The Mathematical Analysis Of The Incompressible Euler And Navier Stokes Equations
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| Author | : Jacob Bedrossian |
| Publisher | : American Mathematical Society |
| Total Pages | : 235 |
| Release | : 2022-09-21 |
| Genre | : Mathematics |
| ISBN | : 1470470497 |
The aim of this book is to provide beginning graduate students who completed the first two semesters of graduate-level analysis and PDE courses with a first exposure to the mathematical analysis of the incompressible Euler and Navier-Stokes equations. The book gives a concise introduction to the fundamental results in the well-posedness theory of these PDEs, leaving aside some of the technical challenges presented by bounded domains or by intricate functional spaces. Chapters 1 and 2 cover the fundamentals of the Euler theory: derivation, Eulerian and Lagrangian perspectives, vorticity, special solutions, existence theory for smooth solutions, and blowup criteria. Chapters 3, 4, and 5 cover the fundamentals of the Navier-Stokes theory: derivation, special solutions, existence theory for strong solutions, Leray theory of weak solutions, weak-strong uniqueness, existence theory of mild solutions, and Prodi-Serrin regularity criteria. Chapter 6 provides a short guide to the must-read topics, including active research directions, for an advanced graduate student working in incompressible fluids. It may be used as a roadmap for a topics course in a subsequent semester. The appendix recalls basic results from real, harmonic, and functional analysis. Each chapter concludes with exercises, making the text suitable for a one-semester graduate course. Prerequisites to this book are the first two semesters of graduate-level analysis and PDE courses.
| Author | : Andrew J. Majda |
| Publisher | : Cambridge University Press |
| Total Pages | : 562 |
| Release | : 2002 |
| Genre | : Mathematics |
| ISBN | : 9780521639484 |
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.
| Author | : S. Friedlander |
| Publisher | : Gulf Professional Publishing |
| Total Pages | : 627 |
| Release | : 2003-03-27 |
| Genre | : Science |
| ISBN | : 008053354X |
The Handbook of Mathematical Fluid Dynamics is a compendium of essays that provides a survey of the major topics in the subject. Each article traces developments, surveys the results of the past decade, discusses the current state of knowledge and presents major future directions and open problems. Extensive bibliographic material is provided. The book is intended to be useful both to experts in the field and to mathematicians and other scientists who wish to learn about or begin research in mathematical fluid dynamics. The Handbook illuminates an exciting subject that involves rigorous mathematical theory applied to an important physical problem, namely the motion of fluids.
| Author | : Roger Temam |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 426 |
| Release | : 2001-04-10 |
| Genre | : Mathematics |
| ISBN | : 0821827375 |
Originally published in 1977, the book is devoted to the theory and numerical analysis of the Navier-Stokes equations for viscous incompressible fluid. On the theoretical side, results related to the existence, the uniqueness, and, in some cases, the regularity of solutions are presented. On the numerical side, various approaches to the approximation of Navier-Stokes problems by discretization are considered, such as the finite dereference method, the finite element method, and the fractional steps method. The problems of stability and convergence for numerical methods are treated as completely as possible. The new material in the present book (as compared to the preceding 1984 edition) is an appendix reproducing a survey article written in 1998. This appendix touches upon a few aspects not addressed in the earlier editions, in particular a short derivation of the Navier-Stokes equations from the basic conservation principles in continuum mechanics, further historical perspectives, and indications on new developments in the area. The appendix also surveys some aspects of the related Euler equations and the compressible Navier-Stokes equations. The book is written in the style of a textbook and the author has attempted to make the treatment self-contained. It can be used as a textbook or a reference book for researchers. Prerequisites for reading the book include some familiarity with the Navier-Stokes equations and some knowledge of functional analysis and Sololev spaces.
| Author | : Oscar Gonzalez |
| Publisher | : American Mathematical Society |
| Total Pages | : 228 |
| Release | : 2022-12-05 |
| Genre | : Mathematics |
| ISBN | : 147046991X |
The analysis and interpretation of mathematical models is an essential part of the modern scientific process. Topics in Applied Mathematics and Modeling is designed for a one-semester course in this area aimed at a wide undergraduate audience in the mathematical sciences. The prerequisite for access is exposure to the central ideas of linear algebra and ordinary differential equations. The subjects explored in the book are dimensional analysis and scaling, dynamical systems, perturbation methods, and calculus of variations. These are immense subjects of wide applicability and a fertile ground for critical thinking and quantitative reasoning, in which every student of mathematics should have some experience. Students who use this book will enhance their understanding of mathematics, acquire tools to explore meaningful scientific problems, and increase their preparedness for future research and advanced studies. The highlights of the book are case studies and mini-projects, which illustrate the mathematics in action. The book also contains a wealth of examples, figures, and regular exercises to support teaching and learning. The book includes opportunities for computer-aided explorations, and each chapter contains a bibliography with references covering further details of the material.
| Author | : Tristan Buckmaster |
| Publisher | : Springer Nature |
| Total Pages | : 169 |
| Release | : 2020-09-28 |
| Genre | : Mathematics |
| ISBN | : 3030548996 |
This volume brings together four contributions to mathematical fluid mechanics, a classical but still highly active research field. The contributions cover not only the classical Navier-Stokes equations and Euler equations, but also some simplified models, and fluids interacting with elastic walls. The questions addressed in the lectures range from the basic problems of existence/blow-up of weak and more regular solutions, to modeling and aspects related to numerical methods. This book covers recent advances in several important areas of fluid mechanics. An output of the CIME Summer School "Progress in mathematical fluid mechanics" held in Cetraro in 2019, it offers a collection of lecture notes prepared by T. Buckmaster, (Princeton), S. Canic (UCB) P. Constantin (Princeton) and A. Kiselev (Duke). These notes will be a valuable asset for researchers and advanced graduate students in several aspects of mathematicsl fluid mechanics.
| Author | : William Layton |
| Publisher | : SIAM |
| Total Pages | : 220 |
| Release | : 2008-01-01 |
| Genre | : Mathematics |
| ISBN | : 0898718902 |
Introduction to the Numerical Analysis of Incompressible Viscous Flows treats the numerical analysis of finite element computational fluid dynamics. Assuming minimal background, the text covers finite element methods; the derivation, behavior, analysis, and numerical analysis of Navier-Stokes equations; and turbulence and turbulence models used in simulations. Each chapter on theory is followed by a numerical analysis chapter that expands on the theory. This book provides the foundation for understanding the interconnection of the physics, mathematics, and numerics of the incompressible case, which is essential for progressing to the more complex flows not addressed in this book (e.g., viscoelasticity, plasmas, compressible flows, coating flows, flows of mixtures of fluids, and bubbly flows). With mathematical rigor and physical clarity, the book progresses from the mathematical preliminaries of energy and stress to finite element computational fluid dynamics in a format manageable in one semester. Audience: this unified treatment of fluid mechanics, analysis, and numerical analysis is intended for graduate students in mathematics, engineering, physics, and the sciences who are interested in understanding the foundations of methods commonly used for flow simulations.
| Author | : Frank Duzaar |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 135 |
| Release | : 2011 |
| Genre | : Mathematics |
| ISBN | : 0821849670 |
The authors establish a series of optimal regularity results for solutions to general non-linear parabolic systems $ u_t- \mathrm{div} \ a(x,t,u,Du)+H=0,$ under the main assumption of polynomial growth at rate $p$ i.e. $ a(x,t,u,Du) \leq L(1+ Du ^{p-1}), p \geq 2.$ They give a unified treatment of various interconnected aspects of the regularity theory: optimal partial regularity results for the spatial gradient of solutions, the first estimates on the (parabolic) Hausdorff dimension of the related singular set, and the first Calderon-Zygmund estimates for non-homogeneous problems are achieved here.
| Author | : Julio González-Díaz |
| Publisher | : American Mathematical Society |
| Total Pages | : 432 |
| Release | : 2023-12-05 |
| Genre | : Mathematics |
| ISBN | : 1470475634 |
Game theory provides a mathematical setting for analyzing competition and cooperation in interactive situations. The theory has been famously applied in economics, but is relevant in many other sciences, such as psychology, computer science, artificial intelligence, biology, and political science. This book presents an introductory and up-to-date course on game theory addressed to mathematicians and economists, and to other scientists having a basic mathematical background. The book is self-contained, providing a formal description of the classic game-theoretic concepts together with rigorous proofs of the main results in the field. The theory is illustrated through abundant examples, applications, and exercises. The style is distinctively concise, while offering motivations and interpretations of the theory to make the book accessible to a wide readership. The basic concepts and results of game theory are given a formal treatment, and the mathematical tools necessary to develop them are carefully presented. In this second edition, the content on cooperative games is considerably strengthened, with a new chapter on applications of cooperative games and operations research, including some material on computational aspects and applications outside academia.
| Author | : Henk Bruin |
| Publisher | : American Mathematical Society |
| Total Pages | : 481 |
| Release | : 2023-01-20 |
| Genre | : Mathematics |
| ISBN | : 1470469847 |
Symbolic dynamics is essential in the study of dynamical systems of various types and is connected to many other fields such as stochastic processes, ergodic theory, representation of numbers, information and coding, etc. This graduate text introduces symbolic dynamics from a perspective of topological dynamical systems and presents a vast variety of important examples. After introducing symbolic and topological dynamics, the core of the book consists of discussions of various subshifts of positive entropy, of zero entropy, other non-shift minimal action on the Cantor set, and a study of the ergodic properties of these systems. The author presents recent developments such as spacing shifts, square-free shifts, density shifts, $mathcal{B}$-free shifts, Bratteli-Vershik systems, enumeration scales, amorphic complexity, and a modern and complete treatment of kneading theory. Later, he provides an overview of automata and linguistic complexity (Chomsky's hierarchy). The necessary background for the book varies, but for most of it a solid knowledge of real analysis and linear algebra and first courses in probability and measure theory, metric spaces, number theory, topology, and set theory suffice. Most of the exercises have solutions in the back of the book.
