Cauchy Problem For Differential Operators With Double Characteristics

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Cauchy Problem for Differential Operators with Double Characteristics

Author : Tatsuo Nishitani
Publisher : Springer
ISBN 13 : 3319676121
Total Pages : 215 pages
Book Rating : 4.3/5 (196 download)

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Book Synopsis Cauchy Problem for Differential Operators with Double Characteristics by : Tatsuo Nishitani

Download or read book Cauchy Problem for Differential Operators with Double Characteristics written by Tatsuo Nishitani and published by Springer. This book was released on 2017-11-24 with total page 215 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Notas E Communicações de Matemática

Author : Universidade Federal de Pernambuco. Departamento de Matemática
Publisher :
ISBN 13 :
Total Pages : 44 pages
Book Rating : 4.3/5 (243 download)

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Book Synopsis Notas E Communicações de Matemática by : Universidade Federal de Pernambuco. Departamento de Matemática

Download or read book Notas E Communicações de Matemática written by Universidade Federal de Pernambuco. Departamento de Matemática and published by . This book was released on 1965 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Cauchy Problem for Noneffectively Hyperbolic Operators

Author : Tatsuo Nishitani
Publisher :
ISBN 13 : 9784864970181
Total Pages : 0 pages
Book Rating : 4.9/5 (71 download)

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Book Synopsis Cauchy Problem for Noneffectively Hyperbolic Operators by : Tatsuo Nishitani

Download or read book Cauchy Problem for Noneffectively Hyperbolic Operators written by Tatsuo Nishitani and published by . This book was released on 2013-06 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Annotation At a double characteristic point of a differential operator with real characteristics, the linearization of the Hamilton vector field of the principal symbol is called the Hamilton map and according to either the Hamilton map has non-zero real eigenvalues or not, the operator is said to be effectively hyperbolic or noneffectively hyperbolic. For noneffectively hyperbolic operators, it was proved in the late of 1970s that for the Cauchy problem to be C well posed the subprincipal symbol has to be real and bounded, in modulus, by the sum of modulus of pure imaginary eigenvalues of the Hamilton map. It has been recognized that what is crucial to the C well-posedness is not only the Hamilton map but also the behavior of orbits of the Hamilton flow near the double characteristic manifold and the Hamilton map itself is not enough to determine completely the behavior of orbits of the flow. Strikingly enough, if there is an orbit of the Hamilton flow which lands tangentially on the double characteristic manifold then the Cauchy problem is not C well posed even though the Levi condition is satisfied, only well posed in much smaller function spaces, the Gevrey class of order 1 s 5 and not well posed in the Gevrey class of order s 5. In this lecture, we provide a general picture of the Cauchy problem for noneffectively hyperbolic operators, from the view point that the Hamilton map and the geometry of orbits of the Hamilton flow completely characterizes the well/not well-posedness of the Cauchy problem, exposing well/not well-posed results of the Cauchy problem with detailed proofs. Book jacket.

Uniqueness and Non-Uniqueness in the Cauchy Problem

Author : Zuily
Publisher : Springer Science & Business Media
ISBN 13 : 1489966560
Total Pages : 184 pages
Book Rating : 4.4/5 (899 download)

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Book Synopsis Uniqueness and Non-Uniqueness in the Cauchy Problem by : Zuily

Download or read book Uniqueness and Non-Uniqueness in the Cauchy Problem written by Zuily and published by Springer Science & Business Media. This book was released on 2013-11-21 with total page 184 pages. Available in PDF, EPUB and Kindle. Book excerpt:

The Cauchy Problem for Higher Order Abstract Differential Equations

Author : Ti-Jun Xiao
Publisher : Springer
ISBN 13 : 3540494790
Total Pages : 314 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis The Cauchy Problem for Higher Order Abstract Differential Equations by : Ti-Jun Xiao

Download or read book The Cauchy Problem for Higher Order Abstract Differential Equations written by Ti-Jun Xiao and published by Springer. This book was released on 2013-12-11 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main purpose of this book is to present the basic theory and some recent de velopments concerning the Cauchy problem for higher order abstract differential equations u(n)(t) + ~ AiU(i)(t) = 0, t ~ 0, { U(k)(O) = Uk, 0 ~ k ~ n-l. where AQ, Ab . . . , A - are linear operators in a topological vector space E. n 1 Many problems in nature can be modeled as (ACP ). For example, many n initial value or initial-boundary value problems for partial differential equations, stemmed from mechanics, physics, engineering, control theory, etc. , can be trans lated into this form by regarding the partial differential operators in the space variables as operators Ai (0 ~ i ~ n - 1) in some function space E and letting the boundary conditions (if any) be absorbed into the definition of the space E or of the domain of Ai (this idea of treating initial value or initial-boundary value problems was discovered independently by E. Hille and K. Yosida in the forties). The theory of (ACP ) is closely connected with many other branches of n mathematics. Therefore, the study of (ACPn) is important for both theoretical investigations and practical applications. Over the past half a century, (ACP ) has been studied extensively.

Spectral Properties of Differential Operators with Vanishing Coefficients

Author : Daniel Jordon
Publisher :
ISBN 13 :
Total Pages : 208 pages
Book Rating : 4.:/5 (854 download)

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Book Synopsis Spectral Properties of Differential Operators with Vanishing Coefficients by : Daniel Jordon

Download or read book Spectral Properties of Differential Operators with Vanishing Coefficients written by Daniel Jordon and published by . This book was released on 2013 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this thesis is to ascertain whether linear differential operators with vanishing coefficients make suitable operators for Cauchy problems. Well-posedness for linear Cauchy problems - characterized by existence, uniqueness, and continuous dependence on the initial data - depends on a ray being in the spectrum of the operator and an estimate for the resolvent operator along this ray. This was originally shown by Hille and Yosida for operators when every positive real number is in the resolvent set, and later generalized by Feller, Miyadera, and Phillips. We restrict our attention to the setting where the differential operator acts on functions that depend on a spatial variable that takes values from a bounded subset of the real line. We establish ill-posedness of the Cauchy problem by analyzing the spectral properties of the differential operator and prove the spectrum is the entire complex plane for a wide variety of differential operators with vanishing coefficients. If the differential operator is the product of a polynomial in the derivative with a scaler function that has roots of finite multiplicity, we develop simple criteria for establishing ill-posedness of the Cauchy problem. We establish point spectral results when the function is real-valued with only simple roots, in particular we show the point spectrum is the entire complex plane. Much less is known when the coefficients of the differential operator depend on time. For these non-autonomous Cauchy problems (NCPs) only sufficient conditions for well-posedness are known, with necessary conditions still lacking. In this thesis we make strides with establishing necessary spectral conditions for well-posed NCPs in the case where the family of operators is continuous.

Pseudo Differential Operators

Author : M. Taylor
Publisher : Springer
ISBN 13 : 3540372660
Total Pages : 160 pages
Book Rating : 4.5/5 (43 download)

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Book Synopsis Pseudo Differential Operators by : M. Taylor

Download or read book Pseudo Differential Operators written by M. Taylor and published by Springer. This book was released on 2006-12-08 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes are based on the lectures given on partial differential equations at the University of Michigan during the winter semester of 1972, with some extensions. The students to whom these lectures were addressed were assumed to have knowledge of elementary functional analysis, the Fourier transform, distribution theory, and Sobolev spaces, and such tools are used without comment. In this monography, we develop one tool, the calculus of pseudo differential operators, and apply it to several of the main problems of partial differential equations.

The Analysis of Linear Partial Differential Operators II

Author : Lars Hörmander
Publisher : Springer Science & Business Media
ISBN 13 : 9783540225164
Total Pages : 416 pages
Book Rating : 4.2/5 (251 download)

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Book Synopsis The Analysis of Linear Partial Differential Operators II by : Lars Hörmander

Download or read book The Analysis of Linear Partial Differential Operators II written by Lars Hörmander and published by Springer Science & Business Media. This book was released on 2004-11-17 with total page 416 pages. Available in PDF, EPUB and Kindle. Book excerpt: Author received the 1962 Fields Medal Author received the 1988 Wolf Prize (honoring achievemnets of a lifetime) Author is leading expert in partial differential equations

Hyperbolic Differential Operators And Related Problems

Author : Vincenzo Ancona
Publisher : CRC Press
ISBN 13 : 9780203911143
Total Pages : 390 pages
Book Rating : 4.9/5 (111 download)

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Book Synopsis Hyperbolic Differential Operators And Related Problems by : Vincenzo Ancona

Download or read book Hyperbolic Differential Operators And Related Problems written by Vincenzo Ancona and published by CRC Press. This book was released on 2003-03-06 with total page 390 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presenting research from more than 30 international authorities, this reference provides a complete arsenal of tools and theorems to analyze systems of hyperbolic partial differential equations. The authors investigate a wide variety of problems in areas such as thermodynamics, electromagnetics, fluid dynamics, differential geometry, and topology. Renewing thought in the field of mathematical physics, Hyperbolic Differential Operators defines the notion of pseudosymmetry for matrix symbols of order zero as well as the notion of time function. Surpassing previously published material on the topic, this text is key for researchers and mathematicians specializing in hyperbolic, Schrödinger, Einstein, and partial differential equations; complex analysis; and mathematical physics.

Partial Differential Equations IV

Author : Yu.V. Egorov
Publisher : Springer Science & Business Media
ISBN 13 : 3662092077
Total Pages : 247 pages
Book Rating : 4.6/5 (62 download)

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Book Synopsis Partial Differential Equations IV by : Yu.V. Egorov

Download or read book Partial Differential Equations IV written by Yu.V. Egorov and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt: A two-part monograph covering recent research in an important field, previously scattered in numerous journals, including the latest results in the theory of mixed problems for hyperbolic operators. The book is hence of immense value to graduate students and researchers in partial differential equations and theoretical physics.

The Analysis of Linear Partial Differential Operators III

Author : Lars Hörmander
Publisher : Springer Science & Business Media
ISBN 13 : 3540499377
Total Pages : 537 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis The Analysis of Linear Partial Differential Operators III by : Lars Hörmander

Download or read book The Analysis of Linear Partial Differential Operators III written by Lars Hörmander and published by Springer Science & Business Media. This book was released on 2007-03-15 with total page 537 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the reviews: "Volumes III and IV complete L. Hörmander's treatise on linear partial differential equations. They constitute the most complete and up-to-date account of this subject, by the author who has dominated it and made the most significant contributions in the last decades.....It is a superb book, which must be present in every mathematical library, and an indispensable tool for all - young and old - interested in the theory of partial differential operators." L. Boutet de Monvel in Bulletin of the American Mathematical Society, 1987. "This treatise is outstanding in every respect and must be counted among the great books in mathematics. It is certainly no easy reading (...) but a careful study is extremely rewarding for its wealth of ideas and techniques and the beauty of presentation." J. Brüning in Zentralblatt MATH, 1987.

Patterns and Waves

Author : T. Nishida
Publisher : Elsevier
ISBN 13 : 0080875394
Total Pages : 709 pages
Book Rating : 4.0/5 (88 download)

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Book Synopsis Patterns and Waves by : T. Nishida

Download or read book Patterns and Waves written by T. Nishida and published by Elsevier. This book was released on 2011-09-22 with total page 709 pages. Available in PDF, EPUB and Kindle. Book excerpt: Part I of this volume surveys the developments in the analysis of nonlinear phenomena in Japan during the past decade, while Part II consists of up-to-date original papers concerning qualitative theories and their applications.Dealt with here are nonlinear problems related to general analysis, fluid dynamics, mathematical biology and computer sciences, and their underlying mathematical structures, e.g. nonlinear waves and propagations, bifurcation phenomena, chaotic phenomena, and fractals.The volume is dedicated to Professor Masaya Yamaguti in celebration of his 60th birthday.

Lectures on Cauchy's Problem in Linear Partial Differential Equations

Author : Jacques Hadamard
Publisher :
ISBN 13 :
Total Pages : 344 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Lectures on Cauchy's Problem in Linear Partial Differential Equations by : Jacques Hadamard

Download or read book Lectures on Cauchy's Problem in Linear Partial Differential Equations written by Jacques Hadamard and published by . This book was released on 1923 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Theory of Differential Equations

Author : I. M. Gel'fand
Publisher : Academic Press
ISBN 13 : 1483262324
Total Pages : 233 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis Theory of Differential Equations by : I. M. Gel'fand

Download or read book Theory of Differential Equations written by I. M. Gel'fand and published by Academic Press. This book was released on 2014-05-12 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: Generalized Functions, Volume 3: Theory of Differential Equations focuses on the application of generalized functions to problems of the theory of partial differential equations. This book discusses the problems of determining uniqueness and correctness classes for solutions of the Cauchy problem for systems with constant coefficients and eigenfunction expansions for self-adjoint differential operators. The topics covered include the bounded operators in spaces of type W, Cauchy problem in a topological vector space, and theorem of the Phragmén-Lindelöf type. The correctness classes for the Cauchy problem, systems that are Petrovski?-correct, and generalized eigenfunctions of self-adjoint operators are also reviewed. This text likewise covers the differentiation of functionals of strongly and weakly bounded variation. This volume is beneficial to students and researchers interested in the theory of differential equations.

Unpublished Manuscripts

Author : Lars Hörmander
Publisher : Springer
ISBN 13 : 3319698508
Total Pages : 368 pages
Book Rating : 4.3/5 (196 download)

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Book Synopsis Unpublished Manuscripts by : Lars Hörmander

Download or read book Unpublished Manuscripts written by Lars Hörmander and published by Springer. This book was released on 2018-06-19 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents, for the first time, the unpublished manuscripts of Lars Hörmander, written between 1951 and 2007. Hörmander himself organised the manuscripts and also wrote the notes explaining their origins, presenting the material in the form he fully intended it to be published in. As his daughter, Sofia Broström, mentions in the Foreword, towards the end of his life, Hörmander "carefully went through his unpublished manuscripts, checking and revising each of them with his very critical eye, deciding what should be kept for posterity and what should be thrown out". He also compiled the complete bibliography of all his published mathematical works that is included at the end of the present book. Of both historical and mathematical value, the contents of this book will undoubtedly inspire mathematicians of different horizons.

Homogenization of Differential Operators and Integral Functionals

Author : V.V. Jikov
Publisher : Springer Science & Business Media
ISBN 13 : 3642846599
Total Pages : 583 pages
Book Rating : 4.6/5 (428 download)

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Book Synopsis Homogenization of Differential Operators and Integral Functionals by : V.V. Jikov

Download or read book Homogenization of Differential Operators and Integral Functionals written by V.V. Jikov and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 583 pages. Available in PDF, EPUB and Kindle. Book excerpt: It was mainly during the last two decades that the theory of homogenization or averaging of partial differential equations took shape as a distinct mathe matical discipline. This theory has a lot of important applications in mechanics of composite and perforated materials, filtration, disperse media, and in many other branches of physics, mechanics and modern technology. There is a vast literature on the subject. The term averaging has been usually associated with the methods of non linear mechanics and ordinary differential equations developed in the works of Poincare, Van Der Pol, Krylov, Bogoliubov, etc. For a long time, after the works of Maxwell and Rayleigh, homogeniza tion problems for· partial differential equations were being mostly considered by specialists in physics and mechanics, and were staying beyond the scope of mathematicians. A great deal of attention was given to the so called disperse media, which, in the simplest case, are two-phase media formed by the main homogeneous material containing small foreign particles (grains, inclusions). Such two-phase bodies, whose size is considerably larger than that of each sep arate inclusion, have been discovered to possess stable physical properties (such as heat transfer, electric conductivity, etc.) which differ from those of the con stituent phases. For this reason, the word homogenized, or effective, is used in relation to these characteristics. An enormous number of results, approximation formulas, and estimates have been obtained in connection with such problems as electromagnetic wave scattering on small particles, effective heat transfer in two-phase media, etc.

Seminar on Singularities of Solutions of Linear Partial Differential Equations. (AM-91), Volume 91

Author : Lars Hörmander
Publisher : Princeton University Press
ISBN 13 : 1400881587
Total Pages : 296 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis Seminar on Singularities of Solutions of Linear Partial Differential Equations. (AM-91), Volume 91 by : Lars Hörmander

Download or read book Seminar on Singularities of Solutions of Linear Partial Differential Equations. (AM-91), Volume 91 written by Lars Hörmander and published by Princeton University Press. This book was released on 2016-03-02 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: Singularities of solutions of differential equations forms the common theme of these papers taken from a seminar held at the Institute for Advanced Study in Princeton in 1977-1978. While some of the lectures were devoted to the analysis of singularities, others focused on applications in spectral theory. As an introduction to the subject, this volume treats current research in the field in such a way that it can be studied with profit by the non-specialist.