Advances in Mathematical Fluid Mechanics : Lecture Notes of the Sixth International School Mathematical Theory in Fluid Mechanics Paseky Czech Republic Sept. 19–26 1999 This book consists of six survey contributions focusing on several open problems of theoretical fluid mechanics both for incompressible and compressible fluids. The following topics are studied intensively within the book: global in time qualitative properties of solutions to compressible fluid models; fluid mechanics limits as compressible incompressible kinetic macroscopic viscous inviscid; adaptive Navier Stokes solver via wavelets; well posedness of the evolutionary Navier Stokes equations in 3D; existence theory for the incompressible Navier Stokes equations in exterior and aperture domains. All six articles present significant results and provide a better understanding of the problems in areas that enjoy long lasting attention of researchers dealing with fluid mechanics PDEs. Although the papers have the character of detailed summaries their central parts contain the newest results achieved by the authors who are experts in the topics they present