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."
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