Advances in Mathematical Fluid Mechanics: Lecture Notes of the Sixth International School Mathematical Theory in Fluid Mechanics , Paseky, Czech Republic, Sept. 19-26, 1999 by Springer-Verlag Berlin and Heidelberg GmbH & Co. KG (Paperback, 2000)
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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 Canne 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 nlinear evolutionary problems. Particular attention is paid to the critical spaces that are invariant under the self-similar transform. For sufficiently small Reylds numbers, the conditional stability in the sense of Lyapuv 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.
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
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Springer-Verlag Berlin and Heidelberg GmbH & Co. K