This volume consists of five research articles, each dedicated to a significant topic in the mathematical theory of the Navier-Stokes equations, for compressible and incompressible fluids, and to related questions. All results given here are new and represent a noticeable contribution to the subject. One of the most famous predictions of the Kolmogorov theory of turbulence is the so-called Kolmogorov-obukhov five-thirds law. As is known, this law is heuristic and, to date, there is no rigorous justification. The article of A. Biryuk deals with the Cauchy problem for a multi-dimensional Burgers equation with periodic boundary conditions. Estimates in suitable norms for the corresponding solutions are derived for "large" Reynolds numbers, and their relation with the Kolmogorov-Obukhov law are discussed. Similar estimates are also obtained for the Navier-Stokes equation. In the late sixties J. L. Lions introduced a "perturbation" of the Navier­ Stokes equations in which he added in the linear momentum equation the hyper­ dissipative term (-Ll),Bu, f3 ~ 5/4, where Ll is the Laplace operator. This term is referred to as an "artificial" viscosity. Even though it is not physically moti­ vated, artificial viscosity has proved a useful device in numerical simulations of the Navier-Stokes equations at high Reynolds numbers. The paper of of D. Chae and J. Lee investigates the global well-posedness of a modification of the Navier­ Stokes equation similar to that introduced by Lions, but where now the original dissipative term -Llu is replaced by (-Ll)O:u, 0 S Ct < 5/4.

This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge.

This book is a translation of the 6th to 8th edition of the author's Kleines Lehrbuch der Physik. The circle of readers to which it hopes to appeal and the author's purpose in writing it have been set out in the Preface to the first German edition, published in 1948. The present book consistently follows the principles of the theory of quantities, the beginnings of which date back to James C. Maxwell. This means that in all equations in this book the symbols invariably stand for physical quantities and not for the numerical values of quantities. Only then are the equations gene rally valid and independent of the choice of units used in their evaluation. The units used are always the "metric" units which have been gaining ground increasingly also in the English-speaking countries. A conversion table for some of the more important Anglo-American units is given on page XIV. I would like to record my sincere gratitude to Mr. Ewald Osers for his pains­ taking work in making this translation and to Mr. P. C. Banbury, Ph. D., of the Department of Physics, University of Reading, England, both for the advice he has given hirn throughout and for devising the problems specially for this edition.

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