Carlo Marchioro - Böcker

This book deals with fluid dynamics of incompressible non-viscous fluids. The main goal is to present an argument of large interest for physics, and applications in a rigorous logical and mathematical setup, therefore avoiding cumbersome technicalities. Classical as well as modern mathematical developments are illustrated in this book, which should fill a gap in the present literature. The book does not require a deep mathematical knowledge. The required background is a good understanding of classical arguments of mathematical analysis, including the basic elements of ordinary and partial differential equations, measure theory and analytic functions, and a few notions of potential theory and functional analysis. The contents of the book begins with the Euler equation, construction of solutions, stability of stationary solutions of the Euler equation. It continues with the vortex model, approximation methods, evolution of discontinuities, and concludes with turbulence.

Fluid dynamics is an ancient science incredibly alive today. Modern technol­ ogy and new needs require a deeper knowledge of the behavior of real fluids, and new discoveries or steps forward pose, quite often, challenging and diffi­ cult new mathematical {::oblems. In this framework, a special role is played by incompressible nonviscous (sometimes called perfect) flows. This is a mathematical model consisting essentially of an evolution equation (the Euler equation) for the velocity field of fluids. Such an equation, which is nothing other than the Newton laws plus some additional structural hypo­ theses, was discovered by Euler in 1755, and although it is more than two centuries old, many fundamental questions concerning its solutions are still open. In particular, it is not known whether the solutions, for reasonably general initial conditions, develop singularities in a finite time, and very little is known about the long-term behavior of smooth solutions. These and other basic problems are still open, and this is one of the reasons why the mathe­ matical theory of perfect flows is far from being completed. Incompressible flows have been attached, by many distinguished mathe­ maticians, with a large variety of mathematical techniques so that, today, this field constitutes a very rich and stimulating part of applied mathematics.

In this monograph we present a review of a number of recent results on the motion of a classical body immersed in an infinitely extended medium and subjected to the action of an external force.