Nonlinear Dispersive Equations: Local and Global Analysis, Issue 106
Among nonlinear PDEs, dispersive and wave equations form an important class of equations. These include the nonlinear Schrodinger equation, the nonlinear wave equation, the Korteweg de Vries equation, and the wave maps equation. This book is an introduction to the methods and results used in the modern analysis (both locally and globally in time) of the Cauchy problem for such equations. Starting only with a basic knowledge of graduate real analysis and Fourier analysis, the textfirst presents basic nonlinear tools such as the bootstrap method and perturbation theory in the simpler context of nonlinear ODE, then introduces the harmonic analysis and geometric tools used to control linear dispersive PDE. These methods are then combined to study four model nonlinear dispersiveequations. Through extensive exercises, diagrams, and informal discussion, the book gives a rigorous theoretical treatment of the material, the real-world intuition and heuristics that underlie the subject, as well as mentioning connections with other areas of PDE, harmonic analysis, and dynamical systems. As the subject is vast, the book does not attempt to give a comprehensive survey of the field, but instead concentrates on a representative sample of results for a selected set of equations,ranging from the fundamental local and global existence theorems to very recent results, particularly focusing on the recent progress in understanding the evolution of energy-critical dispersive equations from large data. The book is suitable for a graduate course on nonlinear PDE.
What people are saying - Write a review
We haven't found any reviews in the usual places.
Constant coefficient linear dispersive equations
Semilinear dispersive equations
The Korteweg de Vries equation
Energycritical semilinear dispersive equations
Other editions - View all
Airy equation apply argument asymptotic behaviour blowup blowup solution bound classical solution component conclude conservation law conserved quantities coordinates Coulomb gauge datum defined defocusing derivative differential dimensions dispersive Duhamel Duhamel's formula energy establish Exercise finite speed Fourier transform function gauge global existence global wellposedness Hamiltonian Hamiltonian flow harmonic map heuristic high frequencies Hint hypothesis identity initial data instance integration interaction interval invariance Klein-Gordon equation Lax pair Lemma linear Lipschitz localised low frequencies method Morawetz inequality Noether's theorem non-negative nonlinear norm obeys obtain perturbation principle proof Proposition pseudoconformal R1+d scalar scaling Schrodinger equation Schwartz Schwartz functions Section Show smooth Sobolev embedding Sobolev spaces soliton solves spacetime spatial speed of propagation stress-energy tensor Strichartz estimates subcritical supercritical symmetry symplectic term theorem uniqueness vector field wave equation wave map equation wellposedness theory Xs'b zero