The book presents topics of science and engineering, which occur in nature or are part of our daily lives. It describes phenomena which are modelled by partial differential equations, relating to physical variables like mass, velocity and energy, etc. to their spatial and temporal variations.
Typically, these equations are highly nonlinear, in many cases they are also vectorial systems, and they represent a challenge even for the most modern and sophisticated mathematical-analytical and mathematical-numerical techniques.
The topics chosen reflect the longtime scientific interests of the author. They include flow of fluids and gases, granular flows, biological processes like pattern formation on animal skins, kinetics of rarified gases and semiconductor devices. Each topic is briefly presented in its scientific or engineering context, followed by an introduction of the mathematical models in the form of partial differential equations with a discussion of their basic mathematical properties.
Each chapter is illustrated by a series of high quality photographs, taken by the author. They demonstrate in an allegoric way that partial differential equations can be used to address a large variety of phenomena occuring in and influencing our daily lives.
Introduction 1 Kinetic Equations: From Newton to Boltzmann 2 The Navier?Stokes and Euler Equations: Fluid and Gas Dynamics 3 Granular Material Flows 4 Chemotactic Cell Motion and Biological Pattern Formation 5 Semiconductor Modeling 6 Free Boundary Problems and Phase Transitions 7 Reaction-Diffusion Equations - Homogeneous and Heterogeneous Environments 8 Optimal Transportation andMonge?AmpereEquations 9 Wave Equations 10 Digital Image Processing and Analysis ? PDEs and Variational Tools 11 Socio-Economic Modelling
Prof. Peter Markowich is the recipient of the Wittgenstein Award 2000, the highest ranking Austrian Science Prize.