Blowup for Nonlinear Hyperbolic Equations

Inhaltsverzeichnis

• 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.
• 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.
• 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.
• 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.