Blowup for nonlinear hyperbolic equations

著者

• Alinhac, S. (Serge)

書誌事項

Blowup for nonlinear hyperbolic equations

Serge Alinhac

（Progress in nonlinear differential equations and their applications / editor, Haim Brezis, v. 17）

Birkhäuser, c1995

• : us
• : sz
• : pbk

注記

Includes bibliographical references (p. [107]-112) and index

内容説明・目次

巻冊次

: us ISBN 9780817638108

内容説明

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.

目次

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 x 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.

巻冊次

: pbk ISBN 9781461275886

内容説明

The content of this book corresponds to a one-semester course taught at the University of Paris-Sud (Orsay) in the spring 1994. It is accessible to students or researchers with a basic elementary knowledge of Partial Dif- ferential Equations, especially of hyperbolic PDE (Cauchy problem, wave operator, energy inequality, finite speed of propagation, symmetric systems, etc.). This course is not some final encyclopedic reference gathering all avail- able results. We tried instead to provide a short synthetic view of what we believe are the main results obtained so far, with self-contained proofs. In fact, many of the most important questions in the field are still completely open, and we hope that this monograph will give young mathe- maticians the desire to perform further research. The bibliography, restricted to papers where blowup is explicitly dis- cussed, is the only part we tried to make as complete as possible (despite the new preprints circulating everyday) j the references are generally not mentioned in the text, but in the Notes at the end of each chapter. Basic references corresponding best to the content of these Notes are the books by Courant and Friedrichs [CFr], Hormander [HoI] and [Ho2], Majda [Ma] and Smoller [Sm], and the survey papers by John [J06], Strauss [St] and Zuily [Zu].

目次

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 x 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.

巻冊次

: sz ISBN 9783764338107

内容説明

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.

目次

• Part 1 The two basic blowup mechanisms: the ODE mechanism - systms of ODE, strictly hyperbolic semilinear systems in the plane, semilinear wave equations
• the geometric blowup mechanism - Burgers' equation and the method of characteristics, blowup of a quasilinear system, blowup solutions, how to solve the blowup system, how ...u blows up, singular solutions and explosive solutions
• combinations of the two mechanisms - which mechanism takes place first?, simultaneous occurrence of the two mechanisms. Part 2 First concepts on global Cauchy problems: short time existence
• lifespan and blowup criterion
• blow-up or not? functional methods
• Burgers' equation
• semilinear wave equation
• the Euler system
• blowup or not? comparison and averaging methods. Part 3 Semilinear wave equations: semilinear blowup criterion
• maximal influence domain
• maximal influence domain for weak solutions
• blowup rates at the boundary of the maximal influence domain
• an example of a sharp estimate of the lifespan. Part 4 Quasilinear equations in one space dimension: the scalar case
• Riemann invariants, simple waves and L1-boundedness
• the case of 2x2 systems
• general systems with small data
• rotationally invariant wave equations. Part 5 Nonlinear geometrical optics and applications: quasilinear systems in one space dimension - formal analysis, slow time and reduced equations, existence, approximation and blowup
• quasilinear wave equations - formal analysis, slow time and reduced equations, existence, null conditions, blowup
• further results on the wave equation - formal analysis near the boundary of the light cone, slow time and reduced equations, a local blowup problem, asymptotic lifespan for the two-dimensional wave equation.

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詳細情報

• NII書誌ID(NCID): BA24873803
• ISBN:
  • 9780817638108
  • 3764338105
  • 9781461275886
• LCCN: 95002743
• 出版国コード: us
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Boston ; Basel
• ページ数/冊数: xiv, 112 p.
• 大きさ: 25 cm
• 分類:
  • LCC : QA377
  • DC20 : 515/.353
• 件名:
  • LCSH : Differential equations, Hyperbolic -- Numerical solutions
  • LCSH : Cauchy problem
  • LCSH : Blowing up (Algebraic geometry)
• 親書誌ID:
  • BA07303999