Microlocal Analysis and Nonlinear Waves

1. Front Matter

   Pages i-xiii

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2. On the Interactions of Conormal Waves for Semilinear Wave Equations

   • Antônio Sá Barreto

   Pages 1-7

3. Regularity of Nonlinear Waves Associated with a Cusp

   • Michael Beals

   Pages 9-27

4. Evolution of a Punctual Singularity in an Eulerian Flow

   • Jean Yves Chemin

   Pages 29-36

5. Water Waves, Hamiltonian Systems and Cauchy Integrals

   • Walter Craig

   Pages 37-45

6. Infinite Gain of Regularity for Dispersive Evolution Equations

   • W. Craig, T. Kappeler, W. Strauss

   Pages 47-50

7. On the Fully Non-Linear Cauchy Problem with Small Data. II.

   • Lars Hörmander

   Pages 51-81

8. Interacting Weakly Nonlinear Hyperbolic and Dispersive Waves

   • John K. Hunter

   Pages 83-111

9. Nonlinear Resonance Can Create Dense Oscillations

   • Jean-Luc Joly, Jeffrey Rauch

   Pages 113-123

10. Lower Bounds of the Life-Span of Small Classical Solutions for Nonlinear Wave Equations

    • Li Ta-Tsien

    Pages 125-136

11. Propagation of Stronger Singularities of Solutions to Semilinear Wave Equations

    • Liu Linqi

    Pages 137-154

12. Conormality, Cusps and Non-Linear Interaction

    • Richard B. Melrose

    Pages 155-166

13. Quasimodes for the Laplace Operator and Glancing Hypersurfaces

    • Georgi S. Popov

    Pages 167-178

14. A Decay Estimate for the Three-Dimensional Inhomogeneous Klein-Gordon Equation and Global Existence for Nonlinear Equations

    • Thomas C. Sideris

    Pages 179-183

15. Interaction of Singularities and Propagation into Shadow Regions in Semilinear Boundary Problems

    • Mark Williams

    Pages 185-199

This IMA Volume in Mathematics and its Applications MICROLOCAL ANALYSIS AND NONLINEAR WAVES is based on the proceedings of a workshop which was an integral part of the 1988- 1989 IMA program on "Nonlinear Waves". We thank Michael Beals, Richard Melrose and Jeffrey Rauch for organizing the meeting and editing this proceedings volume. We also take this opportunity to thank the National Science Foundation whose financial support made the workshop possible. A vner Friedman Willard Miller, Jr. PREFACE Microlocal analysis is natural and very successful in the study of the propagation of linear hyperbolic waves. For example consider the initial value problem Pu = f E e'(RHd), supp f C {t ;::: O} u = 0 for t < o. If P( t, x, Dt,x) is a strictly hyperbolic operator or system then the singular support of f gives an upper bound for the singular support of u (Courant-Lax, Lax, Ludwig), namely singsupp u C the union of forward rays passing through the singular support of f.

• Laplace operator
• Microlocal analysis
• Oscillation
• Smooth function
• Volume
• behavior
• form
• hamiltonian system
• information
• integral
• online
• operator
• singularity
• wave
• wave equation

• Department of Mathematics, Rutgers University, New Brunswick, USA

  Michael Beals

• Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA

  Richard B. Melrose

• Department of Mathematics, University of Michigan, Ann Arbor, USA

  Jeffrey Rauch

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