Arick Shao  邵崇哲 
Research / PublicationsEvery once in a while, some of my research gets published. Papers, Preprints, and PresentationsBelow are all my preprints and journal articles, along with any corresponding slides from talks given. Unique continuation from infinity in asymptotically Antide Sitter spacetimes II: Nonstatic boundariesJoint with: Gustav Holzegel We generalize our results from the preceding paper (below) to linear and nonlinear KleinGordon equations on asymptotically Antide Sitter spacetimes with nonstatic 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 nonstatic boundary case. Our uniqueness results will have applications toward studying rigidity and holography properties on asymptotically Antide Sitter spacetimes.
Unique continuation from infinity in asymptotically Antide Sitter spacetimesJoint with: Gustav Holzegel We consider the unique continuation properties from infinity of asymptotically Antide Sitter spacetimes by studying KleinGordontype 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 wellposedness 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 blowup points for subconformal focusing nonlinear wavesJoint 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 blowup 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 Carlemantype 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 wavesJoint 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 powerlaw 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. Unique continuation from infinity for linear wavesJoint 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 positivemass spacetimesincluding 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 KleinGordon 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.
Bounds on the Bondi energy by a flux of curvatureJoint with: Spyros Alexakis Using recent results on the analysis of nearMinkowski and nearSchwarzschild 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 areanormalized 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 boundsJoint with: Spyros Alexakis The goal of this paper is to control the geometry of geodesically foliated truncated null cones extending to infinity in an Einsteinvacuum 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. Hamiltonian dynamics of a particle interacting with a wave fieldJoint 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 BoseEinstein condensate in the meanfield limit. In this paper, we establish two main results:
The differential equations modeling this system are similar to those previously studied, except here we deal with a positive speed of sound. New tensorial estimates in Besov spaces for timedependent (2+1)dimensional problemsThis technical paper establishes various bilinear product estimates and elliptic estimates on a oneparameter 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 Einsteinvacuum 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. On breakdown criteria for nonvacuum Einstein equationsThis paper provides a summary of the work done in my Ph.D. dissertation (see below).
A generalized representation formula for systems of tensor wave equationsIn 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 firstorder terms in the wave equations. A new and more illustrative proof for this type of formula that weakens the required assumptions is also given.
OtherThis section contains some other misellaneous researchrelated documents. Breakdown Criteria for Nonvacuum Einstein EquationsThis was my Ph.D. dissertation, written at Princeton University. The general problem of interest is to determine when an existing finitetime 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 Einsteinvacuum spacetimes, to Einsteinscalar and EinsteinMaxwell spacetimes.
