Applied Nonlinear Semigroups

A. Belleni-Morante and A.C. McBride

John Wiley and Sons, 1998. 273 pages

A Concise Guide to Semigroups and Evolution Equations

Aldo Belleni-Morante

World Scientific, 1994. 164 pages

Anyone who has been regularly attending the ICTT meetings during the past few years is well aware of the prominent role which rigorous (or quasi-rigorous) mathematics has come to play in transport theory. Invariably one hears at these meetings talks on existence theory for nonlinear (and in some cases linear) equations; asymptotics; numerical analysis; dynamical systems; and similar topics, all of which can be classified as mathematics but which are supposed to be accessible and useful to the general audience. This audience includes of course, in addition to mathematicians, engineers of various types, physicists, sometimes chemists, etc.

One of the strengths of transport theory is that it is an interdisciplinary science. The ICTT meetings have, indeed, fostered that venue (note that all sessions of the sixteen meetings held to date have been plenary; conservatives, such as your author, have vigorously opposed attempts to introduce parallelism.) This means that transport theorists with backgrounds in physics or mathematics are supposed to be able to listen to, understand, and profit from talks with strong mathematical bias. This goal has been admirably achieved, with many of the attendees whose training was not in mathematics making commendable efforts to learn enough mathematics (not to mention mathematics jargon) to be full participants in those sessions slanted towards rigor. At the risk of being redundant, I want to reiterate how this approach to transport theory has been the strength of the field, and has contributed to making the ICTT meetings as well as this journal paragons of interdisciplinarism.

The two books being reviewed here are authored (or in one case co-authored) by a perfect example of the interdisciplinary endeavor. Aldo Belleni-Morante, trained as a mathematician but now professor of engineering at the University of Firenze, is well known for his many contributions to transport theory throughout a long career. What the transport theory community may not be so well aware of is his many contributions to applied mathematics, especially in the area of linear and nonlinear semigroups. The two books under review are examples of this work; in both books the illustrative examples stress transport problems, and one can only assume that Aldo had in mind the transport theory community as an important subset of his readership.

Let me quote from the preface to Applied Nonlinear Semigroups:

Mathematical prerequisites have been kept to a minimum. Knowledge of the basics of real analysis and vector spaces should be sufficient. The book is intended to be largely self-contained; in particular, no prior knowledge of functional analysis is required, with all the necessary concepts being introduced in the extensive [62 page] first chapter.
In this preface, the authors go on to give a review of their own book which I can do no better than to paraphrase. Chapter 1 deals with spaces and operators, in particular Banach and Hilbert spaces over the reals. Operators and the functional calculus are introduced, with some of the applications involving the Lesbegue integral. (The basic concepts of measure theory and Lesbegue integration are given in Appendix A.) A physical presence is maintained throughout the chapter by attaching meaning to the various norms; the L1 norm might represent the number of particles in a system, while the L?norm might express the maximum value of the temperature, etc. Also, mathematical concepts which are often difficult for non-mathematicians to grasp are explained through simple illustrations. For example, early in the chapter the space of finite sequences is shown to be not complete and the importance of dealing with complete spaces is elucidated. Later in the chapter the Banach fixed-point theorem is stated and proved while the more useful Schauder theorem is stated without proof. (Throughout the book references are given to missing proofs.)

The theory of strongly continuous semigroups of linear operators is presented in Chapter 2. Motivation is provided and understanding enhanced through the introduction of several simple examples after which the Hille-Yosida theorem is quoted. As in Chapter 1, relatively simple examples are used to provide both motivation and understanding. Important topics such as dissipativity and perturbation of linear semigroups are also clearly presented.

Chapter 3 introduces the reader to semilinear semigroup theory (that is, the nonlinear term is treated as a perturbation of the generator of a strongly continuous semigroup). Then the concepts of strong, mild and weak solutions are introduced and differentiated. Lipschitz conditions, Frechet derivatives and a priori bounds are also explained and illustrative examples given.

Chapters 4 and 5 represent the "meat" of the book in that full nonlinear problems are discussed. In Chapter 4, dissipative operators are treated, and are connected physically to dissipative processes such as heat conduction, particle transport with absorption and Newton's laws with friction. Then in Chapter 5 the theory of Abstract Cauchy Problems involving nonlinear operators is developed.

Finally Chapter 6 is devoted to a number of important examples and applications, all but one involving particle transport. The exception is a study of the combustion of a solid fuel (which is actually a diffusion process).

Useful and challenging problems are given at the end of the chapters, making the book suitable as a text for an advanced course. It can also be read profitably by non-mathematicians wishing to make contact, either as spectators or laborers, in the field of nonlinear transport theory. This is a field I entered some ten years ago, under the tutelage of Reinhard Illner and Horst Lange. How much easier my task, and theirs, would have been had this book been available at that time.

The second book under review, A Concise GuideÖ(ACG), is oriented towards a somewhat different readership, namely one which begins with more mathematical sophistication. For example, four pages are devoted (Appendix A) to the topic of Lebesgue integration as contrasted with nine pages in Applied Nonlinear Semigroups (ANS). Similarly, Appendix B covers the topic of Banach spaces in eight pages as contrasted with the 63 pages in ANS.

Another major difference between the two books is that while ANS is primarily devoted to the study of nonlinear equations ACG is, except for the last (eighth) chapter and a brief example of a semilinear example in Chapter 1, concerned only with linear problems. Futhermore, Chapter 8 deals only with semilinear equations.

Chapter 1 is devoted to two examples of evolution problems as a way of setting the stage. Both are transport problems, the first linear, the second, as already mentioned, semilinear.

Thus motivated, the reader is propelled, in Chapter 2, into the study of semigroups of bounded, linear operators. (Here Gronwall's inequality is introduced on page 25, whereas in ANS it makes its first appearance on page 130.) Chapter 3 generalizes the results of Chapter 2 to the case of unbounded generators, where domain questions have to be studied in detail. A specific regularity condition is imposed on (the resolvents of) the generator A, namely that A be an element of the set G(1,0;X). In Chapter 4 the cases where A belongs to G(M,? ,X) and G'(1,0;X) are treated. (If you aren't familiar with these sets donít despair; they are carefully defined in the text.)

Chapter 5 deals with bounded perturbations to the generator and Chapter 6 with holomorphic semigroups. Then, in Chapter 7 an abstract version of the (linear) particle transport problem introduced in Chapter 1 as an example is studied. The author concludes this chapter with the statement that the method exemplified in this chapter "is a standard procedure when semigroup theory is used to study problems from applied sciences."

Each of these books can be studied profitably by different classes of this journal's readership. ANS would be of primary interest to engineers and scientists attempting to understand what those mathematicians are doing, while ACG would be more of interest to mathematicians wondering what scientists and engineers are up to. Both books are well written, without obvious errors. As already mentioned, ANS, which contains problems, would be appropriate as a text book, while CGS has no exercises included, being intended primarily as a research handbook.

Paul F. Zweifel