Arick Shao 邵崇哲

# Research / Publications

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

## Papers, Preprints, and Presentations

The following is a list of my preprints and journal articles.

### On Carleman and observability estimates for wave equations on time-dependent domains

We establish new Carleman estimates for the wave equation, which we then apply to derive novel observability inequalities for a general class of linear wave equations. The main features of these inequalities are that (a) they apply to a fully general class of time-dependent domains, with timelike moving boundaries, (b) they apply to linear wave equations in any spatial dimension and with general time-dependent lower-order coefficients, and (c) they allow for significantly smaller time-dependent regions of observations than allowed from existing Carleman estimate methods. As a standard application, we establish exact controllability for general linear waves, again in the setting of time-dependent domains and regions of control.

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

Joint with: Gustav Holzegel

We generalize our unique continuation results recently established for a class of linear and nonlinear wave equations $$\Box_g \phi + \sigma \phi = \mathcal{G} ( \phi, \partial \phi )$$ on asymptotically anti-de Sitter (aAdS) spacetimes to aAdS spacetimes admitting non-static boundary metrics. The new Carleman estimates established in this setting constitute an essential ingredient in proving unique continuation results for the full nonlinear Einstein equations, which will be addressed in forthcoming papers. Key to the proof is a new geometrically adapted construction of foliations of pseudoconvex hypersurfaces near the conformal boundary.

• Journal (2017, Communications in Partial Differential Equations)
• arXiv (2016, 2017)

### Unique continuation from infinity in asymptotically Anti-de Sitter spacetimes

Joint with: Gustav Holzegel

We consider the unique continuation properties of asymptotically Anti-de Sitter spacetimes by studying Klein-Gordon-type equations $$\Box_g \phi + \sigma \phi = \mathcal{G} ( \phi, \partial \phi )$$, $$\sigma \in \mathbb{R}$$, on a large class of such spacetimes. Our main result establishes that if $$\phi$$ vanishes to sufficiently high order (depending on $$\sigma$$) on a sufficiently long time interval along the conformal boundary $$\mathcal{I}$$, then the solution necessarily vanishes in a neighborhood of $$\mathcal{I}$$. In particular, in the $$\sigma$$-range where Dirichlet and Neumann conditions are possible on $$\mathcal{I}$$ for the forward problem, we prove uniqueness if both these conditions are imposed. The length of the time interval can be related to the refocusing time of null geodesics on these backgrounds and is expected to be sharp. Some global applications as well a uniqueness result for gravitational perturbations are also discussed. The proof is based on novel Carleman estimates established in this setting.

• Journal (2016, Communications in Mathematical Physics)
• arXiv (2015)

### 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 $$H^1$$-norm of the solution inside cones that terminate at the singularity. We derive bounds that restrict how this $$H^1$$-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 $$L^{p+1}$$-norm in the interior of timelike cones by their $$H^1$$-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.

• Journal (2017, Transactions of the American Mathematical Society)
• arXiv (2014, 2017)

### Global uniqueness theorems for linear and nonlinear waves

Joint with: Spyros Alexakis

We prove a unique continuation from infinity theorem for regular waves of the form $$[ \Box + \mathcal{V} (t, x) ]\phi=0$$. 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. The "no radiation" assumption is captured in a specific, finite rate of decay which in general depends on the $$L^\infty$$-profile of the potential $$\mathcal{V}$$. We show that the 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. These results are obtained using a new family of global Carleman-type estimates on the exterior of a null cone. A companion paper to this one explores further applications of these new estimates to such nonlinear waves.

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

### Unique continuation from infinity for linear waves

Joint with: Spyros Alexakis and Volker Schlue

We prove various uniqueness results from null infinity, for linear waves on asymptotically flat space-times. Assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities, we derive that the solution must vanish in an open set in the interior. We find that the parts of infinity where we must impose a vanishing condition depend strongly on the background geometry. In particular, for backgrounds with positive mass (such as Schwarzschild or Kerr), the required assumptions are much weaker than the ones in the Minkowski space-time. The results are nearly optimal in many respects. They can be considered analogues of uniqueness from infinity results for second order elliptic operators. This work is partly motivated by questions in general relativity.

• Journal (2016, Advances in Mathematics)
• arXiv (2013, 2014)

### Bounds on the Bondi energy by a flux of curvature

Joint with: Spyros Alexakis

We consider smooth null cones in a vacuum spacetime that extend to future null infinity. For such cones that are perturbations of shear-free outgoing null cones in Schwarzschild spacetimes, we prove bounds for the Bondi energy, momentum, and rate of energy loss. The bounds depend on the closeness between the given cone and a corresponding cone in a Schwarzschild spacetime, measured purely in terms of the differences between certain weighted $$L^2$$-norms of the spacetime curvature on the cones, and of the geometries of the spheres from which they emanate. A key step in this paper is the construction of a family of asymptotically round cuts of our cone, relative to which the Bondi energy is measured.

• Journal (2016, Journal of the European Mathematical Society)
• arXiv (2013, 2016)

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

Joint with: Spyros Alexakis

The main objective of this paper is to control the geometry of a future outgoing truncated null cone extending smoothly toward infinity in an Einstein-vacuum spacetime. In particular, we wish to do this under minimal regularity assumptions, namely, at the (weighted) $$L^2$$-curvature level. We show that if the curvature flux and the data on an initial sphere of the cone are sufficiently close to the corresponding values in a standard Minkowski or Schwarzschild null cone, then we can obtain quantitative bounds on the geometry of the entire infinite cone. The same bounds also imply the existence of limits at infinity, along the null cone, of the naturally scaled geometric quantities. In our sequel paper, we will apply these results in order to control various physical quantities—e.g., the Bondi energy and (linear and angular) momenta—associated with such infinite null cones in vacuum spacetimes.

• 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 of a heavy tracer particle interacting with a dense weakly interacting Bose-Einstein condensate in the classical (mean-field) limit. Solutions describing ballistic subsonic motion of the particle through the condensate are constructed. We establish asymptotic stability of ballistic subsonic motion.

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

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

In this paper, we consider various tensorial estimates in geometric Besov-type norms on a one-parameter foliation of surfaces with evolving geometries. Moreover, we wish to accomplish this with only very weak control on these geometries. Several of these estimates were established in previous works by S. Klainerman and I. Rodnianski, but in very specific settings. A primary objective of this paper is to significantly simplify and make more robust the proofs of the estimates. Another goal is to generalize these estimates to more abstract settings. In upcoming papers (joint with S. Alexakis), we will apply these estimates in order study truncated null cones in an Einstein-vacuum spacetime extending to infinity. This analysis will then be used to study and to control the Bondi mass and the angular momentum under minimal conditions.

• Journal (2014, Journal of Hyperbolic Differential Equations)
• arXiv (2012, 2015)

### On breakdown criteria for nonvacuum Einstein equations

The recent "breakdown criterion" result of S. Klainerman and I. Rodnianski stated roughly that an Einstein-vacuum spacetime, given as a CMC foliation, can be further extended in time if the second fundamental form and the derivative of the lapse of the foliation are uniformly bounded. This theorem and its proof were extended to Einstein-scalar and Einstein-Maxwell spacetimes in the author's PhD thesis. In this paper, we state the main results of the thesis, and we summarize and discuss their proofs. In particular, we will discuss the various issues resulting from nontrivial Ricci curvature and the coupling between the Einstein and the field equations.

• Journal (2011, Annales Henri Poincaré)
• arXiv (2010, 2011)

### A generalized representation formula for systems of tensor wave equations

In this paper, we generalize the Kirchhoff-Sobolev parametrix of Klainerman and Rodnianski to systems of tensor wave equations with additional first-order terms. We also present a different derivation, which better highlights that such representation formulas are supported entirely on past null cones. This generalization is a key component for extending Klainerman and Rodnianski's breakdown criterion result for Einstein-vacuum spacetimes to Einstein-Maxwell and Einstein-Yang-Mills spacetimes.

• Journal (2011, Communications in Mathematical Physics)
• arXiv (2010)

## 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.

We generalize a recent "breakdown criterion" result of S. Klainerman and I. Rodnianski, which states roughly that an Einstein vacuum spacetime, given as a CMC foliation, can be extended if the second fundamental form and the derivative of the lapse of the foliation are uniformly bounded. We adapt this theorem and its proof to Einstein-scalar and Einstein-Maxwell spacetimes. In particular, we deal with additional issues resulting from nontrivial Ricci curvature and the coupling between the Einstein and the field equations. The results we prove can be directly extended to Einstein-Klein-Gordon and Einstein-Yang-Mills spacetimes.