7R4. Systems of Conservation Laws 2: Geometric Structures, Oscillations and Mixed Problems. - D Serre (Ecole Normale Superieure, Lyon, France). Cambridge UP, Cambridge, UK. 2000. 269 pp. ISBN 0-521-63330-3. $74.95.
Reviewed by Y Horie (Los Alamos Natl Lab, Group X-7, MS D413, Los Alamos NM 87545).
Conservation laws arise in many areas of continuum physics or mechanics and are described by certain types of nonlinear partial differential equations. Solving these PDEs for non-dissipative systems with appropriate initial and boundary values is known as the Cauchy problem or the mixed problem. This two-volume set of books is concerned with the modern mathematical treatment of the mixed problem. According to the author, the coverage of the book is comparable to that of Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (Springer, Berlin, 1984), but the subject matters are dealt with “with more detail, but less animation.” Though important, the phenomenon of relaxation and kinetic formulation are not included.
Volume 1 dealt with the fundamentals, including such topics as hyperbolicity, Lax shock, the Glimm scheme, and viscosity solution. The second volume, consisting of Chapters 8 through 15, is concerned with questions such as: the convergence of the viscosity method, stability for the Cauchy problem, admissible boundary conditions, and the multi-dimensional mixed problem. But the author cautions that “the materials in the second volume represent a significant part of the active research in the last fifteen years in the field, and are intended more for research workers and less for students.” Whether dealing with fundamentals or the state-of-the art research, this book is definitely written for post-graduate students and active researchers in mathematics. All the basic ideas are discussed fully, but for nonspecialists guiding from well-versed instructors may be helpful to wade through the materials. The readers are expected to have a good mathematical background in contemporary functional analysis. Keywords such as hyperbolic systems, shock waves, viscosity, and initial and boundary value problems are familiar to physical scientists and engineers, but the questions treated in the book are primarily mathematical: existence, uniqueness, convergence, stability, and consistency, etc.
The book, however, has something illuminating to offer even to those who may not have the appetite or requisite mathematics to follow theorems and lemmas in depth. It shows what it means to think with the equations from the mathematical point of view. Half-baked knowledge is always dangerous, but this reviewer, who is not a mathematician, was still able to gain new appreciation of and fresh insight into the kinds of questions he does not normally worry about. Good examples are the treatment of stability types and admissible boundary conditions on a finite domain in mathematical terms. His formal training ends at the level of classic books by such people as Courant and Friedrichs, Whitham, and Courant and Hilbert, but he could not help wonder at the mathematical transformation of the questions one used to think intuitively about on a physical basis. At the same time, as marked by Courant (Methods of Mathematical Physics, Volume 1), one fears an increasing gap between contemporary mathematics and the interest of many scientists and engineers.
Systems of Conservation Laws 2 is primarily for researchers in mathematics who are interested in the current state of knowledge and students in advanced graduate courses in PDEs. The book is also recommended to non-mathematicians as a means of getting a broad brush view of the conservation equations, nonlinear wave propagation, and shock waves from the mathematical point of view. The book is definitely recommended for university libraries.