The Hyperbolic Cauchy Problem
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The Hyperbolic Cauchy Problem
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Author : Kunihiko Kajitani
language : en
Publisher: Springer
Release Date : 2006-11-15
The Hyperbolic Cauchy Problem written by Kunihiko Kajitani and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-15 with Mathematics categories.
The approach to the Cauchy problem taken here by the authors is based on theuse of Fourier integral operators with a complex-valued phase function, which is a time function chosen suitably according to the geometry of the multiple characteristics. The correctness of the Cauchy problem in the Gevrey classes for operators with hyperbolic principal part is shown in the first part. In the second part, the correctness of the Cauchy problem for effectively hyperbolic operators is proved with a precise estimate of the loss of derivatives. This method can be applied to other (non) hyperbolic problems. The text is based on a course of lectures given for graduate students but will be of interest to researchers interested in hyperbolic partial differential equations. In the latter part the reader is expected to be familiar with some theory of pseudo-differential operators.
The Hyperbolic Cauchy Problem
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Author : Kunihiko Kajitani
language : en
Publisher:
Release Date : 2014-01-15
The Hyperbolic Cauchy Problem written by Kunihiko Kajitani and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-15 with categories.
Hyperbolic Systems Of Conservation Laws
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Author : Alberto Bressan
language : en
Publisher: Oxford University Press, USA
Release Date : 2000
Hyperbolic Systems Of Conservation Laws written by Alberto Bressan and has been published by Oxford University Press, USA this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Mathematics categories.
This book provides a self-contained introduction to the mathematical theory of hyperbolic systems of conservation laws, with particular emphasis on the study of discontinuous solutions, characterized by the appearance of shock waves. This area has experienced substantial progress in very recent years thanks to the introduction of new techniques, in particular the front tracking algorithm and the semigroup approach. These techniques provide a solution to the long standing open problems of uniqueness and stability of entropy weak solutions. This volume is the first to present a comprehensive account of these new, fundamental advances. It also includes a detailed analysis of the stability and convergence of the front tracking algorithm. A set of problems, with varying difficulty is given at the end of each chapter to verify and expand understanding of the concepts and techniques previously discussed. For researchers, this book will provide an indispensable reference to the state of the art in the field of hyperbolic systems of conservation laws.
Cauchy S Problem For Hyperbolic Equations
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Author : Lars Gårding
language : en
Publisher:
Release Date : 1958
Cauchy S Problem For Hyperbolic Equations written by Lars Gårding and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1958 with Cauchy problem categories.
On The Cauchy Problem
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Author : Sigeru Mizohata
language : en
Publisher: Academic Press
Release Date : 2014-05-10
On The Cauchy Problem written by Sigeru Mizohata and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-05-10 with Mathematics categories.
Notes and Reports in Mathematics in Science and Engineering, Volume 3: On the Cauchy Problem focuses on the processes, methodologies, and mathematical approaches to Cauchy problems. The publication first elaborates on evolution equations, Lax-Mizohata theorem, and Cauchy problems in Gevrey class. Discussions focus on fundamental proposition, proof of theorem 4, Gevrey property in t of solutions, basic facts on pseudo-differential, and proof of theorem 3. The book then takes a look at micro-local analysis in Gevrey class, including proof and consequences of theorem 1. The manuscript examines Schrödinger type equations, as well as general view-points on evolution equations. Numerical representations and analyses are provided in the explanation of these type of equations. The book is a valuable reference for mathematicians and researchers interested in the Cauchy problem.
The Cauchy Problem For Hyperbolic Operators
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Author : Karen Yagdjian
language : en
Publisher: De Gruyter Akademie Forschung
Release Date : 1997
The Cauchy Problem For Hyperbolic Operators written by Karen Yagdjian and has been published by De Gruyter Akademie Forschung this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with Mathematics categories.
Hyperbolic Systems With Analytic Coefficients
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Author : Tatsuo Nishitani
language : en
Publisher: Springer
Release Date : 2013-11-19
Hyperbolic Systems With Analytic Coefficients written by Tatsuo Nishitani and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-19 with Mathematics categories.
This monograph focuses on the well-posedness of the Cauchy problem for linear hyperbolic systems with matrix coefficients. Mainly two questions are discussed: (A) Under which conditions on lower order terms is the Cauchy problem well posed? (B) When is the Cauchy problem well posed for any lower order term? For first order two by two systems with two independent variables with real analytic coefficients, we present complete answers for both (A) and (B). For first order systems with real analytic coefficients we prove general necessary conditions for question (B) in terms of minors of the principal symbols. With regard to sufficient conditions for (B), we introduce hyperbolic systems with nondegenerate characteristics, which contain strictly hyperbolic systems, and prove that the Cauchy problem for hyperbolic systems with nondegenerate characteristics is well posed for any lower order term. We also prove that any hyperbolic system which is close to a hyperbolic system with a nondegenerate characteristic of multiple order has a nondegenerate characteristic of the same order nearby.
Cauchy Problem For Differential Operators With Double Characteristics
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Author : Tatsuo Nishitani
language : en
Publisher: Springer
Release Date : 2017-11-24
Cauchy Problem For Differential Operators With Double Characteristics written by Tatsuo Nishitani and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-24 with Mathematics categories.
Combining geometrical and microlocal tools, this monograph gives detailed proofs of many well/ill-posed results related to the Cauchy problem for differential operators with non-effectively hyperbolic double characteristics. Previously scattered over numerous different publications, the results are presented from the viewpoint that the Hamilton map and the geometry of bicharacteristics completely characterizes the well/ill-posedness of the Cauchy problem. A doubly characteristic point of a differential operator P of order m (i.e. one where Pm = dPm = 0) is effectively hyperbolic if the Hamilton map FPm has real non-zero eigen values. When the characteristics are at most double and every double characteristic is effectively hyperbolic, the Cauchy problem for P can be solved for arbitrary lower order terms. If there is a non-effectively hyperbolic characteristic, solvability requires the subprincipal symbol of P to lie between −Pμj and Pμj , where iμj are the positive imaginary eigenvalues of FPm . Moreover, if 0 is an eigenvalue of FPm with corresponding 4 × 4 Jordan block, the spectral structure of FPm is insufficient to determine whether the Cauchy problem is well-posed and the behavior of bicharacteristics near the doubly characteristic manifold plays a crucial role.
Lectures On Cauchy S Problem In Linear Partial Differential Equations
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Author : Jacques Hadamard
language : en
Publisher: Courier Corporation
Release Date : 2014-08-25
Lectures On Cauchy S Problem In Linear Partial Differential Equations written by Jacques Hadamard and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-08-25 with Mathematics categories.
Would well repay study by most theoretical physicists." — Physics Today "An overwhelming influence on subsequent work on the wave equation." — Science Progress "One of the classical treatises on hyperbolic equations." — Royal Naval Scientific Service Delivered at Columbia University and the Universities of Rome and Zürich, these lectures represent a pioneering investigation. Jacques Hadamard based his research on prior studies by Riemann, Kirchhoff, and Volterra. He extended and improved Volterra's work, applying its theories relating to spherical and cylindrical waves to all normal hyperbolic equations instead of only to one. Topics include the general properties of Cauchy's problem, the fundamental formula and the elementary solution, equations with an odd number of independent variables, and equations with an even number of independent variables and the method of descent.
Cauchy Problem For Quasilinear Hyperbolic Systems
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Author : Liu Fagui
language : en
Publisher:
Release Date : 2006
Cauchy Problem For Quasilinear Hyperbolic Systems written by Liu Fagui and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with categories.