1.1. From the Foreword.

This book contains a series of lectures on the modelling of collisions delivered at a summer school which was held in September 1994 in Saint-Malo. These lectures were delivered by a physicist, Alain Decoster, and two Peter Markowich and Benoît Perthame. A Decoster discussed during three lectures the modelling of collision terms in fluid models of plasmas. On the other hand, B Perthame devoted his two lectures to the study of collision models in dilute gases, while P Markowich developed quantum hydrodynamic aspects during one lecture.

1.2. Review by: Reinhard Illner.
Mathematical Reviews MR1650315 (2000b:76092).

This book consists of three parts, in which three (groups of) authors give introductions to special parts of kinetic theory and summaries of recent original results.
...
Part 2 of the book, authored by Perthame and containing also contributions by Desvillettes, contains a concise yet interesting introduction into the modern theory of the Boltzmann equation, including a discussion of the classical properties, a Fourier-transformation-based proof of the uniqueness of Maxwellian equilibria, and a statement on the global existence of renormalised solutions. A discussion of derivations of kinetic equations from hierarchies (including the Boltzmann-Grad limit) is also given. The most original part of this section of the book is the material on other collision models, such as the generalisation to polyatomic gases, the Enskog equation, the case of inelastic collisions and the inclusion of rotational degrees of freedom. The Euler limits in the polyatomic and Enskog scenarios are also discussed.

2.1. From the Publisher.

This book gives a general presentation of the mathematical and numerical connections kinetic theory and conservation laws based on several earlier works with P L Lions and E Tadmor, as well as on more recent developments. The kinetic formalism approach allows the reader to consider Partial Differential Equations, such as some nonlinear conservation laws, as linear kinetic (or semi-kinetic) equations acting on a nonlinear quantity. It also aids the reader with using Fourier transform, regularisation, and moments methods to provide new approaches for proving uniqueness, regularising effects, and a priori bounds.

Special care has been given to introduce basic tools, including the classical Boltzmann formalism to derive compressible fluid dynamics, the study of oscillations through the kinetic defect measure, and an elementary construction of solutions to scalar conservation laws. More advanced material contains regularising effects through averaging lemmas, existence of global large solutions to isentropic gas dynamics, and a new uniqueness proof for scalar conservation laws. Sections are also devoted to the derivation of numerical approaches, the 'kinetic schemes', and the analysis of their theoretical properties.

2.2. From the Preface.

In these lectures, we survey various relations between some hyperbolic conservation laws and kinetic equations, i.e. the first-order partial differential equations in which the advection velocity is a free variable, as they arise classically in kinetic physics (Boltzmann and Vlasov theories). The long-term motivation behind the study of such relations is to prove the compressible limit of the Boltzmann equation.

2.3. From the Overview.

We begin with a rapid and informal overview of the kinetic approach to conservations laws. It is intended to present quickly the examples where the same approach can be developed, and to present the main ideas behind the theory. We also try to show its applications and limitations. More complete proofs and consequences in the hyperbolic case are developed later, as well as a more gradual introduction of the tools.

The idea we would like to emphasise mainly is that the kinetic approach has two levels; the kinetic formulations where a full description is given when a large enough family of entropies is available, and the more general kinetic representation, which is based on a single entropy.

The other point we emphasise is that the kinetic approach not only applies to hyperbolic equations but also to the parabolic case and to some variational problems. Therefore the analogy with the Boltzmann equation is somewhat limited and we prefer to proceed directly through entropies.

2.4. Review by: Michael Junk.
zbMATH 1030.35002.

With this book, the author gives an overview of the mathematical connections between kinetic theory and conservation laws. The main advantage of such a connection is that tools from linear theory can be applied to nonlinear problems because the kinetic approach allows to rewrite the original nonlinear equation as a linear equation acting on a nonlinear quantity. Tools like Fourier transform, moment methods, and regularisation by convolution can then be applied and either yield new results (like regularity or a priori bounds for the solution of the nonlinear equation), or known results are recovered with a different proof (like existence and uniqueness for scalar conservation laws). Apart from giving a new perspective on the theory of conservation laws, the author also shows how the kinetic approach can be used to construct numerical methods with interesting features.

A nice feature of the book is the rapid introduction into kinetic formulations presented in the first chapter. By skipping many technical details, the reader quickly learns about typical tools and applications. This motivates to read further and to collect the missing details in the subsequent chapters. The different kinetic techniques are explained using the following examples: scalar conservation laws, isentropic gas dynamics and 2 × 2 systems, multidimensional Saint-Venant system, multidimensional gas dynamics, Ginzburg-Landau line energies.

The second chapter is devoted to the study of a particular indicator function which appears as equilibrium distribution in the kinetic approach to scalar conservation laws. It is shown how the indicator function can be used to characterise weak limits of nonlinear functions which links the approach to the concept of Young measures.

In chapters three to five, the paradigmatic example of multidimensional scalar conservation laws is carefully presented and examined. The central tool in chapters three and four is the hydrodynamical limit which connects the conservation law with a particular semilinear relaxation-type kinetic equation. This equation plays a similar role as the viscous extension of the conservation law but has the advantage that the type of the equation is not changed. It allows to obtain extensions of the classical Kruzkov existence and uniqueness results. In chapter five, the kinetic formulation is used to prove various regularity effects for conservation laws and chapter six concentrates on the extension of the kinetic formulation to the finite volume method.

In chapter seven, the approach is extended to the study of the 2 × 2 systems system of isentropic gas dynamics in a single space dimension. An important difference to the case of scalar conservation laws is the loss of a purely kinetic transport operator in the kinetic formulation which now depends also on the macroscopic velocity. Even though fewer tools are available for such semi-kinetic equations, it is still possible to use the approach to prove a priori bounds, regularising effects and time decay. The presentation closes with a short chapter on kinetic schemes for equations of gas dynamics.

Altogether, the book is a good introduction into the theory of kinetic formulations of conservation laws written by the leading expert in the field. A list of open problems and the extensive bibliography also makes it a starting point for further research.

2.5. Review by: Manuel Portilheiro.
Mathematical Reviews MR2064166 (2005d:35005).

In this book the author presents the theory of kinetic formulation of conservation laws which the author and his collaborators developed in the last decade ...

This is a beautiful theory with many applications. It consists of introducing a (so-called) kinetic variable (by analogy with the Boltzmann equation) to transform a multidimensional conservation law into a linear transport equation, the associated kinetic equation. The right-hand side of the kinetic equation is a measure, but the equation is now linear. This is a great advantage in many respects, namely bringing new techniques to the field of conservation laws.

After a brief overview, the author develops the theory and tools of the kinetic formulation. The author shows the equivalence between entropy solutions of the conservation law and those of the kinetic equation. Using the kinetic formulation, he then obtains an existence result and a uniqueness result ... for the entropy solutions of conservation laws.

The book is written in a very clear way. Some of the proofs have been simplified from their original publication, which gives the work an added interest. The author manages to show how beautiful and simple the theory is and yet how powerful it can be. It is doubtlessly an essential title for anyone interested in conservation laws.

3.1. From the Publisher.

This book presents models written as partial differential equations and originating from various questions in population biology, such as physiologically structured equations, adaptive dynamics, and bacterial movement. Its purpose is to derive appropriate mathematical tools and qualitative properties of the solutions (long time behaviour, concentration phenomena, asymptotic behaviour, regularising effects, blow-up or dispersion). Original mathematical methods described are, among others, the generalised relative entropy method - a unique method to tackle most of the problems in population biology, the description of Dirac concentration effects using a new type of Hamilton-Jacobi equations, and a general point of view on chemotaxis including various scales of description leading to kinetic, parabolic or hyperbolic equations.

3.2. From the Preface.

These lecture notes are based on several courses and lectures given at different places (University Pierre et Marie Curie, University of Bordeaux, CNRS research groups GRIP and CHANT, University of Roma I) for an audience of mathematicians. The main motivation is indeed the mathematical study of Partial Differential Equations that arise from biological studies. Among them, parabolic equations are the most popular and also the most numerous (one of the reasons is that the small size, at the cell level, is favourable to large viscosities). Many papers and books treat this subject, from modelling or analysis points of view. This oriented the choice of subjects for these notes towards less classical models based on integral equations (where PDEs arise in the asymptotic analysis), transport PDEs (therefore of hyperbolic type), kinetic equations and their parabolic limits.

The first goal of these notes is to mention (and describe very roughly) various fields of biology where PDEs are used; the book therefore contains many examples without mathematical analysis. In some other cases complete mathematical proofs are detailed, but the choice has been a compromise between technicality and ease of interpretation of the mathematical result. It is usual in the field to see mathematics as a black box where to enter specific models, often at the expense of simplifications. Here, the idea is different; the mathematical proof should be close to the 'natural' structure of the model and reflect somehow its meaning in terms of applications.

Dealing with first order PDEs, one could think that these notes are relying on the burden of using the method of characteristics and of defining weak solutions. We rather consider that, after the numerous advances during the 1980s, it is now clear that 'solutions in the sense of distributions' (because they are unique in a class exceeding the framework of the Cauchy-Lipschitz theory) is the correct concept. They allow for abstract manipulations, which we justify in the first section of the chapter 'General mathematical tools', and we use them freely throughout the text. Then one can concentrate on the intimate mathematical structure of the models.

3.3. Review by: Reinhard Illner.
Mathematical Reviews MR2270822 (2007j:35004).

Mathematical biology has grown to be a very large field; its expansion is driven by the new "golden age" of research in the biological sciences, such as microbiology, biochemistry, genomics and proteomics, epidemiology (in itself driven by the appearance of new diseases), and others. Mathematics, as a tool and as the natural language for modelling complex dynamics, is following and paralleling this research in a systematic way as biologists turn to mathematicians to model and analyse, and mathematicians turn to biologists in search of new, exciting applications of their methodology.

Many traditional biology contexts lead to systems of ODEs or systems of delay-differential equations, for which one then investigates the existence and stability of steady states, limit cycles, or the occurrence of chaos. More detailed modelling efforts may lead to equations which include diffusion or transport terms. The former are commonly included, usually with constant diffusivities, where random ("diffusive") dispersion of a population in space is assumed; the latter can arise from a variety of modelling assumptions, and not necessarily from spatial transport processes alone.
...
This wonderful 200-page monograph by Benoît Perthame, based on his series of lectures, provides a clear and concise, yet comprehensive treatment of biological scenarios in which transport terms enter the modelling cycle, and of the tools for their analytical and numerical treatment. The book contains six chapters, loosely divided by types of transport but united by the mathematical methodology.
...
This book is an excellent, state-of-the-art survey of, and introduction to, transport problems arising in biology; at the same time the methodology is relevant for transport equations arising in other contexts. The text is well suited for a graduate course or a seminar.

4.1. From the Publisher.

This book presents several fundamental questions in mathematical biology such as Turing instability, pattern formation, reaction-diffusion systems, invasion waves and Fokker-Planck equations. These are classical modelling tools for mathematical biology with applications to ecology and population dynamics, the neurosciences, enzymatic reactions, chemotaxis, invasion waves etc. The book presents these aspects from a mathematical perspective, with the aim of identifying those qualitative properties of the models that are relevant for biological applications. To do so, it uncovers the mechanisms at work behind Turing instability, pattern formation and invasion waves. This involves several mathematical tools, such as stability and instability analysis, blow-up in finite time, asymptotic methods and relative entropy properties. Given the content presented, the book is well suited as a textbook for master-level coursework.

4.2. From the Preface.

This book is based on lecture notes written for a course I gave at Université Pierre et Marie Curie at the master level (second year); these notes have been extended with more advanced material. They are an attempt to show why Partial Differential Equations (PDEs in short) are used in biology and how one can analytically derive qualitative properties of the solutions, which are relevant for applications. Of course, many types of PDEs are used and these notes are restricted to parabolic equations, while transport equations were already presented in [B Perthame, Transport Equations Arising in Biology].

To give some order to the content, an organising principle was needed. The leading idea has been to explain in which circumstances the solutions of parabolic equations can exhibit interesting patterns. Indeed, smoothing effects and time decay usually lead to 'flat' solutions. I classify the parabolic equations of interest in Chap. 1, according to two main mathematical structures; Lotka-Volterra equations or reaction-diffusion equations, as they appear in models of chemical reactions. This idea was already present in a course taught by Jost [Mathematical Methods in Biology and Neurobiology] in École Normale Supérieure some years ago and it also leads my progression throughout these notes.

What ingredients are required to explain patterns in reaction-diffusion systems as they are observed in nature? I begin by explaining when patterns cannot occur (when dissipation and relaxation dominates), then I explain gradually the most standard theories of pattern formation. Because of the smoothing properties of the linear heat equation, its solutions cannot exhibit interesting patterns. Nonlinearities are not always sufficient as is shown in Chap. 2; small Lipschitz bounds in bounded domains or entropy properties are major obstacles to pattern formation. The manipulations here are formal; to justify these, careful attention is brought to the notion of weak solutions in Chap. 3.

A first class of problems, leading to interesting behaviours, occurs in the full space because long range propagation phenomena can occur. Already in one space dimension travelling wave solutions are remarkable examples. Avery elementary presentation is given in Chap. 4 for the famous models à la Fisher/KPP. The subject has been studied since a long time and more complete texts are [P C Fife, Mathematical Aspects of Reacting and Diffusing Systems, A I Volpert and A V Volpert, Traveling Wave Solutions of Parabolic Systems]. Spikes, spots (in higher dimensions), and pulses are other related patterns, which I present in Chap. 5; when these spots move with instabilities, one can observe various types of patterns that are also presented.

Too large nonlinearities are not an option to generate patterns because blow-up in finite time can occur. This is the matter of Chap. 6 where I present several methods to prove blow-up.

The main method to obtain patterns remains the Turing instability, one of the most counterintuitive results in the theory of PDEs. When the ordinary differential system is stable, diffusion can turn it unstable when diffusion coefficients of the system components are very different. This matter is presented in Chap. 7, with several examples and illustrations. Fokker-Planck equations are a class of PDEs, which extend the field of reaction-diffusion systems by including drift terms, and which are important for their use in many fields. In particular, because the example of chemotaxis (Keller-Segel system) appears several times in these notes, a general presentation is given in Chap. 8 These are Kolmogorov equations for Stochastic Differential Equations. When jump processes are considered (these also appear in several areas of biology), one arrives at a simpler class of integral equations, which can be handled rigorously; these integral equations also serve to prove existence of solutions of the Fokker-Planck equations by an asymptotic procedure for small jumps. This matter is presented in Chap. 9.

In addition to the derivation of Fokker-Planck equations, small parameters play an important role in pattern formation and lead to an asymptotic analysis of what happens when these small parameters vanish. The mathematical tools to handle this subject are often more advanced (such as viscosity solutions to Hamilton-Jacobi equations [G Barles, Solutions de viscosite des equations de Hamilton-Jacobi]) and these notes mention only a few singular perturbation problems. A single example is treated in detail, namely the derivation of the Stefan problem from reaction-diffusion systems. This is performed in Chap. 10.

Besides these singular perturbation problems, and among many others, the use of parabolic equations in tissue growth modelling, in developmental biology and in neurosciences is other important topic that uses material beyond the scope of the present notes.

The material for these notes comes from very different sources, articles, books, courses, conference talks, and discussions. I tried to limit the number of references and I did not try to be exhaustive. I indicate in footnotes some basic articles, hoping that they are sufficient to initiate a search in each particular subject.

4.3. Review by: Jonathan Zinsl.
zbMATH 1333.35001.

This book presents a variety of phenomena arising in the analysis of partial differential equations modelling of biological, physical and chemical processes. Specifically (and in contrast to the author's book [Transport equations in biology (2007)]), it is focussed on (in many cases nonlinear) parabolic equations and phenomena related to pattern formation such as Turing instability or travelling waves. In addition, this book gives a concise introduction to more fundamental topics in the theory of partial differential equations, for example, notions of weak solutions or blow-up of solutions. The specific modelling aspects of each set of equations are presented in detail; furthermore, the mathematical results are put into connection with their biological background and in most cases illustrated by simulations. This book can well serve as a textbook for a course on master's level. Exercise problems are given in each chapter.

5.1. From the Publisher.

This book presents the state of the art in mathematical research on modelling the mechanics of biological systems - a science at the intersection between biology, mechanics and mathematics known as mechanobiology. The book gathers comprehensive surveys of the most significant areas of mechanobiology: cell motility and locomotion by shape control (Antonio DeSimone); models of cell motion and tissue growth (Benoît Perthame); numerical simulation of cardiac electromechanics (Alfio Quarteroni); and power-stroke-driven muscle contraction (Lev Truskinovsky).

Each section is self-contained in terms of the biomechanical background, and the content is accessible to all readers with a basic understanding of differential equations and numerical analysis. The book disentangles the phenomenological complexity of the biomechanical problems, while at the same time addressing the mathematical complexity with invaluable clarity. The book is intended for a wide audience, in particular graduate students and applied mathematicians interested in entering this fascinating field.

5.2. Note.

Benoît Perthame contributed Chapter 2, Models of Cell Motion and Tissue Growth.

5.3. Abstract to Chapter 2.

The mathematical description of cell movement, from the individual scale to the collective motion, is a rich and complex domain of biomathematics which leads to several types of questions and partial differential equations. For instance, bacteria move by run-and-tumble movement, which is well described, at the cell scale, by a kinetic equation in the phase coordinates. At the population scale, chemotactic effects lead to the famous parabolic Keller-Segel system, and the many improvements of it that have been addressed recently.

When considering living tissues, concepts issued from mechanics arise. Notions of pressure, phases, incompressibility are used in systems which carry the typical parabolic and hyperbolic characters of fluid mechanics. Their complexity is directly related to the details in the biological description and opens numerous mathematical questions which are poorly understood.
The various process involved in cell movements can be considered at the cell scale, at the population scale and, for tissues, at the organ scale. This leads to study singular perturbation problems of various types. For tumour growth, the tumour boundary can appear as a free boundary or as an internal layer.

5.4. From the Introduction to Chapter 2.

The mathematical description of cell movement from the individual scale to the collective motion, is a rich and complex domain of biomathematics which has been treated for a long time and with a rich variety of Partial Differential Equations. The goal, in this Chapter, is to give a flavour of the domain, showing the miscellaneous types of equations that can be encountered, the qualitative behaviour of solutions and some mathematical aspects. There are two general views which lead the line of the chapter:

Present the domain of cell populations along the scales. We depart from the cell individual behaviour, with the kinetic (mesoscopic) scale. This leads to the various forms of chemotaxis of swimming bacteria. Then, compressible models are presented as well as their incompressible limit. In some sense, this plan follows the traditional view of fluid mechanics.

Explain how individual cells, which can be bacteria, interacting in a rather elementary way with their environment (secreting chemical substances and reacting to them) can generate collective behaviours which are complex and seem to result from organised strategies.

Several examples of reaction-diffusion equations, where the mechanism is due
to interaction of cells with the environment through nutrients or other physical effects, are presented in [other works], showing mechanisms which underlie the pattern formation ability of such mechanisms. Reaction-diffusion equations are indeed usually encountered when considering populations of cells. However, many more types of equations, with very diverse nonlinearities can occur depending of the scale of interest.

At the individual scale, bacteria swim by a series of straight jumps followed by a fast reorganisation of their flagella which lead to a new direction of jump. This trajectory is called run-and-tumble and is well described by a kinetic (linear Boltzman) equation in the phase space (position, velocity). A major issue here is to explain the modulation of these changes of direction depending on the environment. This environment may be characterised by the concentration of a chemoattractant and this results in a dynamics called chemotaxis where cells move preferentially toward the chemoattractant. Cells can emit the chemoattractant themselves, and this leads to nonlinear systems.

Multiscale analysis allows to see the process at the population size and we arrive at a Fokker-Planck equation for cell number density coupled to the diffusion equation for the chemoattractant concentration. The resulting set of equations is called the Keller-Segel system. This famous system has attracted a lot of attention because of the complex patterns exhibited by solutions and in particular the blow-up phenomena which leads the cells to a pointwise concentration. Also, many improvements have been addressed recently which better fit some experimental observations.

When considering living tissues, which are denser ensembles of cells, fibres, liquids and molecules, these concepts are not enough, even if chemotaxis is important also. Then, the description uses concepts issued from mechanics and multiphase flows. Notions of pressure, Darcy's law, incompressibility are used in systems which carry parabolic and hyperbolic characters. Their complexity is directly related to the details in the biological description and opens numerous mathematical questions. A recent and comprehensive presentation of tissue mechanics can be found in [51]. The case of tumour growth has been attracted a huge literature recently and several surveys are available.

The domain of cell movement at the population scale is so vast that many aspects cannot be treated in this chapter. For instance, we do not touch the question of interaction between cell movements and the surrounding fluid, or chemotaxis in fluids neither how cilia or flagella direct the motion of cells. The internal description of crawling cells are also wide subjects of present interest which are treated in the other chapters of this Lecture Note.