Shock Waves and Reaction Diffusion Equations

The purpose of this book is to make easily available the basics of the theory of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by Charles Conley.

Author: Joel Smoller

Publisher: Springer

ISBN: UOM:39015016355649

Category: Science

Page: 618

View: 284


The purpose of this book is to make easily available the basics of the theory of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by Charles Conley. It presents the modern ideas in these fields in a way that is accessible to a wider audience than just mathematicians.The book is divided into four main parts: linear theory, reaction-diffusion equations, shock-wave theory, and the Conley index. For the second edition numerous typographical errors and other mistakes have been corrected and a new chapter on recent results has been added. The new chapter contains discussions of the stability of traveling waves, symmetry-breaking bifurcations, compensated compactness, viscous profiles for shock waves, and general notions for construction traveling-wave solutions for systems of nonlinear equations.

Shock Waves and Reaction Diffusion Equations

[CG] Conley, C. and R. Gardner An application of the generalized Morse index to travelling wave solutions of a competitive reactiondiffusion model. Preprint. [CS] Conley, C. and J. Smoller 1. Viscosity matrices for two-dimensional ...

Author: Joel Smoller

Publisher: Springer Science & Business Media

ISBN: 9781468401523

Category: Science

Page: 581

View: 904


. . . the progress of physics will to a large extent depend on the progress of nonlinear mathe matics, of methods to solve nonlinear equations . . . and therefore we can learn by comparing different nonlinear problems. WERNER HEISENBERG I undertook to write this book for two reasons. First, I wanted to make easily available the basics of both the theory of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by C. Conley. These important subjects seem difficult to learn since the results are scattered throughout the research journals. 1 Second, I feel that there is a need to present the modern methods and ideas in these fields to a wider audience than just mathe maticians. Thus, the book has some rather sophisticated aspects to it, as well as certain textbook aspects. The latter serve to explain, somewhat, the reason that a book with the title Shock Waves and Reaction-Diffusion Equations has the first nine chapters devoted to linear partial differential equations. More precisely, I have found from my classroom experience that it is far easier to grasp the subtleties of nonlinear partial differential equations after one has an understanding of the basic notions in the linear theory. This book is divided into four main parts: linear theory, reaction diffusion equations, shock wave theory, and the Conley index, in that order. Thus, the text begins with a discussion of ill-posed problems.

Matched Asymptotic Expansions in Reaction Diffusion Theory

36, 479-500 (1992) Needham, D.J., Merkin, J.H.: The development of travelling waves in a simple ... A Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Springer, Berlin (1989) Snita, D., Seveikova, M., Marek, M., Merkin, ...

Author: J.A. Leach

Publisher: Springer Science & Business Media

ISBN: 9780857293961

Category: Mathematics

Page: 290

View: 312


This volume contains a wealth of results and methodologies applicable to a wide range of problems arising in reaction-diffusion theory. It can be viewed both as a handbook, and as a detailed description of the methodology. The authors present new methods based on matched asymptotic expansions.

Reaction diffusion Equations And Their Applications And Computational Aspects Proceedings Of The China japan Symposium

[5] P.C.Fife, Boundary and Interior Transition Layer Phenomena for pairs of Second-Order Differential Equations, J.Math. Anal.Appl.,54 (1976), 497–521. [6] J.Smoller, Shock Waves and Reaction-Diffusion Equations, Springer - Verlag, ...

Author: Li Ta-tsien

Publisher: World Scientific

ISBN: 9789814547840


Page: 240

View: 586


This unique work looks at the relationship between psychopathology and world politics. What happens when the brain/mind ceases to function properly? How does this impinge on world affairs? What is to be done, for example, when a leader ceases to act in a seemingly sane fashion and yet still commands the loyalty of those who maintain him or her in office? What is to be done when a leader's advisers seem rational but are clearly not? Indeed, what is to be done when a whole society goes insane? This is to raise more questions than a single work can adequately answer.In lieu of a comprehensive account, which would be beyond the scope of one study, what this book does is first describe psychopathology in general terms and its relationship to world affairs. It then looks at denial in particular and at “speaking the truth” as a potentially therapeutic antidote, especially in relation to nuclear weapons. It follows this by looking at delusion in general and at what being “in touch” with reality might entail with regard to a so-called “failed state”. With topics ranging from Hitler's mental health to the continuing threat of nuclear Armageddon, this book is a valuable contribution to the field of international politics and modern psychology, and will hopefully lead to a better understanding of contemporary world affairs and the global issue of conflict resolution.

Travelling Waves in Nonlinear Diffusion Convection Reaction

|273] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983. |274] Song B., Eristence, uniqueness and properties of the solutions of a degenerate parabolic equation with diffusion-advection-absorption ...

Author: Brian H. Gilding

Publisher: Birkhäuser

ISBN: 9783034879644

Category: Mathematics

Page: 210

View: 298


This monograph has grown out of research we started in 1987, although the foun dations were laid in the 1970's when both of us were working on our doctoral theses, trying to generalize the now classic paper of Oleinik, Kalashnikov and Chzhou on nonlinear degenerate diffusion. Brian worked under the guidance of Bert Peletier at the University of Sussex in Brighton, England, and, later at Delft University of Technology in the Netherlands on extending the earlier mathematics to include nonlinear convection; while Robert worked at Lomonosov State Univer sity in Moscow under the supervision of Anatolii Kalashnikov on generalizing the earlier mathematics to include nonlinear absorption. We first met at a conference held in Rome in 1985. In 1987 we met again in Madrid at the invitation of Ildefonso Diaz, where we were both staying at 'La Residencia'. As providence would have it, the University 'Complutense' closed down during this visit in response to student demonstra tions, and, we were very much left to our own devices. It was natural that we should gravitate to a research topic of common interest. This turned out to be the characterization of the phenomenon of finite speed of propagation for nonlin ear reaction-convection-diffusion equations. Brian had just completed some work on this topic for nonlinear diffusion-convection, while Robert had earlier done the same for nonlinear diffusion-absorption. There was no question but that we bundle our efforts on the general situation.

Advances in the Theory of Shock Waves

[19] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer Verlag, 1983. [20] J. Smoller and B. Temple, Global solutions of the relativistic Euler equations, Commun. Math. Phys., 157(1993), pp. 67–99.

Author: Heinrich Freistühler

Publisher: Springer Science & Business Media

ISBN: 9781461201939

Category: Mathematics

Page: 520

View: 967


In the field known as "the mathematical theory of shock waves," very exciting and unexpected developments have occurred in the last few years. Joel Smoller and Blake Temple have established classes of shock wave solutions to the Einstein Euler equations of general relativity; indeed, the mathematical and physical con sequences of these examples constitute a whole new area of research. The stability theory of "viscous" shock waves has received a new, geometric perspective due to the work of Kevin Zumbrun and collaborators, which offers a spectral approach to systems. Due to the intersection of point and essential spectrum, such an ap proach had for a long time seemed out of reach. The stability problem for "in viscid" shock waves has been given a novel, clear and concise treatment by Guy Metivier and coworkers through the use of paradifferential calculus. The L 1 semi group theory for systems of conservation laws, itself still a recent development, has been considerably condensed by the introduction of new distance functionals through Tai-Ping Liu and collaborators; these functionals compare solutions to different data by direct reference to their wave structure. The fundamental prop erties of systems with relaxation have found a systematic description through the papers of Wen-An Yong; for shock waves, this means a first general theorem on the existence of corresponding profiles. The five articles of this book reflect the above developments.

Shock Wave Interactions in General Relativity

S. Kind and J. Ehlers, Initial boundary value problem for spherically symmetric Einstein equations for a perfect fluid, Class. Quantum Grav., 18(1957), ... J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer Verlag, 1983.

Author: Jeffrey Groah

Publisher: Springer Science & Business Media

ISBN: 9780387446028

Category: Mathematics

Page: 152

View: 431


This monograph presents a self contained mathematical treatment of the initial value problem for shock wave solutions of the Einstein equations in General Relativity. It has a clearly outlined goal: proving a certain local existence theorem. Concluding remarks are added and commentary is provided throughout. The author is a well regarded expert in this area.

Shock Waves in Conservation Laws with Physical Viscosity

MR1470318 (98j:35121) T.-P. Liu and S.-H Yu, Green's function for Boltzmann equation, 3-D waves, Bulletin, Inst. Math. ... MR1466143(98e:73148) [Sm] Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., ...

Author: Tai-Ping Liu

Publisher: American Mathematical Soc.

ISBN: 9781470410162

Category: Conservation laws (Mathematics)

Page: 168

View: 330


The authors study the perturbation of a shock wave in conservation laws with physical viscosity. They obtain the detailed pointwise estimates of the solutions. In particular, they show that the solution converges to a translated shock profile. The strength of the perturbation and that of the shock are assumed to be small but independent. The authors' assumptions on the viscosity matrix are general so that their results apply to the Navier-Stokes equations for the compressible fluid and the full system of magnetohydrodynamics, including the cases of multiple eigenvalues in the transversal fields, as long as the shock is classical. The authors' analysis depends on accurate construction of an approximate Green's function. The form of the ansatz for the perturbation is carefully constructed and is sufficiently tight so that the author can close the nonlinear term through Duhamel's principle.

Mathematical Analysis of Shock Wave Reflection

J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer, New York, 1994) 10. S. Canic, B.L. Keyfitz, G.M. Lieberman, A proof of existence of perturbed steady transonic shocks via a free boundary problem. Comm.

Author: Shuxing Chen

Publisher: Springer Nature

ISBN: 9789811577529

Category: Mathematics

Page: 251

View: 566


This book is aimed to make careful analysis to various mathematical problems derived from shock reflection by using the theory of partial differential equations. The occurrence, propagation and reflection of shock waves are important phenomena in fluid dynamics. Comparing the plenty of studies of physical experiments and numerical simulations on this subject, this book makes main efforts to develop the related theory of mathematical analysis, which is rather incomplete so far. The book first introduces some basic knowledge on the system of compressible flow and shock waves, then presents the concept of shock polar and its properties, particularly the properties of the shock polar for potential flow equation, which are first systematically presented and proved in this book. Mathematical analysis of regular reflection and Mach reflection in steady and unsteady flow are the most essential parts of this book. To give challenges in future research, some long-standing open problems are listed in the end. This book is attractive to researchers in the fields of partial differential equations, system of conservation laws, fluid dynamics, and shock theory.

Handbook of Shock Waves Three Volume Set

Sidilkover, D. (1999) Factorizable schemes for the equations of ̄uid ̄ow. ICASE Report No. 99±20, June. Smoller, J. (1983). Shock Waves and Reaction-Diffusion Equations, New York: Springer-Verlag. Spekreijse, S.P. (1987) Multigrid ...

Author: Gabi Ben-Dor

Publisher: Elsevier

ISBN: 9780080533728

Category: Technology & Engineering

Page: 2102

View: 111


The Handbook of Shock Waves contains a comprehensive, structured coverage of research topics related to shock wave phenomena including shock waves in gases, liquids, solids, and space. Shock waves represent an extremely important physical phenomena which appears to be of special practical importance in three major fields: compressible flow (aerodynamics), materials science, and astrophysics. Shock waves comprise a phenomenon that occurs when pressure builds to force a reaction, i.e. sonic boom that occurs when a jet breaks the speed of sound. This Handbook contains experimental, theoretical, and numerical results which never before appeared under one cover; the first handbook of its kind. The Handbook of Shock Waves is intended for researchers and engineers active in shock wave related fields. Additionally, R&D establishments, applied science & research laboratories and scientific and engineering libraries both in universities and government institutions. As well as, undergraduate and graduate students in fluid mechanics, gas dynamics, and physics. Key Features * Ben-Dor is known as one of the founders of the field of shock waves * Covers a broad spectrum of shock wave research topics * Provides a comprehensive description of various shock wave related subjects * First handbook ever to include under one separate cover: experimental, theoretical, and numerical results