Math 290: PDE Learning Seminar (Fall 2021)
Nonlinear wave equations and general relativity

The goal of the first part of this learning seminar is to cover the vector field method and its application to the proof of the global nonlinear stability of the Minkowski spacetime. Afterwards, we will cover a selection of topics in nonlinear wave equations relevant to general relativity, which may include: integrated local energy decay (Morawetz estimate) and the r-p method; Price's law; null hypersurfaces and Penrose's incompleteness theorem; double null foliation and the formation of trapped surfaces, etc.

For UC Berkeley students: To enroll in this seminar, please use CCN#15383(27)


In general, we will meet on Wednesdays from 1:10 pm to 2 pm at:

(see the up-to-date syllabus below for the precise schedule).

Mailing list & shared files

If you would like to join the seminar mailing list, please visit:

The relevant files will be available at (access restricted to the group):

Syllabus (will be updated)
Date Topic Refs. Speaker
9/8 Organizational meeting
9/15 Introduction to the vector field method [1] Sung-Jin Oh
9/22 Global existence of $\Box \phi = Q(\partial \phi, \partial \phi)$ with null structure in $\mathbb{R}^{1+3}$[1] Jason Zhao
9/29 Einstein's equation and Choquet-Bruhat's theorem [1, 2] Ely Sandine
10/6 Lindblad-Rodnianski's proof of the global nonlinear stability of the Minkowski spacetime, 1 [1, 3] Yuchen Mao, Mitchell Taylor
10/13 Lindblad-Rodnianski's proof of the global nonlinear stability of the Minkowski spacetime, 2 [1, 3]
10/20 Introduction to local energy decay (Morawetz) estimates TBD Ovidiu Avadanei
10/27 The $r^p$ method [4, 5] James Rowan
11/3 Price's law [6 or 7] Ben Pineau
11/10 Geometry of null hypersurfaces and Penrose's incompleteness theorem [8] Xiaoyu Huang
11/17 No talk
11/24 Thanksgiving week
12/1 Formation of trapped surface, 1 [9, 11] Zhongkai Tao
12/8 Formation of trapped surface, 2 [9, 10] Shi Zhuo Looi
References (will be updated)
1 J. Luk. Lecture notes on nonlinear wave equations
2 H. Ringström, The Cauchy problem in general relativity, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House
3 H. Lindblad and I. Rodnianski, The global stability of Minkowski space-time in harmonic gauge, Annals of Mathematics, 171 (2010), 1401-1477
4 M. Dafermos and I. Rodnianski, A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, arXiv:0910.4957
5 G. Moschidis, The $r^p$-weighted energy method of Dafermos and Rodnianski in general asymptotically flat spacetimes and applications, Annals of PDE, (2016) 2:6
6 D. Tataru, Local decay of waves on asymptotically flat stationary space-times, American Journal of Mathematics, 135 (2013), 361-401
7 J. Metcalfe, D. Tataru and M. Tohaneanu, Price’s law on nonstationary space–times, Advances in Mathematics 230 (2012) 995–1028
8 S. Aretakis, Hyperbolic PDEs and Applications in Relativity, Chapters 4 and 5
9 D. Christodoulou, The Formation of Black Holes in General Relativity, arXiv:0805.3880
10 J. Li and H. Mei, A construction of collapsing spacetimes in vacuum, arXiv:2005.01249
11 S. Klainerman, J. Luk and I. Rodnianski, A fully anisotropic mechanism for formation of trapped surfaces in vacuum, Inventiones Mathematicae, 198 (2014), 1-26

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