Nonlinear evolution equations : kinetic approach / Nina B. MaslovaType de document : MonographieCollection : Series on advances in mathematics for applied sciences, 10Langue : anglais.Pays : Singapour.Éditeur : Singapore : World Scientific, 1993Description : 1 vol. (193 p.) ; 23 cmISBN : 9810211627.ISSN : 1793-0901.Bibliographie : Bibliogr..Sujet MSC : 76P05, Rarefied gas flows, Boltzmann equation in fluid mechanics
82C40, Statistical mechanics, structure of matter, Kinetic theory of gases in time-dependent statistical mechanics
35L60, PDEs - Hyperbolic equations and hyperbolic systems, First-order nonlinear hyperbolic equations
35Q35, PDEs of mathematical physics and other areas of application, PDEs in connection with fluid mechanics
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The main purpose of this excellent book is to present a collection of results that clarify the role of kinetic equations in nonlinear dynamics. The starting point is the connection between the fluid dynamic and the kinetic levels. Several recent investigations gave an unexpected marginal result that fluid dynamical equations can be approximated by kinetic models, not motivated physically at all and relatively simple from the mathematical and computational point of view. (MathSciNet).
The author called this book a book on nonlinear evolution equations, but this is really a book on the Boltzmann equation and its linearized cousin, with a few side remarks on applications of kinetic models. The book is a little gem which contains quite a few results that were difficult to find before, because they were published either not at all or only in Russian. Thus the job of reviewing became rather exciting as this reviewer worked his way through the book. The central and strongest part of the monograph is the detailed and lengthy discussion of steady boundary value problems for linear and nonlinear Boltzmann equation. Nina Maslova demonstrates great skill in the design and use of suitable function spaces for these problems. The weaknesses of the book are not so much related to content but rather to form. The notation is very compressed and combines with a sometimes confusing enumeration scheme. The English is rough and the proofs are brief, making for challenging reading. Nevertheless, this book should be on the shelf of anybody doing research in kinetic theory. (Zentralblatt)