The Vector Field Method and Its Applications to Nonlinear Evolution Equations PDF Download
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Author: Leonardo Enrique Abbrescia
Publisher:
ISBN:
Category : Electronic dissertations
Languages : en
Pages : 229
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Book Description
The vector field method was introduced in the 1980s by Sergiu Klainerman to analyze the decay properties of the linear wave equation. Since its historical debut, the vector field method has been at the forefront of several breakthrough results including the global stability of Minkowski space, the dynamical formation of black holes, and shock formation in 3D compressible fluids.This work showcases how the vector field method can be used in a systematic way to derive a priori estimates for nonlinear evolution equations. For nonlinear dispersive equations, these estimates can be married to the decay properties enjoyed by the solutions to derive quantitative asymptotics. This is done in this work through the lens of three concrete problems: a nonlocal kinetic model, the wave maps equation, and the relativistic membrane equation. For the kinetic model, the vector field method is paired with dispersive decay properties of the spatial density to prove global wellposedness of small data. This can be interpreted physically as "stability" of the trivial solution. For the wave maps equation, a stability result is proven for a "non-trivial" ODE geodesic solution. For the relativistic membrane equation, the vector field method is used to prove stability of large simple-traveling-waves. For the wave map and membrane equations, we intimately use several structural properties known as null conditions that preclude singular behavior.
Author: Leonardo Enrique Abbrescia
Publisher:
ISBN:
Category : Electronic dissertations
Languages : en
Pages : 229
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Book Description
The vector field method was introduced in the 1980s by Sergiu Klainerman to analyze the decay properties of the linear wave equation. Since its historical debut, the vector field method has been at the forefront of several breakthrough results including the global stability of Minkowski space, the dynamical formation of black holes, and shock formation in 3D compressible fluids.This work showcases how the vector field method can be used in a systematic way to derive a priori estimates for nonlinear evolution equations. For nonlinear dispersive equations, these estimates can be married to the decay properties enjoyed by the solutions to derive quantitative asymptotics. This is done in this work through the lens of three concrete problems: a nonlocal kinetic model, the wave maps equation, and the relativistic membrane equation. For the kinetic model, the vector field method is paired with dispersive decay properties of the spatial density to prove global wellposedness of small data. This can be interpreted physically as "stability" of the trivial solution. For the wave maps equation, a stability result is proven for a "non-trivial" ODE geodesic solution. For the relativistic membrane equation, the vector field method is used to prove stability of large simple-traveling-waves. For the wave map and membrane equations, we intimately use several structural properties known as null conditions that preclude singular behavior.
Author: Sandra Carillo
Publisher: Springer Science & Business Media
ISBN: 3642840396
Category : Science
Languages : en
Pages : 247
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Book Description
Nonlinear Evolution Equations and Dynamical Systems (NEEDS) provides a presentation of the state of the art. Except for a few review papers, the 40 contributions are intentially brief to give only the gist of the methods, proofs, etc. including references to the relevant litera- ture. This gives a handy overview of current research activities. Hence, the book should be equally useful to the senior resercher as well as the colleague just entering the field. Keypoints treated are: i) integrable systems in multidimensions and associated phenomenology ("dromions"); ii) criteria and tests of integrability (e.g., Painlev test); iii) new developments related to the scattering transform; iv) algebraic approaches to integrable systems and Hamiltonian theory (e.g., connections with Young-Baxter equations and Kac-Moody algebras); v) new developments in mappings and cellular automata, vi) applications to general relativity, condensed matter physics, and oceanography.
Author: N. U. Ahmed
Publisher: Springer Nature
ISBN: 3031372603
Category : Mathematics
Languages : en
Pages : 236
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Book Description
This book offers the first comprehensive presentation of measure-valued solutions for nonlinear deterministic and stochastic evolution equations on infinite dimensional Banach spaces. Unlike traditional solutions, measure-valued solutions allow for a much broader class of abstract evolution equations to be addressed, providing a broader approach. The book presents extensive results on the existence of measure-valued solutions for differential equations that have no solutions in the usual sense. It covers a range of topics, including evolution equations with continuous/discontinuous vector fields, neutral evolution equations subject to vector measures as impulsive forces, stochastic evolution equations, and optimal control of evolution equations. The optimal control problems considered cover the existence of solutions, necessary conditions of optimality, and more, significantly complementing the existing literature. This book will be of great interest to researchers in functional analysis, partial differential equations, dynamic systems and their optimal control, and their applications, advancing previous research and providing a foundation for further exploration of the field.
Author: M Boiti
Publisher: World Scientific
ISBN: 981455541X
Category :
Languages : en
Pages : 474
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Book Description
The Workshop NEEDS '91 brought together, from all over the world, scientists engaged in research on nonlinear systems, either their underlying mathematical properties or their physical applications. Accordingly, many talks were devoted to present methods of solution (like spectral transform) and to the investigation of structural (geometrical and/or algebraic) properties of (continuous and discrete) nonlinear evolution equations. Peculiar nonlinear systems, such as cellular automata, were also discussed. Applications to various fields of physics, namely, quantum field theory, fluid dynamics, general relativity and plasma physics were considered.
Author: H Beirao Da Veiga
Publisher: World Scientific
ISBN: 981455166X
Category :
Languages : en
Pages : 230
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Book Description
This book provides an introduction for graduate students and advanced undergraduate students to the field of astrophysical fluid dynamics. Although sometimes ignored, fluid dynamical processes play a central role in virtually all areas of astrophysics.No previous knowledge of fluid dynamics is assumed. After establishing the basic equations of fluid dynamics and the physics relevant to an astrophysical application, a variety of topics in the field are addressed. There is also a chapter introducing the reader to numerical methods. Appendices list useful physical constants and astronomical quantities, and provide handy reference material on Cartesian tensors, vector calculus in polar coordinates, self-adjoint eigenvalue problems and JWKB theory./a
Author: Reinhard Racke
Publisher: Birkhäuser
ISBN: 3319218735
Category : Mathematics
Languages : en
Pages : 315
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Book Description
This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behaviour of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial boundary value problems and for open questions are provided. In this second edition, initial-boundary value problems in waveguides are additionally considered.
Author: Gheorghe Morosanu
Publisher: Springer Science & Business Media
ISBN: 9789027724861
Category : Science
Languages : en
Pages : 362
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Book Description
Author: David Ellwood
Publisher: American Mathematical Soc.
ISBN: 0821868616
Category : Mathematics
Languages : en
Pages : 587
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Book Description
This volume is a collection of notes from lectures given at the 2008 Clay Mathematics Institute Summer School, held in Zürich, Switzerland. The lectures were designed for graduate students and mathematicians within five years of the Ph.D., and the main focus of the program was on recent progress in the theory of evolution equations. Such equations lie at the heart of many areas of mathematical physics and arise not only in situations with a manifest time evolution (such as linear and nonlinear wave and Schrödinger equations) but also in the high energy or semi-classical limits of elliptic problems. The three main courses focused primarily on microlocal analysis and spectral and scattering theory, the theory of the nonlinear Schrödinger and wave equations, and evolution problems in general relativity. These major topics were supplemented by several mini-courses reporting on the derivation of effective evolution equations from microscopic quantum dynamics; on wave maps with and without symmetries; on quantum N-body scattering, diffraction of waves, and symmetric spaces; and on nonlinear Schrödinger equations at critical regularity. Although highly detailed treatments of some of these topics are now available in the published literature, in this collection the reader can learn the fundamental ideas and tools with a minimum of technical machinery. Moreover, the treatment in this volume emphasizes common themes and techniques in the field, including exact and approximate conservation laws, energy methods, and positive commutator arguments. Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).
Author: Robert M. Conte
Publisher: Springer Science & Business Media
ISBN: 9783540200871
Category : Science
Languages : en
Pages : 306
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Book Description
Many physical phenomena are described by nonlinear evolution equation. Those that are integrable provide various mathematical methods, presented by experts in this tutorial book, to find special analytic solutions to both integrable and partially integrable equations. The direct method to build solutions includes the analysis of singularities à la Painlevé, Lie symmetries leaving the equation invariant, extension of the Hirota method, construction of the nonlinear superposition formula. The main inverse method described here relies on the bi-hamiltonian structure of integrable equations. The book also presents some extension to equations with discrete independent and dependent variables. The different chapters face from different points of view the theory of exact solutions and of the complete integrability of nonlinear evolution equations. Several examples and applications to concrete problems allow the reader to experience directly the power of the different machineries involved.
Author: J. Kral
Publisher: Springer Science & Business Media
ISBN: 1461344255
Category : Mathematics
Languages : en
Pages : 138
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Book Description
Preface.- Gottfried Anger: Direct and inverse problems in potential theory.- Viorel Barbu: Regularity results for sane differential equations associated with maximal monotone operators in Hilbert spaces.- Haim Brezis: Classes d'interpolation associées à un opérateur monotone et applications.- Siegfried Dnümmel: On inverse problems for k-dimensional potentials.- Jozef Ka?ur: Application of Rothe's method to nonlinear parabolic boundary value problems.- Josef Král: Potentials and removability of singularities.- Vladimir Lovicar: Theorem of Fréchet and asymptotically almost periodid solutions of.
