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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 consists of six articles, each treating an important
topic in the theory ofthe Navier-Stokes equations, at the research
level. Some of the articles are mainly expository, putting
together, in a unified setting, the results of recent research
papers and conference lectures. Several other articles are devoted
mainly to new results, but present them within a wider context and
with a fuller exposition than is usual for journals. The plan to
publish these articles as a book began with the lecture notes for
the short courses of G.P. Galdi and R. Rannacher, given at the
beginning of the International Workshop on Theoretical and
Numerical Fluid Dynamics, held in Vancouver, Canada, July 27 to
August 2, 1996. A renewed energy for this project came with the
founding of the Journal of Mathematical Fluid Mechanics, by G.P.
Galdi, J. Heywood, and R. Rannacher, in 1998. At that time it was
decided that this volume should be published in association with
the journal, and expanded to include articles by J. Heywood and W.
Nagata, J. Heywood and M. Padula, and P. Gervasio, A. Quarteroni
and F. Saleri. The original lecture notes were also revised and
updated.
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 set of six papers, written by eminent experts in the field, is
concerned with that part of fluid mechanics that seeks its
foundation in the rigorous mathematical treatment of the
Navier-Stokes equations. In particular, an overview is given on
state of research regarding the global existence of smooth
solutions, for which uniqueness and continuous dependence on the
data can be proven. Then, the book moves on to a discussion of
recent developments of the finite element Galerkin method, with an
emphasis on a priori and a posteriori error estimation and adaptive
mesh refinement. A further article elaborates on spectral Galerkin
methods and their extension to domains with complicated geometries
by employing the techniques of domain decomposition. The rigorous
explanation of bifurcation phenomena in fluids has long been a
central topic in the theory of Navier-Stokes equations. Here,
bifurcation theory is introduced in a general setting that is
particularly convenient for application to such problems. Finally,
the extension of Navier-Stokes theory to compressible viscous
flows, studied in two more papers, opens up a fascinating panorama
of theoretical and numerical problems. While some of the
contributions are expository, others primarily present new results
within a wider context and fuller exposition than is usual for
research papers. The book is meant to introduce researchers and
advanced students to the research level on some of the most
important topics of the field.
V.A. Solonnikov, A. Tani: Evolution free boundary problem for
equations of motion of viscous compressible barotropic liquid.- W.
Borchers, T. Miyakawa: On some coercive estimates for the Stokes
problem in unbounded domains.- R. Farwig, H. Sohr: An approach to
resolvent estimates for the Stokes equations in L(q)-spaces.- R.
Rannacher: On Chorin's projection method for the incompressible
Navier-Stokes equations.- E. S}li, A. Ware: Analysis of the
spectral Lagrange-Galerkin method for the Navier-Stokes equations.-
G. Grubb: Initial value problems for the Navier-Stokes equations
with Neumann conditions.- B.J. Schmitt, W. v.Wahl: Decomposition of
solenoidal fields into poloidal fields, toroidal fields and the
mean flow. Applications to the Boussinesq-equations.- O. Walsh:
Eddy solutions of the Navier-Stokesequations.- W. Xie: On a
three-norm inequality for the Stokes operator in nonsmooth domains.
These proceedings contain original (refereed) research articles by
specialists from many countries, on a wide variety of aspects of
Navier-Stokes equations. Additionally, 2 survey articles intended
for a general readership are included: one surveys the present
state of the subject via open problems, and the other deals with
the interplay between theory and numerical analysis.
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