Arick Shao 邵崇哲

Research / Publications

Every once in a while, some of my research gets published.


Papers, Preprints, and Presentations

Below are all my preprints and journal articles, along with any corresponding slides from talks given.

Unique continuation from infinity in asymptotically Anti-de Sitter spacetimes II: Non-static boundaries

Joint with: Gustav Holzegel

We generalize our results from the preceding paper (below) to linear and nonlinear Klein-Gordon equations on asymptotically Anti-de Sitter spacetimes with non-static boundary metrics. The main novelty is a modification of our previous Carleman estimates using a more refined construction of pseudoconvex hypersurfaces that is applicable to this non-static boundary case. Our uniqueness results will have applications toward studying rigidity and holography properties on asymptotically Anti-de Sitter spacetimes.

Unique continuation from infinity in asymptotically Anti-de Sitter spacetimes

Joint with: Gustav Holzegel

We consider the unique continuation properties from infinity of asymptotically Anti-de Sitter spacetimes by studying Klein-Gordon-type equations on these spacetimes. Our main result establishes that if a solution vanishes to sufficiently high order on a sufficiently long time interval along the conformal boundary, then the solution necessarily vanishes in a neighborhood of the boundary. In particular, for cases where there is a well-posedness theory, we prove uniqueness whenever both vanishing Dirichlet and Neumann boundary conditions are imposed. The proof is based on novel Carleman estimates established for this setting.

On the profile of energy concentration at blow-up points for subconformal focusing nonlinear waves

Joint with: Spyros Alexakis

We consider singularities of the focusing subconformal nonlinear wave equation and some generalizations of it. At noncharacteristic points on the singularity surface, Merle and Zaag have identified the rate of blow-up of the energy norm of the solution inside cones that terminate at the singularity. We derive bounds that restrict how this energy can be distributed inside such cones. Our proof relies on new localized estimates, obtained using Carleman-type inequalities, for such nonlinear waves. These bound the solution in the interior of timelike cones by their energy norm near the boundary of the cones. Such estimates can also be applied to obtain certain integrated decay estimates for globally regular solutions to such equations, in the interior of time cones.

Global uniqueness theorems for linear and nonlinear waves

Joint with: Spyros Alexakis

We prove unique continuation from infinity theorems for regular linear and nonlinear wave equations with potential. Under the assumption of no incoming and no outgoing radiation on specific halves of past and future null infinities, we show that the solution must vanish everywhere. In the linear case, the "no radiation" assumption is captured in a specific, finite rate of decay which in general depends on the size of the potential. We also show that this result is optimal in many regards. These results are then extended to certain power-law type nonlinear wave equations, where the order of decay one must assume is independent of the size of the nonlinear term. All these results are obtained using a new family of global Carleman estimates on the exterior of a null cone.

  • Journal (2015, Journal of Functional Analysis)
  • arXiv (2014)

Unique continuation from infinity for linear waves

Joint with: Spyros Alexakis and Volker Schlue

We prove various unique continuation results from infinity for geometric wave equations on various asymptotically Minkowski spacetimes. Roughly speaking, we show that if such a solution vanishes sufficiently at a portion of null infinity, then the solution must also vanish in a neighborhood in the interior. In terms of the background, our results apply to general perturbations of Minkowski spacetimes and to a rather general class of positive-mass spacetimes---including both the Schwarzschild and Kerr families. In terms of the wave equation, we prove results pertaining both to equations with vanishing potential and to equations with bounded potential (e.g., the Klein-Gordon equations). Our results can be closely related to analogous "unique continuation from infinity" results for elliptic equations and is motivated by problems in general relativity.

  • Journal (2015, Advances in Mathematics)
  • arXiv (2013, 2014)
  • Presentation slides (.pdf)
    • (Geometry and Analysis Seminar, Imperial College London; 11/2014)
    • (Seminar of Analysis and Applications, EPFL; 11/2014)
    • (Geometric Analysis and PDE Seminar, Cambridge University; 10/2014)
    • (Analysis Seminar, University of Warwick; 10/2014)
    • (Seminar on Mathematical General Relativity, Université Pierre et Marie Curie, 09/2014)
    • (Geometric Analysis Colloquium, Fields Institute; 12/2013)

Bounds on the Bondi energy by a flux of curvature

Joint with: Spyros Alexakis

Using recent results on the analysis of near-Minkowski and near-Schwarzschild geodesically foliated null cones extending to infinity, we proceed to control the Bondi energy, angular momentum, and rate of energy loss associated with such null cones. In particular, we construct on such a cone a family of spherical cuts going to infinity for which their area-normalized metrics become asymptotically round. We then measure and control the above physical quantities at infinity associated with this family.

On the geometry of null cones to infinity under curvature flux bounds

Joint with: Spyros Alexakis

The goal of this paper is to control the geometry of geodesically foliated truncated null cones extending to infinity in an Einstein-vacuum spacetime by its curvature flux. In particular, we consider null cones which are close, at the curvature flux level, to standard Minkowski and Schwarzschild null cones (on the other hand, we make no such global assumptions on the spacetime). The analysis is based on methods developed by Klainerman and Rodnianski, but the proof here is significantly simplified, mostly due to the use of newly developed techniques.

  • Journal (2014, Classical and Quantum Gravity)
  • arXiv (2013, 2014)

Hamiltonian dynamics of a particle interacting with a wave field

Joint with: Daniel Egli, Jürg Fröhlich, Israel Michael Sigal, Gang Zhou

We study the Hamiltonian equations of motion associated with a heavy tracer particle in a dense weakly interacting Bose-Einstein condensate in the mean-field limit. In this paper, we establish two main results:

  1. We show the existence of subsonic and sonic traveling wave solutions representing inertial motion of the particle.
  2. We prove the asymptotic stability of subsonic traveling wave solutions.

The differential equations modeling this system are similar to those previously studied, except here we deal with a positive speed of sound.

  • Journal (2013, Communications in Partial Differential Equations)
  • arXiv (2012)

New tensorial estimates in Besov spaces for time-dependent (2+1)-dimensional problems

This technical paper establishes various bilinear product estimates and elliptic estimates on a one-parameter foliation of compact surfaces with evolving geometries, while assuming only very weak control on how these geometries evolve. Such estimates were proved by Klainerman and Rodnianski in a very specific setting: truncated geodesic foliated null cones in Einstein-vacuum spacetimes. Here, we significantly simplify the existing proofs, while simultaneously extending the result to more general settings. The results of this paper will be used in upcoming projects in order to study null cones extending to infinity.

  • Journal (2014, Journal of Hyperbolic Differential Equations)
  • arXiv (2012, 2013)

On breakdown criteria for nonvacuum Einstein equations

This paper provides a summary of the work done in my Ph.D. dissertation (see below).

  • Journal (2011, Annales Henri Poincaré)
  • arXiv (2010, 2011)
  • Presentation slides (.pdf)
    • (Analysis and Applied Math Seminar, University of Toronto; 10/2011)
    • (Analysis Seminar, Princeton University; 12/2009)

A generalized representation formula for systems of tensor wave equations

In this paper, we derive a fully covariant representation formula for tensor wave equations on curved spacetimes. This extends the previous results of Klainerman and Rodnianski, as it also allows for first-order terms in the wave equations. A new and more illustrative proof for this type of formula that weakens the required assumptions is also given.


Other

This section contains some other misellaneous research-related documents.

Breakdown Criteria for Nonvacuum Einstein Equations

This was my Ph.D. dissertation, written at Princeton University. The general problem of interest is to determine when an existing finite-time solution of the Einstein equations, here in the constant mean curvature gauge, can be further extended in time. This dissertation is a detailed extension of a previous result of S. Klainerman and I. Rodnianski, which treated Einstein-vacuum spacetimes, to Einstein-scalar and Einstein-Maxwell spacetimes.

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