7R8. Advances in the Theory of Shock Waves. Progress in Nonlinear Differential Equations and Their Applications, Vol 47. - H Freistuhler (Max Planck Inst for Math in the Sci, Leipzig, 04103, Germany) and A Szepessy (Dept of Math, Royal Inst of Tech, Stockholm, 100 44, Sweden). Birkhauser Boston, Cambridge MA. 2001. 516 pp. ISBN 0-8176-4187-4. $79.95.
Reviewed by Y Horie (Los Alamos Natl Lab, Group X-7, MS D413, Los Alamos NM 87545).
The genesis of this book is the summer school for young specialists, organized at Kochel am See, Germany in May 1999, and held in the context of the European research network “Hyperbolic Systems of Conservations Laws.” The five articles in this book are related to lectures at the school and are concerned with the mathematical theory of shock waves. But, most of the articles go far beyond the level of detail expected for lecture notes. For example, articles by J Smoller and B Temple, “Shock wave solutions of the Einstein equations: a general theory with examples,” and by K Zumbrun, “Multidimensional stability of planar viscous shock waves,” are book length: 153 and 208 pages, respectively, out of 506 pages for the book. They are a comprehensive summation of their recent works. The remaining three articles are “Well-posedness theory for hyperbolic systems of conservations laws” (T-P Liu), “Stability of multidimensional shocks” (G Me´tivier), and “Basic aspects of hyperbolic relaxation systems” (W-A Yong).
The book is primarily intended for specialists in mathematics and applied mathematics in the theory of shock waves. However, materials are not always academic (this characterization really depends on one’s interest and background). Selective technical applications are discussed, eg, in Yong and Zumbrun. Also, the Zumbrun’s section on open problems is of interest even to the non-specialists. The book is obviously of interest to technical people. One outstanding example is the article by Smoller and Temple where the primary interest is shock wave solutions of the Einstein equation. This is a very readable article even for non-specialists and can be appreciated with introductory graduate level background in general relativity.
The book collects very exciting and unexpected (according to the editors) developments in the field. Me´tivier discusses the stability problem for multidimensional inviscid shock waves in a novel treatment through use of paradifferential calculus. Liu introduces new entropy (distance) functionals for consideration of the well-posedness of conservation systems. These functionals compare solutions to different data by direct reference to their wave structure. Yong provides a systematic description of the fundamental properties of systems with source terms divided by a small parameter. He finds several basic structural conditions aiming at the existence of a well-behaved limit as the parameter tends to zero. Also, he demonstrates a first general theorem on the existence of a traveling wave solution. Zumbrun presents a comprehensive treatment of shock stability for systems including both multidimensional and regularizing effects based on the Evans function technique. No simple summary of this article is adequate for the amount of materials presented in it. The main contribution, by his own word, is to reduce the study of viscous shock stability to the study of generalized spectrum conditions: the first a purely ODE problem and the second, mainly linear algebraic. The latter, augmented with a viscous condition, is verified as a necessary condition for stability, and furnishes a myriad of examples of both multi- and one-dimensional viscous instability through the literature on inviscid shock dynamics. The former allows the numerical detection of Poincare-Hopf or other interesting bifurcations not captured by inviscid analysis.
The article by Smoller and Temple on classes of shock wave solutions to the Einstein equations is exciting and provocative. The characterization by the editors that it constitutes a whole new area of research is not far from the truth. They looked at the question that many of us pondered about, but did not (or could not) do anything about it: that is, could the big bang be a shock wave? If so, what are the mathematical and physical conditions that allow such an interpretation? It is provocative, because it offers alternative views of the standard model for cosmology—that there is a shock wave present at the leading edge of the universe. It is speculative in that it assumes a state outside the shock wave (that is, outside the edge of the universe) and that “the space-time before the explosion occurred took a long time getting into the pre-explosion configuration.” Plus, because of irreversibility in the shock, “it becomes impossible to reconstruct the details of the early explosion from present data, at least at the level of the continuum model.”
Because of the “expansive” article by Smoller and Temple, the book, Advances in the Theory of Shock Waves, is recommended not only for the specialists in the mathematical theory of shock waves, but also those in cosmology as well as anyone whose thinking extends to the leading edge of the universe.
