Blowup for Nonlinear Hyperbolic Equations

ISBN-13: 9781461275886 / Angielski / Miękka / 2011 / 113 str.

Serge Alinhac

Opis

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].

Spis treści

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.