April 28th – May 15th, 2026
Gran Sasso Science Institute, L'Aquila (IT)
The course will introduce basic properties of linear waves and some global results for nonlinear wave equations via energy methods. Part of the course will discuss connections with global problems for wave equations arising from physics, such as the nonlinear stability of Minkowski space as a solution to Einstein's equations of general relativity. The course will be intended for a broad audience. Some exposure to the analysis of partial differential equations (as, for instance, the material covered in the first-semester courses at GSSI) will be assumed.
Calendar
Lecture 1 (28/04, Zenith Conference Room -1, 10:45am--12:15pm)
Introduction.
Lecture 2 (30/04, Zenith Conference Room -1, 10:45am--12:15pm)
Linear wave equation: Conservation of energy, uniqueness and finite speed of propagation via energy methods.
Lecture 3 (05/05, Main Lecture Hall, 10:45am--12:15pm)
Linear wave equation: Dispersion via energy methods (statement of Klainerman--Sobolev inequality and corollaries).
Lecture 4 (07/05, Main Lecture Hall, 10:45am--12:15pm)
Linear wave equation: Proof of Klainerman--Sobolev inequality.
Lecture 5 (08/05, Main Lecture Hall, 2--3:30pm)
Existence for general linear wave equations and local theory for nonlinear wave equations.
Lecture 6 (12/05, Zenith Conference Room -1, 10:45am--12:15pm)
Nonlinear wave equations: Global existence for subcritical equations (cubic wave equation in 3+1 dimensions).
Lecture 7 (14/05, Zenith Conference Room -1, 10:45am--12:15pm)
Nonlinear wave equations: Small-data global existence for supercritical equations (wave map equation in 4+1-dimensions).
Lecture 8 (15/05, Zenith Conference Room -1, 10:45am--12:15pm)
Nonlinear wave equations: Small-data global existence for supercritical equations (wave map equation in 3+1-dimensions and null condition).