by $\mu_{i}=a_{i}\mathfrak{p}(u)$,
yielding (1.4) with $\mathbb{M}(u)={\rm diag}(u_{i}/\pi_{i})$. This shows that
the constitutive equations underlying (1.1)–(1.2), and in particular the rank
deficiency of the diffusion matrix, may be subject to different
interpretations: strictly convex free energy and rank-deficient mobility
tensor for (1.4), or full-rank mobility tensor but rank-deficient Hessian for
the free energy (1.5). Interestingly, only the choice
$f_{3}(u)=\mathfrak{p}(u)\log\mathfrak{p}(u)-\mathfrak{p}(u)$ of the free
energy yields the identity
$\mathfrak{p}=-f_{3}+\sum_{i=1}^{n}u_{i}\partial_{u_{i}}f_{3}$, which can be
interpreted as the Gibbs–Duhem equation for the thermodynamic pressure
$\mathfrak{p}$.

In the general case of (1.2), we assume that the matrix

$\displaystyle B=(b_{ij})\in\mathbb{R}^{n\times n}\text{ }$ is symmetric,
positive semidefinite (1.6) $\displaystyle\text{and has rank
}1\leq\mathsf{r}\leq n.$

More generally, our results hold for matrices $B$ for which there exists a
vector $\widehat{\pi}\in\mathbb{R}_{+}^{n}$ such that the product
$B\operatorname{diag}(\widehat{\pi}_{1},\dots,\widehat{\pi}_{n})$ obeys the
properties in (1.6). Indeed, this case can be reduced to (1.6) by the
rescaling $u_{i}\mapsto u_{i}/\widehat{\pi}_{i}$. In particular, the
asymmetric case (1.3) can be covered in this way.

Note that in many physical and biological situations, where the dynamics of
the species is driven by pressure-like quantities or frictional interactions,
it is reasonable to further impose the non-negativity of the coefficients
$b_{ij}$. However, our analysis shall reveal that positivity is a requirement
only for $(b_{ij})$ as an operator.

System (1.1)–(1.2) with (1.6) admits the free energy functions

$\displaystyle\widetilde{f}_{1}(u)=\sum_{i=1}^{n}(u_{i}\log
u_{i}-u_{i}),\quad\widetilde{f}_{2}(u)=\frac{1}{2}\sum_{i,j=1}^{n}b_{ij}u_{i}u_{j}.$

In the context of mathematical biology, the functional associated with the
quadratic function $\widetilde{f}_{2}$ is known as the Rao entropy [24]. Rank-
deficiency can for example be justified if the driving functional measures
some dissimilarity between the species (genetic characters,
$\widetilde{f}_{2}(u)=\sum_{i,j=1}^{n}(u_{i}-u_{j})^{2}$). Recently, in the
context of electrochemistry, Eisenberg and Lin proposed a similar quadratic
correction of the standard entropy to describe repulsive interactions between
ions crowded in small channels; see [20, eq. (2.11)] with
$b_{ij}=\epsilon_{ij}(a_{i}+a_{j})^{12}$, the interaction energies
$\epsilon_{ij}\geq 0$ and the diameter $a_{i}>0$ of the $i^{\rm th}$ ion
species. More generally, rank-deficiency for the Hessian of the free energy
function is a singular limit, which can be motivated from general
representation theorems for the free energy function (see equations (30), (32)
in [5]), when the behaviour of the system is widely dominated by its response
under pressure, hence by the equation of state rather than by the statistical
entropy of mixing.

### Motivation and key aspects

With the choice (1.3), the second-order system (1.1)–(1.2) does not enjoy full
parabolicity, and even for $u_{i}$ strictly positive, the system is not
parabolic in the sense of Petrovskii. Thus, the theories by Amann or
Solonnikov do not apply in the present setting and we cannot expect the
classical parabolic smoothing properties. Therefore, even the local-in-time
well-posedness in spaces of high regularity is a non-trivial question, which
has hardly been investigated in the literature for rank-deficient cross-
diffusions. The present article aims to provide an answer to this question for
system (1.1)–(1.2) under conditions (1.6). Our key idea is to separate the
degenerate variables from the parabolic evolution in such a way that the
resulting system has a (symmetrisable) hyperbolic–parabolic structure. After
identification of such a proper set of variables, we are able to obtain closed
higher-order a priori estimates for short times by means of classical energy
methods, which then allow for the construction of local-in-time strong
solutions by iteration. We emphasize that for general second-order quasilinear
symmetric systems, the commutator terms arising upon higher-order spatial
differentiation in the equation for $\partial^{\alpha}u_{i}$,
$\alpha\in\mathbb{N}^{d}$, may not be controlled and standard parabolic
estimates generally fail in the rank-deficient case.

For positive densities, the diffusion matrix associated with system
(1.1)–(1.3) (resp. (1.1), (1.2) with (1.6)) has rank one (resp. rank
$\mathsf{r}$). Thus, we are looking for a change of variables that transforms
the system into a composite problem coupling a Friedrichs-symmetrisable
hyperbolic system for $n{-}1$ (resp. for $n{-}\mathsf{r}$) components to a
scalar parabolic equation (resp. to a parabolic system for $\mathsf{r}$
components). We recall that Friedrichs-symmetrisable first-order systems allow
for energy estimates in a similar way as scalar equations and that, as is well
known, symmetrisability is a direct consequence of the existence of a strictly
convex entropy [9, 12]. While the entropic structure of the present models
allows us to recast the equations in the form of a (degenerate) second-order
symmetric system (see the introductory comments in Section 5), it is not
obvious why symmetrisability should be inherited by any first-order subsystem
that might be obtained after changing coordinates.

The main results of this article can be summarized as follows:

  * •

We provide transformations leading to a normal form of symmetric
hyperbolic–parabolic type for the rank-one systems (Rk1).

  * •

We derive a transformation that allows us to write system (1.1), (1.2) with
(1.6) for an arbitrary rank $\mathsf{r}\in\\{1,\dots,n\\}$ in a symmetric
hyperbolic–parabolic form.

  * •

We show that such systems, for any rank $\mathsf{r}\in\\{1,\dots,n\\}$, admit
unique local-in-time classical solutions provided the initial densities lie in
$H^{s}(\mathbb{T}^{d})$ for some $s>d/2+1$ and are strictly positive.

### Comparison to existing literature on hyperbolic–parabolic systems

Normal forms of symmetric hyperbolic–parabolic type have first been introduced
in the context of viscous compressible fluid dynamics. In their seminal work
[17], Kawashima and Shizuta rely on entropic variables and a nullspace
invariance condition of the matrix associated with the second-order spatial
differential operator in order to derive an appropriate composite form. In our
class of systems, this invariance condition amounts to a state-independence of
the kernel or range of the underlying mobility tensor $\mathbb{M}(u)$, a
property which is not fulfilled in the present cross-diffusive problem. Thus,
the results by Kawashima and Shizuta, including their extensions in [11, 25,
26], are not directly applicable.

The normal forms that arise here (see equation (2.10)) are structurally
different from those appearing in the fluid dynamics context. They also seem
to be new with respect to more recent work on hyperbolic–parabolic problems
considered, for instance, in thermoviscoelasticity [7]. A trivial version of
our normal form is obtained for (Rk1) when setting $k_{i}=1$ for all $i$ (or
equal to any other fixed constant). Then, if $n=2$, the change of variables
$w_{1}=u_{1}/(u_{1}+u_{2})$, $w_{2}=a_{1}u_{1}+a_{2}u_{2}$, as first employed
in [4], leads to the pure transport equation $\partial_{t}w_{1}=\nabla
w_{2}\cdot\nabla w_{1}$ whose velocity field is given by the negative gradient
of the solution to the porous-medium equation
$\partial_{t}w_{2}=\operatorname{div}(w_{2}\nabla w_{2})$.

From a technical point of view, the difficulties in the Cauchy problem
associated with our class of normal forms are not very different from those
arising in (viscous) systems of conservations laws. The symmetrisability of
the hyperbolic subsystem allows us to adopt an $L^{2}$-approach, which was
first applied by Kawashima to symmetric hyperbolic–parabolic systems,
requiring initial data in $H^{s}$ for $s>d/2+2$ [16]. This regularity
condition was improved by Serre [25], who only required $s>d/2+1$ and obtained
strong solutions. With regard to the classical solutions constructed here, the
condition $s>d/2+1$ in the $L^{2}$-framework should essentially be optimal, as
it just yields the continuous differentiability of the hyperbolic components
as well as the spatial Lipschitz continuity of the ‘velocity field’. We obtain
classical solutions by exploiting a modest regularising effect in the
quasilinear strongly parabolic subsystem, whose coefficient matrix has limited
regularity due to its dependence on the hyperbolic components. It turns out
that the regularity is sufficient to deduce classical solvability for small
positive times.

Finally, we point out that specific situations may allow for improved results
and alternative methods. Besides the one-dimensional case $d=1$, the binary
case $n=2$ and, more generally, the case $\mathsf{r}=n-1$ are special, since
then the hyperbolic part of the system is scalar and can thus be treated by
the method of characteristics for quasilinear first-order equations. Boundary-
value problems may also be treated more easily in this case.

### Outline

The remaining part of this article is structured as follows. In Section 2, we
formulate our main results on the rank-one system (Rk1) and the general
problem (1.2), (1.6). We provide in Section 3 an explicit transformation
leading to a simple normal form for (Rk1). Subsequently, we establish the
existence of a unique local-in-time classical solution for systems of that
form; see Section 4. Section 5 is concerned with general rank-$\mathsf{r}$
systems. Here, a different change of variables is used leading to a somewhat
different, though structurally similar normal form. In Appendix A, we briefly
discuss possible alternative transformations in the rank-one case.

### Notation

For $s\in\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$, we equip the space
$H^{s}:=H^{s}(\mathbb{T}^{d})$ with the canonical norm
$\|v\|_{H^{s}}=(\sum_{|\alpha|\leq
s}\int_{\mathbb{T}^{d}}|\partial^{\alpha}v|^{2}\mathrm{d}x)^{1/2}$. We do not
distinguish between the spaces $H^{s}(\mathbb{T}^{d})$ and
$H^{s}(\mathbb{T}^{d};\mathbb{R}^{n})$ if no confusion arises. For any vector
$v=(v_{1},\ldots,v_{n})^{\mathsf{T}}\in\mathbb{R}^{n}$, we introduce
$v^{\prime}=(v_{1},\ldots,v_{n-1})^{\mathsf{T}}$. Moreover, given
$\mathsf{r}\in\\{1,\dots,n\\}$, we write
$v=(v_{\mathrm{I}},v_{\mathrm{II}})^{\mathsf{T}}$, where
$v_{\mathrm{I}}:=(v_{1},\dots,v_{n-\mathsf{r}})^{\mathsf{T}}$ and
$v_{\mathrm{II}}:=(v_{n-\mathsf{r}+1},\dots,v_{n})^{\mathsf{T}}$ with the
understanding that $v=v_{\mathrm{II}}$ if $\mathsf{r}=n$. We denote by
$\mathbb{R}^{N\times N}_{\rm sym}$ the space of symmetric real $(N\times
N)$-matrices and by $\mathbb{R}^{N\times N}_{\rm spd}$ the space of symmetric
positive definite real matrices. Finally, we set $\mathbb{R}_{+}=(0,\infty)$
and $\mathbb{R}_{+}^{N}:=(\mathbb{R}_{+})^{N}$.

##  2\. Main results

###  2.1. Main results in the rank-one case

We abbreviate $n^{\prime}=n{-}1$ and let
$\mathcal{D}:=\mathbb{R}^{n^{\prime}}\times\mathbb{R}_{+}$. Given
$\mathbb{Y}=(\mathbb{Y}_{i\ell})\in
C^{\infty}(\mathcal{D};\mathbb{R}^{n^{\prime}\times n^{\prime}})$,
$Y_{n}=(Y_{ni})\in C^{\infty}(\mathcal{D};\mathbb{R}^{n^{\prime}})$ and $a\in
C^{\infty}(\mathcal{D};\mathbb{R}_{+})$, we consider for
$w:=(w^{\prime},w_{n}):=(w_{1},\dots,w_{n-1},w_{n})$ PDE systems of the form

$\displaystyle\partial_{t}w^{\prime}$ $\displaystyle=\nabla
w_{n}\cdot\mathbb{Y}(w)\nabla w^{\prime}+Y_{n}(w)|\nabla w_{n}|^{2},$ (2.1a)
$\displaystyle\partial_{t}w_{n}$ $\displaystyle=\operatorname{div}(a(w)\nabla
w_{n}),$ (2.1b)

where we abbreviated

$\displaystyle(\nabla w_{n}\cdot\mathbb{Y}(w)\nabla
w^{\prime})_{i}:=\sum_{\nu=1}^{d}\partial_{x_{\nu}}w_{n}\bigg{(}\sum_{\ell=1}^{n^{\prime}}\mathbb{Y}_{i\ell}(w)\partial_{x_{\nu}}w_{\ell}\bigg{)},\quad
i=1,\dots,n^{\prime}.$ (2.2)

######  Definition 2.1 (Symmetrisable hyperbolic part).

We say that system (2.1) has a symmetrisable hyperbolic part if there exists a
smooth mapping $\mathbb{A}_{0}\in
C^{\infty}(\mathcal{D};\mathbb{R}^{n^{\prime}\times
n^{\prime}}_{\mathrm{sym}})$ with the following properties:

  1. (i)

$\mathbb{A}_{0}(u)\in\mathbb{R}^{n^{\prime}\times n^{\prime}}_{\rm spd}$ for
all $u\in\mathcal{D}$, or equivalently (given the smooth dependence of
$\mathbb{A}_{0}$ on $u$), for each $\mathcal{K}\Subset\mathcal{D}$ there
exists $c_{\mathcal{K}}>0$ such that

$\displaystyle\xi^{\mathsf{T}}\mathbb{A}_{0}(v)\xi\geq
c_{\mathcal{K}}|\xi|^{2}\quad\text{for all
}\xi\in\mathbb{R}^{n^{\prime}}\text{ and }v\in\mathcal{K}.$ (2.3)

  2. (ii)

$\mathbb{A}_{0}$ symmetrises (2.1a) in the sense that

$\displaystyle\mathbb{A}_{0}(v)\mathbb{Y}(v)\in\mathbb{R}^{n^{\prime}\times
n^{\prime}}_{\mathrm{sym}}\quad\text{for every $v\in\mathcal{D}$.}$ (2.4)

The above definition is motivated by the structural properties of system (Rk1)
after a suitable change of variables and the fact that, under reasonable
hypotheses, quasilinear Friedrichs-symmetrisable hyperbolic systems admit
local smooth solutions.

Our first result asserts the existence of such a change of variables.

######  Theorem 2.2 (Normal form).

Suppose that $k_{i}$, $a_{i}>0$ and $\max_{i\not=n}k_{i}<k_{n}$. Then there
exists a smooth diffeomorphism $\Phi:\mathbb{R}_{+}^{n}\to\mathcal{D}$,
$u\mapsto w$, such that system (Rk1) in the $w$-variables takes the form (2.1)
with a symmetrisable hyperbolic part. The transformation $\Phi$ can be chosen
in such a way that $w_{n}=\mathfrak{p}(u)$.

The theorem is proved in Section 3. We note that the condition
$\max_{i\not=n}k_{i}<k_{n}$ is not restrictive in the sense that it does not
appear in Theorem 2.4 on the local existence and uniqueness for (Rk1). The
symmetriser and the normal form derived in the proof of Theorem 2.2 are
explicit and take a relatively simple form; see Section 3.2. For example, if
$n=2$, the change of variables $u\mapsto w$ reads as

$\displaystyle
w_{1}=\log\bigg{(}\frac{u_{1}^{1/k_{1}}}{u_{2}^{1/k_{2}}}\bigg{)},\quad
w_{2}=\mathfrak{p}(u).$ (2.5)

Other transformations leading to somewhat different, albeit structurally
similar normal forms are possible. For instance, one may alternatively
consider

$\displaystyle
w_{1}=\frac{u_{1}^{1/k_{1}}}{u_{1}^{1/k_{1}}+u_{2}^{1/k_{2}}},\quad
w_{2}=\mathfrak{p}(u);$ (2.6)

see Appendix A for details.

Our second result asserts the local-in-time existence of classical solutions
for symmetrisable systems of the form (2.1) for data in the Sobolev space
$H^{s}=H^{s}(\mathbb{T}^{d})$, $s>d/2+1$.

######  Theorem 2.3 (Local classical solutions for systems in normal form).

Suppose that system (2.1) has a symmetrisable hyperbolic part. Let $s>d/2+1$
and $w^{\mathrm{in}}\in H^{s}$ with
$r:=\min_{\mathbb{T}^{d}}w^{\mathrm{in}}_{n}>0$. Then there exists a time
$T=T(\|w^{\mathrm{in}}\|_{H^{s}},r)>0$ and a unique classical solution
$w=(w^{\prime},w_{n})\in C([0,T];H^{s})$ to system (2.1) in
$(0,T)\times\mathbb{T}^{d}$ satisfying $w_{|t=0}=w^{\mathrm{in}}$,
$\inf_{(0,T)\times\mathbb{T}^{d}}w_{n}\geq r$ and

$\displaystyle\partial_{t}w^{\prime}\in C([0,T];H^{s-1}),\quad w^{\prime}\in
C^{1}([0,T]\times\mathbb{T}^{d}),$ (2.7)
$\displaystyle\partial_{t}w_{n},\,\nabla^{2}w_{n}\in L^{2}(0,T;H^{s-1})\cap
C_{\mathrm{loc}}((0,T]\times\mathbb{T}^{d}).$ (2.8)

See Section 4 for the proof of this theorem. The local existence of classical
solutions to our original cross-diffusion system is now essentially obtained
as a corollary.

######  Theorem 2.4 (Local classical solutions for (Rk1)).

Suppose that $s>d/2+1$ and let $k_{i}$, $a_{i}>0$ for $i=1,\dots,n$. Then, for
every initial condition $u^{\mathrm{in}}\in
H^{s}(\mathbb{T}^{d};\mathcal{D}_{0})$, where
$\mathcal{D}_{0}\Subset\mathbb{R}_{+}^{n}$, there exists a time
$T=T(\|u^{\mathrm{in}}\|_{H^{s}},\mathcal{D}_{0})>0$ such that system (Rk1)
has a unique classical solution
$u:[0,T]\times\mathbb{T}^{d}\to\mathbb{R}_{+}^{n}$ in the regularity class
$u\in C([0,T];H^{s})$,

$\displaystyle\partial_{t}u,\,\nabla^{2}\mathfrak{p}(u)\in
L^{2}(0,T;H^{s-1}(\mathbb{T}^{d}))\cap
C_{\mathrm{loc}}((0,T]\times\mathbb{T}^{d})$ (2.9)

that satisfies $u_{|t=0}=u^{\mathrm{in}}$.

We refer to Section 4.6 for the proof of this theorem.

###  2.2. The general case of incomplete diffusion

Given the above results, it is natural to ask whether a similar theory can be
obtained for the rank-$\mathsf{r}$ generalisations with
$\mathsf{r}\in\\{1,\dots,n\\}$, as introduced in Section 1. Thus, given a
symmetric positive semidefinite matrix $B=(b_{ij})\in\mathbb{R}_{\rm
spd}^{n\times n}$ such that $\operatorname{rank}B=\mathsf{r}$, we aim to
identify an appropriate normal form for the induced equations of motion,

$\displaystyle\partial_{t}u_{i}-\operatorname{div}\bigg{(}u_{i}\nabla\sum_{j=1}^{n}b_{ij}u_{j}\bigg{)}=0,\quad
i=1,\dots,n.$ ($\mathsf{B}$)

Our main results on this problem may be summarised as follows.

######  Theorem 2.5 (Normal form).

Let $\mathsf{r}\in\\{1,\dots,n\\}$. There exists a domain
$\mathcal{D}_{1}\subset\mathbb{R}^{n}$ and a smooth diffeomorphism
$\Phi:\mathbb{R}_{+}^{n}\to\mathcal{D}_{1}$, $u\mapsto
w=(w_{\mathrm{I}},w_{\mathrm{II}})$, where
$w_{\mathrm{I}}=(w_{1},\dots,w_{n-\mathsf{r}})$ and
$w_{\mathrm{II}}=(w_{n-\mathsf{r}+1},\dots,w_{n})$, such that system
($\mathsf{B}$) in the $w$-variables can be recast in the symmetric
hyperbolic–parabolic form

$\displaystyle\mathbb{A}_{0}^{\mathrm{I}}(w)\partial_{t}w_{\mathrm{I}}$
$\displaystyle=\sum_{\nu=1}^{d}\mathbb{A}_{1}^{\mathrm{I}}(w,\partial_{x_{\nu}}w_{\mathrm{II}})\partial_{x_{\nu}}w_{\mathrm{I}}+f^{\mathrm{I}}(w,\nabla
w_{\mathrm{II}}),$ (2.10a)
$\displaystyle\mathbb{A}_{0}^{\mathrm{II}}\partial_{t}w_{\mathrm{II}}$
$\displaystyle=\operatorname{div}\big{(}\mathbb{A}_{1}^{\mathrm{II}}(w)\nabla
w_{\mathrm{II}}\big{)},$ (2.10b)

where
$\mathbb{A}_{0}^{\mathrm{I}}:\mathcal{D}_{1}\to\mathbb{R}^{(n-\mathsf{r})\times(n-\mathsf{r})}_{\rm
spd}$ and
$\mathbb{A}_{1}^{\mathrm{II}}:\mathcal{D}_{1}\to\mathbb{R}^{\mathsf{r}\times\mathsf{r}}_{\rm
spd}$ are smooth mappings and
$\mathbb{A}_{0}^{\mathrm{II}}\in\mathbb{R}^{\mathsf{r}\times\mathsf{r}}_{\rm
spd}$ is a constant diagonal matrix. The smooth map
$\mathbb{A}^{\mathrm{I}}_{1}:\mathcal{D}_{1}\times\mathbb{R}^{\mathsf{r}}\to\mathbb{R}^{(n-\mathsf{r})\times(n-\mathsf{r})}_{\rm
sym}$ is linear in the second argument, and
$f^{\mathrm{I}}:\mathcal{D}_{1}\times\mathbb{R}^{\mathsf{r}}\to\mathbb{R}^{n-\mathsf{r}}$
is smooth and quadratic in the second argument.

The precise form of the coefficient matrices appearing in Theorem 2.5 is
described in Proposition 5.2. The transformation $\Phi$ is built upon an
orthonormal basis of constant eigenvectors associated with $B$ and slightly
differs from that in Theorem 2.2. We emphasise that normal forms are generally
not unique, and in different applications different choices might be
preferable.

The local existence of a unique classical solution to the symmetric
hyperbolic–parabolic systems of the form (2.10) will be established in Section
5.3, cf. Theorem 5.5. As a consequence of the analysis in Section 5.3, we
obtain the local existence of a unique positive classical solution to the
cross-diffusion system ($\mathsf{B}$) for smooth positive initial data.

######  Theorem 2.6 (Local classical solutions in original variables).

Let $B\in\mathbb{R}^{n\times n}$ fulfil the conditions (1.6) and let
$s>d/2+1$. Then, for every initial condition $u^{\mathrm{in}}\in
H^{s}(\mathbb{T}^{d};\mathcal{D}_{0})$, where
$\mathcal{D}_{0}\Subset\mathbb{R}_{+}^{n}$, there exists a time
$T=T(\|u^{\mathrm{in}}\|_{H^{s}},\mathcal{D}_{0})>0$ such that system
($\mathsf{B}$) has a unique classical solution
$u:[0,T]\times\mathbb{T}^{d}\to\mathbb{R}_{+}^{n}$ in the regularity class
$u\in C([0,T];H^{s})$,

$\displaystyle\partial_{t}u,\nabla^{2}Bu\in
L^{2}(0,T;H^{s-1}(\mathbb{T}^{d}))\cap
C_{\mathrm{loc}}((0,T]\times\mathbb{T}^{d})$ (2.11)

that satisfies $u_{|t=0}=u^{\mathrm{in}}$.

######  Remark 2.7.

By a rescaling, we see that our results apply more generally to systems

$\displaystyle\partial_{t}u_{i}-\operatorname{div}\big{(}u_{i}\nabla
p_{i}(u)\big{)}=0,\quad p_{i}(u)=\sum_{j=1}^{n}a_{ij}u_{j},$

for which the matrix $A=(a_{ij})\in\mathbb{R}^{n\times n}$ is such that for
some $\widehat{\pi}\in\mathbb{R}_{+}^{n}$ the product
$B:=A\operatorname{diag}(\widehat{\pi}_{1},\ldots,\widehat{\pi}_{n})$
satisfies the properties in (1.6).

##  3\. An explicit normal form for rank-one systems

In this section, we provide the proof of Theorem 2.2. Observe that if a
diffeomorphism $\Phi$ has the required properties in the case $a_{i}=1$ for
$i=1,\ldots,n$ in (1.3), then the transformation $\Phi\circ\Gamma$ with
$\Gamma_{i}(u)=a_{i}u_{i}$ enjoys the properties needed in the general case
$a_{i}>0$ for all $i$. We therefore assume that $a_{i}=1$ for $i=1,\ldots,n$
in (1.3).

###  3.1. Change of variables

Between the domains $\mathcal{\widehat{D}}:=\mathbb{R}_{+}^{n}$ and
$\mathcal{D}:=\mathbb{R}^{n^{\prime}}\times\mathbb{R}_{+}$, where
$n^{\prime}:=n-1$, we consider the transformation
$\Phi:\mathcal{\widehat{D}}\to\mathcal{D}$,

$\displaystyle\Phi(u)=\begin{pmatrix}\log\big{(}u_{1}^{1/k_{1}}\big{/}u_{n}^{1/k_{n}}\big{)}\\\
\vdots\\\
\log\big{(}u_{n^{\prime}}^{1/k_{n^{\prime}}}\big{/}u_{n}^{1/k_{n}}\big{)}\\\\[3.0pt]
\sum_{j=1}^{n}u_{j}\end{pmatrix}\quad\mbox{for }u\in\widehat{\mathcal{D}}.$
(3.1)

Its Jacobian

$\mathrm{D}\Phi(u)=\begin{pmatrix}\frac{1}{k_{1}u_{1}}&0&\dots&0&-\frac{1}{k_{n}u_{n}}\\\
0&\frac{1}{k_{2}u_{2}}&0&\dots&-\frac{1}{k_{n}u_{n}}\\\
\vdots&&\ddots&&\vdots\\\
0&\dots&0&\frac{1}{k_{n^{\prime}}u_{n^{\prime}}}&-\frac{1}{k_{n}u_{n}}\\\
1&1&1&\dots&1\end{pmatrix}$ (3.2)

has a non-vanishing determinant,

$\det\mathrm{D}\Phi(u)=\sum_{\ell=1}^{n}\prod_{i\not=\ell}\frac{1}{k_{i}u_{i}}>0.$
(3.3)

######  Lemma 3.1.

The map $\Phi:\widehat{\mathcal{D}}\to\mathcal{D}$ is a
$C^{\infty}$-diffeomorphism.

###### Proof.

By definition, $\Phi\in C^{\infty}$, and by the implicit function theorem and
(3.3), the map $\Phi$ is locally invertible with smooth inverse. To show that
$\Phi$ is a bijection from $\widehat{\mathcal{D}}$ to $\mathcal{D}$, we
observe that the equation $\Phi(u)=w$ for $u\in\widehat{\mathcal{D}}$ and
$w\in\mathcal{D}$ is equivalent to

$\displaystyle u_{j}=\exp(k_{j}w_{j}){u_{n}}^{k_{j}/k_{n}},\
j=1,\dots,n^{\prime},\quad\text{and}\quad\sum_{j=1}^{n}u_{j}=w_{n}.$ (3.4)

Inserting the first relation into the second one, we find that

$\displaystyle
g_{w^{\prime}}(u_{n}):=u_{n}+\sum_{j=1}^{n^{\prime}}\exp(k_{j}w_{j}){u_{n}}^{k_{j}/k_{n}}=w_{n}.$
(3.5)

For fixed $w=(w^{\prime},w_{n})\in\mathcal{D}$, the function
$g_{w^{\prime}}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is strictly increasing and
continuous with $\lim_{s\downarrow 0}g_{w^{\prime}}(s)=0$ and
$\lim_{s\uparrow+\infty}g_{w^{\prime}}(s)=+\infty$. Hence, there exists a
unique solution $u_{n}\in\mathbb{R}_{+}$ of the equation
$g_{w^{\prime}}(u_{n})=w_{n}$, which in turn uniquely determines
$u_{j}\in\mathbb{R}_{+}$, $j=1,\dots,n^{\prime}$, by means of the first
condition in (3.4). ∎

###  3.2. PDE system in new variables

We now formulate our original equation in the new variables $w=\Phi(u)$. We
recall that the original PDE system takes the form

$\displaystyle\partial_{t}u_{i}=\operatorname{div}(k_{i}u_{i}\nabla
w_{n}),\quad i=1,\dots,n,$ (3.6)

and let $\Psi(w)=u$ denote the inverse of $\Phi$. Summing up the $n$ equations
(3.6) leads to

$\displaystyle\partial_{t}w_{n}=\operatorname{div}(a(w)\nabla w_{n}),$ (3.7)

where $a(w):=\widehat{a}(\Psi(w))$ with
$\widehat{a}(u)=\sum_{i=1}^{n}k_{i}u_{i}$. For $u=\Psi(w)$, we have
$\widehat{a}(u)\geq(\min_{i}k_{i})\sum_{i=1}^{n}u_{i}=(\min_{i}k_{i})w_{n}$.
Since $\Psi\in C^{\infty}(\mathcal{D};\widehat{\mathcal{D}})$ and
$\inf_{w\in\mathcal{K}}w_{n}>0$ for every $\mathcal{K}\Subset\mathcal{D}$, we
find that $a\in C^{\infty}(\mathcal{D};\mathbb{R}_{+})$.

Next, let $i=1,\dots,n^{\prime}$. We deduce from (3.2) and (3.6) that

$\displaystyle\partial_{t}w_{i}$
$\displaystyle=\sum_{\ell=1}^{n}\partial_{u_{\ell}}\Phi_{i}(u)\partial_{t}u_{\ell}=\frac{\partial_{t}u_{i}}{k_{i}u_{i}}-\frac{\partial_{t}u_{n}}{k_{n}u_{n}}=\frac{1}{u_{i}}\operatorname{div}(u_{i}\nabla
w_{n})-\frac{1}{u_{n}}\operatorname{div}(u_{n}\nabla w_{n})$
$\displaystyle=\nabla(\log\Psi_{i}-\log\Psi_{n})\cdot\nabla
w_{n}=\sum_{\ell=1}^{n}\partial_{w_{\ell}}(\log\Psi_{i}-\log\Psi_{n})\nabla
w_{\ell}\cdot\nabla w_{n}.$

To determine the term $\partial_{w_{\ell}}(\log\Psi_{i}-\log\Psi_{n})$, we use
relation (3.5),

$\partial_{w_{n}}u_{n}=\partial_{w_{n}}\bigg{(}w_{n}-\sum_{j=1}^{n^{\prime}}\exp(k_{j}w_{j})u_{n}^{k_{j}/k_{n}}\bigg{)}=1-\sum_{j=1}^{n^{\prime}}\frac{k_{j}u_{j}}{k_{n}u_{n}}\partial_{w_{n}}u_{n},$

and solve this identity for $\partial_{w_{n}}u_{n}$:

$(\partial_{w_{n}}\Psi_{n})(w)=\partial_{w_{n}}u_{n}=\bigg{(}1+\sum_{j=1}^{n^{\prime}}\frac{k_{j}u_{j}}{k_{n}u_{n}}\bigg{)}^{-1}=\frac{k_{n}u_{n}}{\widehat{a}(u)}.$

In a similar way, it follows for $\ell=1,\ldots,n^{\prime}$ that

$\partial_{w_{\ell}}u_{n}=-k_{\ell}u_{\ell}-\sum_{j=1}^{n^{\prime}}\frac{k_{j}u_{j}}{k_{n}u_{n}}\partial_{w_{\ell}}u_{n}$

and consequently, after solving for $\partial_{w_{\ell}}u_{n}$,

$(\partial_{w_{\ell}}\Psi_{n})(w)=\partial_{w_{\ell}}u_{n}=-k_{\ell}u_{\ell}\bigg{(}1+\sum_{j=1}^{n}\frac{k_{j}u_{j}}{k_{n}u_{n}}\bigg{)}^{-1}=-\frac{k_{n}u_{n}}{\widehat{a}(u)}k_{\ell}u_{\ell}.$

Hence,

$(\partial_{w_{n}}\log\Psi_{n})(\Phi(u))=\frac{k_{n}}{\widehat{a}(u)},\quad(\partial_{w_{\ell}}\log\Psi_{n})(\Phi(u))=-\frac{k_{n}}{\widehat{a}(u)}k_{\ell}u_{\ell}\text{
for }\ell=1,\dots,n^{\prime}.$

Next, we differentiate
$w_{i}=\log(u_{i}^{1/k_{i}}/u_{n}^{1/k_{n}})=(1/k_{i})\log\Psi_{i}-(1/k_{n})\log\Psi_{n}$,
$i=1,\dots,n^{\prime}$, with respect to $w_{\ell}$, $\ell=1,\dots,n^{\prime}$,
which gives
$\delta_{i\ell}=(1/k_{i})\partial_{w_{\ell}}\log\Psi_{i}-(1/k_{n})\partial_{w_{\ell}}\log\Psi_{n}$.
This shows that

$\displaystyle\partial_{w_{\ell}}\log\Psi_{i}-\partial_{w_{\ell}}\log\Psi_{n}=k_{i}\delta_{i\ell}+\bigg{(}\frac{k_{i}}{k_{n}}-1\bigg{)}\partial_{w_{\ell}}\log\Psi_{n}=k_{i}\delta_{i\ell}+\big{(}k_{n}-k_{i}\big{)}\frac{k_{\ell}u_{\ell}}{\widehat{a}(u)},$
(3.8)

Similarly, differentiating
$w_{i}=(1/k_{i})\log\Psi_{i}-(1/k_{n})\log\Psi_{n}$, $i=1,\dots,n^{\prime}$,
with respect to $w_{n}$, we obtain
$0=(1/k_{i})\partial_{w_{n}}\log\Psi_{i}-(1/k_{n})\partial_{w_{n}}\log\Psi_{n}$
or, equivalently, $\partial_{w_{n}}\log\Psi_{i}=k_{i}/\widehat{a}(u)$. Thus,

$\displaystyle\partial_{w_{n}}\log\Psi_{i}-\partial_{w_{n}}\log\Psi_{n}=\frac{k_{i}-k_{n}}{\widehat{a}(u)}.$
(3.9)

In summary, the components $w^{\prime}=(w_{1},\dots,w_{n^{\prime}})$ satisfy
the system

$\displaystyle\partial_{t}w^{\prime}=\sum_{\nu=1}^{d}\partial_{x_{\nu}}w_{n}\mathbb{Y}(w)\partial_{x_{\nu}}w^{\prime}+Y_{n}(w)|\nabla
w_{n}|^{2},$ (3.10)

where $\mathbb{Y}=(\mathbb{Y}_{i\ell})_{i,\ell=1}^{n^{\prime}}$,
$Y_{n}=(Y_{ni})_{i=1}^{n^{\prime}}$, and

$\displaystyle\mathbb{Y}_{i\ell}(w)$
$\displaystyle:=k_{i}\delta_{i\ell}+\big{(}k_{n}-k_{i}\big{)}\frac{k_{\ell}u_{\ell}}{\widehat{a}(u)},\quad$
$\displaystyle i,\ell=1,\dots,n^{\prime},$ (3.11) $\displaystyle Y_{ni}(w)$
$\displaystyle:=\mathbb{Y}_{in}(w):=\frac{k_{i}-k_{n}}{\widehat{a}(u)},\quad$
$\displaystyle i=1,\dots,n^{\prime}.$ (3.12)

###  3.3. Symmetriser for the first-order subsystem

We look for a matrix $\mathbb{A}_{0}(w)\in\mathbb{R}^{n^{\prime}\times
n^{\prime}}_{\mathrm{sym}}$ which is $w$-locally uniformly bounded and
positive definite, such that $\mathbb{A}_{0}(w)\mathbb{Y}(w)$ is symmetric for
all $w$. The ansatz of a diagonal symmetrising matrix

$\displaystyle\mathbb{A}_{0}(w)=\operatorname{diag}(X_{1}(w),\dots,X_{n^{\prime}}(w))$
(3.13)

leads to the conditions

$X_{i}\mathbb{Y}_{ij}=X_{j}\mathbb{Y}_{ji},\quad i,j=1,\dots,n^{\prime},$

for the unknown smooth and positive functions $X_{i}$, $i=1,\dots,n^{\prime}$.
Since, by hypothesis, $0<k_{1}\leq\ldots\leq k_{n^{\prime}}<k_{n}$, an
admissible choice of $X_{i}$ is

$\displaystyle X_{i}=\frac{k_{i}u_{i}}{k_{n}-k_{i}}.$ (3.14)

Thus, (3.13), (3.14) defines an admissible symmetriser for system (3.10). In
view of Lemma 3.1, this completes the proof of Theorem 2.2.

##  4\. Local classical solutions

In this section, we prove Theorem 2.3 and deduce Theorem 2.4 as a corollary.
Thanks to its symmetry, the hyperbolic part can be treated following a well-
established procedure based on $H^{s}$ energy estimates at the linear
approximate level and a Picard iteration, cf. [1, 22] and the references
therein. Hence, our main task is to properly take care of the coupling between
the hyperbolic and the parabolic problem.

We only present the proof for integer $s$. Using basic tools from Fourier
analysis, the extension to fractional $s>d/2+1$ should be straightforward. For
the most part, we avoid using the explicit structure obtained in Section 3. In
this way, we only require a few minor changes to treat the Cauchy problem for
the general rank-$\mathsf{r}$ system ($\mathsf{B}$) once brought into a normal
form.

###  4.1. Preliminaries

Let $\mathbb{A}_{0}$ be a smooth symmetriser for (2.1a). We will construct a
strong solution (within the appropriate regularity class) to the symmetric
hyperbolic–parabolic system

$\displaystyle\mathbb{A}_{0}(w)\partial_{t}w^{\prime}$ $\displaystyle=\nabla
w_{n}\cdot\mathbb{A}_{1}(w)\nabla w^{\prime}+V_{n}(w)|\nabla w_{n}|^{2},$
(4.1a) $\displaystyle\partial_{t}w_{n}$
$\displaystyle=\operatorname{div}(a(w)\nabla w_{n}),$ (4.1b)

with $w|_{t=0}=w^{\mathrm{in}}$, where
$\mathbb{A}_{1}(w)=\mathbb{A}_{0}(w)\mathbb{Y}(w)$ and
$V_{n}(w)=\mathbb{A}_{0}(w)Y_{n}(w)$.

By hypothesis, we can define positive constants

$\displaystyle
r:=\inf_{x\in\mathbb{T}^{d}}w^{\mathrm{in}}_{n}(x)\quad\text{and}\quad
R:=\|w^{\mathrm{in}}\|_{H^{s}}.$

As an immediate consequence, this gives

$\displaystyle\|w^{\mathrm{in}}_{i}\|_{L^{\infty}}\leq LR\quad\text{for
}i=1,\dots,n,$

where $L<\infty$ denotes the constant from the Sobolev embedding
$H^{s}(\mathbb{T}^{d})\hookrightarrow L^{\infty}(\mathbb{T}^{d})$. Defining
$a_{1}:=(r/2)\min_{i}k_{i}>0$, we infer the bound $a(v)\geq a_{1}$ for all

$\displaystyle
v\in\mathcal{D}_{0}:=\big{\\{}\widetilde{v}\in\mathcal{D}:\widetilde{v}_{n}>r/2\text{
and }|\widetilde{v}_{i}|<2LR,\ i=1,\dots,n\big{\\}}.$

We further choose $\lambda_{1}\in(0,1]$ and $\Lambda_{1}\geq 1$ such that

$\displaystyle\Lambda_{1}\mathbb{I}_{n^{\prime}}\geq\mathbb{A}_{0}(v)\geq\lambda_{1}\mathbb{I}_{n^{\prime}}\quad\text{for
all }v\in\mathcal{D}_{0},$ (4.2)

and abbreviate

$\displaystyle K:=2\sqrt{\frac{\Lambda_{1}}{\lambda_{1}}}.$ (4.3)

Note that $a_{1},\lambda_{1},\Lambda_{1}$ and $K$ depend on $w^{\mathrm{in}}$
through $r$ and $R$ only.

To avoid regularity issues during the iteration process, we mollify the
initial data. Let $(\eta_{\ell})_{\ell\in\mathbb{N}_{0}}\subset
C^{\infty}(\mathbb{T}^{d})$ be an approximate identity, that is,
$\int_{\mathbb{T}^{d}}\eta_{\ell}\mathrm{d}x=1$,
$\|\eta_{\ell}\|_{L^{1}}\lesssim 1$ and
$\lim_{\ell\to\infty}\int_{\\{|x|>\delta\\}}|\eta_{\ell}|\mathrm{d}x=0$ for
all $0<\delta\ll 1$. Here, $a^{\ell}\lesssim b^{\ell}$ means that there exists
a constant $C>0$ such that $a^{\ell}\leq Cb^{\ell}$ for all
$\ell\in\mathbb{N}_{0}$. We choose $(\eta_{\ell})$ such that $\eta_{\ell}\geq
0$ for all $\ell$ and introduce the mollified initial datum
$z^{\ell}:=\eta_{\ell}\ast w^{\mathrm{in}}$ for every $\ell\in\mathbb{N}_{0}$.
After possibly passing to a subsequence of $(\eta_{\ell})$, we may assume that

$\displaystyle\|z^{\ell+1}-z^{\ell}\|_{L^{2}}\lesssim 2^{-\ell}R.$ (4.4)

The construction by convolution immediately yields $z^{\ell}_{n}\geq r$ and
$|z^{\ell}_{i}|\leq LR$, $i=1,\dots,n$ and thus,
$z^{\ell}(x)\in\mathcal{D}_{0}$ for all $x\in\mathbb{T}^{d}$ and
$\ell\in\mathbb{N}_{0}$. Moreover,

$\displaystyle\|z^{\ell}\|_{H^{s}}\leq R\quad\text{ for all
}\ell\in\mathbb{N}_{0}.$ (4.5)

Before starting with our analysis, let us recall the following classical
inequalities from calculus [18, 23]: For all $\varphi,\psi\in
C^{\infty}(\mathbb{T}^{d})$ and any multi-index $\alpha\in\mathbb{N}_{0}^{n}$
with $|\alpha|\leq\sigma$,

$\displaystyle\|\partial^{\alpha}(\psi\varphi)\|_{L^{2}}$
$\displaystyle\lesssim\|\psi\|_{L^{\infty}}\|\varphi\|_{H^{\sigma}}+\|\psi\|_{H^{\sigma}}\|\varphi\|_{L^{\infty}},$
(4.6) $\displaystyle\|[\partial^{\alpha},\psi]\varphi\|_{L^{2}}$
$\displaystyle\lesssim\|\nabla\psi\|_{L^{\infty}}\|\varphi\|_{H^{\sigma-1}}+\|\nabla\psi\|_{H^{\sigma-1}}\|\varphi\|_{L^{\infty}},$

where $[A,B]:=AB-BA$ denotes the commutator of two linear operators $A$ and
$B$. Furthermore, for $g\in C^{\infty}(\mathcal{D})$,
$\mathcal{K}\Subset\mathcal{D}$, and $\varphi\in
H^{\sigma}(\mathbb{T}^{d};\mathcal{K})$,

$\displaystyle\|g(\varphi)\|_{H^{\sigma}}\lesssim\|\varphi\|_{H^{\sigma}}+1,$
(4.7)

where the constant associated to this inequality depends on
$\|g\|_{C^{\sigma}(\mathcal{K})}$ and $\|\varphi\|_{L^{\infty}}$. Finally, let
us introduce the Banach space
$X^{\sigma}_{t}:=C([0,t];H^{\sigma}(\mathbb{T}^{d}))$ with the norm

$\displaystyle\|v\|_{X^{\sigma}_{t}}:=\sup_{\tau\in(0,t)}\|v(\tau)\|_{H^{\sigma}}.$
(4.8)

###  4.2. Iteration scheme

We initialise $w^{0}:=z^{0}$ and $t_{0}=\infty$, and consider the following
iteration scheme. Given $v:=w^{\ell-1}\in
C^{\infty}([0,t_{\ell-1}){\times}\mathbb{T}^{d};\mathcal{D})$,
$\ell\in\mathbb{N}$, with $v_{n}\geq r$ and $\|v\|_{X^{s}_{t}}<KR$ for all
$t<t_{\ell-1}$, we let the next iterate $w^{\ell}$ be the solution
$w=(w^{\prime},w_{n})$ to the linear decoupled system

$\displaystyle\mathbb{A}_{0}(v)\partial_{t}w^{\prime}$ $\displaystyle=\nabla
v_{n}\cdot\mathbb{A}_{1}(v)\nabla w^{\prime}+V_{n}(v)|\nabla v_{n}|^{2},$
(4.9a) $\displaystyle\partial_{t}w_{n}$
$\displaystyle=\operatorname{div}(a(v)\nabla w_{n}),$ (4.9b)

supplemented by the initial condition $w|_{t=0}=z^{\ell}$. Notice that
$w^{\ell}$ is well defined and smooth on the entire time interval
$[0,t_{\ell-1})$ thanks to classical theory for linear, uniformly parabolic
equations resp. linear symmetric hyperbolic systems with smooth coefficients
(cf. [19] resp. [1, Chapter 2]) and the fact that there is no coupling between
$w^{\prime}$ and $w_{n}$. Moreover, the maximum principle implies that

$\displaystyle r\leq w_{n}^{\ell}(t,\cdot)\leq LR\quad\text{ for all
}t\in[0,t_{\ell-1}).$ (4.10)

We then let $0<t_{\ell}\leq t_{\ell-1}$ be the maximal time less than or equal
to $t_{\ell-1}$ such that

$\displaystyle\|w^{\ell}\|_{X^{s}_{t}}<KR\quad\text{for all }t<t_{\ell},$
(4.11)

where $K>1$ is the constant in (4.3). In view of (4.5), the time
$t_{\ell}\in(0,t_{\ell-1}]$ is indeed well-defined.

The above construction and the Sobolev embedding $H^{s}\subset L^{\infty}$
(with constant $L$) gives us a first rough control of the values of the
iterates:

$\displaystyle
w^{\ell}(t,x)\in[-LKR,LKR]^{n^{\prime}}\times[r,LR]\quad\text{for all
}(t,x)\in(0,t_{\ell})\times\mathbb{T}^{d}.$ (4.12)

Since $K$ depends on the constants $\Lambda_{1}$ and $\lambda_{1}$, the
control (4.12) needs to be upgraded before we can take advantage of the strict
positive definiteness of $\mathbb{A}_{0}$ in $\mathcal{D}$.

###  4.3. Uniform bounds

######  Lemma 4.1 (Control of values).

There exists a time $T_{1}=T_{1}(R,r^{-1})>0$ such that for all
$\ell\in\mathbb{N}_{0}$,

$\displaystyle w^{\ell}(t,x)\in\mathcal{D}_{0}\quad\text{for all
}(t,x)\in(0,\widehat{t}_{\ell})\times\mathbb{T}^{d},\mbox{ where
}\widehat{t}_{\ell}:=\min\\{t_{\ell},T_{1}\\}.$ (4.13)

###### Proof.

The assertion is true for $\ell=0$ since, by construction,
$z^{0}(x)\in\mathcal{D}_{0}$ for all $x\in\mathbb{T}^{d}$. Let now $\ell\geq
1$. It follows from equation (4.9a) that, for all $\tau<t_{\ell}\leq
t_{\ell-1}$,

$\displaystyle\|\partial_{t}(w^{\ell})^{\prime}\|_{X_{\tau}^{s-1}}$
$\displaystyle\leq
C\big{(}\|w^{\ell-1}\|_{X_{\tau}^{s}},\|(w^{\ell})^{\prime}\|_{X_{\tau}^{s}},R,r^{-1}\big{)}\leq
C_{0},$ (4.14)

where $C_{0}=C_{0}(R,r^{-1})$, and we used inequalities (4.6)–(4.7) and the
control (4.11)–(4.12). Hence, for all $t<t_{\ell}$ and every
$i\in\\{1,\dots,n^{\prime}\\}$,

$\displaystyle\|w^{\ell}_{i}(t)-w^{\ell}_{i}(0)\|_{C(\mathbb{T}^{d})}\leq
tL_{1}C_{0}.$ (4.15)

Here, $L_{1}$ denotes the constant associated with the embedding
$H^{s-1}(\mathbb{T}^{d})\hookrightarrow C(\mathbb{T}^{d})$. Thus, with the
choice $T_{1}:=LR/(2L_{1}C_{0})$, we find that

$|w^{\ell}_{i}(t,x)|\leq|w^{\ell}_{i}(0,x)|+tL_{1}C_{0}\leq
LR+T_{1}L_{1}C_{0}=\frac{3}{2}LR<2LR$

for $i=1,\dots,n^{\prime}$ and all
$(t,x)\in(0,\widehat{t}_{\ell})\times\mathbb{T}^{d}$. Combined with
inequalities (4.10), we deduce the assertion. ∎

######  Lemma 4.2 (Uniform bounds).

For all $\ell\in\mathbb{N}_{0}$, let
$\widehat{t}_{\ell}=\min\\{t_{\ell},T_{1}\\}$, where $T_{1}>0$ denotes the
constant from Lemma 4.1.

  1. (i)

There exists $F(t)=F(t,R,r^{-1})>0$ which is continuous on $[0,\infty)^{3}$
and non-decreasing in each of its arguments such that for all
$\ell\in\mathbb{N}_{+}$,

$\displaystyle\int_{0}^{t}\\!\\!\|\nabla
w_{n}^{\ell}(\tau)\|_{H^{s}}^{2}\mathrm{d}\tau\leq F(t),\quad
t\in[0,\widehat{t}_{\ell-1}).$ (4.16)

  2. (ii)

There exists $T_{*}=T_{*}(R,r^{-1})>0$ such that $\widehat{t}_{\ell}>T_{*}$
for all $\ell\in\mathbb{N}$.

######  Remark 4.3.

We infer immediately from Lemma 4.2 and equations (4.9) a uniform bound on the
time derivative of the iterates: for all $\ell\in\mathbb{N}_{+}$,

$\displaystyle\|\partial_{t}(w^{\ell})^{\prime}\|_{X_{T_{*}}^{s-1}}\leq
C,\quad\|\partial_{t}w_{n}^{\ell}\|_{L^{2}(0,T_{*};H^{s-1})}\leq C({T_{*}}),$
(4.17)

where $C$ and $C(T_{*})$ may depend on $R$ and $r^{-1}$.

###### Proof of Lemma 4.2.

Let $\ell\in\mathbb{N}$ and abbreviate, as before, $w:=w^{\ell}$ and
$v:=w^{\ell-1}$. We will always suppose that $t<\widehat{t}_{\ell-1}$, which
guarantees that $v(\tau,x)\in\mathcal{D}_{0}$ for all
$(\tau,x)\in(0,t)\times\mathbb{T}^{d}$ and $\|v\|_{X_{t}^{s}}\leq KR$.

Re (i): For a multi-index $\alpha\in\mathbb{N}_{0}^{d}$ of order
$0\leq|\alpha|\leq s$, we differentiate (4.9b) by the spatial differential
operator $\partial^{\alpha}$:

$\displaystyle\partial_{t}\partial^{\alpha}w_{n}=\operatorname{div}\big{(}a(v)\partial^{\alpha}\nabla
w_{n}\big{)}+\operatorname{div}\big{(}[\partial^{\alpha},a(v)]\nabla
w_{n}\big{)}.$ (4.18)

Testing this identity with $\partial^{\alpha}w_{n}$, we obtain

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\partial^{\alpha}w_{n}\|_{L^{2}}^{2}+a_{1}\int_{\mathbb{T}^{d}}|\partial^{\alpha}\nabla
w_{n}|^{2}\mathrm{d}x\leq C\|[\partial^{\alpha},a(v)]\nabla
w_{n}\|_{L^{2}}\|\partial^{\alpha}\nabla w_{n}\|_{L^{2}},$

and hence,

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\partial^{\alpha}w_{n}\|_{L^{2}}^{2}+\frac{a_{1}}{2}\|\partial^{\alpha}\nabla
w_{n}\|^{2}_{L^{2}}$ $\displaystyle\leq
C(a_{1})\|[\partial^{\alpha},a(v)]\nabla w_{n}\|_{L^{2}}^{2}.$ (4.19)

Upon taking the sum over $\alpha$, $0\leq|\alpha|\leq s$, integrating in time
and using inequalities (4.6)–(4.7) and the Sobolev embedding
$H^{s-1}\hookrightarrow L^{\infty}$, we arrive at

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\|w_{n}\|_{H^{s}}^{2}+a_{1}\|\nabla
w_{n}\|^{2}_{H^{s}}\leq R^{2}C(R,r^{-1})\|w_{n}\|_{H^{s}}^{2}.$ (4.20)

Gronwall’s lemma implies that

$\displaystyle\|w_{n}(t)\|_{H^{s}}^{2}+a_{1}\int_{0}^{t}\|\nabla
w_{n}\|^{2}_{H^{s}}\mathrm{d}\tau\leq\|w_{n}(0)\|_{H^{s}}^{2}\exp(tC(R,r^{-1}))$
(4.21)

for all $t\in(0,\widehat{t}_{\ell-1})$. The rough bound (4.16) is a
consequence of (4.21) and the initial condition $w_{|t=0}=z^{\ell}$, combined
with the control (4.5) and the fact that the above argument holds for any
$\ell\in\mathbb{N}_{0}$.

Re (ii): In the following, we abbreviate
$w_{\alpha}^{\prime}:=\partial^{\alpha}w^{\prime}$. We first left-multiply
(4.9a) by the matrix $\mathbb{A}_{0}(v)^{-1}$, differentiate the resulting
identity by $\partial^{\alpha}$ and subsequently left-multiply by
$\mathbb{A}_{0}(v)$. This gives

$\displaystyle\mathbb{A}_{0}(v)\partial_{t}w^{\prime}_{\alpha}=\nabla
v_{n}\cdot\mathbb{A}_{1}(v)\nabla
w^{\prime}_{\alpha}+\mathbb{A}_{0}(v)\mathcal{R}_{\alpha},\quad\mbox{where}$
(4.22) $\displaystyle\mathcal{R}_{\alpha}:=\big{[}\partial^{\alpha},\nabla
v_{n}\cdot\mathbb{Y}(v)\big{]}\nabla
w^{\prime}+\partial^{\alpha}\big{(}Y_{n}(v)|\nabla v_{n}|^{2}\big{)}.$ (4.23)

Next, we test this equation with $w_{\alpha}^{\prime}$:

$\displaystyle\frac{1}{2}$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{T}^{d}}(w_{\alpha}^{\prime})^{\mathsf{T}}\mathbb{A}_{0}(v)w^{\prime}_{\alpha}\mathrm{d}x-\int_{\mathbb{T}^{d}}(w_{\alpha}^{\prime})^{\mathsf{T}}\big{(}\mathrm{D}\mathbb{A}_{0}(v)\partial_{t}v\big{)}w_{\alpha}^{\prime}\mathrm{d}x$
$\displaystyle=\frac{1}{2}\int_{\mathbb{T}^{d}}\nabla
v_{n}\cdot\big{\\{}\nabla\big{(}(w_{\alpha}^{\prime})^{\mathsf{T}}\mathbb{A}_{1}(v)w_{\alpha}^{\prime}\big{)}-(w_{\alpha}^{\prime})^{\mathsf{T}}\big{(}\mathrm{D}\mathbb{A}_{1}(v)\nabla
v\big{)}w_{\alpha}^{\prime}\big{\\}}\mathrm{d}x$
$\displaystyle\phantom{xx}{}+\int_{\mathbb{T}^{d}}(w_{\alpha}^{\prime})^{\mathsf{T}}\mathbb{A}_{0}(v)\mathcal{R}_{\alpha}\mathrm{d}x.$

Hence, after a rearrangement and an integration by parts in the second step,

$\displaystyle\frac{1}{2}$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{T}^{d}}(w_{\alpha}^{\prime})^{\mathsf{T}}\mathbb{A}_{0}(v){w}^{\prime}_{\alpha}\mathrm{d}x\leq\frac{1}{2}\int_{\mathbb{T}^{d}}\nabla
v_{n}\cdot\nabla\big{(}(w^{\prime}_{\alpha})^{\mathsf{T}}\mathbb{A}_{1}(v)w^{\prime}_{\alpha}\big{)}\mathrm{d}x$
(4.24)
$\displaystyle\phantom{xx}{}+\frac{1}{2}\int_{\mathbb{T}^{d}}\big{(}|\nabla
v_{n}||\mathrm{D}\mathbb{A}_{1}(v)\nabla
v|+|\mathrm{D}\mathbb{A}_{0}(v)\partial_{t}v|\big{)}|w^{\prime}_{\alpha}|^{2}\mathrm{d}x+\int_{\mathbb{T}^{d}}|\mathcal{R}_{\alpha}||{w}^{\prime}_{\alpha}|\mathrm{d}x$
(4.25) $\displaystyle\lesssim\big{\|}|\Delta v_{n}|+|\nabla
v|^{2}+|\partial_{t}v|\big{\|}_{L^{\infty}}\|w^{\prime}_{\alpha}\|_{L^{2}}^{2}+\|\mathcal{R}_{\alpha}\|_{L^{2}}\|w^{\prime}_{\alpha}\|_{L^{2}},$
(4.26)

where the constant associated to the last inequality depends on $R$ and
$r^{-1}$.

To proceed, we estimate the remainder term $\mathcal{R}_{\alpha}$ using again
inequalities (4.6)–(4.7), the bound $\sup_{t<t_{\ell-1}}\|v\|_{X_{t}^{s}}\leq
KR$ and the embedding $H^{s-1}\hookrightarrow L^{\infty}$:

$\displaystyle\sum_{|\alpha|\leq s}\|\mathcal{R}_{\alpha}\|_{L^{2}}$
$\displaystyle\leq\sum_{|\alpha|\leq s}\Big{(}\|\nabla(\nabla
v_{n}\cdot\mathbb{Y}(v))\|_{L^{\infty}}\|\nabla w^{\prime}\|_{H^{|\alpha|-1}}$
$\displaystyle\phantom{xx}{}+\|\nabla(\nabla
v_{n}\cdot\mathbb{Y}(v))\|_{H^{|\alpha|-1}}\|\nabla
w^{\prime}\|_{L^{\infty}}+\|\partial^{\alpha}\big{(}Y_{n}(v)|\nabla
v_{n}|^{2}\big{)}\|_{L^{2}}\Big{)}$ $\displaystyle\lesssim\big{(}\|\nabla
v_{n}\|_{H^{s}}+1\big{)}\big{(}\|\nabla{w}^{\prime}\|_{H^{s-1}}+1\big{)}.$

To estimate the time derivative $\partial_{t}v$, we use the equation for
$v=w^{\ell-1}$; i.e. (4.9) with $\ell$ replaced by $\ell-1$ supposing that
$\ell\geq 2$ (if $\ell=1$, then $\partial_{t}v\equiv 0$ and there is nothing
to show). This gives

$\displaystyle\|\partial_{t}v\big{\|}_{L^{\infty}}\leq\|\partial_{t}v_{n}\big{\|}_{L^{\infty}}+\|\partial_{t}v^{\prime}\big{\|}_{L^{\infty}}\lesssim\|\Delta
v_{n}\|_{H^{s-1}}+1\lesssim\|\nabla v_{n}\|_{H^{s}}+1.$ (4.27)

In combination, we infer for $\nu(t):=\|\nabla v_{n}(t)\|_{H^{s}}{+}1$ and
$C_{1}=C_{1}(R,r^{-1})$ that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\sum_{|\alpha|\leq
s}\int_{\mathbb{T}^{d}}({w_{\alpha}}^{\prime})^{\mathsf{T}}\mathbb{A}_{0}(v){w}^{\prime}_{\alpha}\mathrm{d}x\leq
C_{1}\nu(t)\|{w}^{\prime}(t)\|_{H^{s}}^{2}+C_{1}\nu(t),$ (4.28)

where we used the elementary estimate $\rho\lesssim\rho^{2}+1$.

Taking into account the equivalence (cf. (4.2) and Lemma 4.1)

$\displaystyle\|w^{\prime}\|_{H^{s}}^{2}\sim\sum_{|\alpha|\leq
s}\int_{\mathbb{T}^{d}}(w_{\alpha}^{\prime})^{\mathsf{T}}\mathbb{A}_{0}(v)w^{\prime}_{\alpha}\mathrm{d}x$
(4.29)

with associated constants $\lambda_{1}$ and $\Lambda_{1}$, and Gronwall’s
inequality, we deduce a bound of the form

$\displaystyle\lambda_{1}\|w^{\prime}(t)\|_{H^{s}}^{2}\leq\big{(}\Lambda_{1}\|w^{\prime}(0)\|_{H^{s}}^{2}+\sqrt{t}\beta\big{)}\exp(\sqrt{t}\beta),\quad\mbox{where}$
(4.30) $\displaystyle\beta:=\bigg{(}\int_{0}^{t}\|\nabla
v_{n}\|_{H^{s}}^{2}\mathrm{d}\tau\bigg{)}^{1/2}C_{1}+\sqrt{t}C_{1}.$ (4.31)

Recalling (4.16), adding inequalities (4.21) and (4.30), inserting the initial
value $w_{|t=0}=z^{\ell}$ and recalling definition (4.3) of $K$, we infer the
bound

$\displaystyle\|w(t)\|_{H^{s}}^{2}\leq
2\frac{\Lambda_{1}}{\lambda_{1}}R^{2}<(KR)^{2}$ (4.32)

for all $t\in[0,T_{*}]$, provided that $T_{*}\in(0,\widehat{t}_{\ell-1})$
(depending on $R$ and $r^{-1}$) is small enough. This implies that
$\widehat{t}_{\ell}>T_{*}$, and inductively we infer (ii). ∎

###  4.4. Convergence

We show that the approximate solutions converge to a strong solution of system
(2.1) as $\ell\to\infty$.

######  Lemma 4.4 (Convergence).

There exists a solution
$w=(w^{\prime},w_{n}):[0,{T_{*}}]\times\mathbb{T}^{d}\to\mathcal{D}_{0}$ to
system (2.1) in $(0,{T_{*}})\times\mathbb{T}^{d}$ satisfying
$w_{|t=0}=w^{\mathrm{in}}$ and the regularity

$\displaystyle w\in L^{\infty}(0,{T_{*}};H^{s})\cap
C_{w}([0,{T_{*}}];H^{s}),\quad w_{n}\in L^{2}(0,{T_{*}};H^{s+1}),$ (4.33a)
$\displaystyle\partial_{t}w_{n}\in
L^{2}(0,{T_{*}};H^{s-1}),\quad\partial_{t}w^{\prime}\in
L^{\infty}(0,{T_{*}};H^{s-1})$ (4.33b)

such that, as $\ell\to\infty$,

$\displaystyle w^{\ell}\to w\quad$ $\displaystyle\text{in
}X_{{T_{*}}}^{\sigma}\text{ for every }\sigma<s,$ (4.34a) $\displaystyle
w^{\ell}(t)\rightharpoonup w(t)\quad$ $\displaystyle\text{in }H^{s}\text{
uniformly in }t\in[0,{T_{*}}],$ (4.34b) $\displaystyle
w^{\ell}_{n}\rightharpoonup w_{n}\quad$ $\displaystyle\text{in
}L^{2}(0,{T_{*}};H^{s+1}),$ (4.34c)
$\displaystyle\partial_{t}w^{\ell}_{n}\rightharpoonup\partial_{t}w_{n}\quad$
$\displaystyle\text{in }L^{2}(0,{T_{*}};H^{s-1}),$ (4.34d)
$\displaystyle\partial_{t}(w^{\ell})^{\prime}\overset{\ast}{\rightharpoonup}\partial_{t}w^{\prime}\quad$
$\displaystyle\text{in }L^{\infty}(0,{T_{*}};H^{s-1}).$ (4.34e)

As a consequence,

$\displaystyle w_{n}\in C([0,{T_{*}}];H^{s}),\quad w^{\prime}\in
C^{1}([0,{T_{*}}]\times\mathbb{T}^{d}),$ (4.35a)
$\displaystyle\partial_{t}w_{n},\,\nabla^{2}w_{n}\in
L^{2}(0,{T_{*}};C(\mathbb{T}^{d})).$ (4.35b)

###### Proof.

We split the proof into several steps.

Step 1. We assert that there exists a function
$p:\mathbb{N}_{+}\to\mathbb{R}_{+}$ of at most polynomial growth such that for
all $\ell\in\mathbb{N}_{+}$,

$\displaystyle\mathsf{N}^{\ell+1}_{{T_{*}}}\leq
2^{-\ell}p(\ell)+2^{-1}\mathsf{N}^{\ell}_{{T_{*}}},\quad\mbox{where}$ (4.36)
$\displaystyle\mathsf{N}^{\ell}_{{T_{*}}}:=\sup_{t\in(0,{T_{*}})}\|w^{\ell}(t){-}w^{\ell-1}(t)\|_{L^{2}}+\|\nabla(w^{\ell}_{n}{-}w^{\ell-1}_{n})\|_{L^{2}(0,T_{*};L^{2}(\mathbb{T}^{d}))}.$
(4.37)

This estimate is the key point of the proof. To verify the assertion, we
subtract the equations for two subsequent iterates, yielding

$\displaystyle\mathbb{A}_{0}(w^{\ell})\partial_{t}(w^{\ell+1}-w^{\ell})^{\prime}$
$\displaystyle=\nabla
w^{\ell}_{n}\cdot\mathbb{A}_{1}(w^{\ell})\nabla(w^{\ell+1}-w^{\ell})^{\prime}+F_{\ell},$
$\displaystyle\partial_{t}(w_{n}^{\ell+1}-w_{n}^{\ell})$
$\displaystyle=\operatorname{div}\big{(}a(w^{\ell})\nabla(w_{n}^{\ell+1}-w_{n}^{\ell})\big{)}+G_{\ell},$

where

$\displaystyle F_{\ell}$ $\displaystyle=V_{n}(w^{\ell})|\nabla
w^{\ell}_{n}|^{2}-V_{n}(w^{\ell-1})|\nabla
w^{\ell-1}_{n}|^{2}\big{(}\mathbb{A}_{0}(w^{\ell})-\mathbb{A}_{0}(w^{\ell-1})\big{)}\partial_{t}(w^{\ell})^{\prime}$
$\displaystyle\phantom{xx}{}+\big{(}\nabla
w^{\ell}_{n}\cdot\mathbb{A}_{1}(w^{\ell})-\nabla
w^{\ell-1}_{n}\cdot\mathbb{A}_{1}(w^{\ell-1})\big{)}\nabla(w^{\ell})^{\prime},$
$\displaystyle G_{\ell}$
$\displaystyle=\operatorname{div}\big{(}(a(w^{\ell})-a(w^{\ell-1}))\nabla
w_{n}^{\ell}\big{)}.$

Energy estimates similar to those in the proof of Lemma 4.2 yield the
following stability estimates for the hyperbolic and the parabolic components,
respectively:

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}$
$\displaystyle\|\mathbb{A}_{0}(w^{\ell})^{1/2}(w^{\ell+1}{-}w^{\ell})^{\prime}\|_{L^{2}}^{2}$
(4.38)
$\displaystyle\lesssim\big{(}1+\|\partial_{t}w_{n}^{\ell}\|_{H^{s-1}}+\|\Delta
w_{n}^{\ell}\|_{H^{s-1}}\big{)}\|(w^{\ell+1}{-}w^{\ell})^{\prime}\|_{L^{2}}^{2}$
(4.39)
$\displaystyle\phantom{xx}{}+\big{(}\|w^{\ell}{-}w^{\ell-1}\|_{L^{2}}+\|\nabla(w^{\ell}_{n}{-}w^{\ell-1}_{n})\|_{L^{2}}\big{)}\|(w^{\ell+1}{-}w^{\ell})^{\prime}\|_{L^{2}},$
(4.40) $\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}$
$\displaystyle\|w_{n}^{\ell+1}-w_{n}^{\ell}\|_{L^{2}}^{2}+\frac{a_{1}}{2}\|\nabla(w_{n}^{\ell+1}-w_{n}^{\ell})\|_{L^{2}}^{2}\lesssim\|w^{\ell}-w^{\ell-1}\|_{L^{2}}^{2}$
(4.41)

with associated constants depending on $R$ and $r^{-1}$. Let us now define the
quantity

$\displaystyle
Q_{\ell}(\varphi,\psi)=\|\mathbb{A}_{0}(w^{\ell})^{1/2}(\varphi-\psi)^{\prime}\|_{L^{2}}^{2}+\|\varphi_{n}-\psi_{n}\|_{L^{2}}^{2}.$
(4.42)

It satisfies $Q_{\ell}(\varphi,\psi)\sim\|\varphi-\psi\|_{L^{2}}^{2}$ for all
$\ell$.

We add (4.38) and (4.41) and use Young’s inequality to find for any
$\delta\in(0,1]$ that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}$ $\displaystyle
Q_{\ell}(w^{\ell+1},w^{\ell})+a_{1}\|\nabla(w_{n}^{\ell+1}-w_{n}^{\ell})\|_{L^{2}}^{2}$
$\displaystyle\leq\big{(}C_{\delta}+C\|\partial_{t}w_{n}^{\ell}\|_{H^{s-1}}+C\|\Delta
w_{n}^{\ell}\|_{H^{s-1}}\big{)}Q_{\ell}(w^{\ell+1},w^{\ell})$
$\displaystyle\phantom{xx}{}+C\,Q_{\ell-1}(w^{\ell},w^{\ell-1})+\delta\|\nabla(w_{n}^{\ell}-w_{n}^{\ell-1})\|_{L^{2}}^{2}.$

Invoking the Gronwall lemma, inserting the bounds (4.16)–(4.17) and applying
(4.2), we conclude that

$\displaystyle\mathsf{n}^{\ell+1}(t)\leq
C_{T_{*}}\exp(C_{\delta}t)\big{(}\|w^{\ell+1}(0){-}w^{\ell}(0)\|_{L^{2}}+\delta\mathsf{n}^{\ell}(t)\big{)},\quad\mbox{where}$
(4.43)
$\displaystyle\mathsf{n}^{\ell}(t):=\|w^{\ell}(t)-w^{\ell-1}(t)\|_{L^{2}}+\bigg{(}\int_{0}^{t}\|\nabla(w_{n}^{\ell}-w_{n}^{\ell-1})\|_{L^{2}}^{2}\mathrm{d}\tau\bigg{)}^{1/2}$
(4.44)

for all $t\in[0,\delta]$ and any $\delta\in(0,T_{*}]$, where
$C_{T_{*}}=C(\sqrt{T_{*}F(T_{*})})$ with a function $F$ as in (4.16).

Let us now fix $\delta=\min\\{1/(4C_{T_{*}}),T_{*}\\}$. By construction,
$\|w^{\ell+1}(0){-}w^{\ell}(0)\|_{L^{2}}\lesssim 2^{-\ell}R$ (cf. (4.4)). If
we choose ${t_{*}}={t_{*}}(\delta,R,r^{-1})\in(0,\delta]$ so small that
$\exp(C_{\delta}{t_{*}})\leq 2$, we deduce an estimate of the form (4.36) with
$T_{*}$ replaced by $t_{*}$ and with $p(\ell)\equiv c=\mathrm{const.}$ It
follows by recursion that $\mathsf{N}^{\ell+1}_{{t_{*}}}\leq
2^{-\ell}c\ell+2^{-\ell}\mathsf{N}^{1}_{{t_{*}}}$ and as a consequence
$\|w^{\ell+1}({t_{*}}){-}w^{\ell}({t_{*}})\|_{L^{2}}\leq 2^{-\ell}C\ell$.
Using this decay property in $\ell$ at the new initial time $t_{*}$, we can
repeat the above argument on the interval $[t_{*},2t_{*}]$ and obtain an
estimate of the form (4.36) with $T_{*}$ replaced by $2t_{*}$ and with
$p(\ell)\equiv C\ell$. Iterating for a total number of $i:=\lfloor
T_{*}/t_{*}\rfloor$ times, we infer (4.36) with the time $T_{*}$ and
$p(\ell)\equiv C\ell^{i}$.

Step 2: Inequality (4.36) implies that
$\sum_{\ell\in\mathbb{N}}\mathsf{N}^{\ell}_{{T_{*}}}<\infty$. As a
consequence, the sequence $(w^{\ell+1}{-}w^{\ell})_{\ell}\subset
X^{0}_{{T_{*}}}$ is summable, and by completeness, there exists $w\in
X^{0}_{{T_{*}}}$ such that $w^{\ell}\to w$ in $X^{0}_{{T_{*}}}$ as
$\ell\to\infty$. The uniform bounds in Lemma 4.2 and Remark 4.3, combined with
classical compactness and interpolation arguments, further yield the
convergence (4.34) as well as the regularity (4.33). We omit the details,
since an exposition of such arguments in a similar context has been provided,
e.g., in [1, Chapter 10.1.1] and [22, p. 39–40].

Step 3: We assert that the limit $w$ further has the regularity (4.35) and is
a strong solution. Indeed, the regularity (4.35a) is an immediate consequence
of (4.33b). Next, the convergence (4.34a) allows us to pass to the limit
$\ell\to\infty$ in equation (4.9a) (where $v=w^{\ell-1}$ and $w=w^{\ell}$),
giving in particular $\partial_{t}w^{\prime}\in
C([0,{T_{*}}]\times\mathbb{T}^{d})$ and showing that $w^{\prime}\in
C^{1}([0,{T_{*}}]\times\mathbb{T}^{d})$ is a classical solution to (4.9a).
Finally, the convergences (4.34a), (4.34c) and (4.34d) imply that equation
(4.9b) is fulfilled in the strong sense, and the regularity (4.35b) follows
from the embedding $H^{s-1}\hookrightarrow C(\mathbb{T}^{d})$. ∎

###  4.5. Regularity

To deduce the temporal continuity of $w^{\prime}$ with values in $H^{s}$, we
need some basic uniqueness properties.

######  Lemma 4.5 (Uniqueness).

Let $T>0$ and $s>d/2+1$. Then the following holds:

  1. (i)

For a given initial value $w^{\mathrm{in}}$, there exists at most one strong
solution $w$ of system (4.1) in $(0,T)\times\mathbb{T}^{d}$ satisfying the
regularity (4.33) (with ${T_{*}}$ replaced by $T$), the initial condition
$w|_{t=0}=w^{\mathrm{in}}$ and $\min_{[0,T]\times\mathbb{T}^{d}}w_{n}>0$.

  2. (ii)

For fixed strictly positive $w_{n}$ satisfying the regularity (4.33b) and a
given initial condition $(w^{\mathrm{in}})^{\prime}$, the hyperbolic subsystem
(4.9a) with the coefficient function $w_{n}$ has at most one classical
solution $w^{\prime}$.

###### Proof.

The assertions can be deduced from energy estimates similar to those in the
proof of Lemma 4.4. ∎

######  Lemma 4.6.

The solution $w$ constructed in Lemma 4.4 satisfies $w^{\prime}\in
C([0,{T_{*}}];H^{s})$.

###### Proof.

In the proof we closely follow [22, Theorem 2.1 (b)]. Since we already know
the weak continuity $w^{\prime}\in C_{w}([0,{T_{*}}];H^{s})$, it is sufficient
to show the continuity of the norm in the Hilbert space $H^{s}$. We first show
the continuity at $t=0$. Equip $H^{s}(\mathbb{T}^{d};\mathbb{R}^{n^{\prime}})$
with the equivalent norm

$\displaystyle\|v\|_{{\widehat{H}^{s}}}:=\bigg{(}\sum_{|\alpha|\leq
s}\int_{\mathbb{T}^{d}}\partial_{\alpha}v^{\mathsf{T}}\mathbb{A}_{0}(w|_{t=0})\partial_{\alpha}v\mathrm{d}x\bigg{)}^{1/2}.$
(4.45)

It then suffices to show that

$\displaystyle\limsup_{t\downarrow
0}\|w^{\prime}(t)\|_{{\widehat{H}^{s}}}\leq\|w^{\prime}(0)\|_{{\widehat{H}^{s}}}.$
(4.46)

To prove this inequality, we recall estimate (4.28), valid for $w=w^{\ell}$
and $v=w^{\ell-1}$. Thanks to the uniform bounds in Lemma 4.2, the right-hand
side of (4.28) can be estimated above by an $\ell$-independent function $f\in
L^{2}([0,{T_{*}}];[0,\infty))$ so that

$\displaystyle\sum_{|\alpha|\leq s}$
$\displaystyle\int_{\mathbb{T}^{d}}\partial_{\alpha}(w^{\ell})^{\prime}(t)^{\mathsf{T}}\mathbb{A}_{0}(w^{\ell-1}(t))\partial_{\alpha}(w^{\ell})^{\prime}(t)\mathrm{d}x$
$\displaystyle\leq\sum_{|\alpha|\leq
s}\int_{\mathbb{T}^{d}}\partial_{\alpha}((z^{\ell})^{\prime})^{\mathsf{T}}\mathbb{A}_{0}(z^{\ell-1})\partial_{\alpha}(z^{\ell})^{\prime}\mathrm{d}x+\int_{0}^{t}f(\tau)\mathrm{d}\tau$

for all $t\in[0,{T_{*}}]$ and $\ell\in\mathbb{N}_{+}$.

It follows from the convergence properties in Section 4.4 and a weak lower
semi-continuity argument that, in the limit $\ell\to\infty$,

$\displaystyle\sum_{|\alpha|\leq s}$
$\displaystyle\int_{\mathbb{T}^{d}}\partial_{\alpha}w^{\prime}(t)^{\mathsf{T}}\mathbb{A}_{0}(w(t))\partial_{\alpha}w^{\prime}(t)\mathrm{d}x$
(4.47) $\displaystyle\leq\sum_{|\alpha|\leq
s}\int_{\mathbb{T}^{d}}\partial_{\alpha}w^{\prime}(0)^{\mathsf{T}}\mathbb{A}_{0}(w(0))\partial_{\alpha}w^{\prime}(0)\mathrm{d}x+\int_{0}^{t}f(\tau)\mathrm{d}\tau.$
(4.48)

Recalling the weak continuity in (4.33a) and taking the $\limsup_{t\downarrow
0}$, we find (4.46), where we used the fact that $\lim_{t\downarrow
0}\|\mathbb{A}_{0}(w(0))-\mathbb{A}_{0}(w(t))\|_{C(\mathbb{T}^{d})}=0$ to
recover the ${\widehat{H}^{s}}$-norm on the left-hand side of (4.46).

The right-continuity of $w^{\prime}$ with values in $H^{s}$ at general
$\widehat{t}\in[0,{T_{*}})$ follows by applying the above result to the time-
shifted problem with initial condition $w(\widehat{t})\in H^{s}$ and
exploiting the fact that $\tau\mapsto w(\widehat{t}+\tau)$ is the unique
solution emanating from $w(\widehat{t})$.

It remains to show the left-continuity of $w^{\prime}$. To this end, we
consider the hyperbolic subsystem with the fixed coefficient function $w_{n}$
and apply the above argument to the time-reversed problem. More precisely, for
establishing an analogue of the crucial $H^{s}$ energy estimate (4.47), one
possibility is to use a Picard iteration for the hyperbolic subsystem, while
approximating the coefficient function $w_{n}$ by the smooth functions
$w_{n}^{\ell}$ from Lemma 4.4. The uniqueness property in Lemma 4.5 (ii) then
implies that the limiting function coincides with the time-reversed version of
$w^{\prime}$. ∎

To complete the proof of Theorem 2.3, it remains to show the regularity

$\displaystyle\partial_{t}w_{n},\nabla^{2}w_{n}\in
C_{\mathrm{loc}}((0,{T_{*}}]\times\mathbb{T}^{d}).$ (4.49)

To this end, let $\beta\in(0,\min\\{s-d/2-1,1\\})$. The regularity
$w^{\prime}\in C([0,{T_{*}}];H^{s})$ and the Sobolev embedding imply that
$w^{\prime}\in C([0,{T_{*}}];C^{1,\beta}(\mathbb{T}^{d}))$. Combined with the
space-uniform temporal Lipschitz regularity of $w^{\prime}$, i.e.
$\partial_{t}w^{\prime}\in C([0,{T_{*}}]\times\mathbb{T}^{d})$, we infer from
[19, Chapter II, Lemma 3.1] a temporal Hölder regularity of the gradient:

$\displaystyle\nabla w^{\prime}\in
C^{\beta/(1+\beta)}([0,{T_{*}}];C(\mathbb{T}^{d})).$ (4.50)

Now, we can use classical regularity results for quasilinear parabolic
equations in divergence form, where $w^{\prime}$ is viewed as a given
function: Thanks to (4.50), $\nabla w^{\prime}$ satisfies a space-time Hölder
condition, which makes Theorem 5.4 in [19, Chapter V] accessible and gives us
an interior space-time Hölder a priori estimate for $\partial_{t}w_{n}$ and
$\nabla^{2}w_{n}$. To conclude (4.49) from the a priori control, we
approximate $w^{\prime}$ by smooth functions whose spatial gradient is
uniformly bounded in some space-time Hölder norm, and exploit the uniqueness
of regular solutions to the parabolic equation in $w_{n}$ (with $w^{\prime}$
acting as a fixed parameter). As a consequence, $\partial_{t}w_{n}$ and
$\nabla^{2}w_{n}$ satisfy a space-time Hölder condition away from $t=0$, which
entails (4.49).

###  4.6. Original variables

We now conclude the existence of classical solutions for the degenerate cross-
diffusion system (Rk1).

###### Proof of Theorem 2.4.

Without loss of generality, after rescaling time, relabelling components and
rescaling $u_{i}\mapsto a_{i}u_{i}$, we can assume that $k_{1}\leq\ldots\leq
k_{n^{\prime}}\leq k_{n}=1$ and $a_{i}=1$ for all $i$. It then suffices to
consider the following two cases.

Case 1: Let $k_{1}\leq\ldots\leq k_{n^{\prime}}<k_{n}$. Then the assertion is
a consequence of Theorems 2.2 and 2.3. The time of existence $T$ can be
bounded below by a positive constant that depends on the datum
$u^{\mathrm{in}}$ only through $\|w^{\mathrm{in}}\|_{H^{s}}$ and
$\min_{\mathbb{T}^{d}}w^{\mathrm{in}}_{n}$, where
$w^{\mathrm{in}}=\Phi(u^{\mathrm{in}})$ with $\Phi$ denoting a diffeomorphism
as in Theorem 2.2 with the property that $w_{n}=\Phi_{n}(u)=\mathfrak{p}(u)$.

Case 2: There exists a minimal $m\in\\{1,\dots,n^{\prime}\\}$ such that
$k_{i}=1$ for $i=m,\dots,n$. In this case, we define
$\widetilde{u}_{i}:=u_{i}$ for $1\leq i\leq m-1$ and
$\widetilde{u}_{m}:=u_{m}+\dots+u_{n}$. The system formulated in terms of
$\widetilde{u}$ satisfies the hypotheses of Case 1 with $n$ replaced by $m$,
which provides a local strong solution $\widetilde{u}$ and in particular a
velocity field ${\bf\widetilde{v}}=-\nabla\sum_{j=1}^{m}\widetilde{u}_{j}$.
Subsequently, we determine the unique solutions to the linear continuity
equations for $u_{m},\dots,u_{n}$ with the fixed velocity field
${\bf\widetilde{v}}$, so that
$u=(\widetilde{u}_{1},\dots,\widetilde{u}_{m},u_{m+1},\dots,u_{n})$ is the
desired classical solution to (Rk1). ∎

##  5\. The general system ($\mathsf{B}$) with incomplete diffusion

In this section, we turn to the general PDE system ($\mathsf{B}$) (see page
$\mathsf{B}$) for symmetric positive semidefinite matrices
$B=(b_{ij})\in\mathbb{R}^{n\times n}$ with rank
$\mathsf{r}\in\\{1,\ldots,n\\}$. In particular, we aim to bring equations
($\mathsf{B}$) into a normal form that makes them accessible to the energy
methods from Section 4.

In the following, we use the notation
$D(v):=\operatorname{diag}(v_{1},\dots,v_{n})\in\mathbb{R}^{n\times n}$ for
$v=(v_{i})\in\mathbb{R}^{n}$. Moreover, we use the convention
$f(v):=(f(v_{1}),\dots,f(v_{n}))^{\mathsf{T}}$ for a function
$f:\mathbb{R}\to\mathbb{R}$ and a vector $v\in\mathbb{R}^{n}$.

###  5.1. Preliminary consideration: the full-rank case

We first note that for symmetric strongly parabolic PDE systems, local strong
solutions to the Cauchy problem in suitable function spaces can be obtained by
means of basic energy estimates in the spirit of Section 4. More generally,
initial-boundary value problems may for instance be treated using Schauder-
type estimates as in [10].

System ($\mathsf{B}$) can be symmetrised in several ways. For instance,
following Kawashima and Shizuta [17], we may consider a change to entropy
variables $\widetilde{v}_{i}=\log u_{i}$ for the strictly convex Shannon
entropy. In this way, we obtain the symmetric system

$\displaystyle
D(e^{\widetilde{v}})\partial_{t}\widetilde{v}=\operatorname{div}\big{(}D(e^{\widetilde{v}})BD(e^{\widetilde{v}})\nabla\widetilde{v}\big{)},$

which is strictly parabolic if and only if $\operatorname{rank}B=n$. Likewise,
we may use the quadratic Rao entropy $H(u)=\frac{1}{2}u^{\mathsf{T}}Bu$, which
is strictly convex if $B$ possesses full rank. In this case, we find
$B^{-1}\partial_{t}v=\operatorname{div}(D(u)\nabla v),$ $u:=B^{-1}v$. Of
course, the classical inversion of $B$ is only possible for
$\operatorname{rank}B=n$. Alternatively, equation ($\mathsf{B}$) may be put in
symmetric form directly by means of the symmetriser $B$ giving
$B\partial_{t}u=\operatorname{div}(BD(u)B\nabla u)$, where we used the fact
that $B$ is constant.

In order to find a symmetrisation suitable to conveniently treat the rank-
deficient case, we use again the fact that $B$ is constant, but somewhat
modify the previous alternative. To this end, denote by
$\xi^{1},\ldots,\xi^{n}$ an orthonormal basis of eigenvectors of $B$ with the
corresponding vector of positive eigenvalues
$\lambda=(\lambda_{1},\ldots,\lambda_{n})$ and define $w_{k}:=\xi^{k}\cdot u$
for $k=1,\ldots,n$. With $\mathsf{O}_{ij}=\xi^{i}_{j}$ for $i,j=1,\ldots,n$,
we obtain $B\mathsf{O}^{\mathsf{T}}=\mathsf{O}^{\mathsf{T}}D(\lambda)$ and
$u=\mathsf{O}^{\mathsf{T}}w$. Hence, we can write ($\mathsf{B}$), i.e.
$\partial_{t}u=\operatorname{div}(D(u)B\nabla u)$, in terms of the variable
$w=(w_{1},\ldots,w_{n})$ as

$D(\lambda)\partial_{t}w=D(\lambda)\mathsf{O}\operatorname{div}(D(u)B\mathsf{O}^{\mathsf{T}}\nabla
w)=\operatorname{div}\big{(}D(\lambda)\mathsf{O}D(u)\mathsf{O}^{\mathsf{T}}D(\lambda)\nabla
w\big{)}.$

If $B$ possesses rank $n$, these equations define a symmetric strongly
parabolic system for $w$ as long as $u$ remains positive componentwise. We
further develop this approach in the next paragraph to derive a normal form in
the case $\operatorname{rank}B=\mathsf{r}$ with $1\leq\mathsf{r}\leq n$.

###  5.2. Normal form of symmetric hyperbolic–parabolic type

We suppose that $\operatorname{rank}B=\mathsf{r}\in\\{1,\dots,n-1\\}$. To ease
the notation, we partition the set of indices $\\{1,\dots,n\\}$ into
${\mathrm{I}}=\\{1,\dots,n-\mathsf{r}\\}$ and
${\mathrm{II}}=\\{n-\mathsf{r}+1,\dots,n\\}$. We choose an orthonormal basis
$\xi^{1},\ldots,\xi^{n}$ of eigenvectors of $B$ with corresponding eigenvalues
$\lambda_{i}=0$ for $i\in{\mathrm{I}}$ and $\lambda_{i}>0$ for
$i\in{\mathrm{II}}$, and further introduce the orthogonal matrix
$\mathsf{O}_{ij}=\xi^{i}_{j}$ for $i,j=1,\ldots,n$ and the rectangular blocks

$\displaystyle\mathsf{Q}_{ij}:=\xi^{i}_{j}\quad\text{for
}i\in{\mathrm{I}},\quad\mathsf{P}_{ij}:=\xi^{i}_{j}\quad\text{for
}i\in{\mathrm{II}},\ j=1,\dots,n.$ (5.1)

For later reference, we note the elementary matrix identities

$\mathsf{Q}\mathsf{Q}^{\mathsf{T}}=\mathbb{I}_{n-\mathsf{r}},\quad\mathsf{P}\mathsf{P}^{\mathsf{T}}=\mathbb{I}_{\mathsf{r}},\quad\mathsf{Q}^{\mathsf{T}}\mathsf{Q}+\mathsf{P}^{\mathsf{T}}\mathsf{P}=\mathbb{I}_{n},\quad\mathsf{P}\mathsf{Q}^{\mathsf{T}}=0,\quad\mathsf{Q}\mathsf{P}^{\mathsf{T}}=0.$
(5.2)

Left-multiplying system ($\mathsf{B}$) by $(\xi^{k})^{\mathsf{T}}$,
$k\in{\mathrm{II}}$, the functions $w_{n-\mathsf{r}+1},$ $\dots,w_{n}$ defined
via $w_{k}:=\xi^{k}\cdot u$ satisfy the parabolic system

$\displaystyle\lambda_{k}\partial_{t}w_{k}=\operatorname{div}\bigg{(}\sum_{\ell\in{\mathrm{II}}}\alpha_{k\ell}(u)\nabla
w_{\ell}\bigg{)},\quad\text{where}$ (5.3)
$\displaystyle\alpha_{k\ell}(u)=D(u)(\lambda_{k}\xi^{k})\cdot(\lambda_{\ell}\xi^{\ell})\quad\text{
for all }k,\ell\in{\mathrm{II}}.$ (5.4)

Moreover, multiplying ($\mathsf{B}$) by $\xi^{k}_{i}/u_{i}$ for
$k\in{\mathrm{I}}$, summing over $i=1,\dots,n$ and using the fact that
$\xi^{k}$ is in the kernel of $B$ yields for $w_{k}:=\xi^{k}\cdot\log u$ the
first-order equation

$\displaystyle\partial_{t}w_{k}=\sum_{i,j=1}^{n}\xi^{k}_{i}b_{ij}\nabla\log
u_{i}\cdot\nabla u_{j}\quad\text{ for all }k\in{\mathrm{I}}.$ (5.5)

This leads us to propose the change of variables $\Phi(u)=w$ defined via

$\displaystyle w_{k}:=\begin{cases}\xi^{k}\cdot\log u&\text{ for
}k\in{\mathrm{I}},\\\ \xi^{k}\cdot u&\text{ for
}k\in{\mathrm{II}}.\end{cases}$ (5.6)

With $w_{\mathrm{I}}:=(w_{1},\ldots,w_{n-\mathsf{r}})$ and
$w_{\mathrm{II}}:=(w_{n-\mathsf{r}+1},\ldots,w_{n})$, we then have
$w_{\mathrm{I}}=\mathsf{Q}\log u$ and $w_{\mathrm{II}}=\mathsf{P}u$. We next
prove a diffeomorphism property for this change of variables.

######  Lemma 5.1.

The map $\Phi$ is smoothly invertible between $\mathbb{R}_{+}^{n}$ and
$\mathcal{D}:=\mathbb{R}^{n-\mathsf{r}}\times\mathsf{P}\mathbb{R}_{+}^{n}$.

###### Proof.

It follows from definition (5.6) and the identity $\log
u_{i}=(\mathsf{O}^{\mathsf{T}}\mathsf{O}\log u)_{i}$ that

$\displaystyle\log u_{i}=\sum_{k=1}^{n}\log
u\cdot\xi^{k}\xi_{i}^{k}=\sum_{k\in{\mathrm{I}}}w_{k}\xi^{k}_{i}+\sum_{k\in{\mathrm{II}}}\log
u\cdot\xi^{k}\xi^{k}_{i}\quad\text{for }i=1,\ldots,n,$ (5.7) $\displaystyle
w_{\ell}=\sum_{i=1}^{n}\xi^{\ell}_{i}u_{i}=\sum_{i=1}^{n}\xi^{\ell}_{i}\exp\bigg{(}\sum_{k\in{\mathrm{I}}}w_{k}\xi^{k}_{i}+\sum_{k\in{\mathrm{II}}}\log
u\cdot\xi^{k}\xi^{k}_{i}\Big{)}\quad\text{for }\ell\in{\mathrm{II}}.$ (5.8)

We assert that for arbitrary $w\in\mathcal{D}$, the last line defines the
$\mathsf{r}$ components $\log u\cdot\xi^{j}$, $j\in{\mathrm{II}}$, implicitly
as functions $X_{j}(w)$. Indeed, we can write the algebraic system (5.8) in
the form $F(X;w_{1},\ldots,w_{n})=0$, where

$\displaystyle
F_{\ell}(X;w):=\sum_{i=1}^{n}\xi^{\ell}_{i}\exp\bigg{(}\sum_{k\in{\mathrm{I}}}w_{k}\xi^{k}_{i}+\sum_{k\in{\mathrm{II}}}X_{k}\xi^{k}_{i}\bigg{)}-w_{\ell},\quad\ell\in{\mathrm{II}}.$

It is readily seen that $F(X;w)=\partial_{X}G(X;w)$ for $G$ given by

$\displaystyle G(X;\,w):=\exp\big{(}\mathsf{Q}^{\sf
T}w_{\mathrm{I}}+\mathsf{P}^{\sf T}X\big{)}\cdot{\bf 1}-w_{{\mathrm{II}}}\cdot
X,$

where we abbreviated ${\bf 1}:=(1,\dots,1)^{\mathsf{T}}$. Moreover, we compute
for $j,\ell\in{\mathrm{II}}$,

$\displaystyle\partial_{X_{j}}F_{\ell}=\sum_{i=1}^{n}\xi^{\ell}_{i}\xi^{j}_{i}\exp\bigg{(}\sum_{k\in{\mathrm{I}}}w_{k}\xi^{k}_{i}+\sum_{k\in{\mathrm{II}}}X_{k}\xi^{k}_{i}\bigg{)}=\sum_{i=1}^{n}\xi^{\ell}_{i}\xi^{j}_{i}u_{i},$

that is $\partial_{X}F=\mathsf{P}D(u)\mathsf{P}^{\sf T}$. Thus,
$\partial_{X}F$ is symmetric positive definite, and $X\mapsto G(X;w)$ is
strictly convex on $\mathbb{R}^{\mathsf{r}}$. The equations $F(X;w)=0$ possess
a unique solution $X=X(w)$ if and only if $X$ is the unique global minimiser
of $G(\cdot;w)$. To prove its existence, a sufficient condition is that
$G(X;w)\rightarrow+\infty$ for $|X|\rightarrow+\infty$. To establish this
property, recall that for
$w_{{\mathrm{II}}}\in\mathsf{P}(\mathbb{R}_{+})^{n}$, there exists
$\zeta\in\mathbb{R}_{+}^{n}$ such that $w_{{\mathrm{II}}}=\mathsf{P}\zeta$,
and then

$\displaystyle G(X;w):=\exp\big{(}\mathsf{Q}^{\sf
T}w_{\mathrm{I}}+\mathsf{P}^{\sf T}X\big{)}\cdot{\bf
1}-\zeta\cdot\mathsf{P}^{\sf T}X.$

For $|X|\rightarrow+\infty$, we must also have $|\mathsf{P}^{\sf
T}X|\rightarrow+\infty$ and we distinguish two cases. If
$\max_{i=1,\ldots,n}(\mathsf{P}^{\sf T}X)_{i}\rightarrow+\infty$, the
exponential term dominates and $G(X;w)$ tends to infinity. Otherwise, if
$\max_{i=1,\ldots,n}(\mathsf{P}^{\sf T}X)_{i}$ is bounded above, then
$\min_{i=1,\ldots,n}(\mathsf{P}^{\sf T}X)_{i}\rightarrow-\infty$ as
$|X|\to+\infty$. The exponential term is bounded, but since
$\min_{i=1,\ldots,n}\zeta_{i}>0$, we see that $-\zeta\cdot\mathsf{P}^{\sf
T}X\rightarrow+\infty$, and $G(X;w)$ again tends to infinity. Thus, we may
recover $u\in\mathbb{R}_{+}^{n}$ from $w\in\mathcal{D}$ via

$\displaystyle
u_{i}=\exp\bigg{(}\sum_{k\in{\mathrm{I}}}w_{k}\xi^{k}_{i}+\sum_{k\in{\mathrm{II}}}X_{k}(w)\xi^{k}_{i}\Big{)}=:\Psi_{i}(w),\quad
i=1,\ldots,n.$ (5.9)

This completes the proof. ∎

For later reference, we note the formula

$\displaystyle(\partial_{X}F)^{-1}=\mathsf{P}[D(u)]^{-1}\mathsf{P}^{\sf
T}-\mathsf{P}[D(u)]^{-1}\mathsf{Q}^{\mathsf{T}}(\mathsf{Q}D(u)^{-1}\mathsf{Q}^{\sf
T})^{-1}\mathsf{Q}D(u)^{-1}\mathsf{P}^{\sf T},$ (5.10)

which can be verified using (5.2), as well as the identity

$\displaystyle\partial_{w_{m}}X(w)=-(\partial_{X}F)^{-1}\mathsf{P}D(u)\xi^{m},\quad
m\in{\mathrm{I}},$ (5.11)

which follows from
$0=\partial_{w_{m}}(F(X;w))=\partial_{X}F\partial_{w_{m}}X+\partial_{w_{m}}F=\partial_{X}F\partial_{w_{m}}X+\mathsf{P}D(u)\xi^{m}$.

We are now in a position to derive an appropriate normal form for system
($\mathsf{B}$). We continue to denote by $\Psi$ the inverse of the
diffeomorphism $\Phi$.

######  Proposition 5.2 (Normal form).

A vector $u=(u_{1},\ldots,u_{n})$ of positive functions is a classical
solution to ($\mathsf{B}$) if and only if the transformed variables
$w=(w_{\mathrm{I}},w_{\mathrm{II}})$ defined via
$w_{\mathrm{I}}=\mathsf{Q}\log u$ and $w_{\mathrm{II}}=\mathsf{P}u$ satisfy,
in the classical sense,

$\displaystyle\mathbb{A}_{0}^{\mathrm{I}}(w)\partial_{t}w_{\mathrm{I}}$
$\displaystyle=\sum_{\nu=1}^{d}\mathbb{A}^{\mathrm{I}}_{1}(w,\partial_{x_{\nu}}w_{\mathrm{II}})\partial_{x_{\nu}}w_{\mathrm{I}}+f^{\mathrm{I}}(w,\nabla
w_{\mathrm{II}}),$ (5.12a) $\displaystyle
D^{\mathrm{II}}(\lambda)\partial_{t}w_{\mathrm{II}}$
$\displaystyle=\operatorname{div}(\mathbb{A}_{1}^{{\mathrm{II}}}(w)\nabla
w_{\mathrm{II}}),$ (5.12b)

where

$\mathbb{A}_{1}^{{\mathrm{II}}}(w)=D^{\mathrm{II}}(\lambda)\mathsf{P}D(\Psi(w))\mathsf{P}^{\sf
T}D^{{\mathrm{II}}}(\lambda),$

$D^{\mathrm{II}}(\lambda)=\operatorname{diag}(\lambda_{n-\mathsf{r}+1},\ldots,\lambda_{n})$
and $D(\Psi(w)):={\rm diag}(\Psi_{1}(w),\ldots,\Psi_{n}(w))$. The maps

$\mathbb{A}_{0}^{\mathrm{I}}:\mathcal{D}\to\mathbb{R}^{(n-\mathsf{r})\times(n-\mathsf{r})}_{\rm
spd},\quad\mathbb{A}^{\mathrm{I}}_{1}:\mathcal{D}\times\mathbb{R}^{\mathsf{r}}\to\mathbb{R}^{(n-\mathsf{r})\times(n-\mathsf{r})}_{\rm
sym},\quad
f^{\mathrm{I}}:\mathcal{D}\times\mathbb{R}^{\mathsf{r}}\to\mathbb{R}^{n-\mathsf{r}}$

are smooth, and $\mathbb{A}^{\mathrm{I}}_{1}$ is linear in the second
argument. More specifically,

$\displaystyle\mathbb{A}_{0}^{\mathrm{I}}(w)$
$\displaystyle=\big{(}\mathsf{Q}D(\Psi(w))^{-1}\mathsf{Q}^{\sf
T}\big{)}^{-1},$
$\displaystyle\mathbb{A}^{\mathrm{I}}_{1}(w,\partial_{x_{\nu}}w_{\mathrm{II}})$
$\displaystyle=\mathsf{Q}\Sigma(\Psi(w))D(\Psi(w))^{-1}D[\mathsf{P}^{\sf
T}\lambda\partial_{x_{\nu}}w_{\mathrm{II}}]\Sigma(\Psi(w))\mathsf{Q}^{\sf T},$

where $D[\mathsf{P}^{\sf T}\lambda\partial_{x_{\nu}}w_{\mathrm{II}}]$ is the
diagonal matrix with diagonal entries given by the vector $\mathsf{P}^{\sf
T}D^{\mathrm{II}}(\lambda)\partial_{x_{\nu}}w_{\mathrm{II}}$. Moreover,

$\displaystyle\Sigma(\Psi(w)):=\mathsf{Q}^{\sf
T}\big{(}\mathsf{Q}D(\Psi(w))^{-1}\mathsf{Q}^{\sf
T}\big{)}^{-1}\mathsf{Q},\quad f^{\mathrm{I}}(w,\nabla
w_{\mathrm{II}})=\mathbb{A}_{0}^{\mathrm{I}}(w)g(w,\nabla w_{\mathrm{II}}),$
(5.13)

and $g=(g_{1},\ldots,g_{n-\mathsf{r}})$ is defined in (5.16).

###### Proof.

We differentiate (5.9) to find that

$\displaystyle\partial_{w_{m}}\log
u_{i}=\xi^{m}_{i}+\sum_{k\in{\mathrm{II}}}\partial_{w_{m}}X_{k}(w)\xi^{k}_{i},\quad
m\in{\mathrm{I}}.$ (5.14)

Introducing $\mu_{i}:=\sum_{j=1}^{n}b_{ij}u_{j}$, it follows that (5.5) is
equivalent to

$\displaystyle\partial_{t}w_{k}=\sum_{i=1}^{n}\xi^{k}_{i}\nabla\log
u_{i}\cdot\nabla\mu_{i}=\sum_{i=1}^{n}\xi^{k}_{i}\sum_{m\in{\mathrm{I}}}\partial_{w_{m}}\log
u_{i}\nabla w_{m}\cdot\nabla\mu_{i}+g_{k},\ k\in{\mathrm{I}},\qquad$ (5.15)

where $\nabla\mu_{i}=\sum_{j\in{\mathrm{II}}}\lambda_{j}\xi^{j}_{i}\nabla
w_{j}$ does not depend on the gradient of the hyperbolic variables, and

$\displaystyle g_{k}=g_{k}(w,\nabla
w_{\mathrm{II}}):=\sum_{i=1}^{n}\xi^{k}_{i}\sum_{m\in{\mathrm{II}}}\partial_{w_{m}}\log
u_{i}\nabla w_{m}\cdot\nabla\mu_{i}$ (5.16)

is a quadratic expression in the second argument. Thus, for $\nu=1,\ldots,d$,
the critical term multiplying $\partial_{x_{\nu}}w_{m}$ in equations (5.15)
equals

$\displaystyle Z_{km}$
$\displaystyle=Z_{km}(w,\partial_{x_{\nu}}w_{\mathrm{II}}):=\sum_{i=1}^{n}\xi^{k}_{i}\partial_{w_{m}}\log
u_{i}\partial_{x_{\nu}}\mu_{i}$
$\displaystyle=\sum_{i=1}^{n}\xi^{k}_{i}\bigg{(}\xi^{m}_{i}+\sum_{j\in{\mathrm{II}}}\partial_{w_{m}}X_{j}(w)\xi^{j}_{i}\bigg{)}\partial_{x_{\nu}}\mu_{i},\quad
k,m\in{\mathrm{I}},$ (5.17)

where the last equality follows from (5.14). Using identity (5.11), we have

$\displaystyle
u_{i}\sum_{j\in{\mathrm{II}}}\partial_{w_{m}}X_{j}(w)\xi^{j}_{i}=e^{i}\cdot\big{[}D(u)\mathsf{P}^{\sf
T}\partial_{w_{m}}X(w)\big{]}=-e^{i}\cdot\big{[}D(u)\mathsf{P}^{\sf
T}(\partial_{X}F)^{-1}\mathsf{P}D(u)\xi^{m}\big{]},$

where $e^{i}$ is the standard $i^{\rm th}$ unit vector of $\mathbb{R}^{n}$. We
combine this result with (5.2):

$\displaystyle Z_{km}$
$\displaystyle=\sum_{i=1}^{n}\xi^{k}_{i}u_{i}\bigg{(}\xi^{m}_{i}+\sum_{j\in{\mathrm{II}}}\partial_{w_{m}}X_{j}(w)\xi^{j}_{i}\bigg{)}\frac{\partial_{x_{\nu}}\mu_{i}}{u_{i}}$
(5.18)
$\displaystyle=\sum_{i=1}^{n}\xi^{k}_{i}e^{i}\cdot\big{[}\big{(}D(u)-D(u)\mathsf{P}^{\sf
T}(\partial_{X}F)^{-1}\mathsf{P}D(u)\big{)}\xi^{m}\big{]}\frac{\partial_{x_{\nu}}\mu_{i}}{u_{i}}$
(5.19)
$\displaystyle=\sum_{i=1}^{n}\xi^{k}_{i}e^{i}\cdot(\Sigma(u)\xi^{m})\frac{\partial_{x_{\nu}}\mu_{i}}{u_{i}},$
(5.20)

where $\Sigma(u):=D(u)-D(u)\mathsf{P}^{\sf
T}(\partial_{X}F)^{-1}\mathsf{P}D(u)$. A computation using (5.10) and the
third identity in (5.2) show that

$\displaystyle\Sigma(u)=\mathsf{Q}^{\sf
T}\big{(}\mathsf{Q}[D(u)]^{-1}\mathsf{Q}^{\sf T}\big{)}^{-1}\mathsf{Q}.$

The symmetric positive definite matrix
$\mathbb{A}_{0}^{\mathrm{I}}(w)\in\mathbb{R}^{(n-\mathsf{r})\times(n-\mathsf{r})}$,
given by

$\displaystyle\mathbb{A}_{0}^{\mathrm{I}}(w)=\mathsf{Q}\Sigma(u)\mathsf{Q}^{\sf
T}=\big{(}\mathsf{Q}D(u)^{-1}\mathsf{Q}^{\sf T}\big{)}^{-1},\quad u:=\Psi(w),$
(5.21)

is our candidate for the symmetriser. We note that the identity
$\mathsf{Q}\mathsf{Q}^{\mathsf{T}}=\mathbb{I}_{n-\mathsf{r}}$ and the form of
$\Sigma(u)$ imply that

$\displaystyle\mathbb{A}_{0}^{\mathrm{I}}(w)\mathsf{Q}=\mathsf{Q}\mathsf{Q}^{\mathsf{T}}\big{(}\mathsf{Q}D(u)^{-1}\mathsf{Q}^{\mathsf{T}}\big{)}^{-1}\mathsf{Q}\mathsf{Q}^{\mathsf{T}}\mathsf{Q}=\mathsf{Q}\mathsf{Q}^{\mathsf{T}}\big{(}\mathsf{Q}D(u)^{-1}\mathsf{Q}^{\mathsf{T}}\big{)}^{-1}\mathsf{Q}=\mathsf{Q}\Sigma(u).\qquad$
(5.22)

For $\ell,m\in{\mathrm{I}}$ and $\nu=1,\ldots,d$, we define

$\displaystyle\big{(}\mathbb{A}^{\mathrm{I}}_{1}(w,\partial_{x_{\nu}}w_{\mathrm{II}})\big{)}_{\ell
m}:=\sum_{j\in{\mathrm{I}}}(\mathbb{A}_{0}^{\mathrm{I}}(w))_{\ell
j}Z_{jm}(u,\partial_{x_{\nu}}w_{\mathrm{II}}).$

Using (5.20) and (5.22), we compute

$\displaystyle\big{(}\mathbb{A}_{1}^{\mathrm{I}}(w,\partial_{x_{\nu}}w_{\mathrm{II}})\big{)}_{\ell
m}$
$\displaystyle=\sum_{i=1}^{n}\sum_{j\in{\mathrm{I}}}(\mathbb{A}_{0}^{\mathrm{I}}(w))_{\ell
j}\mathsf{Q}_{ji}\big{(}e^{i}\cdot(\Sigma(u)\xi^{m})\big{)}\frac{\partial_{x_{\nu}}\mu_{i}}{u_{i}}$
$\displaystyle=\sum_{i=1}^{n}\sum_{j\in{\mathrm{I}}}\mathsf{Q}_{\ell
j}\Sigma(u)_{ji}\big{(}e^{i}\cdot(\Sigma(u)\xi^{m})\big{)}\frac{\partial_{x_{\nu}}\mu_{i}}{u_{i}}$
$\displaystyle=\sum_{i=1}^{n}\big{(}e^{i}\cdot(\Sigma(u)\xi^{\ell})\big{)}\big{(}e^{i}\cdot(\Sigma(u)\xi^{m})\big{)}\frac{\partial_{x_{\nu}}\mu_{i}}{u_{i}},$

which is a symmetric expression in $\ell$ and $m$. In matrix notation, we have

$\displaystyle\mathbb{A}_{1}^{\mathrm{I}}(w,\partial_{x_{\nu}}w_{\mathrm{II}})=\mathsf{Q}\Sigma(u)D(u)^{-1}D[\mathsf{P}^{\sf
T}\lambda\partial_{x_{\nu}}w]\Sigma(u)\mathsf{Q}^{\sf T}.$

Hence, after left-multiplication by $\mathbb{A}_{0}^{\mathrm{I}}(w)$,
equations (5.15) turn into the system

$\displaystyle\mathbb{A}_{0}^{\mathrm{I}}(w)\partial_{t}w_{\mathrm{I}}=\sum_{\nu=1}^{d}\mathbb{A}_{1}^{\mathrm{I}}(w,\partial_{x_{\nu}}w_{\mathrm{II}})\partial_{x_{\nu}}w_{\mathrm{I}}+f^{\mathrm{I}}(w,\nabla
w_{\mathrm{II}}),$ (5.23)

whose principle part is symmetric and where $f^{\mathrm{I}}(w,\nabla
w_{\mathrm{II}}):=\mathbb{A}_{0}^{\mathrm{I}}(w)g(w,\nabla w_{\mathrm{II}})$.
Combining (5.3) and (5.23), we have obtained a composite symmetric
hyperbolic–parabolic normal form as asserted. ∎

######  Remark 5.3.

In some cases, we may derive explicit expressions for the symmetriser. For
instance, if $\mathsf{r}=1$ and $b_{ij}=k_{i}k_{j}$, formula (5.21), the
definition of $\Sigma(u)$ and the fact that
$\mathsf{P}=\pm(\sum_{i=1}^{n}k_{i}^{2})^{-1/2}k^{\mathsf{T}}\in\mathbb{R}^{1\times
n}$ lead to

$\displaystyle\mathbb{A}_{0}^{\mathrm{I}}(w)=\mathsf{Q}\bigg{(}D(u)-\frac{D(u){k}\otimes
D(u){k}}{\sum_{i=1}^{n}k_{i}^{2}u_{i}}\bigg{)}\mathsf{Q}^{\sf T},\quad
u:=\Psi(w),$

where $k=(k_{1},\dots,k_{n})^{\mathsf{T}}$.

######  Remark 5.4.

In Section 3.2, we have obtained a different symmetriser, which takes a simple
diagonal form. On the other hand, the corresponding transformation (3.1) was
not constructed using an orthonormal system. This shows that other choices for
the basis $(\xi^{1},\ldots,\xi^{n})$ might be practically relevant, at least
in the rank-one case. In a similar spirit, the observations in Appendix A show
that more involved nonlinear multipliers might also be considered.

###  5.3. Local classical solutions in the general case

We recall from the preceding section that the change of variables $\Phi$ maps
$\mathbb{R}_{+}^{n}$ diffeomorphically onto
$\mathcal{D}:=\mathbb{R}^{n-\mathsf{r}}\times\mathsf{P}\mathbb{R}_{+}^{n}$.
For $w_{\mathrm{II}}\in\mathsf{P}\mathbb{R}_{+}^{n}$, we define

$\displaystyle\rho(w_{\mathrm{II}}):=\sup_{\mathsf{P}\zeta=w_{\mathrm{II}}}\inf_{i=1,\ldots,n}\zeta_{i}>0.$

######  Theorem 5.5 (Local classical solutions).

Let $s>d/2+1$ and $w^{\mathrm{in}}=(w_{\mathrm{I}},w_{\mathrm{II}})\in
H^{s}(\mathbb{T}^{d})$ with
$\bar{\rho}:=\min_{\mathbb{T}^{d}}\rho(w^{\mathrm{in}}_{\mathrm{II}})>0$. Then
there exists a time $T=T(\|w^{\mathrm{in}}\|_{H^{s}},\bar{\rho})>0$ and a
unique function $w=(w_{\mathrm{I}},w_{\mathrm{II}})\in C([0,T];H^{s})$ with
$\inf_{(0,T)\times\mathbb{T}^{d}}\rho(w_{\mathrm{II}})\geq\bar{\rho}/2$ and

$\displaystyle\partial_{t}w_{\mathrm{I}}\in C([0,T];H^{s-1}),\quad
w_{\mathrm{I}}\in C^{1}([0,T]\times\mathbb{T}^{d}),$ (5.24)
$\displaystyle\partial_{t}w_{\mathrm{II}},\,\nabla^{2}w_{\mathrm{II}}\in
L^{2}(0,T;H^{s-1}(\mathbb{T}^{d}))\cap
C_{\mathrm{loc}}((0,T]\times\mathbb{T}^{d})$ (5.25)

that is a classical solution of system (2.1) in $(0,T)\times\mathbb{T}^{d}$
and satisfies $w(0,\cdot)=w^{\mathrm{in}}$.

###### Proof.

By virtue of Theorem 2.5, we have reduced the question of the local existence
of strong solutions to system ($\mathsf{B}$) to that of constructing strong
solutions to system (5.12). For the latter, we may essentially follow the
proof of Theorem 2.3, and we only describe the necessary modifications. The
main point is that for $\mathsf{r}>1$, we no longer have a maximum principle
for the parabolic problem. In particular, we need to ensure that, on a short
time interval, the approximate solutions stay in an appropriate domain of
uniform parabolicity of the parabolic subsystem. We therefore modify the
definition of $t_{\ell}\in(0,t_{\ell-1}]$, now requiring it to be the maximal
time less than or equal to $t_{\ell-1}$ such that the smooth solution
$w^{\ell}$ to the linear approximate problem analogous to (4.9) satisfies

$\displaystyle\|w^{\ell}\|_{X_{t}^{s}}<KR\quad\text{and}\quad\inf_{(0,t)\times\mathbb{T}^{d}}\rho(w_{\mathrm{II}}^{\ell})>\frac{\bar{\rho}}{4}\quad\text{for
all }t<t_{\ell}.$ (5.26)

In the next step, we derive an estimate analogous to that in Lemma 4.2 (i).
Thanks to the relatively simple symmetric form (5.12b) of the quasilinear
parabolic subproblem, this is achieved essentially in the same way as in the
proof of Lemma 4.2 (i). (Since the matrix $D^{\mathrm{II}}(\lambda)$
multiplying $\partial_{t}w_{\mathrm{II}}$ in (5.12b) is constant, we may even
bring the parabolic subsystem in a canonical form by the change of variables
$w_{\mathrm{II}}\mapsto D^{\mathrm{II}}(\lambda^{1/2})w_{\mathrm{II}}$.) We
deduce an estimate of the form

$\displaystyle\int_{0}^{t}\|\nabla
w_{\mathrm{II}}^{\ell}(\tau)\|^{2}_{H^{s}}\mathrm{d}\tau\leq
C(R,\bar{\rho})\quad\text{for all
}t\in(0,\min\\{t_{\ell-1},1\\}),\;\ell\in\mathbb{N}_{+}.$ (5.27)

Let us emphasise that we do not yet need the improved control of the values to
conclude (5.27).

At this stage, we are in a position to derive a refined control of the values
substituting for Lemma 4.1. For the hyperbolic components $w_{\mathrm{I}}$, we
proceed as before. For the parabolic components, we rely on the following
estimate for all $0\leq t\leq\min\\{t_{\ell},1\\}$

$\displaystyle\|w_{\mathrm{II}}^{\ell}(t)-w_{\mathrm{II}}^{\ell}(0)\|_{C(\mathbb{T}^{d})}\leq
C\int_{0}^{t}\|\partial_{t}w_{\mathrm{II}}^{\ell}(\tau)\|_{H^{s-1}}\mathrm{d}\tau\leq\sqrt{t}C(R,\bar{\rho}),$
(5.28)

where in the second step we used the bound (5.27) in conjunction with the
equation in order to control $\partial_{t}w_{\mathrm{II}}^{\ell}$ by suitable
spatial derivatives. Thus, by choosing $T_{1}\in(0,1]$ small enough depending
on $\bar{\rho}$ and $R$, we can ensure that

$\displaystyle
w^{\ell}(t,x)\in\mathcal{D}_{0}:=\bigg{\\{}\tilde{w}\in\mathcal{D}:|\tilde{w}|<2LR\text{
and }\rho(\tilde{w}_{\mathrm{II}})>\frac{\bar{\rho}}{2}\bigg{\\}}$ (5.29)

for all $(t,x)\in(0,\widehat{t}_{\ell})\times\mathbb{T}^{d}$ and
$\ell\in\mathbb{N}$, where $\widehat{t}_{\ell}:=\min\\{t_{\ell},T_{1}\\}$. At
this point, we may proceed with the proof of the lower bound
$\widehat{t}_{\ell}>T_{*}>0$ along the lines of the proof of Lemma 4.2 (ii).

The convergence in a weaker norm and regularity results analogous to those in
Lemma 4.4 can be deduced as in Section 4.4. The regularity $w_{\mathrm{I}}\in
C([0,{T_{*}}];H^{s})$ is obtained in the same way as in the proof of Lemma
4.6.

It remains to prove the regularity

$\displaystyle\partial_{t}w_{\mathrm{II}},\,\nabla^{2}w_{\mathrm{II}}\in
C_{\mathrm{loc}}((0,{T_{*}}]\times\mathbb{T}^{d}).$ (5.30)

As in the proof of (4.49), we show that the gradient $\nabla w_{\mathrm{I}}$
of the hyperbolic component satisfies a space-time Hölder condition. Moreover,
since $w_{\mathrm{II}}\in W^{1,2}([0,T_{*}];H^{s-1})$, this component also
satisfies a space-time Hölder condition. Applying the linear theory for
strongly parabolic systems in divergence form (see for instance the Schauder-
type estimate in [10, Theorem 2.1]), we find that $\nabla w_{\mathrm{II}}$ and
hence $\nabla w$ satisfy a space-time Hölder condition. Thus, the coefficient
matrix $\mathbb{A}_{1}^{\mathrm{II}}(w)$ of the parabolic subsystem is
sufficiently regular to deduce, by invoking once more classical linear theory,
interior Hölder regularity of $\partial_{t}w_{\mathrm{II}}$ and
$\nabla^{2}w_{\mathrm{II}}$, which implies (5.30). This completes the proof of
Theorem 5.5. ∎

Theorem 2.6 is a consequence of Theorem 5.5 and Proposition 5.2 (with
$\Psi=\Phi^{-1}$ as in Lemma 5.1). The regularity of $Bu$ asserted in Theorem
2.6 follows from the identity

$Bu=\begin{pmatrix}\mathsf{Q}\\\
\mathsf{P}\end{pmatrix}^{\mathsf{T}}\begin{pmatrix}0\\\
D^{\mathrm{II}}(\lambda)w_{\mathrm{II}}\end{pmatrix},$

combined with the regularity of $w_{\mathrm{II}}$ obtained in Theorem 5.5.

##  Appendix A Alternative transformations

###  A.1. A general ansatz

Here, we briefly sketch an ansatz towards a characterisation of the set of
possible diffeomorphisms $w=\Phi(u)$,
$u\in\widehat{\mathcal{D}}:=\mathbb{R}_{+}^{n}$, leading to a normal form of
hyperbolic–parabolic type. For simplicity, we restrict to the rank-one case
$\mathsf{r}=1$ and abbreviate $n^{\prime}:=n-1$. As in Section 5, we introduce
the partition ${\mathrm{I}}=\\{1,\dots,n^{\prime}\\}$ and
${\mathrm{II}}=\\{n\\}$, write
$(w_{\mathrm{I}},w_{\mathrm{II}})=(\Phi_{\mathrm{I}}(u),\Phi_{\mathrm{II}}(u))$
and use the notational conventions introduced above. We further set
$\Psi=\Phi^{-1}$.

For the hyperbolic components, the essential condition for cancelling the
second-order derivatives is $\mathrm{D}\Phi_{\mathrm{I}}(u)D(u)B\equiv 0$. For
problem (1.3), i.e. with $B={k}\otimes{a}$ for vectors
$k=(k_{1},\dots,k_{n})^{\mathsf{T}}$, $a=(a_{1},\dots,a_{n})^{\mathsf{T}}$,
$k_{i}$, $a_{i}>0$, this amounts to requiring that
$\mathrm{D}\Phi_{\mathrm{I}}(u)D(k)u=0\in\mathbb{R}^{n^{\prime}}$. Hence, the
map $\Phi_{\mathrm{I}}$ must be constant along the integral curves
$\gamma_{u_{*}}(t)=D(\mathrm{e}^{kt})u_{*}$, $t\in\mathbb{R}$,
$u_{*}\in\widehat{\mathcal{D}}$, of the vector field $V(u):=D(k)u$. Any
candidate mapping must thus satisfy

$\displaystyle\Phi_{\mathrm{I}}(\gamma_{u_{*}}(t))=c(u_{*})\quad\text{for all
}t\in\mathbb{R},u_{*}\in\widehat{\mathcal{D}}.$ (A.1)

Notice that for the transformation in Section 5, this condition is fulfilled
with

$\displaystyle{\Phi_{\mathrm{I}}}_{i}(\gamma_{u_{*}}(t))=\sum_{j}\mathsf{Q}_{ij}\log(\gamma_{u_{*},j}(t))=\sum_{j}\mathsf{Q}_{ij}(k_{j}t+\log(u_{*,j}))=\sum_{j}\mathsf{Q}_{ij}\log(u_{*,j}),$

where we used the fact that the rows of the matrix $\mathsf{Q}$ defined in
(5.1) are orthogonal to ${k}$. For the transformation (3.1), property (A.1)
follows from a similar calculation. In Section A.2, we will briefly discuss a
different change of coordinates that is subject to (A.1).

With the choice $w_{\mathrm{II}}:=a\cdot u$ as the diffusive variable and
under condition (A.1), system (Rk1) in the new variables $w=\Phi(u)$ takes the
form

$\displaystyle\partial_{t}w_{\mathrm{I}}$ $\displaystyle=\nabla
w_{\mathrm{II}}\cdot\mathbb{Y}(w)\nabla w_{\mathrm{I}}+f(w,\nabla
w_{\mathrm{II}}),$ (A.2) $\displaystyle\partial_{t}w_{\mathrm{II}}$
$\displaystyle=\operatorname{div}\big{(}\mathfrak{a}(w)\nabla
w_{\mathrm{II}}\big{)},$ (A.3)

where $\mathfrak{a}(w)=\sum_{i}a_{i}u_{i}k_{i}$ and

$\displaystyle\mathbb{Y}(w)$
$\displaystyle=\mathrm{D}\Phi_{\mathrm{I}}(u)D(k)D_{w_{\mathrm{I}}}\Psi(w),$
(A.4) $\displaystyle f(w,\nabla w_{\mathrm{II}})$
$\displaystyle=\mathrm{D}\Phi_{\mathrm{I}}(u)D(k)\mathrm{D}_{w_{\mathrm{II}}}\Psi(w)|\nabla
w_{\mathrm{II}}|^{2},$ (A.5)

and we recognise the structure of (2.1).

We note that $\mathbb{Y}$ can be written as

$\displaystyle\mathbb{Y}(w)$
$\displaystyle=\mathrm{D}_{u_{\mathrm{I}}}\Phi_{\mathrm{I}}(u)D^{\mathrm{I}}(k)\mathrm{D}_{w_{\mathrm{I}}}\Psi_{\mathrm{I}}(w)+\mathrm{D}_{u_{\mathrm{II}}}\Phi_{\mathrm{I}}(u)k_{n}\mathrm{D}_{w_{\mathrm{I}}}\Psi_{\mathrm{II}}(w),$

where $D^{\mathrm{I}}(k):=\operatorname{diag}(k_{1},\dots,k_{n^{\prime}})$. If
$\mathrm{D}_{u_{\mathrm{I}}}\Phi_{\mathrm{I}}(u)$ is diagonal, it commutes
with $D^{\mathrm{I}}(k)$, and hence the expression for $\mathbb{Y}(w)$ can be
simplified, using the fact that
$\mathrm{D}\Phi_{\mathrm{I}}(u)\mathrm{D}_{w_{\mathrm{I}}}\Psi(w)=\mathbb{I}_{n^{\prime}}$
in the first term on the right-hand side. This is essentially the technique
used in Section 3. The following section provides an example where
$\mathrm{D}_{u_{\mathrm{I}}}\Phi_{\mathrm{I}}(u)$ takes a more complicated
form.

###  A.2. An example

Consider the system

$\displaystyle\partial_{t}u_{i}=\operatorname{div}(k_{i}u_{i}\nabla(k\cdot
u)),$

which falls into the setting of (Rk1) with the choice $a:=k$. We then let
$\mathcal{\widehat{D}}:=\mathbb{R}_{+}^{n}$,
$\mathcal{D}_{1}:=\mathcal{E}\times\mathbb{R}_{+}$,
$\mathcal{E}:=\\{w_{\mathrm{I}}\in(0,1)^{n^{\prime}}:\sum_{i\in{\mathrm{I}}}w_{i}<1\\}$
and define $\Phi:\mathcal{\widehat{D}}\to\mathcal{D}_{1}$ by

$\displaystyle
w_{i}=\Phi_{i}(u)=\begin{cases}\frac{1}{L(u)}u_{i}^{1/k_{i}},&1\leq i\leq
n^{\prime},\\\ \sum_{j=1}^{n}k_{j}u_{j},&i=n,\end{cases}$ (A.6)

where $L(u):=\sum_{j=1}^{n}u_{j}^{1/k_{j}}$. Definition (A.6) readily shows
that $\Phi$ fulfils condition (A.1). We observe that in the special case where
all $k_{i}$ equal (without loss of generality we may take $k_{i}=1$), the
change of variables (A.6) is more regular near zero and reduces to that used
in [4].

We assert that $\Phi:\mathcal{\widehat{D}}\to\mathcal{D}_{1}$ is a
diffeomorphism. For $i\in{\mathrm{I}}$ and $j=1,\dots,n$, we compute

$\displaystyle\partial_{u_{j}}\Phi_{i}(u)=\frac{\Phi_{i}(u)}{k_{i}u_{i}}\delta_{ij}-\Phi_{i}(u)\frac{u_{j}^{1/k_{j}}}{L(u)}\frac{1}{k_{j}u_{j}}.$

In particular, the $(n^{\prime}\times n^{\prime})$-matrix
$\mathrm{D}_{u_{\mathrm{I}}}\Phi_{\mathrm{I}}$ is the sum of a diagonal and a
rank-one matrix. Moreover,

$\displaystyle\mathrm{D}_{u_{\mathrm{II}}}\Phi_{\mathrm{I}}(u)=-a(u)\Phi_{\mathrm{I}}(u),\quad\mbox{where
\,}a(u)=\frac{u_{n}^{1/k_{n}}}{L(u)}\frac{1}{k_{n}u_{n}},$ (A.7)

and $\mathrm{D}\Phi_{\mathrm{II}}(u)={k}^{\mathsf{T}}$. Using the formula
$\det(M+\zeta\otimes\xi)=\det(M)+\zeta^{\mathsf{T}}(\mathrm{cof}M)\xi$, we see
that

$\det\mathrm{D}_{u_{\mathrm{I}}}\Phi_{\mathrm{I}}=\prod_{\ell\in{\mathrm{I}}}\frac{\Phi_{\ell}(u)}{k_{\ell}u_{\ell}}\bigg{(}1-\sum_{j\in
I}w_{j}\bigg{)}>0.$

Similar calculations show that $\det\mathrm{D}\Phi>0$. The bijectivity of
$\Phi$ from $\widehat{\mathcal{D}}$ to $\mathcal{D}_{1}$ is also elementary to
verify, and we conclude the diffeomorphism property.

Again, let $\Psi$ be the inverse of $\Phi$. Using the following identities,
involving $L=L(u)$,

$\displaystyle u_{j}=(Lw_{j})^{k_{j}},\;j\in{\mathrm{I}},\quad
L=\bigg{(}1-\sum_{\ell\in I}w_{\ell}\bigg{)}^{-1}u_{n}^{1/k_{n}},\quad
w_{n}=\sum_{j\in I}k_{j}L^{k_{j}}w_{j}^{k_{j}}+k_{n}u_{n},$ (A.8)

we compute

$\displaystyle\partial_{w_{i}}\Psi_{j}(w)$
$\displaystyle=k_{j}u_{j}\bigg{(}\frac{\partial_{w_{i}}L}{L}+\frac{1}{w_{j}}\delta_{ij}\bigg{)},\quad
j\in{\mathrm{I}},\;i=1,\dots,n,$ (A.9)
$\displaystyle\partial_{w_{i}}\Psi_{n}(w)$
$\displaystyle=k_{n}u_{n}\bigg{\\{}\frac{\partial_{w_{i}}L}{L}-\bigg{(}1-\sum_{j\in
I}w_{j}\bigg{)}^{-1}\delta_{i{\mathrm{I}}}\bigg{\\}},\quad i=1,\dots,n,$
(A.10) $\displaystyle\frac{\partial_{w_{i}}L}{L}$
$\displaystyle=L\big{(}k_{n}^{2}u_{n}^{1-1/k_{n}}-k_{i}^{2}u_{i}^{1-1/k_{i}}\big{)}\bigg{(}\sum_{j=1}^{n}k_{j}^{2}u_{j}\bigg{)}^{-1},\quad
i\in I,$ (A.11)

where $\delta_{i{\mathrm{I}}}:=1$ if $i\in{\mathrm{I}}$ and
$\delta_{n{\mathrm{I}}}:=0$. Since the transformation (A.6) is still non-
smooth as soon as one of the densities vanishes, this change of variables does
not lead to an improved local existence theory for classical solutions
compared to that based on (3.1).

######  Remark A.1.

In contrast to transformation (3.1),
$\mathrm{D}_{u_{\mathrm{I}}}\Phi_{\mathrm{I}}$ is not diagonal in the present
case. An alternative splitting of $\mathbb{Y}$ such as

$\displaystyle\mathbb{Y}(w)=k_{n}\mathbb{I}_{n^{\prime}}+\mathrm{D}_{u_{\mathrm{I}}}\Phi_{\mathrm{I}}(D(k)-D(k_{n}))\mathrm{D}_{w_{\mathrm{I}}}\Psi_{\mathrm{I}}$
(A.12)

might therefore be favourable for a possible symmetrisation. Thus, it suffices
to find a positive definite matrix
$\mathbb{A}_{0}(w)\in\mathbb{R}^{n^{\prime}\times
n^{\prime}}_{\operatorname{sym}}$ such that the product
$\mathbb{A}_{0}(w)\mathrm{D}_{u_{\mathrm{I}}}\Phi_{\mathrm{I}}(D(k)-D(k_{n}))\mathrm{D}_{w_{\mathrm{I}}}\Psi_{\mathrm{I}}$
is symmetric. A computation yields

$\displaystyle\mathbb{G}(w)_{i\ell}$
$\displaystyle:=\big{(}\mathrm{D}_{u_{\mathrm{I}}}\Phi_{\mathrm{I}}(D(k)-D(k_{n}))\mathrm{D}_{w_{\mathrm{I}}}\Psi_{\mathrm{I}}\big{)}_{i\ell}$
$\displaystyle=(k_{i}-k_{n})\delta_{i\ell}-\Phi_{i}(u)(k_{\ell}-k_{n})+(k_{i}-k_{n}-\lambda(u))\Phi_{i}(u)\frac{\partial_{w_{\ell}}L}{L},$

where $\lambda(u)=\sum_{j\in{\mathrm{I}}}(k_{j}-k_{n})\Phi_{j}(u)$. We observe
that $\mathbb{G}$ is a rank-two perturbation of a diagonal matrix, which means
that the question of symmetrisability is not trivial in general.

## Acknowledgements

The first two authors are grateful to the organizers of the WIAS Days 2022,
where some ideas for this work were initiated. The second author would further
like to thank Dr. Joachim Rehberg for helpful comments. The last author
acknowledges partial support from the Austrian Science Fund (FWF), grants
P33010 and F65. This work has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme, ERC Advanced Grant no. 101018153.

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