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Introduction
2

issues, which can be translated into the language of partial differential equations
and further investigated by mathematical analysis and numerical computations.
The photographs should focus the reader’s attention to real-life/natural prob-
lems, appeal to his esthetic senses and connect directly to the modeling by
partial differential equations. The actual representation of their solutions usu-
ally is done through numerical computations and graphic output algorithms,
but this is NOT the purpose of this book.
Clearly, the choice of the PDE topics in Chapter 1 to Chapter 11 is personally
biased by the author’s mathematical taste, his mathematical experience and
research interests. Some of the chosen topics have been in the center of his
scientific interests and production for many years or even decades (Chapters 1,
4, 5 and 9), some are important sidelines of his research (Chapters 2, 6 and 10)
and the others are in the realm of his passive scientific interests. Completeness
of apresentationofPDEsinapplicationsisnot an issueofthisbookand many
important topics are not covered here (an example is the recent surge in PDE
applications in mathematical finance theory).
The texts are accessible to a broad range of mathematically interested people,
in particular readers with a basic knowledge of differential and integral calculus
in more than one dimension will be able to follow the exposition without diffi-
culty. For example the author believes that advanced undergraduate students of
mathematics, physics or engineering will enjoy the reading and profit from this
book. Also, it could provide motivations and case studies for graduate courses
in applied partial differential equations, with many loose ends which have to be
tied up by further (literature) research. In some instances references are made to
high powered mathematical techniques, which are supposed to be of interest to
themathematicallymoreadvancedreaders,whohaveadirectresearchinterestin
applications and analysis of partial differential equations. Those readers might
learn about some applications which had not crossed their minds before …
Each Chapter is self-contained to a very large extent, with its own bibliogra-
phy. So readers can follow their personal preferences, choose their own sequence
for reading the Chapters or even skip Chapters of lesser interest to them without
loosing the general context.
There is, however, a certain scientifically arguable motivation of the chosen
sequence of topics, albeit to some extent again dictated by the author’s research
background. The first Chapter is on kinetic phase space models, which give rise
to many position space based macroscopic PDE systems, like the Navier–Stokes
and Euler systems describing fluid and gas motion presented in Chapter 2, the
granular flow equations of Chapter 3, the chemotaxis equations of Chapter 4 and
the semiconductor models of Chapter 5. Chapter 6 deals with free boundaries,
Chapter 7 with reaction-diffusion equations (see also the Turing instability
discussed in Chapter 4), Chapter 8 focuses on the Monge–Kantorovich mass
transportation theories which strongly connect to the areas of kinetic theory,
fullynonlinearellipticpartialdifferentialequations,diffusivetheoriesetc.Linear
and nonlinear wave propagation is the topic of Chapter 9, with a strong interplay