Lectures on Nonlinear Hyperbolic Differential Equations

Lars Hormander

• 17 juli 1997
• 9783540629214

Samenvatting:

Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data.

This introduction to the theory of nonlinear hyperbolic differential equations, a revised and extended version of widely circulated lecture notes from 1986, starts from a very elementary level with standard existence and uniqueness theorems for ordinary differential equations, but they are at once supplemented with less well-known material, required later on. A detailed and explicit study of discontinuous solutions of a model equation, Burgers' equation, is then followed by a general study of solutions of conservation laws, with one unknown or one space variable. Asymptotic properties of solutions of the linear wave equation and the Klein-Gordon equation are studied in detail as a preparation for the study of solutions of nonlinear perturbations with small and smooth initial data. Existence of solutions for all times is proved for large space dimensions and lower bounds for the "lifespan" of the solutions are given in low space dimensions. The last four chapters are devoted to microlocal analysis of singularities of solutions of nonlinear differential equations by means of the paradifferential techniques of J.-M. Bony.