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Instabilities and fronts in extended systems. (English) Zbl 0732.35074

Princeton Series in Physics. Princeton, NJ: Princeton University Press. xii, 196 p. $ 29.95 (1990).
Fluid mechanics is one of the richest sources of mathematical equations with interesting nonlinear phenomena and many challenging questions. We summarize the contents of the book. The introductory chapter gives some of the basic equations like Boussinesq’s, Ginsburg-Landau’s and notions as attraction, evolution of infinite-dimensional systems. In chapter 2 the idea of small solutions is developed which involves invariant manifolds and perturbation methods. In chapter 3 we find a systematic exploration of bifurcation theory where the most common bifurcations of a point attractor are classified. This may still play a part in some infinite-dimensional systems. Constant solutions can lead by bifurcation to (quasi-)stationary solutions. Linear stability theory is used a.o. in the Swift-Hohenberg equations (chapter 4). Next is one of the main chapters where the consequences are discussed of linear instability of stationary solutions. This may lead to Eckhaus instability. A brief chapter 6 deals with multi-scaling, the last chapter discusses fronts extensively. It is remarkable how on this difficult subject the authors have presented their material in a concise and at the same time lucid way. This is a very useful addition to the literature.

MSC:

35Q35 PDEs in connection with fluid mechanics
37G99 Local and nonlocal bifurcation theory for dynamical systems
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
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