Energy estimates for non-linear wave equations in mathematical General Relativity

Speaker

Sari Ghanem

Assistant Professor of BIMSA

Time

Wed., 11:00 am–12:30 & 14:30–16:00,

Sept. 17-Dec. 24, 2025

(excluding Oct. 01, 08, 15)

Venue

C658, Shuangqing Complex Building A

Description

This course introduces mathematical tools of analysis for partial differential equations to prove uniform bounds and decay for solutions of non-linear wave equations arising in General Relativity. The course material builds on a series of courses that I gave in Spring--Fall 2014, and in Spring 2025, on the Cauchy problem in mathematical General Relativity, on non-linear wave equations in General Relativity, and on dispersive estimates for non-linear waves in mathematical General Relativity. The goal of this course is to explain the vector field method and how to obtain energy estimates for solutions of tensorial coupled non-linear hyperbolic partial differential equations, in order to prove decay for solutions of non-linear wave equations provided that one exploits the non-linear structure of the wave equations. We shall exhibit how this can be applied to the Einstein equations coupled to non-linear matter such as the Yang-Mills fields, by studying the simpler case of higher dimensions.

Prerequisite：

Basic knowledge from my previous courses on “The Cauchy problem in mathematical General Relativity", on “Non-linear wave equations in General Relativity", and on “Dispersive estimates for non-linear waves in mathematical General Relativity", graduate level knowledge in differential geometry and in Riemannian geometry, and basic knowledge in partial differential equations and analysis.

Target Audience:

Graduate, Postdoc, Researcher

Teaching Language: English