The vector-field method and decay for the wave equation
组织者
拉尔斯•安德森
, Siyuan Ma
, 于品
演讲者
皮特·布鲁
时间
2024年09月18日 10:00 至 12:00
地点
Tsinghua-Jingzhai-105
线上
Zoom 518 868 7656
(BIMSA)
摘要
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 ("$r^p$ 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.