|
Abstract |
The purpose of theses lectures is to explain, first on elementary examples, the basic idea of the BKW method which is here a method of approximation of solution of evolution partial differential equation in the high frequency regime. The role of the Hamilton-Jacobi equation will be highlighted.
Then we treat in full details two important cases : the case of wave equations with variable coefficients and the case of the Schrödinger coefficients with variable coefficients.
The motivation of this will be the proof of some dispersive effects in the two cases of wave and Schrödinger equations. It will allow to solve quasilinear wave and Schrödinger equations for initial data which are less regular than the regularity given by the simple use of energy estimates. |
