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Author: Michael Beals
Publisher: Springer Science & Business Media
ISBN: 1461245540
Category : Mathematics
Languages : en
Pages : 153
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Book Description
This book developed from a series of lectures I gave at the Symposium on Nonlinear Microlocal Analysis held at Nanjing University in October. 1988. Its purpose is to give an overview of the use of microlocal analysis and commutators in the study of solutions to nonlinear wave equations. The weak singularities in the solutions to such equations behave up to a certain extent like those present in the linear case: they propagate along the null bicharacteristics of the operator. On the other hand. examples exhibiting singularities not present in the linear case can also be constructed. I have tried to present a crossection of both the regularity results and the singular examples. for problems on the interior of a domain and on domains with boundary. The main emphasis is on the case of more than one space dimen sion. since that case is treated in great detail in the paper of Rauch-Reed 159]. The results presented here have for the most part appeared elsewhere. and are the work of many authors. but a few new examples and proofs are given. I have attempted to indicate the essential ideas behind the arguments. so that only some of the results are proved in full detail. It is hoped that the central notions of the more technical proofs appearing in research papers will be illuminated by these simpler cases.
Author: Michael Beals
Publisher: Springer Science & Business Media
ISBN: 1461245540
Category : Mathematics
Languages : en
Pages : 153
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Book Description
This book developed from a series of lectures I gave at the Symposium on Nonlinear Microlocal Analysis held at Nanjing University in October. 1988. Its purpose is to give an overview of the use of microlocal analysis and commutators in the study of solutions to nonlinear wave equations. The weak singularities in the solutions to such equations behave up to a certain extent like those present in the linear case: they propagate along the null bicharacteristics of the operator. On the other hand. examples exhibiting singularities not present in the linear case can also be constructed. I have tried to present a crossection of both the regularity results and the singular examples. for problems on the interior of a domain and on domains with boundary. The main emphasis is on the case of more than one space dimen sion. since that case is treated in great detail in the paper of Rauch-Reed 159]. The results presented here have for the most part appeared elsewhere. and are the work of many authors. but a few new examples and proofs are given. I have attempted to indicate the essential ideas behind the arguments. so that only some of the results are proved in full detail. It is hoped that the central notions of the more technical proofs appearing in research papers will be illuminated by these simpler cases.
Author: M K V Murthy
Publisher: CRC Press
ISBN: 9780582087668
Category : Mathematics
Languages : en
Pages : 242
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Book Description
Contains the proceedings of a workshop on nonlinear hyperbolic equations held at Varenna, Italy in June 1990.
Author: Hassane Bougrini
Publisher: Nova Publishers
ISBN: 9781560728788
Category : Mathematics
Languages : en
Pages : 122
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Book Description
The main purpose of this book is to give a self-contained synthesis of different results in the domain of symbolic calculus of conormal singularities of semilinear hyperbolic progressing waves. The authors deal generally with real matrix valued co-efficients and with real vector valued solutions, but the complex case is similar. They consider also N x N first order systems rather than high order scalar equations, because the polarisation properties of symbols are less natural in the latter case. Moreover, although they assume generally that the real characteristics are simple, the methods can give results for symmetric or symmetrisable first order hyperbolic systems.
Author:
Publisher:
ISBN: 9814464996
Category :
Languages : en
Pages :
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Book Description
Author: Satyanad Kichenassamy
Publisher: CRC Press
ISBN: 1000444724
Category : Mathematics
Languages : en
Pages : 297
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Book Description
This work examines the mathematical aspects of nonlinear wave propagation, emphasizing nonlinear hyperbolic problems. It introduces the tools that are most effective for exploring the problems of local and global existence, singularity formation, and large-time behaviour of solutions, and for the study of perturbation methods.
Author: Michael Taylor
Publisher: Springer Science & Business Media
ISBN: 1475741901
Category : Mathematics
Languages : en
Pages : 629
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Book Description
The third of three volumes on partial differential equations, this is devoted to nonlinear PDE. It treats a number of equations of classical continuum mechanics, including relativistic versions, as well as various equations arising in differential geometry, such as in the study of minimal surfaces, isometric imbedding, conformal deformation, harmonic maps, and prescribed Gauss curvature. In addition, some nonlinear diffusion problems are studied. It also introduces such analytical tools as the theory of L Sobolev spaces, H lder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is aimed at graduate students in mathematics, and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis and complex analysis. ^
Author: Jean-Marc Delort
Publisher: Springer
ISBN: 366221539X
Category : Mathematics
Languages : en
Pages : 108
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Book Description
During the last ten years, FBI transformation and second microlocalization have been used by several authors to solve different problems in the theory of linear or nonlinear partial differential equations. The aim of this book is to give an introduction to these topics, in the spirit of the work ofSj strand, and to present their recent application to the propagation of conormal singularities for solutions of seminlinear hyperbolic equations, due to Lebeau. The text is quite self-contained and provides a useful entry to the subject and a bridging link to more specialized papers.
Author: Toshikazu Sunada
Publisher: World Scientific
ISBN: 9810249470
Category : Mathematics
Languages : en
Pages : 633
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Book Description
COntains 55 research and expository articles on a wide range of currently active and interesting areas in pure and applied mathematics.
Author: Michael E. Taylor
Publisher: Springer Science & Business Media
ISBN: 1441970495
Category : Mathematics
Languages : en
Pages : 734
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Book Description
The third of three volumes on partial differential equations, this is devoted to nonlinear PDE. It treats a number of equations of classical continuum mechanics, including relativistic versions, as well as various equations arising in differential geometry, such as in the study of minimal surfaces, isometric imbedding, conformal deformation, harmonic maps, and prescribed Gauss curvature. In addition, some nonlinear diffusion problems are studied. It also introduces such analytical tools as the theory of L Sobolev spaces, H lder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is aimed at graduate students in mathematics, and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis and complex analysis
Author: Jeffrey Rauch
Publisher: American Mathematical Soc.
ISBN: 0821872915
Category : Mathematics
Languages : en
Pages : 386
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Book Description
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.
