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Five leading specialists reflect on different and complementary
approaches to fundamental questions in the study of the Fluid
Mechanics and Gas Dynamics equations. Constantin presents the Euler
equations of ideal incompressible fluids and discusses the blow-up
problem for the Navier-Stokes equations of viscous fluids,
describing some of the major mathematical questions of turbulence
theory. These questions are connected to the
Caffarelli-Kohn-Nirenberg theory of singularities for the
incompressible Navier-Stokes equations that is explained in
Gallavotti's lectures. Kazhikhov introduces the theory of strong
approximation of weak limits via the method of averaging, applied
to Navier-Stokes equations. Y. Meyer focuses on several nonlinear
evolution equations - in particular Navier-Stokes - and some
related unexpected cancellation properties, either imposed on the
initial condition, or satisfied by the solution itself, whenever it
is localized in space or in time variable. Ukai presents the
asymptotic analysis theory of fluid equations. He discusses the
Cauchy-Kovalevskaya technique for the Boltzmann-Grad limit of the
Newtonian equation, the multi-scale analysis, giving the
compressible and incompressible limits of the Boltzmann equation,
and the analysis of their initial layers.
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