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Nonlinear partial differential equations, general relativity, mathematical physics
Part II Linear analysis
I will follow quite closely the lecture notes of Mihalis Dafermos. (Supplementary notes on Theorem 8.2)
Example Sheet 1 Example Sheet 2 Example Sheet 3 Example Sheet 4
Part III Introduction to nonlinear wave equations
Lecture notes Review on Fourier analysis Example Sheet 1 Example Sheet 2 Example Sheet 3
- (with S.-J. Oh and S. Yang) Solutions to the Einstein-scalar-field system in spherical symmetry with large bounded variation norms, preprint.
- (with J. Speck, G. Holzegel and W. W.-Y. Wong) Stable shock formation for nearly plane symmetric waves, to appear in Annals of PDE.
- (with J. Sbierski) Instability results for the wave equation in the interior of Kerr black holes, to appear in J. Funct. Anal.
- (with G. Holzegel, J. Smulevici and C. Warnick) Asymptotic properties of linear field equations in anti-de Sitter space, preprint.
- (with S.-J. Oh) Proof of linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations, to appear in Duke Math. J.
- (with X. An) Trapped surfaces in vacuum arising from mild incoming radiation, to appear in Adv. Theo. Math. Phys..
- (with R. M. Strain) Strichartz estimates and moment bounds for the relativistic Vlasov-Maxwell system, Arch. Rat. Mech. Anal. 219(1):445-552, 2016 (combined from two earlier preprints in the 2D and 2.5D cases and the 3D case).
- (with R. M. Strain ) A new continuation criterion for the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 331:1005-1027, 2014.
- (with S.-J. Oh) Quantitative decay rates for dispersive solutions to the Einstein-scalar field system in spherical symmetry, Analysis and PDE 8(7):1603-1674, 2015.
- Weak null singularities in general relativity, preprint.
- (with S. Klainerman and I. Rodnianski) A fully anisotropic mechanism for formation of trapped surfaces in vacuum, Invent. Math. 194(1):1–26, 2014.
- (with I. Rodnianski) Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations, preprint.
- (with I. Rodnianski) Local propagation of impulsive gravitational waves, Comm. Pure and Appl. Math., 68(4):511–624, 2015.
- On the local existence for the characteristic initial value problem in general relativity, Int. Mat. Res. Notices, 20:4625-4678, 2012.
- The null condition and global existence for nonlinear wave equations on slowly rotating Kerr spacetimes, Journal Eur. Math. Soc., 15(5):1629-1700, 2013.
- A vector field method approach to improved decay for solutions to the wave equation on a slowly rotating Kerr black hole, Analysis and PDE, 5(3):553-625, 2012.
- Improved decay for solutions to the linear wave equation on a Schwarzschild black hole, Annales Henri Poincare, 11:805-880, 2010.
Weak null singularities in general relativity, MSRI, November 2013. Formation of trapped surfaces, IHP, May 2013.