The Navier-Stokes Problem in the 21st Century

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Author: Pierre Gilles Lemarie-Rieusset

Publisher: CRC Press

ISBN: 146656623X

Category: Mathematics

Page: 718

View: 1482

Up-to-Date Coverage of the Navier–Stokes Equation from an Expert in Harmonic Analysis The complete resolution of the Navier–Stokes equation—one of the Clay Millennium Prize Problems—remains an important open challenge in partial differential equations (PDEs) research despite substantial studies on turbulence and three-dimensional fluids. The Navier–Stokes Problem in the 21st Century provides a self-contained guide to the role of harmonic analysis in the PDEs of fluid mechanics. The book focuses on incompressible deterministic Navier–Stokes equations in the case of a fluid filling the whole space. It explores the meaning of the equations, open problems, and recent progress. It includes classical results on local existence and studies criterion for regularity or uniqueness of solutions. The book also incorporates historical references to the (pre)history of the equations as well as recent references that highlight active mathematical research in the field.

The Navier-Stokes Problem in the 21st Century

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Author: Pierre Gilles Lemarie-Rieusset

Publisher: Chapman and Hall/CRC

ISBN: 9781466566217

Category: Mathematics

Page: 300

View: 4505

A devoted discussion on the Navier-Stokes equations in light of the Clay Milennium Prize, this book offers both historical and technical information and provides a self-contained guide to the role of harmonic analysis in the PDEs of fluid mechanics. It looks at the meaning of the equation, problems currently at stake, and recent progress. It includes classical results on local existence, studies criterion for regularity or uniqueness of solutions, and discusses related equations in gas kinetics and fluid mechanics.

Recent developments in the Navier-Stokes problem

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Author: Pierre Gilles Lemarie-Rieusset

Publisher: CRC Press

ISBN: 9781420035674

Category: Mathematics

Page: 408

View: 9232

The Navier-Stokes equations: fascinating, fundamentally important, and challenging,. Although many questions remain open, progress has been made in recent years. The regularity criterion of Caffarelli, Kohn, and Nirenberg led to many new results on existence and non-existence of solutions, and the very active search for mild solutions in the 1990's culminated in the theorem of Koch and Tataru that, in some ways, provides a definitive answer. Recent Developments in the Navier-Stokes Problem brings these and other advances together in a self-contained exposition presented from the perspective of real harmonic analysis. The author first builds a careful foundation in real harmonic analysis, introducing all the material needed for his later discussions. He then studies the Navier-Stokes equations on the whole space, exploring previously scattered results such as the decay of solutions in space and in time, uniqueness, self-similar solutions, the decay of Lebesgue or Besov norms of solutions, and the existence of solutions for a uniformly locally square integrable initial value. Many of the proofs and statements are original and, to the extent possible, presented in the context of real harmonic analysis. Although the existence, regularity, and uniqueness of solutions to the Navier-Stokes equations continue to be a challenge, this book is a welcome opportunity for mathematicians and physicists alike to explore the problem's intricacies from a new and enlightening perspective.

Applied Analysis of the Navier-Stokes Equations

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Author: Charles R. Doering,J. D. Gibbon

Publisher: Cambridge University Press

ISBN: 9780521445689

Category: Mathematics

Page: 217

View: 3117

The Navier-Stokes equations are a set of nonlinear partial differential equations that describe the fundamental dynamics of fluid motion. They are applied routinely to problems in engineering, geophysics, astrophysics, and atmospheric science. This book is an introductory physical and mathematical presentation of the Navier-Stokes equations, focusing on unresolved questions of the regularity of solutions in three spatial dimensions, and the relation of these issues to the physical phenomenon of turbulent fluid motion. The goal of the book is to present a mathematically rigorous investigation of the Navier-Stokes equations that is accessible to a broader audience than just the subfields of mathematics to which it has traditionally been restricted. Therefore, results and techniques from nonlinear functional analysis are introduced as needed with an eye toward communicating the essential ideas behind the rigorous analyses. This book is appropriate for graduate students in many areas of mathematics, physics, and engineering.

Navier-Stokes Equations

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Author: Peter Constantin,Ciprian Foias

Publisher: University of Chicago Press

ISBN: 9780226115498

Category: Mathematics

Page: 190

View: 9731

Both an original contribution and a lucid introduction to mathematical aspects of fluid mechanics, Navier-Stokes Equations provides a compact and self-contained course on these classical, nonlinear, partial differential equations, which are used to describe and analyze fluid dynamics and the flow of gases.

The Three-Dimensional Navier-Stokes Equations

Classical Theory

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Author: James C. Robinson,José L. Rodrigo,Witold Sadowski

Publisher: Cambridge University Press

ISBN: 1107019664

Category: Mathematics

Page: 446

View: 1029

An accessible treatment of the main results in the mathematical theory of the Navier-Stokes equations, primarily aimed at graduate students.

Navier–Stokes Equations

An Introduction with Applications

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Author: Grzegorz Łukaszewicz,Piotr Kalita

Publisher: Springer

ISBN: 331927760X

Category: Mathematics

Page: 390

View: 590

This volume is devoted to the study of the Navier–Stokes equations, providing a comprehensive reference for a range of applications: from advanced undergraduate students to engineers and professional mathematicians involved in research on fluid mechanics, dynamical systems, and mathematical modeling. Equipped with only a basic knowledge of calculus, functional analysis, and partial differential equations, the reader is introduced to the concept and applications of the Navier–Stokes equations through a series of fully self-contained chapters. Including lively illustrations that complement and elucidate the text, and a collection of exercises at the end of each chapter, this book is an indispensable, accessible, classroom-tested tool for teaching and understanding the Navier–Stokes equations. Incompressible Navier–Stokes equations describe the dynamic motion (flow) of incompressible fluid, the unknowns being the velocity and pressure as functions of location (space) and time variables. A solution to these equations predicts the behavior of the fluid, assuming knowledge of its initial and boundary states. These equations are one of the most important models of mathematical physics: although they have been a subject of vivid research for more than 150 years, there are still many open problems due to the nature of nonlinearity present in the equations. The nonlinear convective term present in the equations leads to phenomena such as eddy flows and turbulence. In particular, the question of solution regularity for three-dimensional problem was appointed by Clay Institute as one of the Millennium Problems, the key problems in modern mathematics. The problem remains challenging and fascinating for mathematicians, and the applications of the Navier–Stokes equations range from aerodynamics (drag and lift forces), to the design of watercraft and hydroelectric power plants, to medical applications such as modeling the flow of blood in the circulatory system.

Computational Aerosciences in the 21st Century

Proceedings of the ICASE/LaRC/NSF/ARO Workshop, conducted by the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, The National Science Foundation and the Army Research Office, April 22–24, 1998

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Author: Manuel D. Salas,W. Kyle Anderson

Publisher: Springer Science & Business Media

ISBN: 9401009481

Category: Mathematics

Page: 292

View: 9489

Over the last decade, the role of computational simulations in all aspects of aerospace design has steadily increased. However, despite the many advances, the time required for computations is far too long. This book examines new ideas and methodologies that may, in the next twenty years, revolutionize scientific computing. The book specifically looks at trends in algorithm research, human computer interface, network-based computing, surface modeling and grid generation and computer hardware and architecture. The book provides a good overview of the current state-of-the-art and provides guidelines for future research directions. The book is intended for computational scientists active in the field and program managers making strategic research decisions.

Partial differential equations in general relativity

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Author: Alan D. Rendall

Publisher: Oxford University Press, USA

ISBN: 9780199215409

Category: Mathematics

Page: 279

View: 2305

A graduate level text on a subject which brings together several areas of mathematics and physics: partial differential equations, differential geometry and general relativity. It explains the basics of the theory of partial differential equations in a form accessible to physicists and the basics of general relativity in a form accessible to mathematicians. In recent years the theory of partial differential equations has come to play an ever more important role in research on general relativity. This is partly due to the growth of the field of numerical relativity, stimulated in turn by work on gravitational wave detection, but also due to an increased interest in general relativity among pure mathematicians working in the areas of partial differential equations and Riemannian geometry, who have realized the exceptional richness of the interactions between geometry and analysis which arise. This book provides the background for those wishing to learn about these topics. It treats key themes in general relativity including matter models and symmetry classes and gives an introduction to relevant aspects of the most important classes of partial differential equations, including ordinary differential equations, and material on functional analysis. These elements are brought together to discuss a variety of important examples in the field of mathematical relativity, including asymptotically flat spacetimes, which are used to describe isolated systems, and spatially compact spacetimes, which are of importance in cosmology.

In Pursuit of the Unknown

17 Equations That Changed the World

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Author: Ian Stewart

Publisher: Basic Books

ISBN: 0465029744

Category: Mathematics

Page: 360

View: 647

In In Pursuit of the Unknown, celebrated mathematician Ian Stewart uses a handful of mathematical equations to explore the vitally important connections between math and human progress. We often overlook the historical link between mathematics and technological advances, says Stewart—but this connection is integral to any complete understanding of human history. Equations are modeled on the patterns we find in the world around us, says Stewart, and it is through equations that we are able to make sense of, and in turn influence, our world. Stewart locates the origins of each equation he presents—from Pythagoras's Theorem to Newton's Law of Gravity to Einstein's Theory of Relativity—within a particular historical moment, elucidating the development of mathematical and philosophical thought necessary for each equation's discovery. None of these equations emerged in a vacuum, Stewart shows; each drew, in some way, on past equations and the thinking of the day. In turn, all of these equations paved the way for major developments in mathematics, science, philosophy, and technology. Without logarithms (invented in the early 17th century by John Napier and improved by Henry Briggs), scientists would not have been able to calculate the movement of the planets, and mathematicians would not have been able to develop fractal geometry. The Wave Equation is one of the most important equations in physics, and is crucial for engineers studying the vibrations in vehicles and the response of buildings to earthquakes. And the equation at the heart of Information Theory, devised by Claude Shannon, is the basis of digital communication today. An approachable and informative guide to the equations upon which nearly every aspect of scientific and mathematical understanding depends, In Pursuit of the Unknown is also a reminder that equations have profoundly influenced our thinking and continue to make possible many of the advances that we take for granted.

Partial Differential Equations in Fluid Mechanics

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Author: Charles L. Fefferman,James C. Robinson,José L. Rodrigo

Publisher: Cambridge University Press

ISBN: 1108460968

Category: Mathematics

Page: 336

View: 4998

A selection of survey articles and original research papers in mathematical fluid mechanics, for both researchers and graduate students.

Lectures on Navier-Stokes Equations

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Author: Tai-Peng Tsai

Publisher: American Mathematical Soc.

ISBN: 1470430967

Category: Fluid dynamics

Page: 224

View: 5530

This book is a graduate text on the incompressible Navier-Stokes system, which is of fundamental importance in mathematical fluid mechanics as well as in engineering applications. The goal is to give a rapid exposition on the existence, uniqueness, and regularity of its solutions, with a focus on the regularity problem. To fit into a one-year course for students who have already mastered the basics of PDE theory, many auxiliary results have been described with references but without proofs, and several topics were omitted. Most chapters end with a selection of problems for the reader. After an introduction and a careful study of weak, strong, and mild solutions, the reader is introduced to partial regularity. The coverage of boundary value problems, self-similar solutions, the uniform L3 class including the celebrated Escauriaza-Seregin-Šverák Theorem, and axisymmetric flows in later chapters are unique features of this book that are less explored in other texts. The book can serve as a textbook for a course, as a self-study source for people who already know some PDE theory and wish to learn more about Navier-Stokes equations, or as a reference for some of the important recent developments in the area.

The Millennium Problems

The Seven Greatest Unsolved Mathematical Puzzles of Our Time

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Author: Keith J. Devlin

Publisher: Granta Books

ISBN: 9781862077355

Category: Mathematical recreations

Page: 237

View: 2899

In 2000, the Clay Foundation of Cambridge, Massachusetts, announced a historic competition: Whoever could solve any of seven extraordinarily difficult mathematical problems, and have the solution acknowledged as correct by the experts, would receive $1million in prize money. They encompass many of the most fascinating areas of pure and applied mathematics, from topology and number theory to particle physics, cryptography, computing and even aircraft design. Keith Devlin describes here what the seven problems are, how they came about, and what they mean for mathematics and science. In the hands of Devlin, each Millennium Problem becomes a fascinating window onto the deepest questions in the field.

Vorticity and Incompressible Flow

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Author: Andrew J. Majda,Andrea L. Bertozzi

Publisher: Cambridge University Press

ISBN: 9780521639484

Category: Mathematics

Page: 545

View: 1430

This book is a comprehensive introduction to the mathematical theory of vorticity and incompressible flow ranging from elementary introductory material to current research topics. While the contents center on mathematical theory, many parts of the book showcase the interaction between rigorous mathematical theory, numerical, asymptotic, and qualitative simplified modeling, and physical phenomena. The first half forms an introductory graduate course on vorticity and incompressible flow. The second half comprise a modern applied mathematics graduate course on the weak solution theory for incompressible flow.

Physics in the 21st Century

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Author: K Kikkawa,H Kunitomo,H Ohtsubo

Publisher: World Scientific

ISBN: 9814497584

Category: Science

Page: 212

View: 7173

Towards the close of the 20th century, the world's leading experts in theoretical and experimental physics review the major developments in their respective research areas, and present the prospects for the coming 21st century. The subjects covered in this volume are field theory, string theory, quantum cosmology, solid state physics, physics of complex systems, high energy physics, quark–gluon plasma, nuclear physics and observational cosmology. Contents:Three Stages, Three Modes, and Beyond (Y Nambu)Quantum Transport in Mesoscopic Semiconductor Structures (T Ando)Nonequilibrium Physics and the Origins of Complexity in Nature (J S Langer)New Horizon in Nuclear Physics and Astrophysics Using Radioactive Nuclear Beams (I Tanihata)The Future of Particle Physics (F Wilczek & D J Gross)Status and Prospects of Relativistic Heavy-Ion Experiments (S Nagamiya)Present and Future of High Energy Physics with Hadron Colliders (K Kondo)Cosmological Parameters and Evolution of Galaxies: An Observational Perspective (S Okamura)Quantum Cosmology: Problems for the 21st Century (J B Hartle) Readership: Physicists. keywords:Theoretical Physics;Experimental Physics;Field Theory;String Theory;Quantum Cosmology;Solid State Physics;Quark-Gluon Plasma;High Energy Physics;Nuclear Physics;Cosmology

What's Happening in the Mathematical Sciences

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Author: Barry Cipra

Publisher: American Mathematical Soc.

ISBN: 9780821889985

Category: Science

Page: 51

View: 7328

After rave reviews for last year's issue of What's Happening, volume 2 has been eagerly awaited. Very well written, '' said one reader of volume 1. The writing is brilliant, positively brilliant.'' A terrific publication, '' said another. This is a wonderful tool for showing people what mathematics is about and what mathematicians can do.'' One reader called it a must for all mathematics department reading and coffee lounges.'' Volume 2 of What's Happening features the same lively writing and all new topics. Here you can read about a new class of solitons, the contributions wavelets are making to solving scientific problems, how mathematics is improving medical imaging, and Andrew Wiles's acclaimed work on Fermat's Last Theorem. What's Happening is great for mathematics undergraduates, graduate students, and mathematics clubs---not to mention mathematics faculty, who will enjoy reading about recent developments in fields other than their own. Highlighting the excitement and wonder of mathematics, What's Happening is in a class by itself.

Computational Fluid Dynamics for the 21st Century

Proceedings of a Symposium Honoring Prof. Satofuka on the Occasion of his 60th Birthday, Kyoto, Japan, July 15–17, 2000

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Author: Mohamed Hafez,Koji Morinishi,Jacques Périaux

Publisher: Springer Science & Business Media

ISBN: 3540449590

Category: Technology & Engineering

Page: 413

View: 7656

This volume contains new trends of computational fluid dynamics for the 21st century and consists of papers especially useful to the younger generation of scientists and engineers in this field. Topics include cartesian, gridless and higher-order schemes, and flow-visualization techniques.

Unfolding the Labyrinth: Open Problems in Physics, Mathematics, Astrophysics, and other areas of science

Open Problems in Physics, Mathematics, Astrophysics, and Other Areas of Science,

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Author: Florentin Smarandache,V. Christianto

Publisher: Infinite Study

ISBN: 1599730138

Category: Mathematical physics

Page: 139

View: 7941

Throughout this book, we discuss some open problems in various branches of science, including mathematics, theoretical physics, astrophysics, geophysics etc. It is of our hope that some of the problems discussed in this book will find their place either in theoretical exploration or further experiments, while some parts of these problems may be found useful for scholarly stimulation.The present book is also intended for young physics and mathematics fellows who will perhaps find the unsolved problems described here are at least worth pondering. If this book provides only a few highlights of plausible solutions, it is merely to keep the fun of readers in discovering the answers by themselves. Bon voyage!

Mine Safety and Efficient Exploitation Facing Challenges of the 21st Century

International Mining Forum 2010

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Author: Liu Zegong,Jerzy Kicki,Hua Xinzhu,Yuan Shujie,Dai Guanglong,Eugeniusz J. Sobczyk

Publisher: CRC Press

ISBN: 020309333X

Category: Nature

Page: 300

View: 9333

The International Mining Forum is a meeting of scientists and professionals for exchanging new ideas and experiences, evaluate previously implemented solutions, and discuss fresh ideas that might change the mining industry’s image. In recent years theory and technology in mine safety and efficient exploitation has made considerable progress in China, Poland and other countries, due to the introduction of many new theories and technologies. The International Mining Forum 2010, jointly organized by Anhui University of Science and Technology (China), AGH University of Science and Technology (Poland) and Mineral and Energy Economy Research Institute of the Polish Academy of Sciences (Poland), has provided experts all over the world with an opportunity and platform to exchange information and ideas. Purpose of the forum was to promote research and development of mine safety, efficient exploitation theory and provide theoretical and technical support for mine safety improvement. This book is addressed to researchers and professionals who work in the fields of underground mining technology, rock engineering or mine management.

The Navier-Stokes Equations

An Elementary Functional Analytic Approach

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Author: Hermann Sohr

Publisher: Birkhäuser

ISBN: 3034882556

Category: Mathematics

Page: 367

View: 2911

This book offers an elementary, self-contained approach to the mathematical theory of viscous, incompressible fluid in a domain of the Euclidian space, described by the equations of Navier-Stokes. It is the first to provide a systematic treatment of the subject. It is designed for students familiar with basic tools in Hilbert and Banach spaces, but fundamental properties of, for example, Sobolev spaces, are collected in the first two chapters.