Blowup for Nonlinear Hyperbolic Equations: 17 - Rilegato

Alinhac, Serge

Rilegato

ISBN 10: 0817638105 ISBN 13: 9780817638108

Casa editrice: Birkhauser, 1995

Riassunto

Solutions to partial differential equations or systems often, over specific time periods, exhibit smooth behaviour. Given sufficient time, however, they almost invariably undergo a brutal change in behaviour, and this phenomenon has become known as "blowup". In this book, the author provides an overview of what is known about this situation and discusses many of the open problems concerning it. The book deals with classical solutions of global problems for hyperbolic equations or systems. The approach is based on the display and study of two local blowup mechanisms, which the author calls the "ordinary differential equation mechanism" and the "geometric blowup mechanism". It introduces, via energy methods, the concept of lifespan, related to the nonlinear propagation of regularity (from the past to the future). It addresses specifically the question of whether or not there will be blowup in a solution, and it classifies those methods used to give positive answers to the question. The material corresponds to a one semester course for students or researchers with a basic elementary knowledge of partial differential equations, especially of hyperbolic type including such topics as the Cauchy problem, wave operators, energy inequalities, finite speed of propagation, and symmetric systems. It contains a complete bibliography reflecting the high degree of activity among mathematicians interested in the problem.

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Contenuti

I. The Two Basic Blowup Mechanisms.- A. The ODE mechanism.- 1. Systems of ODE.- 2. Strictly hyperbolic semilinear systems in the plane.- 3. Semilinear wave equations.- B. The geometric blowup mechanism.- 1. Burgers’ equation and the method of characteristics.- 2. Blowup of a quasilinear system.- 3. Blowup solutions.- 4. How to solve the blowup system.- 5. How ?u blows up.- 6. Singular solutions and explosive solutions.- C. Combinations of the two mechanisms.- 1. Which mechanism takes place first?.- 2. Simultaneous occurrence of the two mechanisms.- Notes.- II. First Concepts on Global Cauchy Problems.- 1. Short time existence.- 2. Lifespan and blowup criterion.- 3. Blowup or not? Functional methods.- a. A functional method for Burgers’ equation.- b. Semilinear wave equation.- c. The Euler system.- 4. Blowup or not? Comparison and averaging methods.- Notes.- III. Semilinear Wave Equations.- 1. Semilinear blowup criteria.- 2. Maximal influence domain.- 3. Maximal influence domains for weak solutions.- 4. Blowup rates at the boundary of the maximal influence domain.- 5. An example of a sharp estimate of the lifespan.- Notes.- IV. Quasilinear Systems in One Space Dimension.- 1. The scalar case.- 2. Riemann invariants, simple waves, and L1-boundedness.- 3. The case of 2 × 2 systems.- 4. General systems with small data.- 5. Rotationally invariant wave equations.- Notes.- V. Nonlinear Geometrical Optics and Applications.- 1. Quasilinear systems in one space dimension.- 1.1. Formal analysis.- 1.2. Slow time and reduced equations.- 1.3. Existence, approximation and blowup.- 2. Quasilinear wave equations.- 2.1. Formal analysis.- 2.2. Slow time and reduced equations.- 2.3. Existence, null conditions, blowup.- 3. Further results on the wave equation.- 3.1. Formal analysis near the boundary of the light cone.- 3.2. Slow time and reduced equations.- 3.3. A local blowup problem.- 3.4. Asymptotic lifespan for the two-dimensional wave equation.- Notes.

Product Description

Book by Alinhac Serge

Dettagli dell?edizione

• Editore: Birkhauser
• Data di pubblicazione: 1995
• Lingua: Inglese
• ISBN 10: 0817638105
• ISBN 13: 9780817638108
• Rilegatura: Copertina rigida
• Numero di pagine: 113
• Contatto del produttore: Springer Nature Customer Service Center GmbH; ProductSafety@springernature.com