Advances in Mathematical Fluid Mechanics

1. Front Matter

   Pages I-IX

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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

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.

• Navier-Stokes equation
• Navier-Stokes equations
• fluid mechanics
• fluid models limit
• incompressible and compressible fluid models
• numerical methods
• solvability
• wavelet solver
• fluid- and aerodynamics

• 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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