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Advances in Mathematical Fluid Mechanics - Lecture Notes of the Sixth International School Mathematical Theory in Fluid Mechanics, Paseky, Czech Republic, Sept. 19-26, 1999 (Paperback, 2000 ed.)
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Advances in Mathematical Fluid Mechanics - Lecture Notes of the Sixth International School Mathematical Theory in Fluid Mechanics, Paseky, Czech Republic, Sept. 19-26, 1999 (Paperback, 2000 ed.)
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This book consists of six survey contributions that are focused on
several open problems of theoretical fluid mechanics both for
incompressible and compressible fluids. The first article "Viscous
flows in Besov spaces" by M area Cannone ad dresses the problem of
global existence of a uniquely defined solution to the
three-dimensional Navier-Stokes equations for incompressible
fluids. Among others the following topics are intensively treated
in this contribution: (i) the systematic description of the spaces
of initial conditions for which there exists a unique local (in
time) solution or a unique global solution for small data, (ii) the
existence of forward self-similar solutions, (iii) the relation of
these results to Leray's weak solutions and backward self-similar
solutions, (iv) the extension of the results to further nonlinear
evolutionary problems. Particular attention is paid to the critical
spaces that are invariant under the self-similar transform. For
sufficiently small Reynolds numbers, the conditional stability in
the sense of Lyapunov is also studied. The article is endowed by
interesting personal and historical comments and an exhaustive
bibliography that gives the reader a complete picture about
available literature. The papers "The dynamical system approach to
the Navier-Stokes equa tions for compressible fluids" by Eduard
Feireisl, and "Asymptotic problems and compressible-incompressible
limits" by Nader Masmoudi are devoted to the global (in time)
properties of solutions to the Navier-Stokes equa and three tions
for compressible fluids. The global (in time) analysis of two
dimensional motions of compressible fluids were left open for many
years."
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