Hyperbolic Differential Operators And Related Problems

Author: Vincenzo Ancona,Jean Vaillant

Publisher: CRC Press

ISBN: 9780203911143

Category: Mathematics

Page: 388

View: 5888

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.
Posted in Mathematics

Hyperbolic Partial Differential Equations and Geometric Optics

Author: Jeffrey Rauch

Publisher: American Mathematical Soc.

ISBN: 0821872915

Category: Mathematics

Page: 363

View: 9356

This book introduces graduate students and researchers in mathematics and the sciences to the multifaceted subject of the equations of hyperbolic type, which are used, in particular, to describe propagation of waves at finite speed. Among the topics carefully presented in the book are nonlinear geometric optics, the asymptotic analysis of short wavelength solutions, and nonlinear interaction of such waves. Studied in detail are the damping of waves, resonance, dispersive decay, and solutions to the compressible Euler equations with dense oscillations created by resonant interactions. Many fundamental results are presented for the first time in a textbook format. In addition to dense oscillations, these include the treatment of precise speed of propagation and the existence and stability questions for the three wave interaction equations. One of the strengths of this book is its careful motivation of ideas and proofs, showing how they evolve from related, simpler cases. This makes the book quite useful to both researchers and graduate students interested in hyperbolic partial differential equations. Numerous exercises encourage active participation of the reader. The author is a professor of mathematics at the University of Michigan. A recognized expert in partial differential equations, he has made important contributions to the transformation of three areas of hyperbolic partial differential equations: nonlinear microlocal analysis, the control of waves, and nonlinear geometric optics.
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Phase Space Analysis of Partial Differential Equations

Author: Antonio Bove,Ferruccio Colombini,Daniele Del Santo

Publisher: Springer Science & Business Media

ISBN: 9780817645212

Category: Mathematics

Page: 343

View: 7878

This collection of original articles and surveys treats linear and nonlinear aspects of the theory of partial differential equations. Phase space analysis methods, also known as microlocal analysis, have yielded striking results over the past years and have become one of the main tools of investigation. Equally important is their role in many applications to physics, for example, in quantum and spectral theory. Contributors: H. Bahouri, M. Baouendi, E. Bernardi, M. Bony, A. Bove, N. Burq, J.-Y. Chemin, F. Colombini, T. Colin, P. Cordaro, G. Eskin, X. Fu, N. Hanges, G. Métivier, P. Michor, T. Nishitani, A. Parmeggiani,L. Pernazza, V. Petkov, F. Planchon, M. Prizzi, D. Del Santo, D. Tartakof, D. Tataru, F. Treves, C.-J. Xu, X. Zhang, E. Zuazua
Posted in Mathematics

Cauchy Problem for Differential Operators with Double Characteristics

Non-Effectively Hyperbolic Characteristics

Author: Tatsuo Nishitani

Publisher: Springer

ISBN: 3319676121

Category: Mathematics

Page: 213

View: 1240

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.
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Partial Differential Equations IV

Microlocal Analysis and Hyperbolic Equations

Author: Yu.V. Egorov,M.A. Shubin

Publisher: Springer Science & Business Media

ISBN: 3662092077

Category: Mathematics

Page: 244

View: 8636

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.
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Progress in Partial Differential Equations

Asymptotic Profiles, Regularity and Well-Posedness

Author: Michael Reissig,Michael V. Ruzhansky

Publisher: Springer Science & Business Media

ISBN: 3319001256

Category: Mathematics

Page: 447

View: 7662

Progress in Partial Differential Equations is devoted to modern topics in the theory of partial differential equations. It consists of both original articles and survey papers covering a wide scope of research topics in partial differential equations and their applications. The contributors were participants of the 8th ISAAC congress in Moscow in 2011 or are members of the PDE interest group of the ISAAC society. This volume is addressed to graduate students at various levels as well as researchers in partial differential equations and related fields. The readers will find this an excellent resource of both introductory and advanced material. The key topics are: • Linear hyperbolic equations and systems (scattering, symmetrisers) • Non-linear wave models (global existence, decay estimates, blow-up) • Evolution equations (control theory, well-posedness, smoothing) • Elliptic equations (uniqueness, non-uniqueness, positive solutions) • Special models from applications (Kirchhoff equation, Zakharov-Kuznetsov equation, thermoelasticity)
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Lectures on Nonlinear Hyperbolic Differential Equations

Author: Lars Hörmander

Publisher: Springer Science & Business Media

ISBN: 9783540629214

Category: Mathematics

Page: 289

View: 2994

In this introductory textbook, a revised and extended version of well-known lectures by L. Hörmander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Four chapters are devoted to microanalysis of the singularities of the solutions. This part assumes some familiarity with pseudodifferential operators which are standard in the theory of linear differential operators, but the extension to the more exotic classes of opertors needed in the nonlinear theory is presented in complete detail.
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Differential Operators and Related Topics

Proceedings of the Mark Krein International Conference on Operator Theory and Applications, Odessa, Ukraine, August 18-22, 1997

Author: V. M. Adami͡an

Publisher: Springer Science & Business Media

ISBN: 9783764362874

Category: Mathematics

Page: 420

View: 6830

The present book is the first of the two volume proceedings of the Mark Krein International Conference on Operator Theory and Applications. This conference, which was dedicated to the 90th anniversary of the prominent mathematician Mark Krein, was held in Odessa, Ukraine, from August 18-22, 1997. The conference focused on the main ideas, methods, results, and achievements of M. G. Krein. This first volume is devoted to the theory of differential operators and related topics. It opens with a description of the conference, biographical material and a number of survey papers about the work of M. G. Krein. The main part of the book consists of original research papers presenting the state of the art in the area of differential operators. The second volume of these proceedings, entitled Operator Theory and Related Topics, concerns the other aspects of the conference. The two volumes will be of interest to a wide range of readership in pure and applied mathematics, physics and engineering sciences.
Posted in Mathematics

Partial Differential Equations of Hyperbolic Type and Applications

Author: Giuseppe Geymonat

Publisher: World Scientific

ISBN: 9789971502058

Category: Mathematics

Page: 178

View: 7883

This book introduces the general aspects of hyperbolic conservation laws and their numerical approximation using some of the most modern tools: spectral methods, unstructured meshes and ?-formulation. The applications of these methods are found in some significant examples such as the Euler equations. This book, a collection of articles by the best authors in the field, exposes the reader to the frontier of the research and many open problems.
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Mathematical Reviews

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 6837

Posted in Mathematics

Partial Differential Equations in Clifford Analysis

Author: Elena Obolashvili

Publisher: CRC Press

ISBN: 9780582317499

Category: Mathematics

Page: 160

View: 2487

Clifford analysis represents one of the most remarkable fields of modern mathematics. With the recent finding that almost all classical linear partial differential equations of mathematical physics can be set in the context of Clifford analysis-and that they can be obtained without applying any physical laws-it appears that Clifford analysis itself can suggest new equations or new generalizations of classical equations that may have some physical content. Partial Differential Equations in Clifford Analysis considers-in a multidimensional space-elliptic, hyperbolic, and parabolic operators related to Helmholtz, Klein-Gordon, Maxwell, Dirac, and heat equations. The author addresses two kinds of parabolic operators, both related to the second-order parabolic equations whose principal parts are the Laplacian and d'Alembertian: an elliptic-type parabolic operator and a hyperbolic-type parabolic operator. She obtains explicit integral representations of solutions to various boundary and initial value problems and their properties and solves some two-dimensional and non-local problems. Written for the specialist but accessible to non-specialists as well, Partial Differential Equations in Clifford Analysis presents new results, reformulations, refinements, and extensions of familiar material in a manner that allows the reader to feel and touch every formula and problem. Mathematicians and physicists interested in boundary and initial value problems, partial differential equations, and Clifford analysis will find this monograph a refreshing and insightful study that helps fill a void in the literature and in our knowledge.
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Fundamental Solutions for Differential Operators and Applications

Author: Prem Kythe

Publisher: Springer Science & Business Media

ISBN: 1461241065

Category: Mathematics

Page: 414

View: 5127

A self-contained and systematic development of an aspect of analysis which deals with the theory of fundamental solutions for differential operators, and their applications to boundary value problems of mathematical physics, applied mathematics, and engineering, with the related computational aspects.
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Inverse Problems and Related Topics

Author: Gen Nakamura,Saburou Saitoh,Jin Kean Seo

Publisher: CRC Press

ISBN: 9781584881919

Category: Mathematics

Page: 248

View: 6508

Inverse problems arise in many disciplines and hold great importance to practical applications. However, sound new methods are needed to solve these problems. Over the past few years, Japanese and Korean mathematicians have obtained a number of very interesting and unique results in inverse problems. Inverse Problems and Related Topics compiles papers authored by some of the top researchers in Korea and Japan. It presents a number of original and useful results and offers a unique opportunity to explore the current trends of research in inverse problems in these countries. Highlighting the existence and active work of several Japanese and Korean groups, it also serves as a guide to those seeking future scientific exchange with researchers in these countries.
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Hyperbolic Problems and Regularity Questions

Author: Mariarosaria Padula,Luisa Zanghirati

Publisher: Springer Science & Business Media

ISBN: 3764374519

Category: Mathematics

Page: 231

View: 4898

This book discusses new challenges in the quickly developing field of hyperbolic problems. Particular emphasis lies on the interaction between nonlinear partial differential equations, functional analysis and applied analysis as well as mechanics. The book originates from a recent conference focusing on hyperbolic problems and regularity questions. It is intended for researchers in functional analysis, PDE, fluid dynamics and differential geometry.
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Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems

Author: Songmu Zheng

Publisher: CRC Press

ISBN: 9780582244887

Category: Mathematics

Page: 272

View: 6744

This monograph is devoted to the global existence, uniqueness and asymptotic behaviour of smooth solutions to both initial value problems and initial boundary value problems for nonlinear parabolic equations and hyperbolic parabolic coupled systems. Most of the material is based on recent research carried out by the author and his collaborators. The book can be divided into two parts. In the first part, the results on decay of solutions to nonlinear parabolic equations and hyperbolic parabolic coupled systems are obtained, and a chapter is devoted to the global existence of small smooth solutions to fully nonlinear parabolic equations and quasilinear hyperbolic parabolic coupled systems. Applications of the results to nonlinear thermoelasticity and fluid dynamics are also shown. Some nonlinear parabolic equations and coupled systems arising from the study of phase transitions are investigated in the second part of the book. The global existence, uniqueness and asymptotic behaviour of smooth solutions with arbitrary initial data are obtained. The final chapter is further devoted to related topics: multiplicity of equilibria and the existence of a global attractor, inertial manifold and inertial set. A knowledge of partial differential equations and Sobolev spaces is assumed. As an aid to the reader, the related concepts and results are collected and the relevant references given in the first chapter. The work will be of interest to researchers and graduate students in pure and applied mathematics, mathematical physics and applied sciences.
Posted in Mathematics

Bibliographic Index

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Bibliographical literature

Page: 26

View: 3358

Posted in Bibliographical literature

Asymptotic Methods for Investigating Quasiwave Equations of Hyperbolic Type

Author: Yuri A. Mitropolsky,G. Khoma,M. Gromyak

Publisher: Springer Science & Business Media

ISBN: 9780792345299

Category: Mathematics

Page: 214

View: 3183

The theory of partial differential equations is a wide and rapidly developing branch of contemporary mathematics. Problems related to partial differential equations of order higher than one are so diverse that a general theory can hardly be built up. There are several essentially different kinds of differential equations called elliptic, hyperbolic, and parabolic. Regarding the construction of solutions of Cauchy, mixed and boundary value problems, each kind of equation exhibits entirely different properties. Cauchy problems for hyperbolic equations and systems with variable coefficients have been studied in classical works of Petrovskii, Leret, Courant, Gording. Mixed problems for hyperbolic equations were considered by Vishik, Ladyzhenskaya, and that for general two dimensional equations were investigated by Bitsadze, Vishik, Gol'dberg, Ladyzhenskaya, Myshkis, and others. In last decade the theory of solvability on the whole of boundary value problems for nonlinear differential equations has received intensive development. Significant results for nonlinear elliptic and parabolic equations of second order were obtained in works of Gvazava, Ladyzhenskaya, Nakhushev, Oleinik, Skripnik, and others. Concerning the solvability in general of nonlinear hyperbolic equations, which are connected to the theory of local and nonlocal boundary value problems for hyperbolic equations, there are only partial results obtained by Bronshtein, Pokhozhev, Nakhushev.
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Pseudo-Differential Operators: Analysis, Applications and Computations

Author: Luigi Rodino,M. W. Wong,Hongmei Zhu

Publisher: Springer Science & Business Media

ISBN: 9783034800495

Category: Mathematics

Page: 308

View: 8083

This volume consists of eighteen peer-reviewed papers related to lectures on pseudo-differential operators presented at the meeting of the ISAAC Group in Pseudo-Differential Operators (IGPDO) held at Imperial College London on July 13-18, 2009. Featured in this volume are the analysis, applications and computations of pseudo-differential operators in mathematics, physics and signal analysis. This volume is a useful complement to the volumes “Advances in Pseudo-Differential Operators”, “Pseudo-Differential Operators and Related Topics”, “Modern Trends in Pseudo-Differential Operators”, “New Developments in Pseudo-Differential Operators” and “Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations” published in the same series in, respectively, 2004, 2006, 2007, 2009 and 2010.
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Control Theory for Partial Differential Equations: Volume 2, Abstract Hyperbolic-like Systems Over a Finite Time Horizon

Continuous and Approximation Theories

Author: Irena Lasiecka,Roberto Triggiani

Publisher: Cambridge University Press

ISBN: 9780521584012

Category: Mathematics

Page: 1067

View: 3262

Originally published in 2000, this is the second volume of a comprehensive two-volume treatment of quadratic optimal control theory for partial differential equations over a finite or infinite time horizon, and related differential (integral) and algebraic Riccati equations. Both continuous theory and numerical approximation theory are included. The authors use an abstract space, operator theoretic approach, which is based on semigroups methods, and which unifies across a few basic classes of evolution. The various abstract frameworks are motivated by, and ultimately directed to, partial differential equations with boundary/point control. Volume 2 is focused on the optimal control problem over a finite time interval for hyperbolic dynamical systems. A few abstract models are considered, each motivated by a particular canonical hyperbolic dynamics. It presents numerous fascinating results. These volumes will appeal to graduate students and researchers in pure and applied mathematics and theoretical engineering with an interest in optimal control problems.
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Introduction to Partial Differential Equations and Hilbert Space Methods

Author: Karl E. Gustafson

Publisher: Courier Corporation

ISBN: 9780486612713

Category: Mathematics

Page: 448

View: 3481

This volume offers an excellent undergraduate-level introduction to the main topics, methods, and applications of partial differential equations. Chapter 1 presents a full introduction to partial differential equations and Fourier series as related to applied mathematics. Chapter 2 begins with a more comprehensive look at the principal method for solving partial differential equations — the separation of variables — and then more fully develops that approach in the contexts of Hilbert space and numerical methods. Chapter 3 includes an expanded treatment of first-order systems, a short introduction to computational methods, and aspects of topical research on the partial differential equations of fluid dynamics. With over 600 problems and exercises, along with explanations, examples, and a comprehensive section of answers, hints, and solutions, this superb, easy-to-use text is ideal for a one-semester or full-year course. It will also provide the mathematically inclined layperson with a stimulating review of the subject's essentials.
Posted in Mathematics