twoside=false

# Transport-Reaction Models

(August 16, 2021)

At first, I would like to express my gratitude to my advisors Dr. Christina
Lienstromberg and Prof. Dr. Juan J. L. Velázquez. They always took time to
provide guidance and feedback, which led to many fruitful discussions.

I would also like to thank my parents and my sister. The intense work on my
thesis during the last months would not have been possible without their
continuous support.

###### Contents

  1. 1 Introduction
    1. 1 Function Spaces and Notation
  2. 2 The Transport Semigroup
  3. 3 The Transport-Reaction Semigroup
  4. 4 Spectral Analysis of the Transport-Reaction Semigroup
    1. 2 Spectral Analysis of the Transport Semigroup
    2. 3 Spectral Analysis of the Generator
    3. 4 Weak Spectral Mapping Theorems
  5. 5 Arbitrary Side Lengths and Neumann Boundary Conditions
    1. 5 Arbitrary Side Lengths
    2. 6 Symmetric Models with Neumann Boundary Conditions
  6. 6 Applications: Long-Time Behavior of Hyperbolic Models
    1. 7 A Goldstein-Kac Model
    2. 8 Models with Killing
      1. 8.1 Positive and Mass Conserving Models with Killing
    3. 9 A Reaction Random Walk System
  7. 7 Pattern Formation
    1. 10 Reaction-Diffusion Equations and Turing Patterns
      1. 10.1 Reaction Networks
      2. 10.2 The Full Model and Diffusion-Driven Instabilities
    2. 11 Transport-Driven Instabilities
      1. 11.1 Transport-Driven Instabilities for $N=2$
    3. 12 Supplementary Figures and Simulations
      1. 12.1 Figures of The Spectrum for $N=2$
      2. 12.2 Figures of The Spectrum for $N=3$
      3. 12.3 A Simulation for $N=3$
  8. A Postponed Proofs
  9. B $C_{0}$-Semigroups, Generators and Abstract Cauchy Problems
  10. C Perturbation and Approximation of $C_{0}$-Semigroups
  11. D Spectral Theory and Long-Term Behavior of $C_{0}$-Semigroups

###### List of Figures

  1. 1 Domain Generator $\subsetneq W^{1,p}$: An Illustration of the Idea
  2. 2 Typical Spectrum for $N=2.$ Example 1
  3. 3 Typical Spectrum for $N=2.$ Example 2
  4. 4 Hyperbolic Instabilities for $N=2.$ Example 1
  5. 5 Hyperbolic Instabilities for $N=2.$ Example 2
  6. 6 Eventually Constant Real Parts for $N=2.$ An Example
  7. 7 Spectrum for $N=3$. An Example
  8. 8 Hyperbolic Instabilities for $N=3$. An Example
  9. 9 Turing Patterns for $N=3$. Example 1
  10. 10 Turing Patterns for $N=3$. Example 2
  11. 11 Turing Patterns for $N=3$. Example 3
  12. 12 Turing Patterns for $N=3$. Example 4
  13. 13 A Simulation of Turing Patterns for $N=3$.
  14. 14 A Simulation of the Time Evolution of Turing Patterns
  15. 15 A Simulation of Hyperbolic Instabilities for $N=3$.
  16. 16 A Simulation of the Time Evolution of Hyperbolic Instabilities

##  Chapter 1 Introduction

Transport-reaction equations arise in the mathematical modeling of the
movement of populations. In contrast to parabolic equations, hyperbolic models
realistically account for a finite propagation speed of organisms. Therefore,
scientists increasingly focus on hyperbolic and kinetic models [7, p. 36].
A particularly interesting question is whether mathematical models are able to
describe the variety of patterns observed in nature: rippling waves in
myxobacteria swarms, milling schools of fish, swarms of insects or pedestrian
traffic jams, to name only a few of them [7]. Diffusive-generated patterns
have originally been studied by Turing in his pioneer work [38] and his idea
has been applied many times, for instance in the context of morphogenesis
[13]. Patterns generated by transport on the other hand are less well
understood.

The aim of this master’s thesis is to study the linear transport-reaction
system

$\partial_{t}\begin{pmatrix}u_{1}\\\ \vdots\\\
u_{N}\end{pmatrix}(t,\mathbf{x})+\begin{pmatrix}\mathbf{v_{1}}\cdot\nabla
u_{1}\\\ \vdots\\\ \mathbf{v_{N}}\cdot\nabla
u_{N}\end{pmatrix}(t,\mathbf{x})=B\begin{pmatrix}u_{1}\\\ \vdots\\\
u_{N}\end{pmatrix}(t,\mathbf{x})$ (1)

on the $d$-dimensional hypercube $[0,1]^{d}$ with periodic boundary
conditions, i.e. on the $d$-dimensional torus $\mathbb{T}^{d}.$ Concerning
notation and interpretation, the parameter $N$ is the number of population
subgroups and $\mathbf{v_{1}},\cdots,\mathbf{v_{N}}\in\mathbb{R}^{d}$ are the
transport directions of each component. The matrix $B\in\mathbb{R}^{N\times
N}$ describes linear interactions of the subgroups.
Typically, (1) appears as a linearization of the non-linear partial
differential equation

$\partial_{t}\begin{pmatrix}u_{1}\\\ \vdots\\\
u_{N}\end{pmatrix}(t,\mathbf{x})+\begin{pmatrix}\mathbf{v_{1}}\cdot\nabla
u_{1}\\\ \vdots\\\ \mathbf{v_{N}}\cdot\nabla
u_{N}\end{pmatrix}(t,\mathbf{x})=F(u(t,\mathbf{x})),$ (2)

where $u=(u_{1},\cdots,u_{N})$ and
$F\colon\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$ is a sufficiently smooth
function. There is a large amount of research which deals with the question of
stability of a given equilibrium point of (2); see for instance [18, 29, 31].
Many times, however, only the one dimensional case $d=1$ is studied.
The transport-reaction equation (1) can also be interpreted as a velocity
discretization of

$\partial_{t}u(t,\mathbf{x},\mathbf{v})+\mathbf{v}\cdot\nabla_{x}u(t,\mathbf{x},\mathbf{v})=-\sigma(\mathbf{v})+\int_{V}\kappa(\mathbf{v},\mathbf{v}^{\prime})u(t,\mathbf{x},\mathbf{v})\,d\mathbf{v}^{\prime},$
(3)

where the velocity $\mathbf{v}$ can take values in a set
$V\subseteq\mathbb{R}^{d}.$ The function $\sigma\colon
V\rightarrow\mathbb{R}_{\geq 0}$ describes collisions causing absorption [4]
and corresponds to the diagonal entries of $B$. The so called scattering
kernel $\kappa\colon V\times V\rightarrow\mathbb{R}_{\geq 0}$ governs the
transition of particles with incoming velocity $\mathbf{v}^{\prime}$ to
particles with outgoing velocity $\mathbf{v}$ [4]. It corresponds to the off-
diagonal entries of $B.$ The linear Boltzmann-type equation (3) is, for
example, connected to velocity-jump processes and neutron transport, see [19,
7, 4].

Throughout the whole thesis, we try to be as general as possible in the sense
that we consider the transport-reaction equation (1) on the spaces
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ for all $1\leq p<\infty$ and all
dimensions $d\in\mathbb{N}$ simultaneously. Whenever we need to restrict
ourselves to $p=2$ or $d=1,$ we try to at least indicate how future research
can approach the other cases.
The thesis is structured as follows: Chapter 2 and Chapter 3 deal with well-
posedness of (1) and basic properties such as positivity and mass
conservation. We use a semigroup approach and show that

$-A_{p}+B_{p}=-\begin{pmatrix}\mathbf{v_{1}}\cdot\nabla u_{1}\\\ \vdots\\\
\mathbf{v_{N}}\cdot\nabla u_{N}\end{pmatrix}+B\begin{pmatrix}u_{1}\\\
\vdots\\\ u_{N}\end{pmatrix}$

with an appropriately chosen domain generates a semigroup on
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}),$ which we call the transport-reaction
semigroup $(R_{p}(t))_{t\geq 0}.$ We continue our study with a detailed
spectral analysis of both the generator $-A_{p}+B_{p}$ and the transport-
reaction semigroup $(R_{p}(t))_{t\geq 0}$. Chapter 5 deals with the
incorporation of different side lengths $L>0$ of the torus $\mathbb{T}^{d}$
and Neumann boundary conditions for symmetric models in dimension $d=1$. In
Chapter 6, we apply our results to, for example, a Goldstein-Kac-type equation
[22] and a sophisticated model from [18]. Finally, Chapter 7 is devoted to
pattern formation and transport-driven instabilities.
We tried our best to write every chapter as self-contained as possible and to
motivate all sections of the thesis with a detailed introduction at the
beginning of each chapter.

One main contribution of the thesis is the weak spectral mapping theorem

$\sigma(R_{2}(t))=\overline{e^{t\sigma(-A_{2}+B_{2})}}\quad\text{ for all
}t\in\mathbb{R}$

and a description of $\sigma(-A_{2}+B_{2})$ in terms of roots of polynomials
of order $N$ for all dimensions $d\in\mathbb{N}$. This result allows a very
detailed study of the long-term behavior of solution to (1). To the best of
our knowledge, all current results deal with the one-dimensional case $d=1.$
The proofs are based on Fourier transform and so called matrix multiplication
operators from [21].
The other main contribution is the chapter on pattern formation and transport-
driven instabilities. We introduce the notion of hyperbolic instabilities,
which allows us to prove a dichotomy: every transport-driven instability in
$d=1$ and with arbitrary number of components $N$ is either a Turing pattern
or a hyperbolic instability. Intuitively, hyperbolic instabilities correspond
to increasingly oscillating and chaotic behavior. We do not hold expertise in
biological modeling but we believe that our finding reflects the nature of
movement of populations: either there is a regular pattern or one observes
bustle. As a byproduct of our analysis, we find an algebraic condition in
terms of the transport speeds $v_{1},\cdots,v_{N}$ and $B$ which ensures the
existence of Turing patterns. The mathematical details of this chapter are
based on perturbation theory from [23].

###  1 Function Spaces and Notation

In order to rigorously define standard $L^{p}$ spaces for $1\leq p<\infty$ on
a $d$-dimensional torus $\mathbb{T}^{d}$, we follow the approach from [11, p.
238f.] and consider the quotient map

$q\colon[0,1]^{d}\rightarrow\mathbb{T}^{d},$

which identifies opposite sides of the hypercube. This mapping induces the
measurable space
$(\mathbb{T}^{d},\,q_{\sharp}\mathcal{B}([0,1]^{d}),\,q_{\sharp}\mathcal{L}^{d})$
where $\mathcal{B}([0,1]^{d})$ is the Borel $\sigma$-algebra on the hypercube
and $\mathcal{L}^{d}$ is the $d$-dimensional Lebesgue measure. Moreover, the
canonical norm on $L^{p}(\mathbb{T}^{d},\mathbb{C}^{N})$ is given by

$\|u\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}\coloneqq\left(\sum_{j=1}^{N}\|u_{j}\|_{L^{p}(\mathbb{T}^{d})}^{2}\right)^{\frac{1}{2}}$
(4)

for $u=(u_{1},\cdots,u_{N})\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$ To
give a better visualization, it should be mentioned that functions on
$\mathbb{T}^{d}$ may be considered as periodic functions on $\mathbb{R}^{d}$
or as functions on $[0,1)^{d}.$ The more appropriate point of view will follow
by the context when it matters [11, p. 238f.]. Generally, we are interested in
functions $u\colon\mathbb{R}_{\geq
0}\times\mathbb{T}^{d}\rightarrow\mathbb{C}^{N},$ where the value of $u$ at
$(t,\mathbf{x})$ is for instance the concentration of population subgroups at
time $t$. The reason we are working with complex-valued functions is that we
approach the transport-reaction equation (1) from a semigroup and spectral
theory perspective. This may be counterintuitive at first glance, but
ultimately the only difference is that also complex initial functions are
allowed and that no inconsistencies in the spectral analysis occur. Given real
initial values $u_{0}\colon\mathbb{T}^{d}\rightarrow\mathbb{R}^{N},$ every
solution of the equations we study will also be real valued.

It should also be mentioned that we almost always mean strongly continuous
(semi-)group, when we use the term (semi-)group. A relatively detailed
introduction to semigroup theory is given in Appendix B to Appendix D.
Last but not least, the thesis contains a list of symbols, which can be found
at the very end.

##  Chapter 2 The Transport Semigroup

The first questions regarding the transport-reaction model (1) are to ask for
existence and uniqueness of solutions. This chapter is devoted to answer these
questions in the non-reacting case $B=0$ by using a semigroup approach. In the
process, we study the domain of the generator of the transport semigroup and
identify $C_{c}^{\infty}(\mathbb{T}^{d},\mathbb{C}^{N})$ as a core. The linear
reactions in (1) are incorporated in the next chapter.

Throughout the whole chapter, we assume that the dimension $d\in\mathbb{N}$,
the number of components $N\in\mathbb{N}$, the transport directions
$\mathbf{v_{1}},\cdots,\mathbf{v_{N}}\in\mathbb{R}^{d}$ and $1\leq p<\infty$
are arbitrarily chosen fixed parameters, if not stated otherwise. We work on
the Banach space $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ with norm
$\|\cdot\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}$, which is defined in (4).

We start by introducing the transport semigroup on
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ explicitly via

$(T(t)u)(\mathbf{x})\coloneqq\begin{pmatrix}u_{1}(\mathbf{x}-t\mathbf{v_{1}})\\\
\vdots\\\ u_{N}(\mathbf{x}-t\mathbf{v_{N}})\end{pmatrix}$ (5)

for $t\geq 0$, $N\in\mathbb{N}$ directions
$\mathbf{v_{1}},\cdots,\mathbf{v_{N}}\in\mathbb{R}^{d}$ and
$u=(u_{1},\cdots,u_{N})\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$ Strictly
speaking, we should include the indices $d$ and $p$ into the notation as the
semigroup depends on the Banach space, but we will often drop these two
indices to improve readability.

######  Lemma 2.1.

$(T(t))_{t\geq 0}$ defines a contraction semigroup on
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$

###### Proof.

The lemma with domain $\mathbb{R}^{d}$ instead of $\mathbb{T}^{d}$ is a
standard example for a contraction semigroup and strong continuity follows by
the density of $\mathbb{C}_{c}^{\infty}(\mathbb{R}^{d},\,\mathbb{C}^{N});$ see
for example [11, Proposition 8.5, p. 238]. On a torus, the proof works
similarly. A density argument and Fubini can be used to show

$\int_{\mathbb{T}^{d}}|u_{j}(\mathbf{x})|^{p}\,d\mathbf{x}=\int_{\mathbb{T}^{d}}|u_{j}(\mathbf{x}-t\mathbf{v_{j}})|^{p}\,d\mathbf{x}$

for all $j=1,\cdots,N$ and all $t\geq 0.$ Moreover, the semigroup property is
a consequence of

$((a\text{ mod }1)+b)\text{ mod }1=(a+b)\text{ mod }1\qquad\text{for all
}a,\,b\in\mathbb{R}.$

∎

In the following, we denote the generator of the transport semigroup
$(T(t))_{t\geq 0}$ by $(-A,\,D(-A)).$ Similarly to the semigroup, the
generator depends on $d$ and $p$. This should be kept in mind for later
chapters on the spectral analysis of the generator. In this chapter, however,
the dependence does not play a significant role, which is why we drop the
index. By definition, it holds

$D(-A)=\left\\{u\in
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})\,\colon\,\lim_{t\searrow
0}\frac{u_{j}(\cdot-t\mathbf{v_{j}})-u_{j}(\cdot)}{t}\text{ exists in
}L^{p}(\mathbb{T}^{d})\text{ for all }j=1,\cdots,N\right\\}.$ (6)

For continuously differentiable functions $u$, the limits in
$L^{p}(\mathbb{T}^{d})$ are simply given by

$\lim_{t\searrow
0}\frac{u_{j}(\cdot-t\mathbf{v_{j}})-u_{j}(\cdot)}{t}=-\mathbf{v}_{j}\cdot\nabla
u_{j}$

for all $j=1,\cdots,N.$ We start with a functional analytic approach to extend
this result to weakly differentiable functions $u\in
W^{1,p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$. Studying convergence of difference
quotients with respect to the $L^{p}$-norm is very common and can for example
be found in [9, Part 2; 5.8.2, p. 293f.]. Nevertheless, we decided to include
this approach because in the literature, the arguments are typically only
given for the standard unit vectors as directions. In our case, we have
arbitrary directions $\mathbf{v_{1}},\cdots,\mathbf{v_{N}}\in\mathbb{R}^{d}.$
This functional analytic perspective is somewhat technical in the case $p=1$
since $L^{1}$ is not reflexive. Therefore, these proofs also motivate a pure
semigroup approach: on $\mathbb{T}^{d}$, abstract semigroup theory can be
applied to obtain a shorter and easier proof for the $L^{p}$-convergence of
the difference quotients.

Consider a function $u\in L^{p}(\mathbb{T}^{d})$ and define the difference
quotients

$D_{k}^{h}u(\cdot)\coloneqq\frac{u(\cdot+h\mathbf{e_{k}})-u(\cdot)}{h}$ (7)

for $h\in\mathbb{R}$ and $k=1,\cdots,d.$ In this definition, $\mathbf{e_{k}}$
is the $k$-th standard unit vector of $\mathbb{R}^{d}.$ The next lemma deals
with the limit behavior as $h\rightarrow 0$ (note that $h$ is not assumed to
have a particular sign).

######  Lemma 2.2.

Let $u\in L^{p}(\mathbb{T}^{d})$ and let $k=1,\cdots,d.$

  1. (i)

Let $D_{k}^{h}u\rightharpoonup v$ in $L^{p}(\mathbb{T}^{d})$. Then the $k$-th
weak derivative exists in $L^{p}(\mathbb{T}^{d})$ and $\partial_{k}u=v.$

  2. (ii)

Conversely, if $u\in W^{1,p}(\mathbb{T}^{d})$, it follows that
$D_{k}^{h}u\rightarrow\partial_{k}u$ in $L^{p}(\mathbb{T}^{d})$ for $p>1$ and
$D_{k}^{h}u\rightharpoonup\partial_{k}u$ for $p=1$.

All convergences are with respect to $h\rightarrow 0.$

###### Proof.

The details are given in Appendix A. ∎

Although the directions
$\mathbf{v_{1}},\cdots,\mathbf{v_{N}}\in\mathbb{R}^{d}$ are arbitrary in the
definition of the generator of the transport semigroup (6), a corresponding
result can be shown in the exact same manner. That is, if we define the
directional difference quotient

$D_{\mathbf{v}}^{h}u(\cdot)\coloneqq\frac{u(\cdot+h\mathbf{v})-u(\cdot)}{h}.$
(8)

for $u\in L^{p}(\mathbb{T}^{d}),\,\mathbf{v}\in\mathbb{R}^{d}$ and
$h\in\mathbb{R},$ the following lemma holds true.

######  Lemma 2.3.

Let $1\leq p<\infty,\,u\in W^{1,p}(\mathbb{T}^{d})$ and let
$\mathbf{v}\in\mathbb{R}^{d}.$ Then
$D_{\mathbf{v}}^{h}u\rightarrow\mathbf{v}\cdot\nabla u$ in
$L^{p}(\mathbb{T}^{d})$ for $p>1$ and
$D_{\mathbf{v}}^{h}u\rightharpoonup\mathbf{v}\cdot\nabla u$ in
$L^{p}(\mathbb{T}^{d})$ for $p=1$ as $h\rightarrow 0.$

###### Proof.

Use the inequality

$\|D_{\mathbf{v}}^{h}u\|_{L^{p}(\mathbb{T}^{d})}\leq|\mathbf{v}|\,\||\nabla
u|\|_{L^{p}(\mathbb{T}^{d})}$

for $u\in W^{1,p}(\mathbb{T}^{d})$ and argue as in the proof of Lemma 2.2.
Also the case $p=1$ works analogously. ∎

We define the operator $-\mathcal{A}$ on
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ by

$-\mathcal{A}u=-\begin{pmatrix}\mathbf{v_{1}}\cdot\nabla u_{1}\\\ \vdots\\\
\mathbf{v_{N}}\cdot\nabla u_{N}\end{pmatrix}.$ (9)

Here, we did not specify the domain intentionally. It will vary and we will
specify it later on for different scenarios. With the functional analytic
arguments from above, we obtain the following result concerning the domain of
the generator of $(T(t))_{t\geq 0}.$

######  Lemma 2.4.

For $1<p<\infty,$ the generator $(-A,\,D(-A))$ is an extension of
$(-\mathcal{A},\,W^{1,p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$. In the case
$p=1,$ the generator is an extension of
$(-\mathcal{A},\,C^{1}(\mathbb{T}^{d},\,\mathbb{C}^{N})).$

###### Proof.

The statement for $p=1$ follows immediately from the chain rule and dominated
convergence. For $p>1,$ we can apply Lemma 2.3 componentwise to obtain

$\frac{u_{j}(\cdot-t\mathbf{v_{j}})-u_{j}(\cdot)}{t}=-D_{\mathbf{v_{j}}}^{-t}u_{j}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow-\mathbf{v_{j}}\cdot\nabla
u_{j}$

in $L^{p}(\mathbb{T}^{d})$ for $j=1,\cdots,N.$ ∎

Remarkably enough, the previous functional analytic arguments are insufficient
to show strong convergence in the case $p=1$. An alternative and shorter proof
that uses semigroup theory works for all cases $1\leq p<\infty.$

######  Lemma 2.5.

Let $1\leq p<\infty$. Then $(-A,\,D(-A))$ extends
$(-\mathcal{A},\,W^{1,p}(\mathbb{T}^{d},\,\mathbb{C}^{N})).$

######  Remark 2.6.

This implies in particular, that strong convergence in Lemma 2.3 also holds
for $p=1.$

###### Proof.

Let $1\leq p<\infty$ and let $u\in W^{1,p}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$
Moreover, take an approximation $(u^{(n)})_{n\in\mathbb{N}}\subset
C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ such that $u^{(n)}\rightarrow
u$ in $W^{1,p}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$ By the convergence of the
derivatives, it follows that
$-Au^{(n)}=-\mathcal{A}u^{(n)}\rightarrow-\mathcal{A}u$ in
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$ Since generators of
$C_{0}$-semigroups are closed by Hille-Yosida, see Theorem B.6, we obtain
$u\in D(-A)$ and $-Au=-\mathcal{A}u.$ ∎

At this point, it is unclear, whether the generator of the transport semigroup
is uniquely determined by
$(-\mathcal{A},\,W^{1,p}(\mathbb{T}^{d},\,\mathbb{C}^{N})).$ However, the
semigroup is extremely well behaved in the sense that it is easy to identify
$C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ as a core, i.e.
$C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ is dense in $D(-A)$ w.r.t.
the graph norm $\|\cdot\|_{-A}$.

######  Theorem 2.7.

$C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ is a core for
$(-A,\,D(-A)).$ In particular, $(T(t))_{t\geq 0}$ is the only strongly
continuous semigroup on $L^{p}(\mathbb{T}^{d},\mathbb{C}^{N})$ with a
generator that extends
$(-\mathcal{A},\,C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})).$

###### Proof.

Clearly, we have $T(t)(C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N}))\subset
C_{c}^{\infty}(\mathbb{T}^{d},\mathbb{C}^{N}).$ This implies that
$C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ is a core for $(-A,\,D(-A))$
by [8, II. Proposition 1,7, p. 53].
The addendum is well known but we recall its short and nice proof. Let
$(\widetilde{T}(t))_{t\geq 0}$ be a semigroup with generator
$(-\widetilde{A},\,D(-\widetilde{A}))\supseteq(-\mathcal{A},\,C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$.
The fact that $C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ is a core and
the closedness of $-\widetilde{A}$ due to Hille-Yosida, see Theorem B.6, imply
$(-\widetilde{A},\,D(-\widetilde{A}))\supseteq(-A,\,D(-A))$. Hille-Yosida also
yields existence of some $\lambda\in\rho(-A)\cap\rho(-\widetilde{A})$ and
hence, both the operators

$\lambda+\widetilde{A}\colon
D(-\widetilde{A})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$

and

${\left.\kern-1.2pt\lambda+\widetilde{A}\vphantom{\big{|}}\right|_{D(-A)}}=\lambda+A\colon
D(-A)\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$

are bijective. We obtain $-A=-\widetilde{A}$, so the semigroups $(T(t))_{t\geq
0}$ and $(\widetilde{T}(t))_{t\geq 0}$ coincide by [30, Chapter 1, Theorem
2.6, p. 6].

∎

######  Remark 2.8.

By defining (5) for all $t\in\mathbb{R},$ one can directly see that
$(T(t))_{t\geq 0}$ can be extended to a $C_{0}$-group
$(T(t))_{t\in\mathbb{R}}$ and that

$D(-A)=\left\\{u\in
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})\,\colon\,\lim_{t\rightarrow
0}\frac{u_{j}(\cdot-t\mathbf{v_{j}})-u_{j}(\cdot)}{t}\text{ exists in
}L^{p}(\mathbb{T}^{d})\text{ for all }j=1,\cdots,N\right\\}.$ (10)

The difference is subtle but notice that we have $t\rightarrow 0$ in the above
formula instead of $t\searrow 0$ as in (6). This equality on the domain
follows by an explicit computation because

$\lim_{t\searrow 0}\frac{u_{j}(\cdot-t\mathbf{v_{j}})-u_{j}(\cdot)}{t}\text{
exists in }L^{p}(\mathbb{T}^{d})\text{ for all }j=1,\cdots,N$

if and only if

$\lim_{t\nearrow 0}\frac{u_{j}(\cdot-t\mathbf{v_{j}})-u_{j}(\cdot)}{t}\text{
exists in }L^{p}(\mathbb{T}^{d})\text{ for all }j=1,\cdots,N.$

In particular, the operator $A$ with $D(A)\coloneqq D(-A)$ generates the
$C_{0}$-semigroup $(T(-t))_{t\geq 0}$ given by

$(T(-t)u)(\mathbf{x})\coloneqq\begin{pmatrix}u_{1}(\mathbf{x}+t\mathbf{v_{1}})\\\
\vdots\\\ u_{N}(\mathbf{x}+t\mathbf{v_{N}})\end{pmatrix}$

for $t\geq 0,$ cf. [8, II. 3.a, p. 79].

We now compute the adjoint semigroup $(T_{p}(t)^{*})_{t\geq 0}$ of
$(T_{p}(t))_{t\geq 0}$ on $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})^{*}$ and the
adjoint $-A_{p}^{*}$ of the generator $-A_{p}$, which will be pretty useful
for the spectral analysis performed in Chapter 4. We would like to point out
that the index $p$ does play a crucial role here, in contrast to all other
results in the chapter.
Recall the classical result
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})^{*}=L^{q}(\mathbb{T}^{d},\,\mathbb{C}^{N})$
where $1<p<\infty$ and $q$ is the dual exponent of $p,$ i.e.
$\tfrac{1}{p}+\tfrac{1}{q}=1,$ see [6, IV.8, Theorem 1, p. 286]. Moreover, the
adjoint of the bounded operator $T_{p}(t)$ is defined by

$\langle T_{p}(t)u,\,v\rangle=\langle u,\,T_{p}(t)^{*}v\rangle\quad\text{ for
all }u\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})\text{ and all }v\in
L^{q}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$

Again, the case $p=1$ is somewhat a special case because
$L^{1}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ is not reflexive.

######  Proposition 2.9.

Let $1<p<\infty$ and let $(T_{p}(t))_{t\geq 0}$ be the semigroup generated by
$(-A_{p},\,D(-A_{p})).$ Then $(T_{p}(t)^{*})_{t\geq 0}$ is strongly continuous
on $L^{q}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ and it holds

$\displaystyle(T_{p}(t)^{*})_{t\geq 0}$ $\displaystyle=(T_{q}(-t))_{t\geq 0},$
$\displaystyle(-A_{p}^{*},\,D(-A_{p}^{*}))$
$\displaystyle=(A_{q},\,D(A_{q})).$

###### Proof.

The space $L^{p}(\mathbb{T},\,\mathbb{C}^{N})$ is reflexive by for
$1<p<\infty,$ see [6, IV.8, Corollary 2, p. 288]. Hence, the adjoint semigroup
is again strongly continuous by [8, I 5.14 Proposition, p. 44] and the sun
dual semigroup coincides with the adjoint semigroup, cf. [8, II 2.6, p. 62].
In particular, the generator of the adjoint (strongly continuous) semigroup is
the adjoint of $-A_{p},$ namely $-A_{p}^{*},$ by [8, II 2.6 proposition, p.
63].
It is left to show $-A_{p}^{*}=A_{q}$ and to characterize the domain. Recall
that we have

$\displaystyle D(-A_{p}^{*})=\left\\{v\in
L^{q}(\mathbb{T}^{d},\,\mathbb{C}^{N})\,\colon\,\exists w\in
L^{q}(\mathbb{T}^{d},\,\mathbb{C}^{N})\text{ with }\langle-
A_{p}u,v\rangle=\langle u,w\rangle\text{ for all }u\in D(-A_{p})\right\\}$

and

$-A_{p}^{*}v=w$

for all $v\in D(-A_{p}^{*}).$ Using that
$C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ is a core for
$(-A_{p},\,D(-A_{p}))$ by Theorem 2.7, it follows

$D(-A_{p}^{*})=\left\\{v\in
L^{q}(\mathbb{T}^{d},\,\mathbb{C}^{N})\,\colon\,\exists w\in
L^{q}(\mathbb{T}^{d},\,\mathbb{C}^{N})\text{ with }\langle-
A_{p}u,v\rangle=\langle u,w\rangle\text{ for all }u\in
C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})\right\\}.$

The next step is to show

$(-A_{p}^{*},\,D(-A_{p}^{*}))\supseteq(A_{q},\,C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})).$
(11)

To this end, let $u,v\in C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$ We
use the notation $\mathbf{v_{j,k}}$ for the $k$-th component of
$\mathbf{v_{j}}$ and compute

$\displaystyle\langle-A_{p}u,v\rangle$
$\displaystyle=\sum_{j=1}^{N}\int_{\mathbb{T}^{d}}(-\mathbf{v_{j}}\cdot\nabla
u_{j})\overline{v_{j}}\,d\mathbf{x}=-\sum_{j=1}^{N}\sum_{k=1}^{d}\mathbf{v_{j,k}}\int_{\mathbb{T}^{d}}\partial_{k}u_{j}\overline{v_{j}}\,d\mathbf{x}$
$\displaystyle=\sum_{j=1}^{N}\sum_{k=1}^{d}\mathbf{v_{j,k}}\int_{\mathbb{T}^{d}}u_{j}\partial_{k}\overline{v_{j}}\,d\mathbf{x}=\sum_{j=1}^{N}\int_{\mathbb{T}^{d}}u_{j}\overline{\mathbf{v_{j}}\cdot\nabla
v_{j}}\,d\mathbf{x}=\langle u,\,A_{q}v\rangle.$

This shows (11), i.e. $(T_{p}^{*}(t))_{t\geq 0}$ is a semigroup with a
generator that extends
$(\mathcal{A}_{q},\,C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})).$
On the other hand, $(T_{q}(-t))_{t\geq 0}$ is the unique semigroup on
$L^{q}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ with a generator that extends
$(\mathcal{A}_{q},\,C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$ by
Remark 2.8 and Theorem 2.7, which yields

$(T_{p}(t)^{*})_{t\geq 0}=(T_{q}(-t))_{t\geq 0}.$

In particular, the generators of both semigroups coincide. ∎

The dual characterization
$L^{1}(\mathbb{T}^{d},\,\mathbb{C}^{N})^{*}=L^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$
via the dual exponent also holds for $p=1,$ see [6, IV.8, Theorem 5, p. 289].
The difference to the cases $1<p<\infty$ is that the adjoint semigroup fails
to be strongly continuous, hence we have not considered the generators on
$L^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ yet. We restrict ourselves to
the case $d=1$ with $v_{1},\cdots,v_{N}\neq 0$ and make up this technical
leeway with the definitions

$-A_{\infty}u\coloneqq-\begin{pmatrix}v_{1}u_{1}^{\prime}\\\ \vdots\\\
v_{N}u_{N}^{\prime}\end{pmatrix}$ (12)

and

$D(-A_{\infty})\coloneqq\big{\\{}u\in
L^{\infty}(\mathbb{T},\,\mathbb{C}^{N})\,\colon\,u\text{ is weakly
differentiable, }-A_{\infty}u\in
L^{\infty}(\mathbb{T},\,\mathbb{C}^{N})\big{\\}}.$ (13)

We obtain the following result concerning the dual of $(-A_{1},\,D(-A_{1})).$

######  Proposition 2.10.

Let $p=1$ and let $d=1$. Assume that the transport directions
$v_{1},\cdots,v_{N}$ are non-vanishing, i.e. $v_{1},\cdots,v_{N}\neq 0$. Then
it holds

$\displaystyle D(-A_{\infty})$
$\displaystyle=W^{1,\infty}(\mathbb{T},\,\mathbb{C}^{N}),$
$\displaystyle(-A_{1}^{*},\,D(-A_{1}^{*}))$
$\displaystyle=(A_{\infty},W^{1,\infty}(\mathbb{T},\,\mathbb{C}^{N})).$

###### Proof.

The characterization of the domain follows directly from

$u_{j}^{\prime}=-\frac{1}{v_{j}}(-A_{\infty}u)_{j}$

for all $j=1,\cdots,N$, so we are left to compute the adjoint $-A_{1}^{*}.$
Let $\widetilde{u}\in D(-A_{1}^{*}).$ Then there exists $w\in
L^{\infty}(\mathbb{T},\,\mathbb{C}^{N})$ with

$\langle-A_{1}u,\,\widetilde{u}\rangle=\langle u,\,w\rangle$

for all $u\in D(-A_{1}).$ In particular, given $j=1,\cdots,N,$ we can choose
$\varphi\in\mathbb{C}_{c}^{\infty}(\mathbb{T})$ and

$u_{j}\coloneqq\frac{1}{v_{j}}\varphi,\qquad u_{i}\coloneqq 0\quad\text{ for
}i\neq j$

as the transport directions are non-vanishing. Testing with $u$ implies

$-\int_{\mathbb{T}}\varphi^{\prime}\overline{\widetilde{u}_{j}}\,dx=\frac{1}{v_{j}}\int_{\mathbb{T}}\varphi\overline{w_{j}}\,dx$
(14)

for all test functions $\varphi\in C_{c}^{\infty}(\mathbb{T}).$ We obtain that
$\widetilde{u}$ is weakly differentiable with $\widetilde{u}\in
W^{1,\infty}(\mathbb{T},\,\mathbb{C}^{N})$. Now, given $\widetilde{u}\in
W^{1,\infty}(\mathbb{T},\,\mathbb{C}^{N})$ and $u\in
C_{c}^{\infty}(\mathbb{T},\,\mathbb{C}^{N}),$ we can compute

$\displaystyle\langle-A_{1}u,\,\widetilde{u}\rangle$
$\displaystyle=-\sum_{j=1}^{N}v_{j}\int_{\mathbb{T}}u_{j}^{\prime}\overline{\widetilde{u}_{j}}\,dx=\sum_{j=1}^{N}v_{j}\int_{\mathbb{T}}u_{j}\overline{\widetilde{u}_{j}^{\prime}}\,dx=\langle
u,\,A_{\infty}\widetilde{u}\rangle$

This shows $-A_{1}^{*}\widetilde{u}=A_{\infty}\widetilde{u}$ because
$C_{c}^{\infty}(\mathbb{T},\,\mathbb{C}^{N})$ is a core for
$(-A_{1},\,D(-A_{1}))$ by Theorem 2.7. ∎

In dimension $d=1,$ we can even show
$D(-A)=W^{1,p}(\mathbb{T},\,\mathbb{C}^{N})$ if we assume that the transport
directions are non-vanishing, i.e. $v_{j}\neq 0$ for all $j=1,\cdots,N.$ It
should be noted that the assumption cannot be dropped. By definition (6) of
the domain of the generator, there are no restrictions on $u_{j}$ if $v_{j}=0$
and the component could be an arbitrary function in $L^{p}(\mathbb{T}).$

######  Corollary 2.11.

Let $1\leq p<\infty$ and let $d=1.$ Assume that
$v_{1},\cdots,v_{N}\in\mathbb{R}$ are non-vanishing, i.e.
$v_{1},\cdots,v_{N}\neq 0.$ Then

$(-A,\,D(-A))=(-\mathcal{A},\,W^{1,p}(\mathbb{T},\,\mathbb{C}^{N})).$

###### Proof.

One direction follows from Lemma 2.5. Concerning the other direction, let
$u\in D(-A)$. By Theorem 2.7, we can choose a sequence $(u^{(n)})_{n}\subset
C_{c}^{\infty}(\mathbb{T},\,\mathbb{C}^{N})$ such that $u^{(n)}\rightarrow u$
and $\mathcal{A}u^{(n)}=Au^{(n)}\rightarrow Au$ in
$L^{p}(\mathbb{T},\,\mathbb{C}^{N}).$ Since we assumed non-vanishing transport
directions, it follows that

$\|{u_{j}^{\prime}}^{(n)}-{u_{j}^{\prime}}^{(m)}\|_{L^{p}(\mathbb{T})}=\frac{1}{|v_{j}|}\|(\mathcal{A}u^{(n)})_{j}-(\mathcal{A}u^{(m)})_{j}\|_{L^{p}(\mathbb{T})}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
0\quad\text{as }n,\,m\rightarrow\infty$

holds for all $j=1,\cdots,N$. Hence, the sequence $(u^{(n)})_{n}$ is Cauchy in
$W^{1,p}(\mathbb{T},\,\mathbb{C}^{N}).$ We obtain $u\in
W^{1,p}(\mathbb{T},\,\mathbb{C}^{N})$ and $Au=\mathcal{A}u$ by Lemma 2.5.
∎

For higher dimensions, there are in some sense too many directions. On an
intuitive level, a function $u\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ such
that the components $u_{j}$ “behave nicely” along the directions
$\mathbf{v_{j}}$ has enough regularity to be an element of $D(-A).$ In
general, this should be insufficient for $u$ to be weakly differentiable for
dimensions $d\geq 2$ and one would expect
$W^{1,p}(\mathbb{T}^{d},\,\mathbb{C}^{N})\subsetneq D(-A)$. However, we can
only prove this under the assumption that one component is transport periodic.
Let us clarify what we mean by that.

######  Definition 2.12.

Let $\mathbf{v_{1}},\cdots,\mathbf{v_{N}}\in\mathbb{R}^{d}$ be non-vanishing,
i.e. $\mathbf{v_{1},}\cdots,\mathbf{v_{N}}\neq 0.$ Moreover, let
$j=1,\cdots,N$.

  1. (i)

Component $j$ is transport periodic if there exists $t^{*}>0$ such that
$t^{*}\mathbf{v_{j}}\in\mathbb{Z}^{d}$. In this case,

$\tau_{j}\coloneqq\min\\{t>0\,\colon\,t\mathbf{v_{j}}\in\mathbb{Z}^{d}\\}.$

is called the period of the $j$-th component.

  2. (ii)

The model is jointly periodic if there exists $t^{*}>0$ such that
$t^{*}\mathbf{v_{j}}\in\mathbb{Z}^{d}$ for all $j=1,\cdots,N$. In this case,

$\tau\coloneqq\min\\{t>0\,\colon\,t\mathbf{v_{j}}\in\mathbb{Z}^{d}\text{ for
all }j=1,\cdots,N\\}.$

is called the period of the model.

######  Remark 2.13.

The definition of jointly periodic coincides with the transport semigroup
$(T(t))_{t\geq 0}$ defined in (5) being periodic in the canonical sense of [8,
IV. Definition 2.23, p. 266]. We need a concept to distinguish between the
different components in order to give a relatively weak condition for
$W^{1,p}(\mathbb{T}^{d},\,\mathbb{C}^{N})=D(-A)$ to fail in higher dimensions.

######  Remark 2.14.

In dimension $d=1,$ every component is transport periodic with
$\tau_{j}=1/|v_{j}|,$ assuming that $v_{j}\neq 0.$ This is not the case in
higher dimensions.

Let us characterize transport periodicity algebraically. To this end, we use
the notation $\mathbf{v_{j,k}}$ for the $k$-th component of $\mathbf{v_{j}}.$

######  Lemma 2.15.

Let $\mathbf{v_{1}},\cdots,\mathbf{v_{N}}\in\mathbb{R}^{d}$ be non-vanishing,
i.e. $\mathbf{v_{1},}\cdots,\mathbf{v_{N}}\neq 0.$ Moreover, let
$j=1,\cdots,N.$ The following assertions are equivalent:

  1. (i)

Component $j$ is transport periodic.

  2. (ii)

$\mathbf{v_{j,k}}\in\mathbf{v_{j,k^{*}}}\mathbb{Q}$ for all $k=1,\cdots,d$ and
one (and hence all) $k^{*}$ with $\mathbf{v_{j,k^{*}}}\neq 0.$

  3. (iii)

$\displaystyle\bigcap^{d}_{\begin{subarray}{c}k=1\\\ \mathbf{v_{j,k}}\neq
0\end{subarray}}\mathbf{v_{j,k}}\mathbb{Z}\neq\\{0\\}.$

###### Proof.

The details are given in Appendix A. ∎

Joint periodicity can also be characterized in an algebraic way.

######  Lemma 2.16.

Let $\mathbf{v_{1}},\cdots,\mathbf{v_{N}}\in\mathbb{R}^{d}$ be non-vanishing,
i.e. $\mathbf{v_{1},}\cdots,\mathbf{v_{N}}\neq 0.$ The following assertions
are equivalent:

  1. (i)

The model is jointly periodic.

  2. (ii)

$\mathbf{v_{j,k}}\in\mathbf{v_{j^{*},k^{*}}}\mathbb{Q}$ for all
$j=1,\cdots,N,$ all $k=1,\cdots,d$ and one (and hence all) $j^{*}$ and $k^{*}$
with $\mathbf{v_{j^{*},k^{*}}}\neq 0.$

  3. (iii)

$\displaystyle\bigcap_{j=1}^{N}\displaystyle\bigcap^{d}_{\begin{subarray}{c}k=1\\\
\mathbf{v_{j,k}}\neq 0\end{subarray}}\mathbf{v_{j,k}}\mathbb{Z}\neq\\{0\\}.$

###### Proof.

Notice that the model is jointly periodic if and only if there exists
$t^{*}>0$ such that

$t^{*}\begin{pmatrix}\mathbf{v_{1}}\\\ \vdots\\\
\mathbf{v_{N}}\end{pmatrix}\in\mathbb{Z}^{Nd}.$

The result follows from an application of Lemma 2.15 to the concatenated
vector. ∎

######  Remark 2.17.

For $d\geq 2,$ the above results show that there are very few transport
directions such that one component is transport periodic or the model is
jointly periodic in the sense that these directions are a nullset in
$\mathbb{R}^{d}$ and $\mathbb{R}^{Nd}$ respectively. However, these are the
natural cases a human would choose as an example when trying to show results
for $d\geq 2.$ Even for these ”simple” periodic cases, many properties which
hold true for $d=1$ fail in higher dimensions.

The first negative result for higher dimensions is the next theorem.

######  Theorem 2.18.

Let $d\geq 2$ and let $\mathbf{v_{1}},\cdots,\mathbf{v_{N}}$ be non-vanishing,
i.e. $\mathbf{v_{1}},\cdots,\mathbf{v_{N}}\neq 0$. Assume that there is one
transport periodic component. Then

$(-\mathcal{A},\,W^{1,p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))\subsetneq(-A,\,D(-A)).$

###### Proof.

Let component $j$ be transport periodic with period $\tau_{j}$. We show
existence of a function $u\in D(-A)$ with $u\notin
W^{1,p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ for all $1\leq p<\infty$. The idea
is to take a non-differentiable function $u\in
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ such that $u_{j}$ is constant along
direction $\mathbf{v_{j}}.$
Let $\mathbf{c}=(\frac{1}{2},\cdots,\frac{1}{2})\in\mathbb{T}^{d}$ be the
center of the torus. By assumption, the curve

$\gamma\colon[0,\tau_{j}]\rightarrow\mathbb{T}^{d}\quad\text{with}\quad\gamma(t)\coloneqq\mathbf{c}+t\mathbf{v_{j}}$

fulfills $\gamma(0)=\gamma(\tau_{j}).$ Let $K=\operatorname{rg}(\gamma)$ be
the range of $\gamma$. Then there exists $\delta>0$ with

$U_{\delta}=\\{\mathbf{x}\in\mathbb{T}^{d}\,\colon\,\operatorname{dist}(\mathbf{x},\,K)<\delta\\}\subsetneq\mathbb{T}^{d}.$
(15)

We define $u_{j}\coloneqq\mathbbm{1}_{\text{$U_{\delta}$}}$ and $u_{i}\equiv
0$ for all $i=1,\cdots,N$ with $i\neq j.$ Notice that $u\in
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ and that $u_{j}$ is constant along
direction $\mathbf{v_{j}},$ i.e.

$u_{j}(\cdot-t\mathbf{v_{j}})-u_{j}(\cdot)\equiv 0.$

Hence, $u\in D(-A)$ with $-Au=0$ and it is left to show that $u_{j}\notin
W^{1,p}(\mathbb{T}^{d})$ for all $1\leq p<\infty.$ Assume that for all
$k=1,\cdots,d$ there exist $g_{k}\in
L^{1}_{\operatorname{loc}}(\mathbb{T}^{d})$ such that

$-\int_{\mathbb{T}^{d}}g_{k}\varphi\,d\mathbf{x}=\int_{\mathbb{T}^{d}}u_{j}\partial_{k}\varphi\,d\mathbf{x}=\int_{U_{\delta}}\partial_{k}\varphi\,d\mathbf{x}=\int_{\partial
U_{\delta}}\varphi\mathbf{n_{k}}\,d\mathcal{H}^{d-1}(\mathbf{x})$

holds for all $\varphi\in C_{c}^{\infty}(\mathbb{T}^{d})$ with outer normal
$\mathbf{n}$ of $U_{\delta}.$ Notice that the last step follows from Gauss’s
theorem. Testing with $\varphi\in C_{c}^{\infty}(U_{\delta})$ and $\varphi\in
C_{c}^{\infty}(\overline{U_{\delta}}^{c})$ respectively implies $g_{k}=0$
almost everywhere on $(\partial U_{\delta})^{c}$ by the fundamental lemma of
calculus of variations. The boundary $\partial U_{\delta}$ is a Lebesgue-
nullset and hence, $g_{k}=0$ holds almost everywhere in $\mathbb{T}^{d}$ for
all $k=1,\cdots,d.$ Consequently, $\nabla u_{j}=0$ and $u_{j}$ is constant by
the connectedness of the torus. This contradicts the definition of $u_{j}$. ∎

Figure 1: An illustration of the idea behind the function constructed in (15)
for $d=2$ in the exemplary case $\mathbf{v_{j}}=(\tfrac{1}{4},\tfrac{1}{8}).$
The whole area is the torus $\mathbb{T}^{2}$ and the green area is the support
of $u_{j}$ for $\delta=\tfrac{1}{4\sqrt{5}}$.

After achieving a better understanding of the domain of the generator, it
would be reasonable to turn our attention to the question of well-posedness
for two abstract Cauchy problems. Let $T>0$ and
$I=[0,T)\subseteq\mathbb{R}_{t\geq 0}$ be an interval. The linear hyperbolic
Cauchy problem on $I$ reads

$\begin{cases}\begin{array}[]{rrll}\dot{u}(t)+Au(t)&=&F(t)&\qquad 0<t<T,\\\
u(0)&=&u_{0}\end{array}\end{cases}$ (16)

for $u_{0}\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}),$ an interval and $F\in
C(I,\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$. The semilinear hyperbolic
Cauchy problem on $I$ is given by

$\begin{cases}\begin{array}[]{rrll}\dot{u}(t)+Au(t)&=&F(t,u(t))&\qquad
0<t<T,\\\ u(0)&=&u_{0}\end{array}\end{cases}$ (17)

for $u_{0}\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ and $F\in C(I\times
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}),\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})).$

We decided to postpone the overview on well-posedness results for these
equations to Chapter 3 because the transport semigroup $(T(t))_{t\geq 0}$ with
generator $(-A,\,D(-A))$ is a special case of the transport-reaction semigroup
studied in the next chapter.

##  Chapter 3 The Transport-Reaction Semigroup

As in the previous chapter, we assume that the dimension $d\in\mathbb{N}$, the
number of components $N\in\mathbb{N}$, the transport directions
$\mathbf{v_{1}},\cdots,\mathbf{v_{N}}\in\mathbb{R}^{d}$ and $1\leq p<\infty$
are arbitrarily chosen fixed parameters, if not stated otherwise. We work on
the Banach space $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ with norm
$\|\cdot\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}$, which is defined in (4).

Eventually, our goal is to study the stability and long-time behavior of
solutions to the transport-reaction model (2). Given an equilibrium state
$u=c=\operatorname{const}\in\mathbb{R}^{N}$ with $F(c)=0,$ we follow the
standard approach by defining $u=c+w$ and linearizing the system for the
perturbation $w.$ For a solution of the form $u=c+w$, one has

$\partial_{t}\begin{pmatrix}w_{1}\\\ \vdots\\\
w_{N}\end{pmatrix}+\begin{pmatrix}\mathbf{v_{1}}\cdot\nabla w_{1}\\\ \vdots\\\
\mathbf{v_{N}}\cdot\nabla w_{N}\end{pmatrix}=F(u)=DF(c)w+\mathcal{O}(|w|^{2})$

where $DF(c)\in\mathbb{R}^{N\times N}$ is the Jacobian of $F$ evaluated at
$c.$ The (relabeled) linearized system then reads

$\partial_{t}\begin{pmatrix}u_{1}\\\ \vdots\\\
u_{N}\end{pmatrix}+\begin{pmatrix}\mathbf{v_{1}}\cdot\nabla u_{1}\\\ \vdots\\\
\mathbf{v_{N}}\cdot\nabla u_{N}\end{pmatrix}=DF(c)u.$

For this reason, we will study the equation

$\partial_{t}u+\begin{pmatrix}\mathbf{v_{1}}\cdot\nabla u_{1}\\\ \vdots\\\
\mathbf{v_{N}}\cdot\nabla u_{N}\end{pmatrix}=Bu$ (18)

for an a priori arbitrary matrix $B\in\mathbb{R}^{N\times N}.$ It should be
noted that $B$ is a real matrix and that it induces a bounded linear operator
on $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ by pointwise matrix
multiplication. We will use the letter $B$ for the matrix and the operator.
From a semigroup point of view, there are two different perspectives on (18).
One possibility is to consider the abstract Cauchy problem

$\dot{u}(t)+Au(t)=Bu(t),$ (19)

and to solve it via analyzing the generator $(-A,\,D(-A))$ of the transport
semigroup. This approach can be carried out with the knowledge from Chapter 2
and Appendix B. The other possibility is to consider the abstract Cauchy
problem

$\dot{u}(t)+(A-B)u(t)=0$ (20)

and to study the semigroup generated by $-A+B.$ This section addresses the
second approach and we study the reaction and transport-reaction semigroups
generated by $B$ and $-A+B$ respectively. The main advantage is that standard
perturbation theory gives a detailed description of the perturbed semigroup,
which allows us to prove qualitative properties of the solutions to (18).
An overview of the results from perturbation theory for semigroups we apply,
including references and adaptations to our simple case, can be found in
Appendix C.

As a first step, we take a closer look to the semigroup generated by
$(B,\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$ and characterize positivity.
The main idea for this characterization was found on [5].

######  Definition 3.1.

Let $1\leq p<\infty.$ A function $u\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$
is called non-negative if

$u_{j}(\mathbf{x})\geq 0\text{ for almost all
}\mathbf{x}\in\mathbb{T}^{d}\text{ and all }j=1,\cdots,N.$

In this case, we write $u\geq 0$ and we define
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})_{+}\coloneqq\\{u\in
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})\,\colon\,u\geq 0\\}.$

######  Remark 3.2.

The set $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})_{+}$ is a closed subset of
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ since convergence in
$\|\cdot\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}$ implies pointwise
convergence almost everywhere along a subsequence.

######  Definition 3.3.

A semigroup $(T(t))_{t\geq 0}$ on $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ is
called positive if each operator $T(t)$ is positive, i.e. if

$u\geq 0\implies T(t)u\geq 0$

for all $t\geq 0.$ Similarly, a group $(T(t))_{t\in\mathbb{R}}$ is called
positive if $T(t)$ is positive for all $t\in\mathbb{R}.$

######  Remark 3.4.

These defintions make $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ a Banach
lattice and are just special cases of already existing general theory, cf.
[33]. Postive semigroups arise naturally in biology and physics when $u$
describes non-negative quantities, like for example concentrations. Moreover,
there are strong results concerning specral theory and long-time asymptotics
of positive semigroups, which is why we are interested in a characterization
of all matrices $B$ such that the semigroup generated by $-A+B$ becomes
positive. The fundamentals in a concrete context can be found in [8] and the
detailed theory is presented in [1].

######  Lemma 3.5.

The operator $(B,\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$ generates the
uniformly continuous semigroup $(S(t))_{t\geq 0}=(e^{tB})_{t\geq 0}$ on
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}),$ where $e^{tB}$ is the operator on
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ given by pointwise matrix
multiplication with the matrix $e^{tB}.$ The semigroup is positive if and only
if the off-diagonal entries of $B$ are non-negative.

###### Proof.

Notice that $B$ defines a bounded linear operator on
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ due to the estimate

$\|Bu\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}\lesssim\|B\|_{\infty}\|u\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}.$

Hence, $(B,\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$ generates the uniformly
continuous semigroup $(S(t))_{t\geq 0}=(e^{tB})_{t\geq 0}$ by [30, Chapter 1,
Theorem 1.2, p. 2].
Let us show that this semigroup is positive if the off-diagonal entries of $B$
are non-negative. To this end, we firstly assume that the off-diagonal entries
are positive, i.e.

$b\coloneqq\min_{i\neq j}B_{ij}>0.$ (21)

Let $u\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})_{+}$ and let
$\mathbf{x}\in\mathbb{T}^{d}$ such that $u(\mathbf{x})\geq 0$. By definition,
we have

$(S(t)u)(\mathbf{x})=e^{tB}u(\mathbf{x})=(I_{\mathbb{C}^{N\times
N}}+tB)u(\mathbf{x})+R_{t}u(\mathbf{x})$

for a matrix $R_{t}$ with $\|R_{t}\|_{\infty}=\mathcal{O}(t^{2})$ as
$t\rightarrow 0.$ This implies existence of $t_{1}=t_{1}(B)>0$ and $C=C(B)>0$
such that

$\left(e^{tB}u(\mathbf{x})\right)_{j}\geq\big{(}(I_{\mathbb{C}^{N\times
N}}+tB)u(\mathbf{x})\big{)}_{j}-\|R_{t}\|_{\infty}\|u(\mathbf{x})\|_{\infty}\geq\big{(}(I_{\mathbb{C}^{N\times
N}}+tB)u(\mathbf{x})\big{)}_{j}-Ct^{2}\|u(\mathbf{x})\|_{\infty}$

for all $j=1,\cdots,N$ and $0\leq t\leq t_{1}.$ By our assumption (21), we
also have existence of $t_{2}=t_{2}(B)>0$ with

$\big{(}I_{\mathbb{C}^{N\times N}}+tB\big{)}_{ij}\geq tb>0$

for all $i,j=1,\cdots,N$ and all $0\leq t\leq t_{2}.$ Notice that the identity
matrix dominates the diagonal for small $t.$ Both estimates together imply

$\left(e^{tB}u(\mathbf{x})\right)_{j}\geq
tb\sum_{i=1}^{N}u_{i}(\mathbf{x})-Ct^{2}\|u(\mathbf{x})\|_{\infty}\geq
t(b-Ct)\|u(\mathbf{x})\|_{\infty}\geq 0$

for all $j=1,\cdots,N$ and $t\leq
t^{*}=t^{*}(B)\coloneqq\min\\{t_{1},\,t_{2},\,b/C\\}.$ This implies that the
operators $S(t)$ are positive for $0\leq t\leq t^{*}.$
For $t\in\mathbb{R}_{\geq 0},$ choose $m\in\mathbb{N}_{0}$ and $r\in[0,t^{*})$
such that $t=mt^{*}+r.$ Then

$S(t)=S(r)S(t^{*})^{m}$

is a positive operator, since compositions preserve positivity.
Secondly, let the off-diagonal entries of $B$ be non-negative. In this case,
we take a sequence $(B^{(n)})_{n\in\mathbb{N}}$ of matrices with positive off-
diagonal entries such that $B^{(n)}\rightarrow B$ in $\|\cdot\|_{\infty}.$ Let
$u\geq 0,\,\mathbf{x}\in\mathbb{T}^{d}$ with $u(\mathbf{x})\geq 0$ and $t\geq
0.$ Our first step implies

$(S(t)u)(\mathbf{x})=e^{tB^{(n)}}u(\mathbf{x})+(e^{tB}-e^{tB^{(n)}})u(\mathbf{x})\geq-\|e^{tB}-e^{tB^{(n)}}\|_{\infty}\|u(\mathbf{x})\|_{\infty}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
0$

as $n\rightarrow\infty,$ i.e. $(S(t))_{t\geq 0}$ defines a positive semigroup.
Conversely, if there exists one negative off-diagonal entry $B_{ij}<0$, we
define the non-negative function $u\equiv e_{j}\in
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ for the standard unit vector
$e_{j}\in\mathbb{C}^{N}.$ Similar to before, we obtain the estimate

$(S(t)u)_{i}(\mathbf{x})\equiv\left(e^{tB}e_{j}\right)_{i}\leq\big{(}(I_{\mathbb{C}^{N\times
N}}+tB)e_{j}\big{)}_{i}+Ct^{2}=t(B_{ij}+Ct)<0$

for some constant $C=C(B)>0$ and small $t.$ The last inequality is a
consequence of the negativity of $B_{ij}$. In particular, $S(t)$ is not a
positive operator for small enough $t>0$ and the semigroup $(S(t))_{t\geq 0}$
cannot be positive. ∎

Given this knowledge, we can use perturbation theory to show that
$(-A+B,\,D(-A))$ generates a strongly continuous semigroup, which we call the
transport-reaction semigroup $(R(t))_{t\geq 0}.$

######  Theorem 3.6.

The operator $(-A+B,\,D(-A))$ generates a $C_{0}$-semigroup $(R(t))_{t\geq 0}$
on $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$. Moreover, the following
properties hold:

  1. (i)

$\|R(t)\|_{\mathcal{L}(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))}\leq
e^{\|B\|_{\mathcal{L}(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))}t}$

  2. (ii)

$R(t)u=T(t)u+\int_{0}^{t}T(t-s)BR(s)u\,ds$

  3. (iii)

$R(t)u=T(t)u+\int_{0}^{t}R(t-s)BT(s)\,ds$

  4. (iv)

$R(t)=\sum_{k=0}^{\infty}R_{k}(t)$

with

$R_{0}(t)=T(t),\quad R_{k+1}(t)u=\int_{0}^{t}T(t-s)BR_{k}(s)u\,ds$

  5. (v)

$R(t)u=\lim_{n\rightarrow\infty}\left(T(\tfrac{t}{n})e^{t/n\,B}\right)^{n}u$

for all $t\geq 0$ and $u\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$ Moreover,
the graph norms of $-A$ and $-A+B$ on $D(-A)$ are equivalent, the series in
$(iv)$ converges in $\mathcal{L}(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$
uniformly on compact subsets of $\mathbb{R}_{\geq 0}$ and for every fixed
$u\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}),$ the convergence in $(v)$ is
uniform on compact subsets of $\mathbb{R}_{\geq 0}.$
In all of the above statements, $(T(t))_{t\geq 0}$ is the transport semigroup
on $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ defined in (5) and with generator
$(-A,\,D(-A))$.

###### Proof.

Let us recall that $(-A,\,D(-A))$ generates the transport semigroup
$(T(t))_{t\geq 0}$ from Chapter 2 and that
$(B,\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$ generates the uniformly
continuous semigroup $(e^{tB})_{t\geq 0},$ see Lemma 3.5.
The fact that $(-A+B,\,D(-A))$ generates a $C_{0}$-semigroup and properties
$(i)-(iii)$ directly follow from Theorem C.1. Property $(iv)$ is a consequence
of Theorem C.2. Also the addenda, except for the uniform convergence of $(v)$,
are consequences of these two theorems.
Furthermore, property $(v)$ and its corresponding addendum can be shown by
applying Theorem C.3. Notice that $(T(t))_{t\geq 0}$ is a contraction
semigroup, so we have the stability bound

$\displaystyle\|(T(\tfrac{t}{n})e^{t/n\,B})^{n}\|_{\mathcal{L}(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))}$
$\displaystyle\leq\left(\|T(\tfrac{t}{n})\|_{\mathcal{L}(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))}\|e^{t/n\,B}\|_{\mathcal{L}(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))}\right)^{n}$
$\displaystyle\leq(e^{t/n\,\|B\|_{\mathcal{L}(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))}})^{n}=e^{t\|B\|_{\mathcal{L}(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))}}.$

Also, we already know that $(-A+B,\,D(-A))$ generates a $C_{0}$-semigroup from
the previous steps. Hence, we can find some
$\lambda>\|B\|_{\mathcal{L}(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))}$ such
that $(\lambda
I_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}-(-A+B))D(-A)=L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$
by Hille-Yosida, see Theorem B.6. Another consequence of Hille-Yosida is that
the closure of $(-A+B,\,D(-A))$ is again $(-A+B,\,D(-A)).$ Therefore, in view
of Theorem C.3, the formula on the right-hand side of $(v)$ is indeed a
formula for the transport-reaction semigroup. ∎

######  Corollary 3.7.

$C_{c}^{\infty}(\mathbb{T}^{d},\mathbb{C}^{N})$ is a core for
$(-A+B,\,D(-A)).$ In particular, $(R(t))_{t\geq 0}$ is the only strongly
continuous semigroup on $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ with a
generator that extends
$(-A+B,\,C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})).$

###### Proof.

We know from Theorem 2.7 that
$C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ is a core for
$(-A,\,D(-A)).$ Therefore, $C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$
is also a core for $(-A+B,\,D(-A))$ because the graph norms of both operators
are equivalent by Theorem 3.6. The uniqueness property is a general result for
cores, cf. Theorem 2.7. ∎

For completeness, we give a brief intermezzo on the well-posedness of two
abstract Cauchy problems. Let $T>0$. The linear hyperbolic Cauchy problem on
the interval $I=[0,T)$ reads

$\begin{cases}\begin{array}[]{rrll}\dot{u}(t)+(A-B)u(t)&=&F(t)&\qquad
0<t<T,\\\ u(0)&=&u_{0}\end{array}\end{cases}$ (22)

for $u_{0}\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ and
$F\colon(0,T)\rightarrow L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$. The
semilinear hyperbolic Cauchy problem on $I$ is given by

$\begin{cases}\begin{array}[]{rrll}\dot{u}(t)+(A-B)u(t)&=&F(t,u(t))&\qquad
0<t<T,\\\ u(0)&=&u_{0}\end{array}\end{cases}$ (23)

for $u_{0}\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ and $F\colon(0,T)\times
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})\rightarrow
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$
The standard solution concepts for equations of type (22) and (23) are
classical and mild solutions, as well as maximal existence times
$T_{\text{max}}(u_{0})$. A detailed overview of all concepts we use, including
precise definitions and references, can be found in Appendix B and we strongly
recommend reading Appendix B for further details on the topic.
Without any a priori bounds of the solution in concrete examples, we can only
apply the standard existence and uniqueness results from Appendix B.

######  Theorem 3.8 (cf. Theorem B.14).

The initial value problem (22) has a unique classical solution $u\in
C(I,\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))\cap
C^{1}((0,T),\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$ on $I$ for every
$u_{0}\in D(-A),$ if $F$ satisfies one of the following two properties:

  1. (i)

$F\in C^{1}([0,T],\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})),$

  2. (ii)

$F\in C((0,T),\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))\cap
L^{1}((0,T),\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))\quad\text{ and }\quad
F(s)\in D(-A)\text{ for all }s\in(0,T)\text{ with }AF(\cdot)\in
L^{1}((0,T),\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})).$

######  Remark 3.9.

The homogeneous equation with $F\equiv 0$ has the global classical solution
$u(\cdot)=R(\cdot)u_{0}\in
C^{1}([0,\infty),\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$ for $u_{0}\in
D(-A),$ see Theorem B.10.

In the semilinear case (23), the existence theorem for mild solutions reads as
follows.

######  Theorem 3.10 (cf. Theorem B.17).

Let $F\colon[0,\infty)\times L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})\rightarrow
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ be continuous in $t$ and locally
Lipschitz continuous in $u$, uniformly in $t$ on bounded intervals. Then for
every $u_{0}\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}),$ there is a time
$T_{\text{max}}=T_{\text{max}}(\|u_{0}\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})})>0,$
possibly $T_{\text{max}}=T,$ such that the initial value problem (23) has a
unique mild solution $u$ on $[0,T_{\text{max}}).$ Moreover, if
$T_{\text{max}}<T,$ then

$\lim_{t\nearrow
T_{\text{max}}}\|u(t)\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}=\infty.$

######  Remark 3.11.

If for each $t^{\prime}>0,$ there exists a constant $C(t^{\prime})>0$ such
that

$\|F(t,u)\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}\leq
C(1+\|u\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})})$

holds for all $t\in[0,t^{\prime}]$ and all $u\in
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$, then (23) has a mild solution on
$[0,\infty),$ i.e. one can choose $T=\infty$ and it holds
$T_{\text{max}}=\infty,$ see B.18.

Under stronger regularity assumptions on the right-hand side and the initial
state, one obtains existence of strong solutions.

######  Theorem 3.12 (cf. Theorem B.19).

Let $F\in C([0,\infty)\times L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}),\,D(-A))$
be uniformly (in t) locally Lipschitz continuous in $u$. Then for every
$u_{0}\in D(-A),$ the initial value problem (23) possesses a unique maximal
strong solution

$u\in C([0,T_{\text{max}}),\,D(-A))\cap
C^{1}((0,T_{\text{max}}),\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})).$

Moreover, the solution depends continuously on the initial data in the sense
of the next theorem.

######  Corollary 3.13 (cf. Corollary B.20).

Let $F\in C([0,\infty)\times
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}),\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$
be globally Lipschitz continuous in $u.$ Then the mild solution depends
Lipschitz continuously on $u_{0}\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}),$
i.e. for all $T>0,$ the mapping $u_{0}\mapstochar\rightarrow u$ is Lipschitz
continuous from $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ into
$C([0,T],\,L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$.

After this summary of well-posedness results for the general transport
reaction equation (2), we again focus on the linear case (18).

Similar to the transport semigroup $(T(t))_{t\geq 0},$ we can extend the
semigroup $(R(t))_{t\geq 0}$ to a strongly continuous group
$(R(t))_{t\in\mathbb{R}}$ and show an analogous result to (10). However, we
have to be a little bit more careful as in Remark 2.8 because the semigroup is
less explicit.

######  Lemma 3.14.

The semigroup $(R(t))_{t\geq 0}$ can be (uniquely) extended to the
$C_{0}$-group $(R(t))_{t\in\mathbb{R}}$ where $(R(-t))_{t\geq 0}$ is the
semigroup $(R_{-}(t))_{t\geq 0}$ generated by $(A-B,\,D(A))$ with
$D(A)=D(-A).$ Moreover, we have

$D(-A)=\left\\{u\in
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})\,\colon\,\lim_{t\rightarrow
0}\frac{R(t)u-u}{t}\textup{ exists in
}L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})\right\\}.$ (24)

###### Proof.

Both $(-A+B,\,D(-A))$ and $(A-B,\,D(A))$ generate the $C_{0}$-semigroups
$(R(t))_{t\geq 0}$ and $(R_{-}(t))_{t\geq 0}$ respectively by Theorem 3.6,
since the change of signs simply changes the direction of movement and the
sign of $B,$ i.e. the model is still of the type covered in Theorem 3.6.
Now, [8, II 3.11, p. 79] implies that $(-A+B,\,D(-A))$ generates a strongly
continuous group $(\widetilde{R}(t))_{t\in\mathbb{R}}$ with

$D(-A)=\left\\{u\in
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})\,\colon\,\lim_{t\rightarrow
0}\frac{\widetilde{R}(t)u-u}{t}\text{ exists in
}L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})\right\\}.$ (25)

Notice that the generator theorems in [8] are insufficient to show that
$(\widetilde{R}(t))_{t\in\mathbb{R}}$ is in fact the $C_{0}$-group we would
expect. We only have that $(\widetilde{R}(t))_{t\geq 0}$ is a semigroup with a
generator that extends $(-A+B,\,D(-A))$ and that $(\widetilde{R}(-t))_{t\geq
0}$ is a semigroup whose generator extends $(A-B,\,D(A)).$ But this has the
consequence that the generators of these semigroups extend
$(-A+B,\,C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$ and
$(A-B,\,C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$ respectively.
However, $C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ is a core of
$(-A+B,\,D(-A))$ and $(A-B,\,D(A))$ by Corollary 3.7, so we obtain

$(\widetilde{R}(t))_{t\geq 0}=(R(t))_{t\geq 0}\quad\text{ and
}\quad(\widetilde{R}(-t))_{t\geq 0}=(R_{-}(t))_{t\geq 0}.$

That is, $(\widetilde{R}(t))_{t\in\mathbb{R}}=(R(t))_{t\in\mathbb{R}}$ is an
extension of $(R(t))_{t\geq 0}$. Given this equality, (24) follows from (25),
which completes the proof. ∎

Alternatively, one can prove the above result in a natural way by explicitly
defining the extension $(R(t))_{t\in\mathbb{R}}$ and showing the group
property. In this case, one can use a comparable argument with cores to show
(24). The logic is interesting in itself and we also present this approach.

###### A second proof of Lemma 3.14.

Let $(R_{-}(t))_{t\geq 0}$ be the semigroup generated by $(A-B,\,D(A)).$ We
define

$R(t)=\begin{cases}R(t)&t\geq 0,\\\ R_{-}(-t)&t\leq 0.\end{cases}$

This defines a strongly continuous group by $R(-t)=R(t)^{-1}.$ The latter
follows from

$\frac{d}{dt}R(-t)R(t)u=(A-B)R(-t)R(t)u+R(-t)(-A+B)R(t)u=0$

for all $u\in D(-A)$ and density of $D(-A)$ in
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$. Let
$(\widetilde{A},\,D(\widetilde{A}))$ be the generator of the $C_{0}$-group
$(R(t))_{t\in\mathbb{R}}$ in the sense of [8, II 3.11, p. 79], i.e.

$D(\widetilde{A})=\left\\{u\in
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})\,\colon\,\lim_{t\rightarrow
0}\frac{R(t)u-u}{t}\text{ exists in
}L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})\right\\}$

and

$\widetilde{A}u=\lim_{t\rightarrow 0}\frac{R(t)u-u}{t}$

for $u\in D(\widetilde{A}).$ It is left to show that
$(\widetilde{A},\,D(\widetilde{A}))=(-A+B,\,D(-A))$ holds. The direction
“$\subseteq$” follows directly from the definition of a generator, see
Definition B.3. For the other direction, notice that
$(\widetilde{A},\,D(\widetilde{A}))$ generates a strongly continuous semigroup
$(\widetilde{R}(t))_{t\geq 0}$ by [8, II 3.11, p. 79] with
$C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})\subseteq D(\widetilde{A})$
because $C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ lies in the domains
of the generators of $(R(t))_{t\geq 0}$ and $(R_{-}(t))_{t\geq 0}$ by Lemma
2.5. Hence, $(\widetilde{A},\,D(\widetilde{A}))$ is closed by Theorem B.6 and
extends $(-A+B,\,C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})).$ The set
$C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ is a core of
$(-A+B,\,D(-A))$, see Corollary 3.7, and as a consequence, we have
$(\widetilde{A},\,D(\widetilde{A}))\supseteq(-A+B,\,D(-A))$ by the closedness
of $\widetilde{A}.$ ∎

The next chapter deals with the spectral properties of the generators
$(-A_{p}+B_{p},\,D(-A_{p}))$ and the generated semigroups $(R_{p}(t))_{t\geq
0}$ on $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$, mostly in the case $p=2.$
We also present methods which enable some extensions of the results to $1\leq
p<\infty.$ At this point, it is unclear whether the spectra are independent of
$p$ and a key strategy will be to start by showing equality of the spectra for
all $p\geq 2.$ In a second step, one can extend the results to $1\leq p\leq 2$
by using a duality argument. To this end, it is crucial to study the adjoint
transport-reaction semigroup $(R_{p}(t)^{*})_{t\geq 0}$ on
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})^{*}.$ Recall the classical result
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})^{*}=L^{q}(\mathbb{T}^{d},\,\mathbb{C}^{N})$,
where $1<p<\infty$ and $q$ is the dual exponent of $p,$ i.e.
$\tfrac{1}{p}+\tfrac{1}{q}=1,$ see [6, IV.8, Theorem 1, p. 286]. Moreover, the
adjoint of the bounded operator $R_{p}(t)$ is defined by

$\langle R_{p}(t)u,\,v\rangle=\langle u,\,R_{p}(t)^{*}v\rangle\quad\text{ for
all }u\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})\text{ and all }v\in
L^{q}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$

Similar to Proposition 2.9 and Proposition 2.10, we split our analysis into
the cases $1<p<\infty$ and $p=1.$

######  Proposition 3.15.

Let $1<p<\infty$ and let $(R_{p}(t))_{t\geq 0}$ be the semigroup on
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ generated by
$(-A_{p}+B_{p},\,D(-A_{p})).$ Then $(R_{p}(t)^{*})_{t\geq 0}$ is a strongly
continuous semigroup on $L^{q}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ with
generator $(A_{q}+B_{q}^{T},\,D(A_{q}))$. Moreover, it holds

$((-A_{p}+B_{p})^{*},\,D((-A_{p}+B_{p})^{*}))=(A_{q}+B_{q}^{T},\,D(A_{q}))^{.}$

###### Proof.

The space $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ is reflexive, see [6, IV.8,
Corollary 2, p. 288]. Hence, the adjoint semigroup is again strongly
continuous by [8, I 5.14 Proposition, p. 44] and the sun dual semigroup
coincides with the adjoint semigroup, cf. [8, II 2.6, p. 62]. In particular,
the generator of the adjoint (strongly continuous) semigroup is the adjoint
$(-A_{p}+B_{p})^{*}$ by [8, II 2.6 proposition, p. 63].
It is left to show $(-A_{p}+B_{p})^{*}=A_{q}+B_{q}^{T}$. First of all, it
should be noted that

$\displaystyle\langle B_{p}u,\,v\rangle$
$\displaystyle=\sum_{j=1}^{N}\int_{\mathbb{T}^{d}}(Bu(\mathbf{x}))_{j}\overline{v_{j}(\mathbf{x})}\,d\mathbf{x}=\int_{\mathbb{T}^{d}}\langle
Bu(\mathbf{x}),\,v(\mathbf{x})\rangle_{\mathbb{C}^{N}}\,d\mathbf{x}=\int_{\mathbb{T}}\langle
u(\mathbf{x}),\,B^{T}v(\mathbf{x})\rangle_{\mathbb{C}^{N}}\,d\mathbf{x}$
$\displaystyle=\sum_{j=1}^{N}\int_{\mathbb{T}}u_{j}(\mathbf{x})\overline{(B^{T}v(\mathbf{x}))_{j}}\,d\mathbf{x}=\langle
u,\,B_{q}^{T}v\rangle$

holds for all $u\in L^{p}(\mathbb{T},\,\mathbb{C}^{N})$ and all $v\in
L^{q}(\mathbb{T},\,\mathbb{C}^{N}),$ i.e. $B_{p}^{*}=B_{q}^{T}.$
Secondly, recall that we have already shown
$(-A_{p}^{*},\,D(-A_{p}^{*}))=(A_{q},\,D(A_{q}))$ in Proposition 2.9. Finally,
the boundedness of $B_{p}$ implies

$((-A_{p}+B_{p})^{*},\,D((-A_{p}+B_{p})^{*}))=(-A_{p}^{*}+B_{p}^{*},\,D(-A_{p}^{*}))=(A_{q}+B_{q}^{T},\,D(A_{q})).$

∎

The dual characterization
$L^{1}(\mathbb{T}^{d},\,\mathbb{C}^{N})^{*}=L^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$
via the dual exponent also holds for $p=1,$ see [6, IV.8, Theorem 5, p. 289].
The difference to the cases $1<p<\infty$ is that the adjoint semigroup fails
to be strongly continuous. As in Chapter 2, we restrict ourselves to the case
$d=1$ with $v_{1},\cdots,v_{N}\neq 0$ and make up this technical leeway with
the definitions

$(-A_{\infty}+B_{\infty})u\coloneqq-\begin{pmatrix}v_{1}u_{1}^{{}^{\prime}}\\\
\vdots\\\ v_{N}u_{N}^{{}^{\prime}}\end{pmatrix}+Bu$ (26)

and

$D(-A_{\infty}+B_{\infty})\coloneqq D(-A_{\infty})=\big{\\{}u\in
L^{\infty}(\mathbb{T},\,\mathbb{C}^{N})\,\colon\,u\text{ is weakly
differentiable, }-A_{\infty}u\in
L^{\infty}(\mathbb{T},\,\mathbb{C}^{N})\big{\\}}.$ (27)

We obtain the following result concerning the dual of
$(-A_{1}+B_{1},\,D(-A_{1})).$

######  Proposition 3.16.

Let $p=1$ and let $d=1$. Assume that the transport directions
$v_{1},\cdots,v_{N}$ are non-vanishing, i.e. $v_{1},\cdots,v_{N}\neq 0$. Then
it holds

$\displaystyle D(-A_{\infty}+B_{\infty})$
$\displaystyle=W^{1,\infty}(\mathbb{T},\,\mathbb{C}^{N}),$
$\displaystyle((-A_{1}+B_{1})^{*},\,D((-A_{1}+B_{1})^{*}))$
$\displaystyle=(A_{\infty}+B_{\infty}^{T},\,W^{1,\infty}(\mathbb{T},\,\mathbb{C}^{N})).$

###### Proof.

The characterization of the domain was proven in Proposition 2.10. Moreover,
we have $B_{1}^{*}=B_{\infty}^{T},$ which can be obtained just like in
Proposition 3.15. Proposition 2.10 also yields

$((-A_{1}+B_{1})^{*},\,D((-A_{1}+B_{1})^{*}))=(-A_{1}^{*}+B_{1}^{*},\,D(-A_{1}^{*}))=(A_{\infty}+B_{\infty}^{T},\,W^{1,\infty}(\mathbb{T},\,\mathbb{C}^{N}))$

because $B_{1}$ is bounded. ∎

We will now focus our attention on two important qualitative properties of
transport-reaction semigroups: positivity and mass conservation. It turns out
that both properties can be fully characterized by conditions on the matrix
$B$. These conditions are the same as for the ODE model

$\begin{cases}\begin{array}[]{rrll}\dot{y}(t)&=&By(t)&\qquad t>0,\\\
y(0)&=&y_{0}\end{array}\end{cases}$ (28)

with given initial state $y_{0}\in\mathbb{C}^{N},$ which is very reasonable
because transport should not have any impact on the positivity of the
concentrations or the total mass of the system.

######  Theorem 3.17.

The semigroup $(R(t))_{t\geq 0}$ on $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$
generated by $(-A+B,\,D(A))$ is positive if and only if the off-diagonal
entries of $B$ are non-negative.

###### Proof.

Let us begin by showing that $(R(t))_{t\geq 0}$ is positive if the off-
diagonal entries of $B$ are non-negative. The strategy is to apply $(v)$ from
Theorem 3.6. Recall that $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})_{+}$ is a
closed subset of $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$, see Remark 3.2. Let
$u\geq 0$ and let $t\geq 0.$ The result simply follows from the fact that the
transport semigroup $(T(t))_{t\geq 0}$ and $(S(t))_{t\geq 0}=(e^{tB})_{t\geq
0}$ are positive by Lemma 3.5. Hence,

$\left(T(\tfrac{t}{n})e^{t/n\,B}\right)^{n}u\geq 0$

for all $n\in\mathbb{N}$ and the limit $n\rightarrow\infty$ implies $R(t)u\geq
0$.
Conversely, assume that the semigroup $(R(t))_{t\geq 0}$ is a positive
semigroup. This time, the idea is to conclude that $B=(-A+B)+A$ generates a
positive semigroup. If this is the case, the statement will follow from Lemma
3.5. Some caution is required, since the perturbation of the generator $-A+B$
of the positive semigroup $(R(t))_{t\geq 0}$ is unbounded. Nevertheless, we
can apply Theorem C.3. More precisely, we know that $(-A+B,\,D(-A))$ and
$(A,\,D(A))$ generate the positive $C_{0}$-semigroups $(R(t))_{t\geq 0}$ and
$(T(-t))_{t\geq 0}$ respectively. The latter is a consequence of the fact that
$(-A,\,D(-A))$ generates a positive $C_{0}$-group, cf. Remark 2.8. We have the
stability bound

$\displaystyle\|(R(\tfrac{t}{n})T(-\tfrac{t}{n}))^{n}\|_{\mathcal{L}(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))}\leq\|R(\tfrac{t}{n})\|_{\mathcal{L}(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))}^{n}\leq
e^{\|B\|_{\mathcal{L}(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))}t},$

since $(T(-t))_{t\geq 0}$ is a $C_{0}$-semigroup of contractions and by an
application of statement $(i)$ from Theorem 3.6. Next, we notice that
$B\in\mathcal{L}(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$ implies that

$(\lambda
I_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}-B)L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})=L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$

holds for
$\lambda>\|B\|_{\mathcal{L}(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}))}$ and
therefore, a standard density argument implies density of $(\lambda
I_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}-B)D(-A)$ by the density of $D(-A)$
and the continuity of $B$. We can finally apply Theorem C.3 and conclude that
$(B,\,D(-A))$ has a closure which generates a $C_{0}$-semigroup
$(\widetilde{S}(t))_{t\geq 0}$. This is unsurprising as the closure of
$(B,\,D(-A))$ is the operator $B$ with full domain. As a consequence,
$(\widetilde{S}(t))_{t\geq 0}=(S(t))_{t\geq 0}$ by [8, II. 1.4, p. 51]. More
importantly, we obtain the formula

$S(t)u=\lim_{n\rightarrow\infty}\left(R(\tfrac{t}{n})T(-\tfrac{t}{n})\right)^{n}u$

for all $u\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ and all $t\geq 0.$ It
now follows similarly to the first part of the proof that $(S(t))_{t\geq 0}$
is a positive semigroup, i.e. $B$ generates a positive semigroup. ∎

######  Corollary 3.18.

The group $(R(t))_{t\in\mathbb{R}}$ on
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ generated by $(-A+B,\,D(A))$ is
positive if and only if $B$ is a diagonal matrix.

###### Proof.

By Lemma 3.14, the group $(R(t))_{t\in\mathbb{R}}$ is positive if and only if
the two semigroups $(R(t))_{t\geq 0}$ and $(R_{-}(t))_{t\geq 0}$, which are
generated by $(-A+B,\,D(-A))$ and $(A-B,\,D(A))$ respectively, are positive
semigroups. Theorem 3.17 implies that this is the case if and only if the
diagonal entries of $B$ and $-B$ are non-negative, i.e. iff $B$ is a diagonal
matrix. ∎

######  Definition 3.19.

We stay that $y\in\mathbb{C}^{N}$ is associated to conservation if the
quantity

$\langle
y,\,\overline{\int_{\mathbb{T}^{d}}(R(t)u)(\mathbf{x})\,d\mathbf{x}}\rangle_{\mathbb{C}^{N}}\in\mathbb{C}$

is independent of $t\geq 0$ for all $u\in
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$ The model is called mass conserving
if $(1,\cdots,1)^{T}$ is associated to conservation.

######  Remark 3.20.

Mass conservation essentially means

$\sum_{j=1}^{N}\int_{\mathbb{T}^{d}}u_{j}(t,\mathbf{x})\,d\mathbf{x}=\sum_{j=1}^{N}\int_{\mathbb{T}^{d}}u_{0}(\mathbf{x})\,d\mathbf{x}$

for all $t>0$, the solution $u$ of

$\partial_{t}u+\begin{pmatrix}\mathbf{v_{1}}\cdot\nabla u_{1}\\\ \vdots\\\
\mathbf{v_{N}}\cdot\nabla u_{N}\end{pmatrix}=Bu$

and “any” initial value $u_{0}.$ This definition via the PDE has the
disadvantage that one needs to specify an appropriate set of admissible
initial functions, whereas the semigroup is defined on the whole space
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$

We now characterize all vectors $y\in\mathbb{C}^{N}$ associated to
conservation. Interestingly enough, the proof relies on the structure of the
semigroup and does not use $\tfrac{d}{dt}R(t)u=(-A+B)R(t)u$ for $u\in D(-A).$

######  Lemma 3.21.

Let $(R(t))_{t\geq 0}$ be the semigroup on
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ generated by $(-A+B,\,D(-A))$. Then

$\int_{\mathbb{T}^{d}}(R(t)u)(\mathbf{x})\,d\mathbf{x}=e^{tB}\int_{\mathbb{T}^{d}}u(\mathbf{x})\,d\mathbf{x}$

holds for all $u\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ and all $t\geq 0.$

###### Proof.

The two key ingredients to this observation are

$\int_{\mathbb{T}^{d}}(T(t)u)(\mathbf{x})\,d\mathbf{x}=\int_{\mathbb{T}^{d}}u(\mathbf{x})\,dx$
(29)

and

$\int_{\mathbb{T}^{d}}(e^{tB}u)(\mathbf{x})\,d\mathbf{x}=e^{tB}\int_{\mathbb{T}^{d}}u(\mathbf{x})\,d\mathbf{x}$
(30)

for all $u\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ and all $t\geq 0.$ They
follow from the independence of integrals of periodic functions w.r.t.
translations and the definition of the operator $e^{tB}$ as a pointwise matrix
multiplication. The equations (29) and (30) imply

$\displaystyle\int_{\mathbb{T}^{d}}(R(t)u)(\mathbf{x})\,d\mathbf{x}$
$\displaystyle=\lim_{n\rightarrow\infty}\int_{\mathbb{T}^{d}}\big{(}(T(\tfrac{t}{n})e^{t/n\,B})^{n}u\big{)}(\mathbf{x})\,d\mathbf{x}=\lim_{n\rightarrow\infty}\int_{\mathbb{T}^{d}}\big{(}e^{t/n\,B}(T(\tfrac{t}{n})e^{t/n\,B})^{n-1}u\big{)}(\mathbf{x})\,d\mathbf{x}$
$\displaystyle=\lim_{n\rightarrow\infty}e^{t/n\,B}\int_{\mathbb{T}^{d}}\big{(}(T(\tfrac{t}{n})e^{t/n\,B})^{n-1}u\big{)}(\mathbf{x})\,d\mathbf{x}=e^{tB}\int_{\mathbb{T}^{d}}u(\mathbf{x})\,d\mathbf{x}.$

for all $u\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ and all $t\geq 0$ by an
application of property $(v)$ from Theorem 3.6 and a simple induction. ∎

######  Remark 3.22.

If $B$ is a stable matrix, i.e.
$\sigma(B)\subseteq\mathbb{C}_{-}=\\{\lambda\in\mathbb{C}\,\colon\,\operatorname{Re}(\lambda)<0\\}$,
the above lemma shows that the averages of the solution converge to $0$.

######  Theorem 3.23.

A vector $y\in\mathbb{C}^{N}$ is associated to conservation if and only if
$B^{T}y=0.$

###### Proof.

$``\impliedby"$ Let $y\in\mathbb{C}^{N}$ with $B^{T}y=0$ and let $u\in
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$ We can use Lemma 3.21 to compute

$\displaystyle\frac{d}{dt}\langle
y,\,\overline{\int_{\mathbb{T}^{d}}(R(t)u)(\mathbf{x})\,d\mathbf{x}}\rangle_{\mathbb{C}^{N}}$
$\displaystyle=\frac{d}{dt}\langle
y,\,e^{tB}\overline{\int_{\mathbb{T}^{d}}u(\mathbf{x})\,d\mathbf{x}}\rangle_{\mathbb{C}^{N}}=\langle
y,\,Be^{tB}\overline{\int_{\mathbb{T}^{d}}u(\mathbf{x})\,d\mathbf{x}}\rangle_{\mathbb{C}^{N}}$
$\displaystyle=\langle
B^{T}y,\,e^{tB}\overline{\int_{\mathbb{T}^{d}}u(\mathbf{x})\,d\mathbf{x}}\rangle_{\mathbb{C}^{N}}=0$

for $t>0.$ The continuity of $t\mapstochar\rightarrow R(t)u$ on $[0,\infty)$
yields

$\langle
y,\,\overline{\int_{\mathbb{T}^{d}}(R(t)u)(\mathbf{x})\,d\mathbf{x}}\rangle_{\mathbb{C}^{N}}=\langle
y,\,\overline{\int_{\mathbb{T}^{d}}u(\mathbf{x})\,d\mathbf{x}}\rangle_{\mathbb{C}^{N}}$
(31)

for all $t\geq 0.$

$``\implies"$ Let $y\in\mathbb{C}^{N}$ be associated to conservation, i.e.

$\langle
y,\,\overline{\int_{\mathbb{T}^{d}}(R(t)u)(\mathbf{x})\,d\mathbf{x}}\rangle_{\mathbb{C}^{N}}=\langle
y,\,\overline{\int_{\mathbb{T}^{d}}u(\mathbf{x})\,d\mathbf{x}}\rangle_{\mathbb{C}^{N}}$

holds for all $u\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ and all $t\geq 0.$
Differentiating with respect to $t$ implies

$\langle
B^{T}y,\,e^{tB}\overline{\int_{\mathbb{T}^{d}}u(\mathbf{x})\,d\mathbf{x}}\rangle_{\mathbb{C}^{N}}=0$

for all $t>0$ because of the computation done in $``\impliedby"$ and the fact
that the right-hand side is independent of $t$. Letting $t\searrow 0$ results
in

$\langle
B^{T}y,\,\overline{\int_{\mathbb{T}^{d}}u(\mathbf{x})\,d\mathbf{x}}\rangle_{\mathbb{C}^{N}}=0.$

Finally, taking $u\equiv\overline{z}$ for $z\in\mathbb{C}^{N}$ yields $\langle
B^{T}y,\,z\rangle_{\mathbb{C}^{N}}=0$ for all $z\in\mathbb{C}^{N}$, which
finishes the proof. ∎

##  Chapter 4 Spectral Analysis of the Transport-Reaction Semigroup

As always, we assume that the dimension $d\in\mathbb{N}$, the number of
components $N\in\mathbb{N}$, the transport directions
$\mathbf{v_{1}},\cdots,\mathbf{v_{N}}\in\mathbb{R}^{d}$ and $1\leq p<\infty$
are arbitrarily chosen fixed parameters, if not stated otherwise. We work on
the Banach space $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ with norm
$\|\cdot\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}$, which was defined in
(4).

Our long-term goal is to study the stability of solutions of the transport-
reaction model (2). Let us recall from the introduction of Chapter 3 that the
linearized equation reads as

$\partial_{t}u+\begin{pmatrix}\mathbf{v_{1}}\cdot\nabla u_{1}\\\ \vdots\\\
\mathbf{v_{N}}\cdot\nabla u_{N}\end{pmatrix}=Bu$

for an a priori arbitrary matrix $B\in\mathbb{R}^{N\times N}.$ In Chapter 3,
we considered the abstract Cauchy problem

$\dot{u}(t)+(A-B)u(t)=0$ (32)

on $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$, where $(-A,\,D(-A))$ is the
generator of the transport semigroup introduced in Chapter 2 and $B$ is the
bounded linear operator on $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ induced by
the matrix $B$. Notice that we slightly abuse notation and again use the
letter $B$ for the matrix and the operator. Distinction between both of them
will always follow within the context of usage.

Semigroup and spectral theory offer a great tool to study stability properties
both qualitatively and quantitatively with precise convergence rates. The most
important definitions and results we will apply can be found in Appendix D,
including additional references. Roughly speaking, if $(R(t))_{t\geq 0}$ is
the semigroup generated by $(-A+B,\,D(-A))$, certain spectral properties of
the operators $R(t)$ for $t\geq 0$ imply stability. It is then natural to try
to find a connection between the spectrum of the generator $(-A+B,\,D(-A))$
and the spectrum of $R(t)$ for $t\geq 0.$ This connection does indeed exist
via the spectral mapping theorem

$``\quad\sigma(R(t))=e^{t\sigma(-A+B)}\quad".$ (33)

Standard results in the literature write $\sigma(R(t))\backslash\\{0\\}$ in
(33). But we have seen that $(R(t))_{t\geq 0}$ extends to a strongly
continuous group, which implies $0\in\rho(R(t))$ for all $t\in\mathbb{R},$ see
Lemma 3.14. However, the equation is put in quotation marks as it does not
always hold. In our case, the PDE is hyperbolic and the semigroup
$(R(t))_{t\geq 0}$ does not have powerful properties like analyticity. It is
in general not even eventually norm continuous and one cannot apply Corollary
D.12. In fact, (33) fails for the transport semigroup $(T(t))_{t\geq 0}$ and
the best one can hope for is a weaker version of (33), the so called weak
spectral mapping theorem

$\sigma(R(t))=\overline{e^{t\sigma(-A+B)}}$ (34)

The objective of the whole chapter is to show (34) for the transport-reaction
semigroups $(R(t))_{t\geq 0}$ on $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$.
Unfortunately, we often need to restrict ourselves to the cases $p=2$ or
$d=1.$

This chapter is structured as follows: we start by performing a complete
spectral analysis of the transport semigroup $(T(t))_{t\geq 0}$ for $d=1$.
These computations mainly serve as a motivating example and show that (33)
cannot be expected to hold for the transport-reaction semigroups. Beyond that,
the computations indicate that a weak spectral mapping theorem should hold.
We continue by characterizing the spectrum of the generator
$(-A_{2}+B_{2},\,D(-A_{2}))$ for arbitrary dimension $d\in\mathbb{N}$.
Afterwards, we study the spectra of the generators
$(-A_{p}+B_{p},\,D(-A_{p}))$ for dimension $d=1$ and arbitrary $1\leq
p<\infty$. It may seem counter intuitive to start this way, since these
results are insufficient to perform a stability analysis for the model without
knowing that (34) holds. However, a great understanding of the spectrum of the
generator for all $1\leq p<\infty$ is required to extend a weak spectral
mapping theorem from $p=2$ to different $p$. In particular, independence of
$\sigma(-A_{p}+B_{p})$ of $p$ plays a crucial role. When studying the spectrum
for $p\neq 2$, we assume $d=1$ because in this case, the generator has compact
resolvent if the transport directions are non-vanishing. This implies that the
spectrum consists only of eigenvalues. For higher dimensions, we show that
$(-A+B,\,D(-A))$ does not have compact resolvent if one component $j$ is
transport periodic, see Definition 2.12. Rigorously proving stability results
then turns into a difficult task for $p\neq 2$ and $d\geq 2$. A computation of
the eigenvalues of $(-A+B,\,D(-A))$ could yield an incomplete picture without
further work on the problem $\sigma(-A+B)=\sigma_{p}(-A+B)$.
We finally show (34) for $p=2$ and all dimensions $d\in\mathbb{N}$. For $d=1,$
we can partially extend the result to other $p$ by using the main ideas from
[25]. More precisely, we show that the weak spectral mapping theorem all
$1\leq p<\infty$ is basically equivalent to the weak spectral mapping theorem
for $p=1$, cf. Corollary 4.27 for the precise statement. In particular, (34)
extends to all $1\leq p<\infty$ if the group $(R(t))_{t\in\mathbb{R}}$ is
positive but unfortunately, the model is quite unspectacular if $B$ is a
diagonal matrix, cf. Corollary 3.18.

We conjecture that (34) holds for all dimensions $d\in\mathbb{N}$ and all
$1\leq p<\infty$ but proving this seems to be a difficult task.

###  2 Spectral Analysis of the Transport Semigroup

The next example is a complete spectral analysis in the non-reacting case in
dimension $d=1$ and with non-vanishing transport directions
$v_{1},\cdots,v_{N}\neq 0$. It shows that one cannot expect (33) to hold, even
in the most simplest scenario. Essentially, the example we formulate is a
detailed and generalized version of [8, IV. Example 2.6 (iv), p. 254] and [34,
Example 4.12, p. 100]. The main ideas are already presented there.

######  Example 4.1 ($N=1$).

Let us consider $d=1,\,N=1$ and $B=0$ with non-vanishing transport direction
$v\neq 0.$ In this case, we have

$-Au=-vu^{\prime}\quad\text{and}\quad(T(t)u)(x)=u(x-tv)$

with $D(-A)=W^{1,p}(\mathbb{T})$ by Corollary 2.11. Clearly, the semigroup is
periodic with period $\tau=1/|v|.$ By the spectral inclusion Theorem D.11, we
obtain

$e^{\tau\sigma(-A)}\subseteq\sigma(T(\tau))=\sigma(I_{L^{p}(\mathbb{T},\,\mathbb{C}^{N})})=\\{1\\}$

and thus $\sigma(-A)\subseteq 2\pi iv\mathbb{Z}.$ In fact, there holds
equality: for $\lambda\in 2\pi iv\mathbb{Z}$ we define the function

$u(x)=e^{-\tfrac{\lambda}{v}x}.$

Then

$-Au=-vu^{\prime}=\lambda u$

and $u\in W^{1,p}(\mathbb{T})$ as it fulfills the periodic boundary
conditions. That is, we have shown

$\sigma(-A)=\sigma_{p}(-A)=2\pi iv\mathbb{Z}$

explicitly.
In the following, we compute the spectrum of $T(t)$ for $t\geq 0.$ To this
end, let us introduce some notation. For a rational number $p/q\in\mathbb{Q}$,
we choose $p\in\mathbb{Z},\,q\in\mathbb{N}$ and define the $q$-th roots of
unit $\Gamma_{q}\coloneqq\\{z\in\mathbb{C}\,\colon\,z^{q}=1\\}.$ The goal is
to show

$\sigma(T(t))=\begin{cases}\mathbb{S}^{1}\qquad
tv\in\mathbb{R}\backslash\mathbb{Q},\\\ \Gamma_{q}\qquad tv=\frac{p}{q}\text{
with }p\text{ and }q\text{ coprime}\end{cases}.$

Recall that $(T(t))_{t\geq 0}$ defines a group of isometries and hence

$\displaystyle
r(T(t))=\max\\{|\lambda|\,\colon\,\lambda\in\sigma(T(t))\\}=\lim_{n\rightarrow\infty}\|T(t)^{n}\|^{\frac{1}{n}}=\lim_{n\rightarrow\infty}\|T(nt)\|^{\frac{1}{n}}=1,$
(35)

where $r(T(t))$ is the spectral radius of $T(t),$ see Definition D.3, and the
second equality follows from [35, Theorem 1.16, p. 7]. Equation (35) holds for
all $t\in\mathbb{R}$ and we particularly obtain

$r(T(t)^{-1})=r(T(-t))=1.$

Now, note that boundedness and invertibility of $T(t)$ implies

$\sigma(T(-t))=\frac{1}{\sigma(T(t))}.$

Therefore, we also get

$1=r(T(-t))=\max\\{|\lambda|\,\colon\,\lambda\in\sigma(T(-t))\\}=\max\left\\{\tfrac{1}{|\lambda|}\,\colon\,\lambda\in\sigma(T(t))\right\\},$
(36)

so (35) and (36) imply $\sigma(T(t))\subseteq\mathbb{S}^{1}$ for all $t\geq
0.$
At this point, we again use the spectral inclusion Theorem D.11 to deduce

$e^{t\sigma(-A)}\subseteq\sigma(T(t))$ (37)

for all $t\geq 0.$ Recall $\sigma(-A)=2\pi iv\mathbb{Z}$. In the case
$tv\in\mathbb{R}\backslash\mathbb{Q},$ it is well known and relatively easy to
see that

$e^{t2\pi iv\mathbb{Z}}\subseteq\mathbb{S}^{1}$

is a dense inclusion. By the closedness of spectra, it follows

$\mathbb{S}^{1}=\overline{e^{t\sigma(-A)}}\subseteq\overline{\sigma(T(t))}=\sigma(T(t))\subseteq\mathbb{S}^{1}.$

In the second case, let $tv=\tfrac{p}{q}\in\mathbb{Q}$ for coprime
$p\in\mathbb{Z}$ and $q\in\mathbb{N}.$ We can use the spectral mapping theorem
[35, Theorem 5.3, p. 84] for bounded linear operators and the holomorphic map
$z\mapstochar\rightarrow z^{q}$ to obtain

$\sigma(T(t))^{q}=\sigma\big{(}T(\tfrac{p}{qv})\big{)}^{q}=\sigma\big{(}T(\tfrac{p}{qv})^{q}\big{)}=\sigma\big{(}T(\tfrac{1}{v})^{p}\big{)}=\sigma(T(\operatorname{sgn}(v)\tau)^{p})=\sigma(I_{L^{p}(\mathbb{T},\,\mathbb{C}^{N})})=\\{1\\},$

i.e. $\sigma(T(t))\subseteq\Gamma_{q}.$ On the other hand,

$\Gamma_{q}=e^{2\pi i\tfrac{p}{q}\mathbb{Z}}=e^{t2\pi
iv\mathbb{Z}}=e^{t\sigma(-A)}\subseteq\sigma(T(t))$

by (37) and the fact that $p$ and $q$ are coprime ($e^{2\pi ip/q}$ is a
primitive $q$-th root if and only if $p$ and $q$ are coprime, which is a
classical result from Algebra).

######  Remark 4.2.

The result on the spectrum of $T(t)$ for $t\geq 0$ is an interesting result in
itself. It gives precise information on the solvability of a class of
difference equations on $L^{p}(\mathbb{T}),$ i.e. solvability in
$L^{p}(\mathbb{T})$ of the equation

$\lambda u(x)-u(x-tv)=\big{(}(\lambda-T(t))u\big{)}(x)=v(x)\quad\text{ for
almost all }x\in\mathbb{T},$

given $\lambda\in\mathbb{C}$ and $v\in L^{p}(\mathbb{T}).$ Note that also the
question of uniqueness of the solution can be answered precisely due to
$\sigma_{p}(T(t))=e^{t\sigma_{p}(-A)}=e^{t\sigma(-A)}$ by Theorem D.10.

######  Example 4.3 (arbitrary $N\in\mathbb{N}$).

The result from Example 4.1 can be naturally extended to the case
$d=1,\,N\in\mathbb{N}$ and $B=0$ with non-vanishing transport directions
$v_{j}\neq 0$ for $j=1,\cdots,N.$ That is, $N$ species move along a circle
without reacting with each other. We have

$-Au=-\begin{pmatrix}v_{1}u_{1}^{\prime}\\\ \vdots\\\
v_{N}u_{N}^{\prime}\end{pmatrix}\quad\text{ and
}\quad(T(t)u)(x)=\begin{pmatrix}u_{1}(x-tv_{1})\\\ \vdots\\\
u_{N}(x-tv_{N})\end{pmatrix}$

with $D(-A)=W^{1,p}(\mathbb{T},\,\mathbb{C}^{N})$ by Corollary 2.11. Both the
generator and the semigroup act on the $N$ different components independently
and therefore, we get

$\sigma(-A)=\sigma_{p}(-A)=2\pi
i\bigcup_{\begin{subarray}{c}j=1\end{subarray}}^{N}v_{j}\mathbb{Z}$

and

$\sigma(T(t))=\begin{cases}\mathbb{S}^{1}&\exists j=1,\cdots,N\text{ with
}tv_{j}\in\mathbb{R}\backslash\mathbb{Q}\\\
\cup_{j=1}^{N}\Gamma_{q_{j}}&tv_{j}=\tfrac{p_{j}}{q_{j}}\text{ with
}\,p_{j}\text{ and }q_{j}\text{ coprime}\end{cases}.$ (38)

If we additionally assume that the semigroup is periodic, the first condition
in (38) reduces to a condition on $tv_{1}$ because $tv_{j}$ is irrational any
$j=1,\cdots,N$ if and only if $tv_{1}$ is irrational, see Lemma 2.16.

Let us emphasize that the above computation verify the weak spectral mapping
theorem

$\sigma(T(t))=\begin{rcases}\begin{dcases}\mathbb{S}^{1}&\exists
j=1,\cdots,N\colon tv_{j}\in\mathbb{R}\backslash\mathbb{Q}\\\
\cup_{j=1}^{N}\Gamma_{q_{j}}&tv_{j}=\tfrac{p_{j}}{q_{j}}\text{ with
}\,p_{j}\text{ and }q_{j}\text{
coprime}\end{dcases}\end{rcases}=\bigcup_{j=1}^{N}\overline{e^{t2\pi
iv_{j}\mathbb{Z}}}=\overline{e^{t\sigma(-A)}}$

and show that the spectral mapping theorem (33) does not hold. The equality in
(33) fails for all $t\geq 0$ such that there exists $j=1,\cdots,N$ with
$tv_{j}\in\mathbb{R}\backslash\mathbb{Q}.$ In addition, we see that the
spectrum of the transport semigroup and its generator are independent of
$1\leq p<\infty.$

###  3 Spectral Analysis of the Generator

As mentioned in the introduction of the chapter, we start with a
characterization of $\sigma(-A_{2}+B_{2})$ for all dimensions
$d\in\mathbb{N}$. The key observation, which will also be crucial for the
proof of the weak spectral mapping theorem, is that Fourier transformation
turns linear differential operators acting on
$L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ to matrix multiplication operators on
$l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})$, the space of the Fourier
coefficients. Then, we can use existing theory of matrix multiplication
operators from [21] and obtain a complete characterization of
$\sigma(-A_{2}+B_{2})$ in terms of roots of polynomials of degree $N$.

Consider the Fourier transformation $\mathcal{F}\colon
L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})\rightarrow
l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})$ with

$(\mathcal{F}u)(k)=\hat{u}(k)\qquad\text{ for }k\in\mathbb{Z}^{d},$ (39)

where $\hat{u}(k)=(\hat{u}_{1}(k),\cdots,\hat{u}_{N}(k))^{T}\in\mathbb{C}^{N}$
are the unique Fourier coefficients

$\hat{u}(k)=\int_{\mathbb{T}^{d}}u(x)e^{-2\pi ik\cdot\mathbf{x}}\,d\mathbf{x}$

with

$u(\mathbf{x})=\sum_{k\in\mathbb{Z}^{d}}\hat{u}(k)e^{2\pi ik\cdot\mathbf{x}}.$
(40)

It is well known that the Fourier transformation defines an isometric
isomorphism and that the Fourier series (40) of $u\in
L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ converges to $u$ in the
$L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ norm, see [11, Theorem 8.20, p. 248].
In particular, equation

$\int_{\mathbb{T}^{d}}|u(\mathbf{x})|^{2}\,d\mathbf{x}=\sum_{k\in\mathbb{Z}^{d}}|\hat{u}(k)|^{2}$
(41)

is called Parseval’s identity.
For our application, it is of great importance that Fourier series of
$C^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ functions are very well-behaved:
for any $u\in C^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ and all
$m\in\mathbb{N},$ there exists a constant $C=C(d,m,u)>0$ such that

$|\hat{u}(k)|\leq C(1+|k|)^{-m}$ (42)

holds, see [15, Theorem 3.3.3, p. 196]. Moreover, for $u\in
C^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N}),$ the Fourier series (40)
converges to $u$ in the $C^{\infty}$ norm by [32, Inversion theorem for
$C^{\infty}$ 5.4, p. 72], i.e. all derivatives of the Fourier series of $u$
converge uniformly to the derivatives of $u$.

At a first glance, the connection of our problem to the abstract setting in
[21] is not evident. We recall the setting from the paper to make it clear and
comprehensible.
Let $(X,\,\Sigma,\,\mu)$ be a $\sigma$-finite measure space, let $1\leq
p<\infty$ and define $E\coloneqq L^{p}(X,\,\mathbb{C}^{N})$. Let $q\colon
X\rightarrow\mathbb{C}^{N\times N}$ be a measurable function.

######  Definition 4.4 ([21, Definition 1]).

An operator $(\mathcal{A}_{q},\,D(\mathcal{A}_{q}))$ defined on
$L^{p}(X,\,\mathbb{C}^{N})$ by $\mathcal{A}_{q}\colon f\mapstochar\rightarrow
qf,$ i.e.

$\mathcal{A}_{q}f(x)=q(x)f(x)$ (43)

for $x\in X$ and for all $f\in D(\mathcal{A}_{q})=\left\\{f\in
L^{p}(X,\,\mathbb{C}^{N})\,\colon\,qf\in L^{p}(X,\,\mathbb{C}^{N})\right\\}$
is called matrix multiplication operator.

######  Proposition 4.5 ([21, Proposition 1]).

If $(\mathcal{A}_{q},\,D(\mathcal{A}_{q}))$ is a matrix multiplication
operator on $L^{p}(X,\,\mathbb{C}^{N})$ with non-void resolvent set
$\rho(\mathcal{A}_{q})$, its spectrum is given by

$\sigma(\mathcal{A}_{q})=\bigcap_{q^{\prime}\in[q]}\overline{\bigcup_{x\in
X}\sigma(q^{\prime}(x))}.$ (44)

Here, $[q]$ is the equivalence class of all measurable functions $\mu$-a.e.
equal to $q$.

In our application, we take
$(X,\Sigma,\,\mu)=(\mathbb{Z}^{d},\mathcal{P}(\mathbb{Z}^{d}),\,\\#)$, where
$\\#$ is the counting measure and $p=2$. Then
$L^{2}(X,\,\mathbb{C}^{N})=l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N}),\,q\colon\mathbb{Z}^{d}\rightarrow\mathbb{C}^{N\times
N}$ and (43) reduces to

$\mathcal{A}_{q}f(k)=q(k)f(k)$ (45)

for $k\in\mathbb{Z}^{d}$ and $f\in D(\mathcal{A}_{q})=\left\\{f\in
l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})\,\colon\,qf\in
l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})\right\\}.$ Secondly, (44) comes down to

$\sigma(\mathcal{A}_{q})=\overline{\bigcup_{k\in\mathbb{Z}^{d}}\sigma(q(k))}$
(46)

because functions on $\mathbb{Z}^{d}$ coinciding $\\#$-a.e. are equal. Notice
that $q(k)$ is a matrix and its spectrum is simply given by the roots of
$\lambda\mapstochar\rightarrow\det(\lambda I_{\mathbb{C}^{N\times N}}-q(k))$.

The following lemma formalizes the above mentioned idea and can for instance
be applied to every reasonable linear differential operator on
$L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ in order to compute its spectrum.

######  Proposition 4.6.

Let $(\mathcal{A},\,D(\mathcal{A}))$ be a densely defined and closed operator
on $L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ with non-void resolvent set
$\rho(\mathcal{A})$. Assume that there exists a function
$q\colon\mathbb{Z}^{d}\rightarrow\mathbb{C}^{N\times N}$ such that
$\mathcal{F}(D(\mathcal{A}))=D(\mathcal{A}_{q})$ and

${L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})}$${D(\mathcal{A})}$${L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})}$${l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})}$${D(\mathcal{A}_{q})}$${l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})}$$\scriptstyle{\mathcal{A}}$$\scriptstyle{\mathcal{F}}$$\scriptstyle{\mathcal{A}_{q}}$$\scriptstyle{\mathcal{F}}$$\scriptstyle{\mathcal{F}^{-1}}$

commutes for its induced matrix multiplication operator
$(\mathcal{A}_{q},\,D(\mathcal{A}_{q})).$ Then it holds

$\sigma(\mathcal{A})=\sigma(\mathcal{A}_{q})=\overline{\bigcup_{k\in\mathbb{Z}^{d}}\sigma(q(k))}.$

###### Proof.

Let $\lambda\in\mathbb{C}.$ By assumption,

$\lambda I_{L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})}-\mathcal{A}\colon
D(\mathcal{A})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$

is invertible if and only if

$\mathcal{F}\big{(}\lambda
I_{L^{2}(\mathbb{T},\,\mathbb{C}^{N})}-\mathcal{A}\big{)}\mathcal{F}^{-1}=\lambda
I_{l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})}-\mathcal{A}_{q}\colon
D(\mathcal{A}_{q})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})$

is invertible. The second equality follows from (46). ∎

Before applying Proposition 4.6 to the generator $(-A_{2}+B_{2},\,D(-A_{2}))$
of the transport-reaction semigroup $(R(t))_{t\geq 0}$, we need to
characterize the domain $D(-A_{2})$ and the application of $-A_{2}+B_{2}$ in
terms of Fourier coefficients. Let us start with the latter.

######  Proposition 4.7.

Let $(-A_{2}+B_{2},\,D(-A_{2}))$ be the generator of the transport-reaction
semigroup $(R_{2}(t))_{t\geq 0}$ on $L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$
Then

$\big{(}(-A_{2}+B_{2})u\big{)}(\mathbf{x})=\sum_{k\in\mathbb{Z}^{d}}M(k)\hat{u}(k)e^{2\pi
ik\cdot\mathbf{x}}$

holds for all $u\in D(-A_{2})$, where $M(k)\in\mathbb{C}^{N\times N}$ are the
matrices defined by

$M(k)\coloneqq-2\pi i\begin{pmatrix}k\cdot\mathbf{v_{1}}&&\\\ &\ddots&\\\
&&k\cdot\mathbf{v_{N}}\end{pmatrix}+B\eqqcolon-2\pi i\mathbf{V}(k)+B.$ (47)

###### Proof.

The first step is to consider $u\in
C^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$ Using Lemma 2.5, a calculation
shows

$((-A_{2}+B_{2})u)(\mathbf{x})=-\begin{pmatrix}\mathbf{v_{1}}\cdot\nabla
u_{1}(\mathbf{x})\\\ \vdots\\\ \mathbf{v_{N}}\cdot\nabla
u_{N}(\mathbf{x})\end{pmatrix}+Bu(\mathbf{x})=\sum_{k\in\mathbb{Z}^{d}}M(k)\hat{u}(k)e^{2\pi
ik\cdot\mathbf{x}}.$ (48)

Every technicality and convergence of the Fourier series works out nicely
because the Fourier series of $u$ converges to $u$ in the $C^{\infty}$ norm,
see (42) and its subsequent comment.
Now, take an arbitrary function $u\in D(-A_{2}).$ By Corollary 3.7, there
exists a sequence $(u^{(n)})_{n\in\mathbb{N}}\subset
C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ such that $u^{(n)}$ converges
to $u$ in the graph norm of $-A_{2}+B_{2}.$ Let $v\coloneqq(-A_{2}+B_{2})u\in
L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ with

$v(\mathbf{x})=\sum_{k\in\mathbb{Z}^{d}}\hat{v}(k)e^{2\pi ik\cdot\mathbf{x}}.$

Parseval’s identity (41) and (48) imply that the convergence of $u^{(n)}$ to
$u$ in the graph norm is equivalent to

$\sum_{k\in\mathbb{Z}^{d}}|\hat{u^{(n)}}(k)-\hat{u}(k)|^{2}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
0$

and

$\sum_{k\in\mathbb{Z}^{d}}|M(k)\hat{u^{(n)}}(k)-\hat{v}(k)|^{2}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
0$

as $n\rightarrow\infty.$ In particular, we obtain

$\displaystyle\hat{u^{(n)}}(k^{*})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\hat{u}(k^{*})\qquad\text{
and }\qquad
M(k^{*})\hat{u^{(n)}}(k^{*})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\hat{v}(k^{*})$

as $n\rightarrow\infty$ for every fixed $k^{*}\in\mathbb{Z}^{d},$ which
implies $\hat{v}(k^{*})=M(k^{*})\hat{u}(k^{*}).$ Since
$k^{*}\in\mathbb{Z}^{d}$ was chosen arbitrarily, we arrive at

$((-A_{2}+B_{2})u)(\mathbf{x})=v(\mathbf{x})=\sum_{k\in\mathbb{Z}^{d}}\hat{v}(k)e^{2\pi
ik\cdot\mathbf{x}}=\sum_{k\in\mathbb{Z}^{d}}M(k)\hat{u}(k)e^{2\pi
ik\cdot\mathbf{x}}.$

∎

With the knowledge from Proposition 4.7, it is easy to characterize the domain
$D(-A_{2}+B_{2})=D(-A_{2})$ in terms of Fourier coefficients.

######  Proposition 4.8.

Let $(-A_{2}+B_{2},\,D(-A_{2}))$ be the generator of the transport-reaction
semigroup $(R_{2}(t))_{t\geq 0}$ on $L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$
Then it holds

$\displaystyle\mathcal{F}(D(-A_{2}))$ $\displaystyle=\bigg{\\{}\hat{u}\in
l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})\,\colon\,\sum_{k\in\mathbb{Z}^{d}}|M(k)\hat{u}(k)|^{2}<\infty\bigg{\\}}$
$\displaystyle=\bigg{\\{}\hat{u}\in
l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})\,\colon\,\sum_{k\in\mathbb{Z}^{d}}|\mathbf{V}(k)\hat{u}(k)|^{2}<\infty\bigg{\\}},$

where $M(k)$ and $\mathbf{V}(k)\in\mathbb{C}^{N\times N}$ are the matrices
given by

$M(k)=-2\pi i\begin{pmatrix}k\cdot\mathbf{v_{1}}&&\\\ &\ddots&\\\
&&k\cdot\mathbf{v_{N}}\end{pmatrix}+B=-2\pi i\mathbf{V}(k)+B.$ (49)

###### Proof.

If $u\in D(-A_{2}),$ the function $(-A_{2}+B_{2})u\in
L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ is given by

$\big{(}(-A_{2}+B_{2})u\big{)}(\mathbf{x})=\sum_{k\in\mathbb{Z}^{d}}M(k)\hat{u}(k)e^{2\pi
ik\cdot\mathbf{x}}$ (50)

due to Proposition 4.7. Therefore, Parseval’s identity (41) yields

$\sum_{k\in\mathbb{Z}^{d}}|M(k)\hat{u}(k)|^{2}<\infty.$

Conversely, let $\hat{u}\in l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})$ with

$\sum_{k\in\mathbb{Z}^{d}}|M(k)\hat{u}(k)|^{2}<\infty.$ (51)

We need to show that the function $u\in
L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ given by

$u(\mathbf{x})=\sum_{k\in\mathbb{Z}^{d}}\hat{u}(k)e^{2\pi ik\cdot\mathbf{x}}$

lies in the domain $D(-A_{2})$ of the generator $-A_{2}+B_{2}.$ To this end,
consider the sequence $(u^{(n)})_{n\in\mathbb{N}}$ defined by

$u^{(n)}(\mathbf{x})\coloneqq\sum_{|k|\leq n}\hat{u}(k)e^{2\pi
ik\cdot\mathbf{x}}.$

Notice that $u^{(n)}\in C^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ converges
to $u$ in $L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ by Parseval’s identity
(41). Moreover,

$\big{(}(-A_{2}+B_{2})u^{(n)}\big{)}(\mathbf{x})=\sum_{|k|\leq
n}M(k)\hat{u}(k)e^{2\pi
ik\cdot\mathbf{x}}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\sum_{k\in\mathbb{Z}^{d}}M(k)\hat{u}(k)e^{2\pi
ik\cdot\mathbf{x}}$

in $L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ as $n\rightarrow\infty$ due to
(51) and Parseval’s identity (41). This yields $u\in D(-A_{2})$ by the
closedness of $(-A_{2}+B_{2},\,D(-A_{2})),$ which is a consequence of the
Hille-Yosida Theorem B.6.
The second equality in the proposition is an immediate consequence of the
triangle inequality in $l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N}).$ ∎

######  Remark 4.9.

Complementary to Theorem 2.18, Proposition 4.8 shows that the domain
$D(-A_{2})$ is in general a larger subspace than
$H^{1}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ for dimensions $d\geq 2,$ even if the
transport directions $\mathbf{v_{1}},\cdots,\mathbf{v_{N}}\in\mathbb{R}^{d}$
are non-vanishing.

Finally, we can rigorously characterize the spectrum of the generator of the
transport-reaction semigroup $(R_{2}(t))_{t\geq 0}.$

######  Theorem 4.10.

Let $(-A_{2}+B_{2},\,D(-A_{2}))$ be the generator of the transport-reaction
semigroup $(R_{2}(t))_{t\geq 0}$ on $L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$
Then it holds

$\sigma(-A_{2}+B_{2})=\overline{\bigcup_{k\in\mathbb{Z}^{d}}\sigma(M(k))},$

where $M(k)\in\mathbb{C}^{N\times N}$ is the matrix

$M(k)=-2\pi i\begin{pmatrix}k\cdot\mathbf{v_{1}}&&\\\ &\ddots&\\\
&&k\cdot\mathbf{v_{N}}\end{pmatrix}+B=-2\pi i\mathbf{V}(k)+B.$ (52)

###### Proof.

Let $(\mathcal{A}_{M},\,D(\mathcal{A}_{M}))$ be the matrix multiplication
operator on $l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})$ induced by
$M=(k\mapstochar\rightarrow M(k))$ (in the sense of Definition 4.4). Notice
that Proposition 4.8 shows $\mathcal{F}(D(-A_{2}))=D(\mathcal{A}_{M})$.
Furthermore, the diagram

${L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})}$${D(-A_{2})}$${L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})}$${l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})}$${D(\mathcal{A}_{M})}$${l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})}$$\scriptstyle{-A_{2}+B_{2}}$$\scriptstyle{\mathcal{F}}$$\scriptstyle{\mathcal{A}_{M}}$$\scriptstyle{\mathcal{F}}$$\scriptstyle{\mathcal{F}^{-1}}$

commutes by Proposition 4.7. The resolvent set $\rho(-A_{2}+B_{2})$ is non-
empty by the Hille-Yosida Theorem B.6 and all properties together imply that
the assumptions of Proposition 4.6 are fulfilled. An application of
Proposition 4.6 finishes the proof. ∎

A simple observation in form of the next lemma shows that the spectrum
$\sigma(-A_{2}+B_{2})$ is symmetric to the real line.

######  Lemma 4.11.

Let $k\in\mathbb{Z}^{d}$ and let $M(k)\in\mathbb{C}^{N\times N}$ be the matrix

$M(k)=-2\pi i\begin{pmatrix}k\cdot\mathbf{v_{1}}&&\\\ &\ddots&\\\
&&k\cdot\mathbf{v_{N}}\end{pmatrix}+B=-2\pi i\mathbf{V}(k)+B.$

Then

$\sigma(M(-k))=\overline{\sigma(M(k))}$

holds for all $k\in\mathbb{Z}^{d},$ where the overline denotes complex
conjugation.

###### Proof.

This is a direct consequence of

$M(-k)=\overline{M(k)}.$

∎

Carrying over the characterization of $\sigma(-A_{2}+B_{2})$ from $p=2$ to
$1\leq p<\infty$ in all dimensions $d\in\mathbb{N}$ seems to be a very
difficult task. In the following, we mostly restrict ourselves to the one
dimensional case $d=1$ and assume that the transport directions
$v_{1},\cdots,v_{N}\in\mathbb{R}$ are non-vanishing, i.e.
$v_{1},\cdots,v_{N}\neq 0.$ All proofs rely on the fact that
$(-A_{p}+B_{p},\,D(-A_{p}))$ has compact resolvent in these simple cases. Our
assumptions are necessary for this approach, since
$(-A_{p}+B_{p},\,D(-A_{p}))$ generally fails to have compact resolvent if one
transport direction vanishes or if the model is considered in higher
dimensions.

Let us recall that we have $D(-A_{p})=W^{1,p}(\mathbb{T},\,\mathbb{C}^{N})$
under the assumptions $d=1$ and non-vanishing transport directions
$v_{1},\cdots,v_{N}\neq 0$ by Corollary 2.11.

######  Theorem 4.12.

Let $1\leq p<\infty.$ Let $d=1$ and let the transport directions
$v_{1},\cdots,v_{N}$ be non-vanishing, i.e. $v_{1},\cdots,v_{N}\neq 0$. Then
the generator $(-A_{p}+B_{p},\,W^{1,p}(\mathbb{T},\,\mathbb{C}^{N}))$ of
$(R_{p}(t))_{t\geq 0}$ has compact resolvent, i.e. $R(\lambda,\,-A_{p}+B_{p})$
is compact for one (and hence all) $\lambda\in\rho(-A_{p}+B_{p}).$
In particular, the following properties hold true:

  1. (i)

$\sigma(-A_{p}+B_{p})=\sigma_{p}(-A_{p}+B_{p})$ contains at most countably
many eigenvalues $\lambda_{j}.$

  2. (ii)

If $\sigma(-A_{p}+B_{p})$ is infinite, then $|\lambda_{j}|\rightarrow\infty$
as $j\rightarrow\infty.$

  3. (iii)

For all $\lambda\in\sigma(-A_{p}+B_{p}),$ the operator
$\lambda_{j}I_{L^{p}(\mathbb{T},\,\mathbb{C}^{N})}-(-A_{p}+B_{p})$ has closed
range and
$\emph{dim}\ker(\lambda_{j}I_{L^{p}(\mathbb{T},\,\mathbb{C}^{N})}-(-A_{p}+B_{p}))=\emph{codim}\operatorname{rg}(\lambda_{j}I_{L^{p}(\mathbb{T},\,\mathbb{C}^{N})}-(-A_{p}+B_{p}))<\infty.$

###### Proof.

We start by showing that the generator has compact resolvent. By [8, II. 4.25,
p. 117], it is equivalent to show that the canonical embedding

$(W^{1,p}(\mathbb{T},\,\mathbb{C}^{N}),\,\|\cdot\|_{-A+B})\lhook\mathrel{\mkern-3.0mu}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}),\,\|\cdot\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})})$

is compact. As the graph norms of
$(-A+B,\,W^{1,p}(\mathbb{T},\,\mathbb{C}^{N}))$ and
$(-A,\,W^{1,p}(\mathbb{T},\,\mathbb{C}^{N}))$ are equivalent by Theorem 3.6,
it is sufficient to show compactness of the embedding

$(W^{1,p}(\mathbb{T},\,\mathbb{C}^{N}),\,\|\cdot\|_{-A})\lhook\mathrel{\mkern-3.0mu}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}),\,\|\cdot\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}).$

Notice that the proof of Corollary 2.11 also implies equivalence of the norms
$\|\cdot\|_{-A}$ and $\|\cdot\|_{W^{1,p}(\mathbb{T},\,\mathbb{C}^{N})}$ on
$W^{1,p}(\mathbb{T},\,\mathbb{C}^{N}).$ The claim now follows from the
classical Rellich-Kondrachov embedding

$W^{1,p}(\mathbb{T},\,\mathbb{C}^{N})\subset\subset
L^{p}(\mathbb{T},\,\mathbb{C}^{N}),$

see [3, Theorem 9.16, p. 285]. Properties $(i)$ to $(iii)$ are a consequence
of [35, Theorem 2.16, p. 27] because the spectrum $\sigma(-A_{p}+B_{p})$ is
non-empty. The easiest way to see is is to take an eigenvalue
$\lambda\in\mathbb{C}$ of $B$ with eigenvector
$z\in\mathbb{C}^{N}\backslash\\{0\\}.$ Then, $u\equiv z\in
W^{1,p}(\mathbb{T},\,\mathbb{C}^{N})$ satisfies $(-A_{p}+B_{p})u\equiv
Bz=\lambda z=\lambda u.$ ∎

######  Remark 4.13.

Below, it will become important that the theorem also holds true for the
generator considered on $L^{\infty}(\mathbb{T},\,\mathbb{C}^{N})$ with
$D(-A_{\infty})=W^{1,\infty}(\mathbb{T},\,\mathbb{C}^{N})$ from Proposition
3.16. This follows from $W^{1,\infty}(\mathbb{T},\mathbb{C}^{N})\subset\subset
L^{\infty}(\mathbb{T},\,\mathbb{C}^{N})$ due to
$W^{1,\infty}(\mathbb{T},\mathbb{C}^{N})\subseteq
W^{1,p}(\mathbb{T},\,\mathbb{C}^{N})\subset\subset
C(\mathbb{T},\,\mathbb{C}^{N})\subseteq
L^{\infty}(\mathbb{T},\,\mathbb{C}^{N})$, where $p>1$ can be chosen
arbitrarily.

The assumption of non-vanishing transport directions $v_{1},\cdots,v_{N}\neq
0$ cannot be dropped. By definition (6) of the domain of the generator, there
are no restrictions on $u_{j}$ if $v_{j}=0$ and the component could be an
arbitrary function in $L^{p}(\mathbb{T}).$ Consequently, the domain of the
operator would no longer be compactly embedded in
$L^{p}(\mathbb{T},\,\mathbb{C}^{N}).$
Concerning higher dimensions $d\geq 2$, we observed in Theorem 2.18 that the
domain of the generator of the semigroup $(R(t))_{t\geq 0}$ is less regular if
one component $j$ is transport periodic in the sense of Definition 2.12. This
suggests that the generator also fails to have compact resolvent. The next
lemma shows this negative result.

######  Proposition 4.14.

Let $1\leq p<\infty.$ Let $d\geq 2$ and let the transport directions
$\mathbf{v_{1}},\cdots,\mathbf{v_{N}}$ be non-vanishing, i.e.
$\mathbf{v_{1}},\cdots,\mathbf{v_{N}}\neq 0$. Assume that one component $j$ is
transport periodic. Then the generator $(-A+B,\,D(-A))$ of $(R(t))_{t\geq 0}$
does not have compact resolvent.

###### Proof.

We show that the embedding

$(D(-A),\,\|\cdot\|_{-A})\lhook\mathrel{\mkern-3.0mu}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow(L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}),\,\|\cdot\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}).$

is not compact. This is equivalent to our statement by [8, II. 4.25, p. 117]
and the equivalence of the graph norms $\|\cdot\|_{-A+B}$ and
$\|\cdot\|_{-A}$, which was shown in Theorem 3.6.
To this end, we construct a bounded sequence
$(u^{(n)})_{n\in\mathbb{N}}\subset(D(-A),\,\|\cdot\|_{-A})$, which fails to
have a convergent subsequence in $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$ The
idea is very similar to the proof of Theorem 2.18. Let
$\mathbf{c}=(\frac{1}{2},\cdots,\frac{1}{2})\in\mathbb{T}^{d}$ be the center
of the torus. By assumption, the curve

$\gamma\colon[0,\tau_{j}]\rightarrow\mathbb{T}^{d}\quad\text{with}\quad\gamma(t)\coloneqq\mathbf{c}+t\mathbf{v_{j}}$

fulfills $\gamma(0)=\gamma(\tau_{j}).$ Let $K=\operatorname{rg}(\gamma)$ be
the range of $\gamma$ and consider the sets $U^{(n)}$ given by

$U^{(n)}=\\{\mathbf{x}\in\mathbb{T}^{d}\,\colon\,\operatorname{dist}(\mathbf{x},\,K)<\tfrac{1}{n}\\}.$

We define

$u_{j}^{(n)}(\mathbf{x})\coloneqq\begin{cases}n^{(d-1)/p}&\mathbf{x}\in
U^{(n)},\\\ 0&\text{otherwise}\end{cases}$

and $u_{i}^{(n)}\equiv 0$ for all $i=1,\cdots,N$ with $i\neq j.$ Notice that
$u\in L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ with

$\|u^{(n)}\|_{L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})}^{p}=n^{d-1}\tau_{j}|\mathbf{v_{j}}|\frac{\pi^{(d-1)/2}}{\Gamma(\tfrac{d+1}{2})}n^{1-d}=\tau_{j}|\mathbf{v_{j}}|\frac{\pi^{(d-1)/2}}{\Gamma(\tfrac{d+1}{2})}$
(53)

for sufficiently large $n\in\mathbb{N}$. The exact formula is a consequence of
the volume of the $(d-1)$-dimensional Euclidean ball with radius $1/n$. In
addition, $u_{j}^{(n)}$ is constant along direction $\mathbf{v_{j}},$ i.e.

$u_{j}^{(n)}(\cdot-t\mathbf{v_{j}})-u_{j}^{(n)}(\cdot)\equiv 0.$

This implies $u^{(n)}\in D(-A)$ with $-Au^{(n)}=0$ for all $n\in\mathbb{N}$
and boundedness of the sequence in $(D(-A),\,\|\cdot\|_{-A}).$
On the other hand, $u^{(n)}\rightarrow 0$ pointwise almost everywhere because
of $d\geq 2$. If a subsequence of $(u^{(n)})_{n\in\mathbb{N}}$ converged in
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$, a subsubsequence would converge to
$0$ in $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$. This contradicts (53). ∎

######  Remark 4.15.

We refer to Theorem 2.18 and Figure 1 for an illustration of the idea.

######  Theorem 4.16.

Let $1\leq p<\infty.$ Let $d=1$ and let the transport directions
$v_{1},\cdots,v_{N}$ be non-vanishing, i.e. $v_{1},\cdots,v_{N}\neq 0.$ Let
$(-A_{p}+B_{p},\,W^{1,p}(\mathbb{T},\,\mathbb{C}^{N}))$ be the generator of
the transport-reaction semigroup $(R_{p}(t))_{t\geq 0}$ on
$L^{p}(\mathbb{T},\,\mathbb{C}^{N}).$ Then it holds

$\sigma(-A_{p}+B_{p})=\sigma_{p}(-A_{p}+B_{p})=\overline{\bigcup_{k\in\mathbb{Z}}\sigma(M(k))},$

where $M(k)\in\mathbb{C}^{N\times N}$ is the matrix

$M(k)=-2\pi ik\begin{pmatrix}v_{1}&&\\\ &\ddots&\\\
&&v_{N}\end{pmatrix}+B\eqqcolon-2\pi ikV+B.$

In particular, the spectrum is independent of $p$.

###### Proof.

Let us recall

$(-A_{p}+B_{p})u=-\begin{pmatrix}v_{1}u_{1}^{\prime}\\\ \vdots\\\
v_{N}u_{N}^{\prime}\end{pmatrix}+Bu$

from Corollary 2.11. We have

${\left.\kern-1.2pt-
A_{p}+B_{p}\vphantom{\big{|}}\right|_{W^{1,p}(\mathbb{T},\,\mathbb{C}^{N})\cap
W^{1,p^{\prime}}(\mathbb{T},\,\mathbb{C}^{N})}}={\left.\kern-1.2pt-
A_{p^{\prime}}+B_{p^{\prime}}\vphantom{\big{|}}\right|_{W^{1,p}(\mathbb{T},\,\mathbb{C}^{N})\cap
W^{1,p^{\prime}}(\mathbb{T},\,\mathbb{C}^{N})}}$ (54)

for all $1\leq p,\,p^{\prime}\leq\infty$ by Corollary 2.11. It should be noted
that we also defined the generator for $p=\infty$ with
$D(-A_{\infty}+B_{\infty})=W^{1,\infty}(\mathbb{T},\,\mathbb{C}^{N})$ in (26)
and (27), cf. Proposition 3.16. The proof includes this case because we apply
an argument with the adjoints of the generators in Step 3. In fact, our proof
shows that the theorem is also valid for $p=\infty$ but we do not need this
result in the sequel.

Step 1: $\sigma(-A_{p^{\prime}}+B_{p^{\prime}})\subseteq\sigma(-A_{p}+B_{p})$
for all $1\leq p\leq p^{\prime}\leq\infty.$
Let $1\leq p\leq p^{\prime}\leq\infty$ and let
$\lambda\in\sigma(-A_{p^{\prime}}+B_{p^{\prime}}).$ We know
$\sigma(-A_{p^{\prime}}+B_{p^{\prime}})=\sigma_{p}(-A_{p^{\prime}}+B_{p^{\prime}})$
from Theorem 4.12 and Remark 4.13. Hence, $\lambda$ is an eigenvalue and there
exists $u\in W^{1,p^{\prime}}(\mathbb{T},\,\mathbb{C}^{N})\backslash\\{0\\}$
with $(-A_{p^{\prime}}+B_{p^{\prime}})u=\lambda u$. The circle $\mathbb{T}$
has finite mass, so we have
$W^{1,p^{\prime}}(\mathbb{T},\,\mathbb{C}^{N})\subseteq
W^{1,p}(\mathbb{T},\,\mathbb{C}^{N})$ and consequently

$(-A_{p}+B_{p})u=(-A_{p^{\prime}}+B_{p^{\prime}})u=\lambda u$

by (54). In words, $\lambda$ is also an eigenvalue of $-A_{p}+B_{p}.$

Step 2: $\sigma(-A_{p}+B_{p})=\sigma(-A_{2}+B_{2})$ for all $2\leq
p\leq\infty.$
Let $2\leq p\leq\infty.$ One direction follows from Step 1. It is left to show
$\sigma(-A_{2}+B_{2})\subseteq\sigma(-A_{p}+B_{p}).$ Theorem 4.10 implies

$\sigma(-A_{2}+B_{2})=\overline{\bigcup_{k\in\mathbb{Z}}\sigma(M(k))}$

where

$M(k)=-2\pi ik\begin{pmatrix}v_{1}&&\\\ &\ddots&\\\
&&v_{N}\end{pmatrix}+B=-2\pi ikV+B.$

The spectrum $\sigma(-A_{p}+B_{p})$ is closed and therefore, we only need to
prove $\sigma(M(k))\subseteq\sigma(-A_{p}+B_{p})$ for all $k\in\mathbb{Z}.$
Let $k\in\mathbb{Z}$ and let $\lambda\in\sigma(M(k)).$ Then there exists
$z\in\mathbb{C}^{N}\backslash\\{0\\}$ with $M(k)z=\lambda z$ and we define

$u\coloneqq ze^{2\pi ikx}.$

Clearly, $u\in W^{1,p}(\mathbb{T},\,\mathbb{C}^{N})\cap
H^{1}(\mathbb{T},\,\mathbb{C}^{N}),\,u\neq 0$ and (54), together with
Proposition 4.7, implies

$(-A_{p}+B_{p})u=(-A_{2}+B_{2})u=M(k)ze^{2\pi ikx}=\lambda ze^{2\pi
ikx}=\lambda u.$

We obtain $\lambda\in\sigma(-A_{p}+B_{p})$ and Step 2 follows.

Step 3: $\sigma(-A_{p}+B_{p})=\sigma(-A_{2}+B_{2})$ for all $1\leq p\leq 2.$
Let $1\leq p\leq 2.$ The whole idea is to apply the classical result that
taking the adjoint does not change the spectrum; see for example [35, Theorem
1.24, p. 12]. Let $q$ be the dual exponent of $p$. We have seen
$(-A_{p}+B_{p})^{*}=A_{q}+B_{q}^{T}$ in Proposition 3.15 and Proposition 3.16
for the cases $1<p\leq 2$ and $p=1$ respectively. For $1\leq p\leq 2,$ the
dual exponent fulfills $2\leq q\leq\infty$ and an application of Step 2 yields

$\displaystyle\sigma(-A_{p}+B_{p})=\sigma((-A_{p}+B_{p})^{*})=\sigma(A_{q}+B_{q}^{T})=\sigma(A_{2}+B_{2}^{T})=\sigma((A_{2}+B_{2}^{T})^{*})=\sigma(-A_{2}+B_{2}).$

Notice that the adjoint generators are still of the class covered throughout
the whole chapter, since the different sign for $A$ just changes the direction
of movement and $B^{T}$ is still a real $N\times N$ matrix. ∎

We continue our study of the spectrum of the generator in Chapter 7, where we
analyze the asymptotics of the eigenvalues of $M(k)$ as
$|k|\rightarrow\infty.$ The results from Chapter 7 give deep insights into the
qualitative behavior of solutions to (18).

###  4 Weak Spectral Mapping Theorems

The idea to use existing theory of matrix multiplication operators is
sufficient to prove a weak spectral mapping theorem for $p=2$ in all
dimensions $d\in\mathbb{N}$. Our strategy is to not only apply Proposition 4.6
for the generator $(-A_{2}+B_{2},\,D(-A_{2}))$ of the semigroup, but also for
each semigroup operator $R_{2}(t)$ itself.
Afterwards, we use the methods from [25] to demonstrate mathematical tools one
can use to extend the result to different $p.$ Given that the spectrum
$\sigma(-A_{p}+B_{p})$ is independent of $1\leq p<\infty$, the ideas from [25]
show that the weak spectral mapping theorem for all $p$ is essentially
equivalent to the weak spectral mapping theorem for $p=1.$

In particular, assuming that the $C_{0}$-group $(R_{p}(t))_{t\in\mathbb{R}}$
on $L^{p}(\mathbb{T},\,\mathbb{C}^{N})$ is positive, abstract theory from [2]
can be used to show a weak spectral mapping theorem for all $1\leq p<\infty$.
Unfortunately, the group is positive if and only if $B$ is a diagonal matrix,
see Corollary 3.18.

A weak spectral mapping theorem for $1\leq p<\infty,\,d\in\mathbb{N}$ and all
matrices $B\in\mathbb{R}^{N\times N}$ remains an open problem.

######  Proposition 4.17.

Let $(R_{2}(t))_{t\geq 0}$ be the transport-reaction semigroup on
$L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ generated by
$(-A_{2}+B_{2},\,D(-A_{2}))$. Then

$(R_{2}(t)u)(\mathbf{x})=\sum_{k\in\mathbb{Z}^{d}}e^{tM(k)}\hat{u}(k)e^{2\pi
ik\cdot\mathbf{x}}$ (55)

holds for all $u\in L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$ Here, $M(k)$ is
the matrix

$M(k)=-2\pi i\begin{pmatrix}k\cdot\mathbf{v_{1}}&&\\\ &\ddots&\\\
&&k\cdot\mathbf{v_{N}}\end{pmatrix}+B=-2\pi i\mathbf{V}(k)+B.$ (56)

###### Proof.

Let us recall that a function $u\in L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ is
given by its Fourier series

$u(\mathbf{x})=\sum_{k\in\mathbb{Z}^{d}}\hat{u}(k)e^{2\pi ik\cdot\mathbf{x}},$

where $(\hat{u}(k))_{k\in\mathbb{Z}^{d}}$ are the unique Fourier coefficients
of $u$. Moreover, we have already proven

$\big{(}(-A_{2}+B_{2})u\big{)}(\mathbf{x})=\sum_{k\in\mathbb{Z}^{d}}M(k)\hat{u}(k)e^{2\pi
ik\cdot\mathbf{x}}$

for all $u\in D(-A_{2})$ in Proposition 4.7 and

$\displaystyle\mathcal{F}(D(-A_{2}))=\bigg{\\{}\hat{u}\in
l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})\,\colon\,\sum_{k\in\mathbb{Z}^{d}}|M(k)\hat{u}(k)|^{2}<\infty\bigg{\\}}$
(57)

in Proposition 4.8. The statement of the proposition is very intuitive but a
rigorous justification is a priori unclear and a little technical.

We start by estimating the matrix norm of $e^{tM(k)}.$ According to
definition, $M(k)=-2\pi i\mathbf{V}(k)+B$. Notice that $\mathbf{V}(k)$ is a
diagonal matrix with real entries, thus $e^{-t2\pi i\mathbf{V}(k)}$ defines an
isometry for all $t\geq 0.$ The Lee-Trotter product formula from Theorem C.3
yields

$\|e^{tM(k)}\|_{\infty}\leq\limsup_{n\rightarrow\infty}\|e^{-t/n\,2\pi
i\mathbf{V}(k)}e^{t/n\,B}\|_{\infty}^{n}\leq\limsup_{n\rightarrow\infty}\|e^{t/n\,B}\|_{\infty}^{n}\leq
e^{t\|B\|_{\infty}}.$ (58)

We obtain that the matrix norms of $e^{tM(k)}$ are uniformly bounded in
$k\in\mathbb{Z}^{d}$. Consequently,

$\bigg{\\{}\hat{u}\in
l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})\,\colon\,\sum_{k\in\mathbb{Z}^{d}}|e^{tM(k)}\hat{u}(k)|^{2}<\infty\bigg{\\}}=l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})$
(59)

and the series in (55) is well defined for all $u\in
L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$ We define the linear operators
$\widetilde{R}(t)$ on $L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ via

$(\widetilde{R}(t)u)(\mathbf{x})\coloneqq\sum_{k\in\mathbb{Z}}e^{tM(k)}\hat{u}(k)e^{2\pi
ik\cdot\mathbf{x}}$

for all $t\geq 0.$ Then, (58) gives the estimate

$\|\widetilde{R}(t)u\|_{L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})}\leq
e^{t\|B\|_{\infty}}\|u\|_{L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})}$ (60)

and the family $(\widetilde{R}(t))_{t\geq 0}$ fulfills the semigroup property.
The next step is to show

$\lim_{t\searrow
0}\left(\frac{\widetilde{R}(t)u-u}{t}\right)(\mathbf{x})=\sum_{k\in\mathbb{Z}^{d}}M(k)\hat{u}(k)e^{2\pi
ik\cdot\mathbf{x}}=((-A_{2}+B_{2})u)(\mathbf{x})$ (61)

for all $u\in D(-A_{2})$ and with convergence in
$L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$. Let us firstly note that the series
is well defined by (57). Secondly, if (61) holds true, (60) and density of
$D(-A_{2})$ in $L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ imply that
$(\widetilde{R}(t))_{t\geq 0}$ is indeed a strongly continuous semigroup.
We now start with the proof of (61). By Parseval’s identity, (61) is
equivalent to

$\lim_{t\searrow
0}\sum_{k\in\mathbb{Z}^{d}}\bigg{|}\bigg{(}\frac{e^{tM(k)}-I_{\mathbb{C}^{N\times
N}}}{t}-M(k)\bigg{)}\hat{u}(k)\bigg{|}^{2}=0.$

The result comes down the question, whether we can interchange the two limits
because

$\lim_{t\searrow 0}\sum_{|k|\leq
n}\bigg{|}\bigg{(}\frac{e^{tM(k)}-I_{\mathbb{C}^{N\times
N}}}{t}-M(k)\bigg{)}\hat{u}(k)\bigg{|}^{2}=0$

holds for all $n\in\mathbb{N}$. The fundamental theorem of calculus implies

$e^{tM(k)}-I_{\mathbb{C}^{N\times
N}}=t\int_{0}^{1}M(k)e^{stM(k)}\,ds=t\bigg{(}\int_{0}^{1}e^{stM(k)}\,ds\bigg{)}M(k)$
(62)

for all $k\in\mathbb{Z}^{d}.$ The last equality follows from the Riemann
definition of the integral, boundedness of $M(k)$ and
$M(k)e^{rM(k)}=e^{rM(k)}M(k)$ for all $r\geq 0.$ As a result, (62) and (58)
show that there is a finite constant $C=C(\|B\|_{\infty},\,1)>0$ with

$\displaystyle\bigg{|}\bigg{(}\frac{e^{tM(k)}-I_{\mathbb{C}^{N\times N}}}{t}$
$\displaystyle-M(k)\bigg{)}\hat{u}(k)\bigg{|}^{2}=\bigg{|}\bigg{(}\int_{0}^{1}e^{stM(k)}-I_{\mathbb{C}^{N\times
N}}\,ds\bigg{)}M(k)\hat{u}(k)\bigg{|}^{2}$
$\displaystyle\leq\bigg{(}\int_{0}^{1}\|e^{stM(k)}-I_{\mathbb{C}^{N\times
N}}\|_{\infty}\,ds\bigg{)}^{2}|M(k)\hat{u}(k)|^{2}\leq C|M(k)\hat{u}(k)|^{2}$

for all $t\leq 1$. Finally, this estimate yields

$\displaystyle\limsup_{t\searrow 0}$
$\displaystyle\sum_{k\in\mathbb{Z}^{d}}\bigg{|}\bigg{(}\frac{e^{tM(k)}-I_{\mathbb{C}^{N\times
N}}}{t}-M(k)\bigg{)}\hat{u}(k)\bigg{|}^{2}$
$\displaystyle\leq\limsup_{t\searrow
0}\sum_{|k|>n}\bigg{|}\bigg{(}\frac{e^{tM(k)}-I_{\mathbb{C}^{N\times
N}}}{t}-M(k)\bigg{)}\hat{u}(k)\bigg{|}^{2}\leq
C\sum_{|k|>n}|M(k)\hat{u}(k)|^{2}$

for all $n\in\mathbb{N}.$ For $u\in D(-A_{2}),$ the limit $n\rightarrow\infty$
implies the desired result by (57).
In summary, we have proven that $(\widetilde{R}(t))_{t\geq 0}$ defines a
strongly continuous semigroup with a generator that extends
$(-A_{2}+B_{2},\,D(-A_{2}))$. In particular, the generator of
$(\widetilde{R}(t))_{t\geq 0}$ extends
$(-A_{2}+B_{2},\,C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$ and
Corollary 3.7 implies $(\widetilde{R}(t))_{t\geq 0}=(R_{2}(t))_{t\geq 0}.$ ∎

######  Remark 4.18.

Equation (55) is an explicit formula of the solution to the abstract Cauchy
problem (32) (in a $L^{2}(\mathbb{T},\,\mathbb{C}^{N})$ setting).

######  Theorem 4.19.

Let $(R_{2}(t))_{t\geq 0}$ be the transport-reaction semigroup on
$L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ generated by
$(-A_{2}+B_{2},\,D(-A_{2}))$. Then the weak spectral mapping theorem

$\sigma(R_{2}(t))=\overline{e^{t\sigma(-A_{2}+B_{2})}}\quad\text{ for all
}t\in\mathbb{R}$

holds.

###### Proof.

Let $t\geq 0.$ Proposition 4.17 allows us to apply the same methods we used
for $(-A_{2}+B_{2},\,D(-A_{2}))$ to $R_{2}(t)$. Let us check the assumptions
of Proposition 4.6 for $(R_{2}(t),\,L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N}))$
and the matrix multiplication operator
$(\mathcal{A}_{e^{tM}},\,D(\mathcal{A}_{e^{tM}}))$ on
$l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})$ induced by
$e^{tM}\coloneqq(k\mapstochar\rightarrow e^{tM(k)}).$
Firstly, we have seen
$D(\mathcal{A}_{e^{tM}})=l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})$ in (59) and
$\mathcal{F}(L^{2}(\mathbb{T},\,\mathbb{C}^{N}))=l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})$
obviously holds true. Secondly, the diagram

${D(R_{2}(t))=L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})}$${L^{2}(\mathbb{T}^{d},\,\mathbb{C}^{N})}$${D(\mathcal{A}_{e^{tM}})=l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})}$${l^{2}(\mathbb{Z}^{d},\,\mathbb{C}^{N})}$$\scriptstyle{R_{2}(t)}$$\scriptstyle{\mathcal{F}}$$\scriptstyle{\mathcal{A}_{e^{tM}}}$$\scriptstyle{\mathcal{F}^{-1}}$

commutes by Proposition 4.17 and, last but not least, the resolvent set
$\rho(R_{2}(t))$ is non-empty because $R_{2}(t)$ is bounded. An application of
Proposition 4.6 implies

$\sigma(R_{2}(t))=\sigma(\mathcal{A}_{e^{tM}})=\overline{\bigcup_{k\in\mathbb{Z}^{d}}\sigma(e^{tM(k)})}.$

The operators $M(k)$ are just matrices, so we obtain

$\sigma(e^{tM(k)})=e^{t\sigma(M(k))}$

by [35, Theorem 5.3, p. 84]. Now, we apply Theorem 4.10 and conclude

$\displaystyle\sigma(R_{2}(t))$
$\displaystyle=\sigma(\mathcal{A}_{e^{tM}})=\overline{\bigcup_{k\in\mathbb{Z}}\sigma(e^{tM(k)})}=\overline{\bigcup_{k\in\mathbb{Z}}e^{t\sigma(M(k))}}\subseteq\overline{\bigcup_{k\in\mathbb{Z}}e^{t\overline{\bigcup_{l\in\mathbb{Z}}\sigma(M(l))}}}$
$\displaystyle=\overline{e^{t\overline{\bigcup_{k\in\mathbb{Z}}\sigma(M(k))}}}=\overline{e^{t\sigma(-A_{2}+B_{2})}}.$

The other direction is the easy one, which follows from the spectral inclusion
theorem

$\sigma(R_{2}(t))\supseteq e^{t\sigma(-A_{2}+B_{2})},$

see Theorem D.11, and the closedness of $\sigma(R_{2}(t))$.
For $t\leq 0,$ the result can be shown with Lemma 3.14 and an application of
the weak spectral mapping theorem to the semigroup $(R_{2}(-t))_{t\geq 0}.$ ∎

We now follow [25] and present methods to extend the weak spectral mapping
theorem to other $1\leq p<\infty$. We present the ideas in detail and adapt
the proofs to our system of PDEs, if necessary. For the rest of the chapter,
we will assume

######  Assumption 1.

  1. (i)

The spectrum $\sigma(-A_{p}+B_{p})$ of the generator of the transport-reaction
semigroup $(R_{p}(t))_{t\geq 0}$ on $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$
is independent of $1\leq p<\infty.$

  2. (ii)

The spectrum $\sigma(A_{p}+B_{p}^{T})$ of $(A_{p}+B_{p}^{T},\,D(A_{p}))$ is
independent of $1\leq p<\infty.$

######  Remarks 4.20.

  1. (i)

Under 1 (i), Proposition 3.15 almost implies 1 (ii) because of

$\sigma(A_{p}+B_{p}^{T})=\sigma(-A_{q}+B_{q})$

for all $1<p<\infty$ with dual exponent $q$. Here, we used that the spectrum
is unchanged when taking the adjoint.

  2. (ii)

The operator $(A_{p}+B_{p}^{T},\,D(A_{p}))$ is still of the class covered in
this thesis. The change of sign for $A_{p}$ just changes the direction of
movement and $B_{p}^{T}$ is a real $N\times N$ matrix.

This assumption is of course only fulfilled in the case $d=1$ with non-
vanishing transport directions $v_{1},\cdots,v_{N}\neq 0,$ see Theorem 4.16.
Nevertheless, we decided to formulate all results under 1 instead of
restricting ourselves to the case $d=1$. The reason for this is to emphasize
the independence of the arguments of the spacial dimension $d\in\mathbb{N}$.

Let us start by formulating some technicalities.

######  Lemma 4.21.

Let $(R_{p}(t))_{t\in\mathbb{R}}$ be the transport-reaction group on
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ generated by
$(-A_{p}+B_{p},\,D(-A_{p}))$. The following properties hold true for all
$1\leq p<p^{\prime}<\infty$ and all $t\in\mathbb{R}:$

  1. (i)

$D(-A_{p}^{\prime})\subseteq D(-A_{p}),$

  2. (ii)

${\left.\kern-1.2pt-
A_{p}+B_{p}\vphantom{\big{|}}\right|_{D(-A_{p}^{\prime})}}=-A_{p^{\prime}}+B_{p^{\prime}}$,

  3. (iii)

${\left.\kern-1.2ptR_{p}(t)\vphantom{\big{|}}\right|_{L^{p^{\prime}}(\mathbb{T}^{d},\,\mathbb{C}^{N})}}=R_{p^{\prime}}(t).$

Under 1 (i), it also holds

  1. (iv)

${\left.\kern-1.2ptR(\lambda,\,-A_{p}+B_{p})\vphantom{\big{|}}\right|_{L^{p^{\prime}}(\mathbb{T}^{d},\,\mathbb{C}^{N})}}=R(\lambda,\,-A_{p^{\prime}}+B_{p^{\prime}})$
for all $\lambda\in\rho(-A+B).$

###### Proof.

We start by showing $(i)$ and $(ii)$ at once. On
$C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N}),$ the generators coincide by
Lemma 2.5 and the definition of $B$ as a pointwise matrix multiplication. Now,
let $u\in D(-A_{p}^{\prime})$. By Corollary 3.7, there exists a sequence
$(u^{(n)})_{n\in\mathbb{N}}\subset
C_{c}^{\infty}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ with $u^{(n)}\rightarrow u$
in the graph norm of $(-A_{p^{\prime}}+B_{p^{\prime}},\,D(-A_{p^{\prime}})).$
We also assumed $p<p^{\prime},$ so $u^{(n)}$ also converges to $u$ in
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ and
$((-A_{p}+B_{p})u^{(n)})_{n\in\mathbb{N}}$ is a Cauchy sequence in
$L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}).$ This yields

$(-A_{p}+B_{p})u^{(n)}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow(-A_{p^{\prime}}+B_{p^{\prime}})u$

in $L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N})$ and the closedness of
$(-A_{p}+B_{p},\,D(-A_{p})),$ see Theorem B.6, implies $u\in D(-A_{p})$ with
$(-A_{p}+B_{p})u=(-A_{p^{\prime}}+B_{p^{\prime}})u.$
Property $(iii)$ follows from the Dyson-Phillips series in Theorem 3.6 and the
fact that the transport groups $(T_{p}(t))_{t\in\mathbb{R}}$ and
$(T_{p^{\prime}}(t))_{t\in\mathbb{R}}$ coincide on
$L^{p^{\prime}}(\mathbb{T}^{d},\,\mathbb{C}^{N})$.
Concerning property $(iv),$ notice that 1(i) justifies the notation
$\rho(-A+B)$ used above. Let $\lambda\in\rho(-A+B)$. For $v\in
L^{p^{\prime}}(\mathbb{T}^{d},\,\mathbb{C}^{N})\subseteq
L^{p}(\mathbb{T}^{d},\,\mathbb{C}^{N}),$ there exist unique $u_{p}\coloneqq
R(\lambda,\,-A_{p}+B_{p})v\in D(-A_{p})$ and $u_{p^{\prime}}\coloneqq
R(\lambda,\,-A_{p^{\prime}}+B_{p^{\prime}})v\in D(-A_{p^{\prime}})$ such that

$\big{(}\lambda-(-A_{p}+B_{p})\big{)}u_{p}=v=\big{(}\lambda-(-A_{p^{\prime}}+B_{p^{\prime}})\big{)}u_{p^{\prime}}$

holds. Property $(ii)$ yields

$\big{(}\lambda-(-A_{p}+B_{p})\big{)}u_{p^{\prime}}=\big{(}\lambda-(-A_{p^{\prime}}+B_{p^{\prime}})\big{)}u_{p^{\prime}}=v,$

which implies $u_{p^{\prime}}=u_{p}$ by the injectivity of
$\lambda-(-A_{p}+B_{p})$. ∎

Before we show equivalence of the weak spectral mapping theorem for all $1\leq