The vector-field method and decay for the wave equation

In this seminar, we will study wave equations particularly nonlinear wave equations, from a geometric perspective, and inspired by the dynamics in general relativity. Among the topics of interest are black holes, stability, long-term dynamics, low-regularity, fluids, and shocks. We will have a combination of external and local speakers, including students. This is coorganised by Pieter Blue, Ma Siyuan, and Yu Pin.

For the first talk, we will have Pieter Blue talk on "The vector-field method and decay for the wave equation"

For more information please see at:

https://bimsa.net/activity/Wavgraandgeo/

Organizers：

Lars Andersson, Pin Yu, Siyuan Ma

Speaker：

Pieter Blue (The University of Edinburgh)

Time：

Wed., 10:00 am -12:00 pm, Sept. 18, 2024

Venue：

Ningzhai（宁斋）104

Online:

Zoom 518 868 7656

Password: BIMSA

Title：

The vector-field method and decay for the wave equation

Abstract：

This talk will look at a method for proving decay of solutions to the linear wave equation and relate this to the study of nonlinear wave equations. Symmetries of an equation can be used to generate conserved quantities associated with solutions and also be used to differentiate solutions to generate new solutions. This leads to conservation of energy and of higher Sobolev norms. By applying the boost symmetries, one finds that weighted energies are bounded, and, from these, the Klainerman-Sobolev estimate gives a rate of decay for solutions of the wave equation. By replacing symmetries by other derivatives, it is possible to prove other decay estimates, which are based on monotonicity rather than conservation. Such estimates include those of Morawetz and the weighted null energies ("rprp method") of Dafermos-Rodnianski, which hold when there are fewer symmetries. These decay estimates can also be related to the null condition on nonlinearities. This will largely be an introductory talk.