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Including previously unpublished, original research material, this
comprehensive book analyses topics of fundamental importance in
theoretical fluid mechanics. The five papers appearing in this
volume are centred around the mathematical theory of the
Navier-Stokes equations (incompressible and compressible) and
certain selected non-Newtonian modifications.
As a satellite conference of the 1998 International Mathematical
Congress and part of the celebration of the 650th anniversary of
Charles University, the Partial Differential Equations Theory and
Numerical Solution conference was held in Prague in August, 1998.
With its rich scientific program, the conference provided an
opportunity for almost 200 participants to gather and discuss
emerging directions and recent developments in partial differential
equations (PDEs). This volume comprises the Proceedings of that
conference. In it, leading specialists in partial differential
equations, calculus of variations, and numerical analysis present
up-to-date results, applications, and advances in numerical methods
in their fields. Conference organizers chose the contributors to
bring together the scientists best able to present a complex view
of problems, starting from the modeling, passing through the
mathematical treatment, and ending with numerical realization. The
applications discussed include fluid dynamics, semiconductor
technology, image analysis, motion analysis, and optimal control.
The importance and quantity of research carried out around the
world in this field makes it imperative for researchers, applied
mathematicians, physicists and engineers to keep up with the latest
developments. With its panel of international contributors and
survey of the recent ramifications of theory, applications, and
numerical methods, Partial Differential Equations: Theory and
Numerical Solution provides a convenient means to that end.
The five papers collected in this volume are the content of a
series of lectures delivered at the Second Winter School in Fluid
Dynamics held in Paseky, Czech Republic, from November 29 to
December 4 1992, concerning different fields in theoretical fluid
mechanics. The lectures present recent results of the authors'
investigations and the majority of the contributions are original
results which are not published elsewhere. Specifically, Galdi
studies the two-dimensional exterior problem for the steady-state
Navier-Stokes equations and Matsumura deals with some basic
questions related to existence and stability of one-dimensional
flow of compressible fluids. Both papers represent a difficult
mathematical approach to solving deep problems. The paper by
Girault furnishes a detailed and comprehensive analysis of the
Stokes problem in exterior domains that has important consequences
on numerical analysis. Litvinov's paper is dedicated to existence
theory for a class of equations describing the motions of certain
non classical fluids. Finally, the contribution from Rajagopal is a
detailed and updated review of non-Newtonian fluid mechanics with
emphasis on the different types of constitutive equations.
This monograph provides a concise treatment of the theory of
nonlinear evolutionary partial differential equations. For scalar
hyperbolic conservation laws, the well posedness of the initial
problem in the whole space as well as the initial boundary value
problem in bounded domains is treated. Further, one of the first
rigorous mathematical treatments of a class of non-Newtonian fluids
is given. The new results, obtained here for both problems, have
applications to many rapidly developing areas of physics, biology
and mechanical engineering. Weak and Measure-valued Solutions to
Evolutionary PDEs will be of interest to researchers and graduate
students in mathematics, theoretical physics and engineering. In
particular, engineers and physicists involved in fluid mechanics
research, and mathematicians interested in PDEs will value this
monograph.
This volume consists of four contributions that are based on a
series of lectures delivered by Jens Frehse. Konstantin Pikeckas,
K.R. Rajagopal and Wolf von Wahl t the Fourth Winter School in
Mathematical Theory in Fluid Mechanics, held in Paseky, Czech
Republic, from December 3-9, 1995. In these papers the authors
present the latest research and updated surveys of relevant topics
in the various areas of theoretical fluid mechanics.
Specifically, Frehse and Ruzicka study the question of the
existence of a regular solution to Navier-Stokes equations in five
dimensions by means of weighted estimates. Pileckas surveys recent
results regarding the solvability of the Stokes and Navier-Stokes
system in domains with outlets at infinity. K.R. Rajagopal presents
an introduction to a continuum approach to mixture theory with the
emphasis on the constitutive equation, boundary conditions and
moving singular surface. Finally, Kaiser and von Wahl bring new
results on stability of basic flow for the Taylor-Couette problem
in the small-gap limit. This volume would be indicated for those in
the fields of applied mathematicians, researchers in fluid
mechanics and theoretical mechanics, and mechanical engineers.
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