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Instability And Non Uniqueness For The 2d Euler Equations After M Vishik
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Book Synopsis Instability and Non-uniqueness for the 2D Euler Equations, after M. Vishik by : Camillo De Lellis
Download or read book Instability and Non-uniqueness for the 2D Euler Equations, after M. Vishik written by Camillo De Lellis and published by Princeton University Press. This book was released on 2024-02-13 with total page 149 pages. Available in PDF, EPUB and Kindle. Book excerpt: An essential companion to M. Vishik’s groundbreaking work in fluid mechanics The incompressible Euler equations are a system of partial differential equations introduced by Leonhard Euler more than 250 years ago to describe the motion of an inviscid incompressible fluid. These equations can be derived from the classical conservations laws of mass and momentum under some very idealized assumptions. While they look simple compared to many other equations of mathematical physics, several fundamental mathematical questions about them are still unanswered. One is under which assumptions it can be rigorously proved that they determine the evolution of the fluid once we know its initial state and the forces acting on it. This book addresses a well-known case of this question in two space dimensions. Following the pioneering ideas of M. Vishik, the authors explain in detail the optimality of a celebrated theorem of V. Yudovich from the 1960s, which states that, in the vorticity formulation, the solution is unique if the initial vorticity and the acting force are bounded. In particular, the authors show that Yudovich’s theorem cannot be generalized to the L^p setting.
Book Synopsis Values of Non-Atomic Games by : Robert J. Aumann
Download or read book Values of Non-Atomic Games written by Robert J. Aumann and published by Princeton University Press. This book was released on 2015-03-08 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: The "Shapley value" of a finite multi- person game associates to each player the amount he should be willing to pay to participate. This book extends the value concept to certain classes of non-atomic games, which are infinite-person games in which no individual player has significance. It is primarily a book of mathematics—a study of non-additive set functions and associated linear operators. Originally published in 1974. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Book Synopsis Mathematical Aspects of Nonlinear Dispersive Equations (AM-163) by : Jean Bourgain
Download or read book Mathematical Aspects of Nonlinear Dispersive Equations (AM-163) written by Jean Bourgain and published by Princeton University Press. This book was released on 2009-01-10 with total page 309 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection of new and original papers on mathematical aspects of nonlinear dispersive equations includes both expository and technical papers that reflect a number of recent advances in the field. The expository papers describe the state of the art and research directions. The technical papers concentrate on a specific problem and the related analysis and are addressed to active researchers. The book deals with many topics that have been the focus of intensive research and, in several cases, significant progress in recent years, including hyperbolic conservation laws, Schrödinger operators, nonlinear Schrödinger and wave equations, and the Euler and Navier-Stokes equations.
Book Synopsis Contributions to the Theory of Partial Differential Equations. (AM-33), Volume 33 by : Lipman Bers
Download or read book Contributions to the Theory of Partial Differential Equations. (AM-33), Volume 33 written by Lipman Bers and published by Princeton University Press. This book was released on 2016-03-02 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: The description for this book, Contributions to the Theory of Partial Differential Equations. (AM-33), Volume 33, will be forthcoming.
Book Synopsis Existence Theorems in Partial Differential Equations. (AM-23), Volume 23 by : Dorothy L. Bernstein
Download or read book Existence Theorems in Partial Differential Equations. (AM-23), Volume 23 written by Dorothy L. Bernstein and published by Princeton University Press. This book was released on 2016-03-02 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: The description for this book, Existence Theorems in Partial Differential Equations. (AM-23), Volume 23, will be forthcoming.
Book Synopsis The Master Equation and the Convergence Problem in Mean Field Games by : Pierre Cardaliaguet
Download or read book The Master Equation and the Convergence Problem in Mean Field Games written by Pierre Cardaliaguet and published by Princeton University Press. This book was released on 2019-08-13 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. This book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. This groundbreaking book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.
Book Synopsis The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations by : Jacob Bedrossian
Download or read book The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations written by Jacob Bedrossian and published by American Mathematical Society. This book was released on 2022-09-22 with total page 235 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Contributions to the Theory of Games (AM-28), Volume II by : Harold William Kuhn
Download or read book Contributions to the Theory of Games (AM-28), Volume II written by Harold William Kuhn and published by Princeton University Press. This book was released on 2016-03-02 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: These two new collections, numbers 28 and 29 respectively in the Annals of Mathematics Studies, continue the high standard set by the earlier Annals Studies 20 and 24 by bringing together important contributions to the theories of games and of nonlinear differential equations.
Book Synopsis Smoothings of Piecewise Linear Manifolds by : Morris W. Hirsch
Download or read book Smoothings of Piecewise Linear Manifolds written by Morris W. Hirsch and published by Princeton University Press. This book was released on 1974-10-21 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: The intention of the authors is to examine the relationship between piecewise linear structure and differential structure: a relationship, they assert, that can be understood as a homotopy obstruction theory, and, hence, can be studied by using the traditional techniques of algebraic topology. Thus the book attacks the problem of existence and classification (up to isotopy) of differential structures compatible with a given combinatorial structure on a manifold. The problem is completely "solved" in the sense that it is reduced to standard problems of algebraic topology. The first part of the book is purely geometrical; it proves that every smoothing of the product of a manifold M and an interval is derived from an essentially unique smoothing of M. In the second part this result is used to translate the classification of smoothings into the problem of putting a linear structure on the tangent microbundle of M. This in turn is converted to the homotopy problem of classifying maps from M into a certain space PL/O. The set of equivalence classes of smoothings on M is given a natural abelian group structure.
Book Synopsis The Geometry and Dynamics of Magnetic Monopoles by : Michael Francis Atiyah
Download or read book The Geometry and Dynamics of Magnetic Monopoles written by Michael Francis Atiyah and published by Princeton University Press. This book was released on 2014-07-14 with total page 143 pages. Available in PDF, EPUB and Kindle. Book excerpt: Systems governed by non-linear differential equations are of fundamental importance in all branches of science, but our understanding of them is still extremely limited. In this book a particular system, describing the interaction of magnetic monopoles, is investigated in detail. The use of new geometrical methods produces a reasonably clear picture of the dynamics for slowly moving monopoles. This picture clarifies the important notion of solitons, which has attracted much attention in recent years. The soliton idea bridges the gap between the concepts of "fields" and "particles," and is here explored in a fully three-dimensional context. While the background and motivation for the work comes from physics, the presentation is mathematical. This book is interdisciplinary and addresses concerns of theoretical physicists interested in elementary particles or general relativity and mathematicians working in analysis or geometry. The interaction between geometry and physics through non-linear partial differential equations is now at a very exciting stage, and the book is a contribution to this activity. Originally published in 1988. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Book Synopsis Degenerate Diffusion Operators Arising in Population Biology by : Charles L. Epstein
Download or read book Degenerate Diffusion Operators Arising in Population Biology written by Charles L. Epstein and published by Princeton University Press. This book was released on 2013-04-07 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the Martingale problem and therefore the existence of the associated Markov process. Charles Epstein and Rafe Mazzeo use an "integral kernel method" to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. Epstein and Mazzeo establish the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. They show that the semigroups defined by these operators have holomorphic extensions to the right half-plane. Epstein and Mazzeo also demonstrate precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.
Book Synopsis The Mathematics of Shock Reflection-Diffraction and von Neumann's Conjectures by : Gui-Qiang G Chen
Download or read book The Mathematics of Shock Reflection-Diffraction and von Neumann's Conjectures written by Gui-Qiang G Chen and published by Princeton University Press. This book was released on 2018-02-27 with total page 832 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a survey of recent developments in the analysis of shock reflection-diffraction, a detailed presentation of original mathematical proofs of von Neumann's conjectures for potential flow, and a collection of related results and new techniques in the analysis of partial differential equations (PDEs), as well as a set of fundamental open problems for further development. Shock waves are fundamental in nature. They are governed by the Euler equations or their variants, generally in the form of nonlinear conservation laws—PDEs of divergence form. When a shock hits an obstacle, shock reflection-diffraction configurations take shape. To understand the fundamental issues involved, such as the structure and transition criteria of different configuration patterns, it is essential to establish the global existence, regularity, and structural stability of shock reflection-diffraction solutions. This involves dealing with several core difficulties in the analysis of nonlinear PDEs—mixed type, free boundaries, and corner singularities—that also arise in fundamental problems in diverse areas such as continuum mechanics, differential geometry, mathematical physics, and materials science. Presenting recently developed approaches and techniques, which will be useful for solving problems with similar difficulties, this book opens up new research opportunities.
Book Synopsis Transport Equations and Multi-D Hyperbolic Conservation Laws by : Luigi Ambrosio
Download or read book Transport Equations and Multi-D Hyperbolic Conservation Laws written by Luigi Ambrosio and published by Springer Science & Business Media. This book was released on 2008-02-17 with total page 141 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of nonlinear hyperbolic equations in several space dimensions has recently obtained remarkable achievements. This volume provides an up-to-date overview of the status and perspectives of two areas of research in PDEs, related to hyperbolic conservation laws. The captivating volume contains surveys of recent deep results and provides an overview of further developments and related open problems. Readers should have basic knowledge of PDE and measure theory.
Book Synopsis Adaptive Control of Parabolic PDEs by : Andrey Smyshlyaev
Download or read book Adaptive Control of Parabolic PDEs written by Andrey Smyshlyaev and published by Princeton University Press. This book was released on 2010-07-01 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces a comprehensive methodology for adaptive control design of parabolic partial differential equations with unknown functional parameters, including reaction-convection-diffusion systems ubiquitous in chemical, thermal, biomedical, aerospace, and energy systems. Andrey Smyshlyaev and Miroslav Krstic develop explicit feedback laws that do not require real-time solution of Riccati or other algebraic operator-valued equations. The book emphasizes stabilization by boundary control and using boundary sensing for unstable PDE systems with an infinite relative degree. The book also presents a rich collection of methods for system identification of PDEs, methods that employ Lyapunov, passivity, observer-based, swapping-based, gradient, and least-squares tools and parameterizations, among others. Including a wealth of stimulating ideas and providing the mathematical and control-systems background needed to follow the designs and proofs, the book will be of great use to students and researchers in mathematics, engineering, and physics. It also makes a valuable supplemental text for graduate courses on distributed parameter systems and adaptive control.
Download or read book Nonplussed! written by Julian Havil and published by Princeton University Press. This book was released on 2010-08-02 with total page 213 pages. Available in PDF, EPUB and Kindle. Book excerpt: Math—the application of reasonable logic to reasonable assumptions—usually produces reasonable results. But sometimes math generates astonishing paradoxes—conclusions that seem completely unreasonable or just plain impossible but that are nevertheless demonstrably true. Did you know that a losing sports team can become a winning one by adding worse players than its opponents? Or that the thirteenth of the month is more likely to be a Friday than any other day? Or that cones can roll unaided uphill? In Nonplussed!—a delightfully eclectic collection of paradoxes from many different areas of math—popular-math writer Julian Havil reveals the math that shows the truth of these and many other unbelievable ideas. Nonplussed! pays special attention to problems from probability and statistics, areas where intuition can easily be wrong. These problems include the vagaries of tennis scoring, what can be deduced from tossing a needle, and disadvantageous games that form winning combinations. Other chapters address everything from the historically important Torricelli's Trumpet to the mind-warping implications of objects that live on high dimensions. Readers learn about the colorful history and people associated with many of these problems in addition to their mathematical proofs. Nonplussed! will appeal to anyone with a calculus background who enjoys popular math books or puzzles.
Book Synopsis Chaotic Transitions in Deterministic and Stochastic Dynamical Systems by : Emil Simiu
Download or read book Chaotic Transitions in Deterministic and Stochastic Dynamical Systems written by Emil Simiu and published by Princeton University Press. This book was released on 2014-09-08 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: The classical Melnikov method provides information on the behavior of deterministic planar systems that may exhibit transitions, i.e. escapes from and captures into preferred regions of phase space. This book develops a unified treatment of deterministic and stochastic systems that extends the applicability of the Melnikov method to physically realizable stochastic planar systems with additive, state-dependent, white, colored, or dichotomous noise. The extended Melnikov method yields the novel result that motions with transitions are chaotic regardless of whether the excitation is deterministic or stochastic. It explains the role in the occurrence of transitions of the characteristics of the system and its deterministic or stochastic excitation, and is a powerful modeling and identification tool. The book is designed primarily for readers interested in applications. The level of preparation required corresponds to the equivalent of a first-year graduate course in applied mathematics. No previous exposure to dynamical systems theory or the theory of stochastic processes is required. The theoretical prerequisites and developments are presented in the first part of the book. The second part of the book is devoted to applications, ranging from physics to mechanical engineering, naval architecture, oceanography, nonlinear control, stochastic resonance, and neurophysiology.
Book Synopsis Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics by : G. F. Roach
Download or read book Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics written by G. F. Roach and published by Princeton University Press. This book was released on 2012-03-04 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: Electromagnetic complex media are artificial materials that affect the propagation of electromagnetic waves in surprising ways not usually seen in nature. Because of their wide range of important applications, these materials have been intensely studied over the past twenty-five years, mainly from the perspectives of physics and engineering. But a body of rigorous mathematical theory has also gradually developed, and this is the first book to present that theory. Designed for researchers and advanced graduate students in applied mathematics, electrical engineering, and physics, this book introduces the electromagnetics of complex media through a systematic, state-of-the-art account of their mathematical theory. The book combines the study of well posedness, homogenization, and controllability of Maxwell equations complemented with constitutive relations describing complex media. The book treats deterministic and stochastic problems both in the frequency and time domains. It also covers computational aspects and scattering problems, among other important topics. Detailed appendices make the book self-contained in terms of mathematical prerequisites, and accessible to engineers and physicists as well as mathematicians.
