Cauchy Problem For Noneffectively Hyperbolic Operators
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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.
Hyperbolic Differential Operators And Related Problems
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Author : Vincenzo Ancona
language : en
Publisher: CRC Press
Release Date : 2003-03-06
Hyperbolic Differential Operators And Related Problems written by Vincenzo Ancona and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-03-06 with Mathematics categories.
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.
Phase Space Analysis Of Partial Differential Equations
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Author : Antonio Bove
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-12-28
Phase Space Analysis Of Partial Differential Equations written by Antonio Bove and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-12-28 with Mathematics categories.
Covers phase space analysis methods, including microlocal analysis, and their applications to physics Treats the linear and nonnlinear aspects of the theory of PDEs Original articles are self-contained with full proofs; survey articles give a quick and direct introduction to selected topics evolving at a fast pace Excellent reference and resource for grad students and researchers in PDEs and related fields
Evolution Equations Of Hyperbolic And Schr Dinger Type
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Author : Michael Ruzhansky
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-08-04
Evolution Equations Of Hyperbolic And Schr Dinger Type written by Michael Ruzhansky and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-08-04 with Mathematics categories.
Evolution equations of hyperbolic or more general p-evolution type form an active field of current research. This volume aims to collect some recent advances in the area in order to allow a quick overview of ongoing research. The contributors are first rate mathematicians. This collection of research papers is centred around parametrix constructions and microlocal analysis; asymptotic constructions of solutions; energy and dispersive estimates; and associated spectral transforms. Applications concerning elasticity and general relativity complement the volume. The book gives an overview of a variety of ongoing current research in the field and, therefore, allows researchers as well as students to grasp new aspects and broaden their understanding of the area.
Studies In Phase Space Analysis With Applications To Pdes
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Author : Massimo Cicognani
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-12
Studies In Phase Space Analysis With Applications To Pdes written by Massimo Cicognani and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-12 with Mathematics categories.
This collection of original articles and surveys, emerging from a 2011 conference in Bertinoro, Italy, addresses recent advances in linear and nonlinear aspects of the theory of partial differential equations (PDEs). Phase space analysis methods, also known as microlocal analysis, have continued to yield striking results over the past years and are now one of the main tools of investigation of PDEs. Their role in many applications to physics, including quantum and spectral theory, is equally important. Key topics addressed in this volume include: *general theory of pseudodifferential operators *Hardy-type inequalities *linear and non-linear hyperbolic equations and systems *Schrödinger equations *water-wave equations *Euler-Poisson systems *Navier-Stokes equations *heat and parabolic equations Various levels of graduate students, along with researchers in PDEs and related fields, will find this book to be an excellent resource. Contributors T. Alazard P.I. Naumkin J.-M. Bony F. Nicola N. Burq T. Nishitani C. Cazacu T. Okaji J.-Y. Chemin M. Paicu E. Cordero A. Parmeggiani R. Danchin V. Petkov I. Gallagher M. Reissig T. Gramchev L. Robbiano N. Hayashi L. Rodino J.Huang M. Ruzhanky D. Lannes J.-C. Saut F. Linares N. Visciglia P.B. Mucha P. Zhang C. Mullaert E. Zuazua T. Narazaki C. Zuily
Advances In Phase Space Analysis Of Partial Differential Equations
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Author : Antonio Bove
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-09-18
Advances In Phase Space Analysis Of Partial Differential Equations written by Antonio Bove and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-09-18 with Mathematics categories.
This collection of original articles and surveys addresses the recent advances in linear and nonlinear aspects of the theory of partial differential equations. The key topics include operators as "sums of squares" of real and complex vector fields, nonlinear evolution equations, local solvability, and hyperbolic questions.
Hyperbolic Equations And Related Topics
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Author : Sigeru Mizohata
language : en
Publisher: Academic Press
Release Date : 2014-05-10
Hyperbolic Equations And Related Topics 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.
Hyperbolic Equations and Related Topics covers the proceedings of the Taniguchi International Symposium, held in Katata, Japan on August 27-31, 1984 and in Kyoto, Japan on September 3-5, 1984. The book focuses on the mathematical analyses involved in hyperbolic equations. The selection first elaborates on complex vector fields; holomorphic extension of CR functions and related problems; second microlocalization and propagation of singularities for semi-linear hyperbolic equations; and scattering matrix for two convex obstacles. Discussions focus on the construction of asymptotic solutions, singular vector fields and Leibniz formula, second microlocalization along a Lagrangean submanifold, and hypo-analytic structures. The text then ponders on the Cauchy problem for effectively hyperbolic equations and for uniformly diagonalizable hyperbolic systems in Gevrey classes. The book takes a look at generalized Hamilton flows and singularities of solutions of the hyperbolic Cauchy problem and analytic and Gevrey well-posedness of the Cauchy problem for second order weakly hyperbolic equations with coefficients irregular in time. The selection is a dependable reference for researchers interested in hyperbolic equations.
Ricerche Di Matematica
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Author :
language : en
Publisher:
Release Date : 2014
Ricerche Di Matematica written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014 with Mathematics categories.
Cauchy Problem For Noneffectively Hyperbolic Operators
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Author : Tatsuo Nishitani
language : en
Publisher:
Release Date : 2013-06
Cauchy Problem For Noneffectively Hyperbolic Operators written by Tatsuo Nishitani and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06 with Mathematics categories.
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.
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Author :
language : en
Publisher:
Release Date : 1994
written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994 with Mathematics categories.