by M Capinski (Jagiellonian Univ., Poland) & N J Cutland (Univ. Hull, UK)

This book is an exposition of a new approach to the Navier-Stokes equations, using powerful techniques provided by nonstandard analysis, as developed by the authors. The topics studied include the existence and uniqueness of weak solutions, statistical solutions and the solution of general stochastic equations.

The authors provide a self-contained introduction to nonstandard analysis, designed with applied mathematicians in mind and concentrated specifically on techniques applicable to the Navier-Stokes equations. The subsequent exposition shows how these new techniques allow a quick and intuitive entrance into the mathematical theory of hydrodynamics, as well as provide a research tool that has proven useful in solving open problems concerning stochastic equations.

• Standard Preliminaries
• Nonstandard Preliminaries
• Loeb Measure and Integration Theory
• Weak Solutions of Navier-Stokes Equations
• Statistical Solutions of Navier-Stokes Equations
• Stochastic Navier-Stokes Equations
• Statistical Solutions of Stochastic Navier-Stokes Equations
• The Euler Equation