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Mathematical analysis and numerical methods for sciences and technology. Vol. 6: Evolution problems II. The Navier-Stokes and transport equations in numerical methods. With the collaboration of Claude Bardos, Michel Cessenat, Alain Kavenoky, Patrick Lascaux, Bertrand Mercier, Olivier Pironneau, Bruno Scheurer, Rémi Sentis. Transl. from the French by Alain Craig. Transl. Ed.: Ian N. Sneddon. (English) Zbl 0802.35001

Berlin etc.: Springer-Verlag. xii, 485 p. DM 205.00; öS 1.599.00; sFr 203.00 /hc (1993).
This book is the final volume of the 6-volume treatise written under the leadership of R. Dautray and J.-L. Lions, with collaboration of a group of well known French mathematicians. It consists of 3 chapters, extensive indices (including a list of physical problems considered in all the 6 volumes, and a 33-page cumulative index).
This volume starts with chapter XIX on the linearized Navier Stokes equations. Here the corresponding functional spaces are introduced, the weak formulations (for the stationary and the evolutionary case) are given and theorems on existence, unicity and of regularity are formulated. Parts of the proofs are shown, or corresponding references cited. In the following chapter XX on numerical methods for evolution problems (of about 170 pages) – after an introduction on convergence, consistency and stability (Lax equivalence theorem with proof) – all basic partial differential equations are considered: heat conduction, wave, advection, first order hyperbolic, and collisional transport equations. The final chapter XXI treats transport problems (for neutrons) in great detail (more than 200 p.), presenting the physical problems, studying existence, unicity, spectral properties and asymptotic behaviour as well as approximation of the neutron transport equation by the diffusion model. In such a great work some shortcomings seem inevitable. Remarkable is in this respect the tacit identification of technology and science with physics; surely one can have different opinions on the selection of cited and proved material. Curious is the definition of stiff problems which refers only to the ratio of the maximal and minimal (by absolute value) eigenvalues; sometimes the typographic differences between the greek letters in the text and in the formulae are disturbing. (However, misprints are rare.)
Interesting is a comparison with the Methods of Mathematical Physics by Courant and Hilbert (the present volume being mostly deductive, using, of course, the modern achievements of functional analysis, not aiming at an introduction into say the theory of partial differential equations; the practical problems for which the theory and numerical methods are developed also show the progress of time – here the atom industry is delivering a visible part of the problems). Yet another great work comes to mind which is not cited here – that of E. Zeidler on Functional Analysis (who gives full proofs, illustrates by mathematical examples, and, by the way, sheds light on mathematical history and culture).
The great value of the whole edition is visible already from this single volume: A wealth of mathematical ideas and techniques is presented, and illustrated on a series of specific physical examples.

MSC:

35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65N99 Numerical methods for partial differential equations, boundary value problems
65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
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