Stanford University Department of Mathematics
Stanford, CA 94305
Nonlinear partial differential equations, general relativity, mathematical physics
I am teaching Math 61CM in the fall. Please refer to the Canvas page for details.
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.
I am organizing the Analysis and PDE seminar with Eugenia Malinnikova. Please contact me if you would like to be added to the e-mail list which contains zoom details.
- (with S. Chaturvedi) Phase mixing for solutions to 1D transport equation in a confining potential, preprint.
- (with S.-J. Oh) Global nonlinear stability of large dispersive solutions to the Einstein equations, preprint.
- (with J. Speck) The stability of simple plane-symmetric shock formation for 3D compressible Euler flow with vorticity and entropy, preprint.
- (with M. Van de Moortel) Nonlinear interaction of three impulsive gravitational waves II: the wave estimates, preprint.
- (with S. Chaturvedi and T. T. Nguyen) The Vlasov-Poisson-Landau system in the weakly collisional regime, preprint.
- (with M. Van de Moortel) Nonlinear interaction of three impulsive gravitational waves I: main result and the geometric estimates, preprint.
- (with I. Rodnianski) High-frequency limits and null dust shell solutions in general relativity, preprint.
- (with G. Fournodavlos) Asymptotically Kasner-like singularities, preprint.
- (with C. Huneau) Trilinear compensated compactness and Burnett's conjecture in general relativity, preprint.
- Stability of vacuum for the Landau equation with moderately soft potentials, Annals of PDE, 5:11, 2019.
- (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, Pure and Appl. Anal., 1(2):263-326, 2019.
- (with C. Huneau) High-frequency backreaction for the Einstein equations under polarized U(1) symmetry, Duke Math. J., 67(18):3315-3402, 2018.
- (with C. Huneau) Einstein equations under polarized U(1) symmetry in an elliptic gauge, Comm. Math. Phys., 361(3):873-949, 2018.
- (with S.-J. Oh and S. Yang) Dynamical black holes with prescribed masses in spherical symmetry, Proceedings of 7th ICCM II, 367-387, 2019.
- (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, Annals of PDE, 5(6), 2019.
- (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, Annals of Math., 190(1):1-111, 2019.
- (with J. Speck) Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity, Invent. Math., 214(1):1-169, 2018.
- (with J. Speck) The hidden null structure of the compressible Euler equations and a prelude to applications, JHDE 14(1):1-60, 2020.
- (with S.-J. Oh and S. Yang) Solutions to the Einstein-scalar-field system in spherical symmetry with large bounded variation norms, Annals of PDE, 4:3, 2018.
- (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, Comm. Math. Phys., 374(2):1125-1178, 2020.
- (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.
High-frequency limits in general relativity and a conjecture of Burnett, ESI, February 2021. Nonlinear interaction of three impulsive gravitational waves, Harvard, February 2021. Global problems in kinetic theory, Penn State University, February 2021. Some singular spacetimes in general relativity, Gravity Initiative, Princeton University, November 2020. On gravitational collapse and spacetime singularities: the many contributions of Sir Roger Penrose, Stanford University, October 2020. High-frequency limits in general relativity and a conjecture of Burnett, BHI, Harvard University, October 2020. Nonlinear interaction of multiple impulsive gravitational waves, ICERM, October 2020. Asymptotic properties of linear field equations in anti-de Sitter space, CMO, August 2019. Mathematical progress in general relativity, GR22, July 2019. Recent progress on the strong cosmic censorship conjecture, BHI, Harvard University, 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.