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The investigation of the role of mechanical and mechano-chemical
interactions in cellular processes and tissue development is a
rapidly growing research field in the life sciences and in
biomedical engineering. Quantitative understanding of this
important area in the study of biological systems requires the
development of adequate mathematical models for the simulation of
the evolution of these systems in space and time. Since expertise
in various fields is necessary, this calls for a multidisciplinary
approach. This edited volume connects basic physical, biological,
and physiological concepts to methods for the mathematical modeling
of various materials by pursuing a multiscale approach, from
subcellular to organ and system level. Written by active
researchers, each chapter provides a detailed introduction to a
given field, illustrates various approaches to creating models, and
explores recent advances and future research perspectives. Topics
covered include molecular dynamics simulations of lipid membranes,
phenomenological continuum mechanics of tissue growth, and
translational cardiovascular modeling. Modeling Biomaterials will
be a valuable resource for both non-specialists and experienced
researchers from various domains of science, such as applied
mathematics, biophysics, computational physiology, and medicine.
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."
This first title in SIAM's Spotlights book series is about the
interplay between modeling, analysis, discretization, matrix
computation, and model reduction. The authors link PDE analysis,
functional analysis, and calculus of variations with matrix
iterative computation using Krylov subspace methods and address the
challenges that arise during formulation of the mathematical model
through to efficient numerical solution of the algebraic problem.
The book's central concept, preconditioning of the conjugate
gradient method, is traditionally developed algebraically using the
preconditioned finite-dimensional algebraic system. In this text,
however, preconditioning is connected to the PDE analysis, and the
infinite-dimensional formulation of the conjugate gradient method
and its discretization and preconditioning are linked together.
This text challenges commonly held views, addresses widespread
misunderstandings, and formulates thought-provoking open questions
for further research.
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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Discovery Miles 4 870
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