Related papers: Controlled regularity at future null infinity from…
We consider a nonlinear Schr\"odinger equation with a bounded local potential in $R^3$. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data are…
Using the Fredholm theory of the linear time-dependent Schr\"odinger equation set up in our previous article arXiv:2201.03140, we solve the final-state problem for the nonlinear Schr\"odinger problem $$ (D_t + \Delta + V) u = N[u], \quad…
The construction of a theory of quantum gravity is an outstanding problem that can benefit from better understanding the laws of nature that are expected to hold in regimes currently inaccessible to experiment. Such fundamental laws can be…
We study semilinear wave equations with Ginzburg-Landau type nonlinearities multiplied by a factor $\epsilon^{-2}$, where $\epsilon>0$ is a small parameter. We prove that for suitable initial data, solutions exhibit energy concentration…
We adapt Luk's analysis of the characteristic initial value problem in General Relativity to the asymptotic characteristic problem for the conformal Einstein field equations to demonstrate the local existence of solutions in a neighbourhood…
We provide a definitive treatment, including sharp decay and the precise late-time asymptotic profile, for generic solutions of linear wave equations with a (singular) inverse-square potential in (3+1)-dimensional Minkowski spacetime. Such…
Expansions of the gravitational field arising from the development of asymptotically Euclidean, time symmetric, conformally flat initial data are calculated in a neighbourhood of spatial and null infinities up to order 6. To this ends a…
The present article considers time symmetric initial data sets for the vacuum Einstein field equations which in a neighbourhood of infinity have the same massless part as that of some static initial data set. It is shown that the solutions…
This article begins with a brief introduction to numerical relativity aimed at readers who have a background in applied mathematics but not necessarily in general relativity. I then introduce and summarise my work on the problem of treating…
This work is concerned with the possibility of proving the boundary null controllability for the degenerate wave equation, developing the asymptotic analysis of a suitable family of state-control pairs $((u_\varepsilon ,…
Systematic numerical investigations of the asymptotics of near Schwarzschild vacuum initial data sets is carried out by inspecting solutions to the parabolic-hyperbolic and to the algebraic-hyperbolic forms of the constraints, respectively.…
In this paper we establish some regularity results concerning the behavior of weak solutions and very weak solutions of the degenerate wave equation near the boundary. For the nondegenerate case, the correponding results were originally…
Using the recent analysis of the output of the low-energy resolvent of Schr\"odinger operators on asymptotically conic manifolds (including Euclidean space) when the potential is short-range, we produce detailed asymptotic expansions for…
We derive new results on radiation, angular momentum at future null infinity and peeling for a general class of spacetimes. For asymptotically-flat solutions of the Einstein vacuum equations with a term homogeneous of degree $-1$ in the…
In this paper, we study the linear wave equations in an asymptotically anti-de Sitter spacetime. We will consider the mixed boundary problem, where the initial data are given on an outgoing null hypersurface and a timelike hypersurface, and…
We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on $\R^d$ with initial data in a scale of weighted Sobolev spaces that allow spatial growth/decay at infinity as $|x|^\beta$ with $\beta<1/2$. In…
In previous work, the numerical solution of the linearized gravitational field equations near space-like and null-infinity was discussed in the form of the spin-2 zero-rest-mass equation for the perturbations of the conformal Weyl…
We study both of the scattering and Cauchy problems for the semilinear wave equation with null quadratic form on the Schwarzschild background. Prescribing the scattering data that are given by the short pulse data on the future null…
I describe the construction of a large class of asymptotically flat initial data with non-vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space-like infinity in terms of…
We consider a family of cylindrical spacetimes endowed with angular momentum that are solutions to the vacuum Einstein equations outside the symmetry axis. This family was recently obtained by performing a complete gauge fixing adapted to…