Ebook: Microlocal Analysis and Nonlinear Waves

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.

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The behavior of linear hyperbolic waves has been analyzed by decomposing the waves into pieces in space-time and into different frequencies. The linear nature of the equations involved allows the reassembling of the pieces in a simple fashion; the individual pieces do not interact. For nonlinear waves the interaction of the pieces seemed to preclude such an analysis, but in the late 1970s it was shown that a similar procedure could be undertaken in this case and would yield important information. The analysis of the decomposed waves, and of waves with special smoothness or size in certain directions, has been fruitful in describing a variety of the properties of nonlinear waves. This volume presents a number of articles on topics of current interest which involves the use of the newer techniques on nonlinear waves. The results established include descriptions of the smoothness of such waves as determined by their geometry, the properties of solutions with high frequency oscillations, and the long-time smoothness and size estimates satisfied by nonlinear waves.

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The behavior of linear hyperbolic waves has been analyzed by decomposing the waves into pieces in space-time and into different frequencies. The linear nature of the equations involved allows the reassembling of the pieces in a simple fashion; the individual pieces do not interact. For nonlinear waves the interaction of the pieces seemed to preclude such an analysis, but in the late 1970s it was shown that a similar procedure could be undertaken in this case and would yield important information. The analysis of the decomposed waves, and of waves with special smoothness or size in certain directions, has been fruitful in describing a variety of the properties of nonlinear waves. This volume presents a number of articles on topics of current interest which involves the use of the newer techniques on nonlinear waves. The results established include descriptions of the smoothness of such waves as determined by their geometry, the properties of solutions with high frequency oscillations, and the long-time smoothness and size estimates satisfied by nonlinear waves.
Content:
Front Matter....Pages i-xiii
On the Interactions of Conormal Waves for Semilinear Wave Equations....Pages 1-7
Regularity of Nonlinear Waves Associated with a Cusp....Pages 9-27
Evolution of a Punctual Singularity in an Eulerian Flow....Pages 29-36
Water Waves, Hamiltonian Systems and Cauchy Integrals....Pages 37-45
Infinite Gain of Regularity for Dispersive Evolution Equations....Pages 47-50
On the Fully Non-Linear Cauchy Problem with Small Data. II.....Pages 51-81
Interacting Weakly Nonlinear Hyperbolic and Dispersive Waves....Pages 83-111
Nonlinear Resonance Can Create Dense Oscillations....Pages 113-123
Lower Bounds of the Life-Span of Small Classical Solutions for Nonlinear Wave Equations....Pages 125-136
Propagation of Stronger Singularities of Solutions to Semilinear Wave Equations....Pages 137-154
Conormality, Cusps and Non-Linear Interaction....Pages 155-166
Quasimodes for the Laplace Operator and Glancing Hypersurfaces....Pages 167-178
A Decay Estimate for the Three-Dimensional Inhomogeneous Klein-Gordon Equation and Global Existence for Nonlinear Equations....Pages 179-183
Interaction of Singularities and Propagation into Shadow Regions in Semilinear Boundary Problems....Pages 185-199

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The behavior of linear hyperbolic waves has been analyzed by decomposing the waves into pieces in space-time and into different frequencies. The linear nature of the equations involved allows the reassembling of the pieces in a simple fashion; the individual pieces do not interact. For nonlinear waves the interaction of the pieces seemed to preclude such an analysis, but in the late 1970s it was shown that a similar procedure could be undertaken in this case and would yield important information. The analysis of the decomposed waves, and of waves with special smoothness or size in certain directions, has been fruitful in describing a variety of the properties of nonlinear waves. This volume presents a number of articles on topics of current interest which involves the use of the newer techniques on nonlinear waves. The results established include descriptions of the smoothness of such waves as determined by their geometry, the properties of solutions with high frequency oscillations, and the long-time smoothness and size estimates satisfied by nonlinear waves.
Content:
Front Matter....Pages i-xiii
On the Interactions of Conormal Waves for Semilinear Wave Equations....Pages 1-7
Regularity of Nonlinear Waves Associated with a Cusp....Pages 9-27
Evolution of a Punctual Singularity in an Eulerian Flow....Pages 29-36
Water Waves, Hamiltonian Systems and Cauchy Integrals....Pages 37-45
Infinite Gain of Regularity for Dispersive Evolution Equations....Pages 47-50
On the Fully Non-Linear Cauchy Problem with Small Data. II.....Pages 51-81
Interacting Weakly Nonlinear Hyperbolic and Dispersive Waves....Pages 83-111
Nonlinear Resonance Can Create Dense Oscillations....Pages 113-123
Lower Bounds of the Life-Span of Small Classical Solutions for Nonlinear Wave Equations....Pages 125-136
Propagation of Stronger Singularities of Solutions to Semilinear Wave Equations....Pages 137-154
Conormality, Cusps and Non-Linear Interaction....Pages 155-166
Quasimodes for the Laplace Operator and Glancing Hypersurfaces....Pages 167-178
A Decay Estimate for the Three-Dimensional Inhomogeneous Klein-Gordon Equation and Global Existence for Nonlinear Equations....Pages 179-183
Interaction of Singularities and Propagation into Shadow Regions in Semilinear Boundary Problems....Pages 185-199
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