Mathematical Aspects of Nonlinear Dispersive Equations (AM-163)

Product description Review "The volume contains valuable contributions to the area of nonlinear PDEs, making it indispensable for all researchers interested in partial differential equations and their applications."---Radu Precup, Mathematica About the Author 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). Excerpt. © Reprinted by permission. All rights reserved. Mathematical Aspects of Nonlinear Dispersive EquationsPRINCETON UNIVERSITY PRESSCopyright © 2007 Princeton University PressAll right reserved.ISBN: 978-0-691-12955-6ContentsPreface................................................................................................................................................................viiChapter 1. On Strichartz's Inequalities and the Nonlinear Schrdinger Equation on Irrational Tori J. Bourgain.........................................................1Chapter 2. Diffusion Bound for a Nonlinear Schrdinger Equation J. Bourgain and W.-M.Wang............................................................................21Chapter 3. Instability of Finite Difference Schemes for Hyperbolic Conservation Laws A. Bressan, P. Baiti, and H. K. Jenssen..........................................43Chapter 4. Nonlinear Elliptic Equations with Measures Revisited H. Brezis, M. Marcus, and A. C. Ponce.................................................................55Chapter 5. Global Solutions for the Nonlinear Schrdinger Equation on Three-Dimensional Compact Manifolds N. Burq, P. Grard, and N. Tzvetkov.........................111Chapter 6. Power Series Solution of a Nonlinear Schrdinger Equation M. Christ........................................................................................131Chapter 7. Eulerian-Lagrangian Formalism and Vortex Reconnection P. Constantin........................................................................................157Chapter 8. Long Time Existence for Small Data Semilinear Klein-Gordon Equations on Spheres J.-M. Delort and J. Szeftel................................................171Chapter 9. Local and Global Wellposedness of Periodic KP-I Equations A. D. Ionescu and C. E. Kenig....................................................................181Chapter 10. The Cauchy Problem for the Navier-Stokes Equations with Spatially Almost Periodic Initial Data Y. Giga, A. Mahalov, and B. Nicolaenko.....................213Chapter 11. Longtime Decay Estimates for the Schrdinger Equation on Manifolds I. Rodnianski and T. Tao...............................................................223Chapter 12. Dispersive Estimates for Schrdinger Operators: A Survey W. Schlag........................................................................................255Contributors...........................................................................................................................................................287Index..................................................................................................................................................................291Chapter One On Strichartz's Inequalities and the Nonlinear Schrdinger Equation on Irrational Tori J. Bourgain1.0 INTRODUCTION Strichartz's inequalities and the Cauchy problem for the nonlinear Schrdinger equation are considerably less understood when the spatial domain is a compact manifold M, compared with the Euclidean situation M = [R.sup.d]. In the latter case, at least the theory of Strichartz inequalities (i.e., moment inequalities for the linear evolution, of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is basically completely understood and is closely related to the theory of oscillatory integral operators. Let M = [T.sup.d] be a flat torus. If M is the usual torus, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.1) a partial Strichartz theorywas developed in [B1], leading to the almost exact counterparts of the Euclidean case for d = 1, 2 (the exact analogues of the p = 6 inequality for d = 1 and p = 4 inequality for d = 2 are false with periodic boundary conditions). Thus, assuming supp [??] [subset] B(0,N), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.2) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.3) For d = 3, we have the inequality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.4) but the issue: Problem.