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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

  • Book
  • © 2000

Overview

Editors:
  1. Josef Málek

    1. Mathematical Institute, Charles University, Prague 8, Czech Republic

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  2. Jindřich Nečas

    1. Mathematical Institute, Charles University, Prague 8, Czech Republic
      Department of Mathematical Sciences, Northern Illinois University, DeKalb, USA

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  3. Mirko Rokyta

    1. Mathematical Institute, Charles University, Prague 8, Czech Republic

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  • A collection of survey articles from top specialists in fluid mechanics

  • Includes supplementary material: sn.pub/extras

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Table of contents (6 chapters)

  1. Front Matter

    Pages I-IX
    Download chapter PDF

  2. Viscous Flows in Besov Spaces

    • Marco Cannone
    Pages 1-34

  3. The Dynamical Systems Approach to the Navier-Stokes Equations of Compressible Fluids

    • Eduard Feireisl
    Pages 35-66

  4. Adaptive Wavelet Solvers for the Unsteady Incompressible Navier-Stokes Equations

    • Michael Griebel, Frank Koster
    Pages 67-118

  5. Asymptotic Problems and Compressible-Incompressible Limit

    • Nader Masmoudi
    Pages 119-158

  6. Weighted Spaces with Detached Asymptotics in Application to the Navier-Stokes Equations

    • Sergueï A. Nazarov
    Pages 159-191

  7. On the Mathematical Theory of Fluid Dynamic Limits to Conservation Laws

    • Athanasios E. Tzavaras
    Pages 192-222
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Keywords

About this book

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.

Editors and Affiliations

  • Mathematical Institute, Charles University, Prague 8, Czech Republic

    Josef Málek, Jindřich Nečas, Mirko Rokyta

  • Department of Mathematical Sciences, Northern Illinois University, DeKalb, USA

    Jindřich Nečas

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