Stanford University Department of Mathematics
Stanford, CA 94305
Nonlinear partial differential equations, general relativity, mathematical physics
Autumn 2018, Math 256A, Partial differential equations Course webpage
I have written a set of notes on nonlinear wave equations, which can be found here. It is complemented by a review on Fourier analysis and the following example sheets (1, 2, 3).
Previous teaching can be found here.
- Stability of vacuum for the Landau equation with moderately soft potentials, preprint.
- (with M. Dafermos) The interior of dynamical vacuum black holes I: The C0-stability of the Kerr Cauchy horizon, preprint.
- (with D. Gajic) The interior of dynamical extremal black holes in spherical symmetry, preprint.
- (with C. Huneau) High-frequency backreaction for the Einstein equations under polarized U(1) symmetry, to appear in Duke Math. J..
- (with C. Huneau) Einstein equations under polarized U(1) symmetry in an elliptic gauge, to appear in Comm. Math. Phys.
- (with S.-J. Oh and S. Yang) Dynamical black holes with prescribed masses in spherical symmetry, preprint.
- (with S.-J. Oh) Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data II. The exterior of the black hole region, preprint.
- (with S.-J. Oh) Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data I. The interior of the black hole region, preprint.
- (with J. Speck) Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity, to appear in Invent. Math.
- (with J. Speck) The hidden null structure of the compressible Euler equations and a prelude to applications, preprint.
- (with S.-J. Oh and S. Yang) Solutions to the Einstein-scalar-field system in spherical symmetry with large bounded variation norms, to appear in Annals of PDE.
- (with J. Speck, G. Holzegel and W. W.-Y. Wong) Stable shock formation for nearly plane symmetric waves, Annals of PDE., 2:10, 2016.
- (with J. Sbierski) Instability results for the wave equation in the interior of Kerr black holes, J. Funct. Anal., 271(7):1948-1995, 2016.
- (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, Duke Math. J., 166(3):437-493, 2017.
- (with X. An) Trapped surfaces in vacuum arising from mild incoming radiation, Adv. Theo. Math. Phys., 21(1):1-120, 2017.
- (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, Journal of AMS. 31:1-63, 2018.
- (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, Cambridge Journal of Math. 5(4): 435-570, 2017.
- (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.
Recent progress on the strong cosmic censorship conjecture, BHI, February 2017. Interior of dynamical vacuum black holes, IHP, November 2015. The stability of the Kerr Cauchy horizon and the strong cosmic censorship conjecture in general relativity, Fields Institute, June 2015. Weak null singularities in general relativity, MSRI, November 2013. Formation of trapped surfaces, IHP, May 2013.