| Arick Shao | 邵崇哲 |
Research / PublicationsEvery once in a while, some of my research gets published. Papers and PreprintsThe following is a list of my preprints and journal articles. Approximate boundary controllability for parabolic equations with inverse square infinite potential wellsJoint with: Bruno Vergara We consider heat operators on a bounded domain \( \Omega \subseteq \mathbb{R}^n \), with a critically singular potential diverging as the inverse square of the distance to \( \partial \Omega \). While null boundary controllability for such operators was recently proved in all dimensions in arXiv:2112.04457, it crucially assumed \( \Omega \) was convex, and the control must be prescribed along the entire boundary. In this article, we prove instead an approximate boundary controllability result for these operators, but we both remove the convexity assumption on \( \Omega \) and allow for the control to be localized near a point \( x_0 \in \partial \Omega \). In addition, we lower the regularity required on \( \partial \Omega \) and the lower-order coefficients. The key novelty is a local Carleman estimate near \( x_0 \), with a carefully chosen weight that takes into account both the appropriate boundary conditions and the local geometry of \( \partial \Omega \). On counterexamples to unique continuation for critically singular wave equationsJoint with: Simon Guisset We consider wave equations with a critically singular potential \( \xi \cdot \sigma^{-2} \) diverging as an inverse square at a hypersurface \( \sigma = 0 \). Our aim is to construct counterexamples to unique continuation from \( \sigma = 0 \) for this equation, provided there exists a family of null geodesics trapped near \( \sigma = 0 \). This extends the classical geometric optics construction of Alinhac-Baouendi (i) to linear differential operators with singular coefficients, and (ii) over non-small portions of \( \sigma = 0 \)—by showing that such counterexamples can be further continued as long as this null geodesic family remains trapped and regular. As an application to relativity and holography, we construct counterexamples to unique continuation from the conformal boundaries of asymptotically Anti-de Sitter spacetimes for some Klein-Gordon equations; this complements the unique continuation results of the second author with Chatzikaleas, Holzegel, and McGill and suggests a potential mechanism for counterexamples to the AdS/CFT correspondence. Controllability of parabolic equations with inverse square infinite potential wells via global Carleman estimatesJoint with: Alberto Enciso, Bruno Vergara We consider heat operators on a convex domain \(\Omega\), with a critically singular potential that diverges as the inverse square of the distance to the boundary of \(\Omega\). We establish a general boundary controllability result for such operators in all dimensions, in particular providing the first such result in more than one spatial dimension. The key step in the proof is a novel global Carleman estimate that captures both the appropriate boundary conditions and the \(H^1\)-energy for this problem. The estimate is derived by combining two intermediate Carleman inequalities with distinct and carefully constructed weights involving non-smooth powers of the boundary distance. (This article replaces a previous one under the same arXiv link with a stronger, and more correct, result.)
The bulk-boundary correspondence for the Einstein equations in asymptotically Anti-de Sitter spacetimesJoint with: Gustav Holzegel In this paper, we consider vacuum asymptotically anti-de Sitter spacetimes \( (\mathcal{M}, g ) \) with conformal boundary \( ( \mathcal{I}, \mathfrak{g} ) \). We establish a correspondence, near \( \mathscr{I} \), between such spacetimes and their conformal boundary data on \( \mathscr{I} \). More specifically, given a domain \( \mathscr{D} \subset \mathscr{I} \), we prove that the coefficients \( \mathfrak{g}^{(0)} = \mathfrak{g} \) and \( \mathfrak{g}^{(n)} \) (the undetermined term or stress energy tensor) in a Fefferman-Graham expansion of the metric \( g \) from the boundary uniquely determine \( g \) near \( \mathscr{D} \), provided \( \mathscr{D} \) satisfies a generalised null convexity condition (GNCC). The GNCC is a conformally invariant criterion on \( \mathscr{D} \), first identified by Chatzikaleas and the second author, that ensures a foliation of pseudoconvex hypersurfaces in \( \mathscr{M} \) near \( \mathscr{D} \), and with the pseudoconvexity degenerating in the limit at \( \mathscr{D} \). As a corollary of this result, we deduce that conformal symmetries of \( ( \mathfrak{g}^{(0)}, \mathfrak{g}^{(n)} ) \) on domains \( \mathscr{D} \subset \mathscr{I} \) satisfying the GNCC extend to spacetimes symmetries near \( \mathscr{D} \). The proof, which does not require any analyticity assumptions, relies on three key ingredients: (1) a calculus of vertical tensor-fields developed for this setting; (2) a novel system of transport and wave equations for differences of metric and curvature quantities; and (3) recently established Carleman estimates for tensorial wave equations near the conformal boundary. A summary of this work is available in the MATRIX Annals: A gauge-invariant unique continuation criterion for waves in asymptotically Anti-de Sitter spacetimesJoint with: Athanasios Chatzikaleas We reconsider the unique continuation property for a general class of tensorial Klein-Gordon equations of the form \[ \Box_g \phi + \sigma \phi = G ( \phi, \nabla \phi ) \text{,} \qquad \sigma \in \mathbb{R} \] on a large class of asymptotically anti-de Sitter spacetimes. In particular, we aim to generalize the previous results of Holzegel, McGill, and Shao (which established the above-mentioned unique continuation property through novel Carleman estimates near the conformal boundary) in the following ways: (1) We replace the so-called null convexity criterion (the key geometric assumption on the conformal boundary needed in previous results to establish the unique continuation properties) by a more general criterion that is also gauge invariant. (2) Our new unique continuation property can be applied from a larger, more general class of domains on the conformal boundary. (3) Similar to previous works, we connect the failure of our generalized null convexity criterion to the existence of certain null geodesics near the conformal boundary. These geodesics can be used to construct counterexamples to unique continuation. Finally, our gauge-invariant criterion and Carleman estimate will constitute a key ingredient in proving unique continuation results for the full nonlinear Einstein-vacuum equations, which will be addressed in a forthcoming paper of Holzegel and Shao. Control of waves on Lorentzian manifolds with curvature boundsJoint with: Vaibhav Jena We prove boundary controllability results for wave equations (with lower-order terms) on Lorentzian manifolds with time-dependent geometry satisfying suitable curvature bounds. The main ingredient is a novel global Carleman estimate on Lorentzian manifolds that is supported in the exterior of a null (or characteristic) cone, which leads to both an observability inequality and bounds for the corresponding constant. The Carleman estimate also yields a unique continuation result on the null cone exterior, which has applications toward inverse problems for linear waves on Lorentzian backgrounds. Global stability of traveling waves for \((1+1)\)-dimensional systems of quasilinear wave equationsJoint with: Dongbing Zha A key feature of \((1+1)\)-dimensional nonlinear wave equations is that they admit left or right traveling waves, under appropriate algebraic conditions on the nonlinearities. In this paper, we prove global stability of such traveling wave solutions for \((1+1)\)-dimensional systems of nonlinear wave equations, given a certain asymptotic null condition and sufficient decay for the traveling wave. We first consider semilinear systems as a simpler model problem; we then proceed to treat more general quasilinear systems. Null Geodesics and Improved Unique Continuation for Waves in Asymptotically Anti-de Sitter SpacetimesJoint with: Alex McGill We consider the question of whether solutions of Klein-Gordon equations on asymptotically Anti-de Sitter spacetimes can be uniquely continued from the conformal boundary. Positive answers were first given by the second author with G. Holzegel, under suitable assumptions on the boundary geometry and with boundary data imposed over a sufficiently long timespan. The key step was to establish Carleman estimates for Klein-Gordon operators near the conformal boundary. In this article, we further improve upon the above-mentioned results. First, we establish new Carleman estimates—and hence new unique continuation results—for Klein-Gordon equations on a larger class of spacetimes, in particular with more general boundary geometries. Second, we argue for the optimality, in many respects, of our assumptions by connecting them to trajectories of null geodesics near the conformal boundary; these geodesics play a crucial role in the construction of counterexamples to unique continuation. Finally, we develop a new covariant formalism that will be useful—both presently and more generally beyond this article—for treating tensorial objects with asymptotic limits at the conformal boundary. The Near-Boundary Geometry of Einstein-Vacuum Asymptotically Anti-de Sitter SpacetimesWe study the geometry of a general class of vacuum asymptotically Anti-de Sitter spacetimes near the conformal boundary. In particular, the spacetime is only assumed to have finite regularity, and it is allowed to have arbitrary boundary topology and geometry. For the main results, we derive limits at the conformal boundary of various geometric quantities, and we use these limits to construct partial Fefferman–Graham expansions from the boundary. The results of this article will be applied, in upcoming papers, toward proving symmetry extension and gravity-boundary correspondence theorems for vacuum asymptotically Anti-de Sitter spacetimes. Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentialsJoint with: Alberto Enciso, Bruno Vergara We establish a new family of Carleman inequalities for wave operators on cylindrical spacetime domains containing a potential that is critically singular, diverging as an inverse square on all the boundary of the domain. These estimates are sharp in the sense that they capture both the natural boundary conditions and the natural \( H^1 \)-energy. The proof is based around three key ingredients: the choice of a novel Carleman weight with rather singular derivatives on the boundary, a generalization of the classical Morawetz inequality that allows for inverse-square singularities, and the systematic use of derivative operations adapted to the potential. As an application of these estimates, we prove a boundary observability property for the associated wave equations. On Carleman and observability estimates for wave equations on time-dependent domainsWe 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. A summary of this work can be found in the proceedings for the Séminaire Laurent Schwartz:
Unique continuation from infinity in asymptotically Anti-de Sitter spacetimes II: Non-static boundariesJoint 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. Unique continuation from infinity in asymptotically Anti-de Sitter spacetimesJoint 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. On the profile of energy concentration at blow-up 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 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. Global uniqueness theorems for linear and nonlinear wavesJoint 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. Unique continuation from infinity for linear wavesJoint with: Spyros Alexakis, 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. Bounds on the Bondi energy by a flux of curvatureJoint 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. On the geometry of null cones to infinity under curvature flux boundsJoint 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. 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 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. New tensorial estimates in Besov spaces for time-dependent (2+1)-dimensional problemsIn 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. On breakdown criteria for nonvacuum Einstein equationsThe 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. A generalized representation formula for systems of tensor wave equationsIn 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. OtherThis section contains some other misellaneous research-related documents. Breakdown Criteria for Nonvacuum Einstein EquationsThis was my PhD 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.
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