Jean Bourgain is Professor of Mathematics at the Institute for Advanced Study in Princeton. In 1994, he won the Fields Medal. He is the author of Green's Function Estimates for Lattice Schrödinger Operators and Applications (Princeton). Carlos E. Kenig is Professor of Mathematics at the University of Chicago. He is a fellow of the American Academy of Arts and Sciences and the author of Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. S. Klainerman is Professor of Mathematics at Princeton University. He is a MacArthur Fellow and Bocher Prize recipient. He is the coauthor of The Global Nonlinear Stability of the Minkowski Space (Princeton).

Preface vii
Chapter 1. On Strichartz's Inequalities and the Nonlinear Schrödinger
Equation on Irrational Tori by J. Bourgain 1
Chapter 2. Diffusion Bound for a Nonlinear Schrödinger Equation by J. Bourgain and W.-M.Wang 21
Chapter 3. Instability of Finite Difference Schemes for Hyperbolic Conservation Laws by A. Bressan, P. Baiti, and H. K. Jenssen 43
Chapter 4. Nonlinear Elliptic Equations with Measures Revisited H. Brezis, M. Marcus, and A. C. Ponce 55
Chapter 5. Global Solutions for the Nonlinear Schrödinger Equation on Three-Dimensional Compact Manifolds by N. Burq, P. Gérard, and N. Tzvetkov 111
Chapter 6. Power Series Solution of a Nonlinear Schrödinger Equation by M. Christ 131
Chapter 7. Eulerian-Lagrangian Formalism and Vortex Reconnection by P. Constantin 157
Chapter 8. Long Time Existence for Small Data Semilinear Klein-Gordon Equations on Spheres by J.-M. Delort and J. Szeftel 171
Chapter 9. Local and GlobalWellposedness of Periodic KP-I Equations by A. D. Ionescu and C. E. Kenig 181
Chapter 10. The Cauchy Problem for the Navier-Stokes Equations with Spatially Almost Periodic Initial Data by Y. Giga, A. Mahalov, and B. Nicolaenko 213
Chapter 11. Longtime Decay Estimates for the Schrödinger Equation
on Manifolds by I. Rodnianski and T. Tao 223
Chapter 12. Dispersive Estimates for Schrödinger Operators: A Survey by W. Schlag 255
Contributors 287
Index 291

This collection of new and original papers on mathematical aspects of nonlinear dispersive equations includes both expository and technical papers that reflect a number of recent advances in the field. The expository papers describe the state of the art and research directions. The technical papers concentrate on a specific problem and the related analysis and are addressed to active researchers.
The book deals with many topics that have been the focus of intensive research and, in several cases, significant progress in recent years, including hyperbolic conservation laws, Schrödinger operators, nonlinear Schrödinger and wave equations, and the Euler and Navier-Stokes equations.