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Nonstandard methods for stochastic fluid mechanics. (English) Zbl 0824.76003
Series on Advances in Mathematics for Applied Sciences 27. Singapore: World Scientific (ISBN 981-02-1710-2). xii, 227 p. (1995).
The book describes new approach to stochastic fluid mechanics. It mainly studies the classical Navier-Stokes equations for an incompressible fluid in a bounded region subject to external forces. Both the initial velocity and external force are assumed random, so the authors study stochastic Navier-Stokes equations. Although the subject of statistical hydromechanics is classic and has been widely developed by Hopf, Foias, Vishik to name the few, the authors look at it from original point of view provided by nonstandard analysis. In order to make the book self- contained, the authors give a detailed introduction to nonstandard analysis with an emphasis on the tools that they need the most (Loeb measure theory). This clearly written exposition of the nonstandard analysis covers about 60 pages. The rest 140 pages contain application of nonstandard technique for solving the classical deterministic Navier- Stokes equations (chapter 4) and stochastic Navier-Stokes equations (chapters 5-7). The final chapter 8 deals with the Euler equations.

76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
76M35 Stochastic analysis applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids