Related papers: Rough solution for the Einstein Vacuum equations
We derive an asymptotic solution of the vacuum Einstein equations that describes the propagation and diffraction of a localized, large-amplitude, rapidly-varying gravitational wave. We compare and contrast the resulting theory of strongly…
This work is a continuation of [E. Bonnetier, D.Bresch, V. Milisic, submitted]; it deals with rough boundaries in the simplified context of a Poisson equation. We impose Dirichlet boundary conditions on the periodic microscopic perturbation…
Exact solutions of Einstein's vacuum equations are considered which describe gravitational waves with distinct wavefronts. A family of such solutions presented recently in which the wavefronts have various geometries and which propagate…
We introduce some analytic relations on the set of partial differential equations of two variables. It relies on a new comparison method to give rough asymptotic estimates for solutions which obey different partial differential equations.…
We study the resolution of discontinuous singularities in gas dynamics via multi-dimensional rarefaction waves. While the mechanism is well-understood in one spatial dimension, the rigorous construction in higher dimensions has remained a…
We introduce a new method for numerically evolving the full Einstein field equations in situations where the spacetime is dominated by a known background solution. The technique leverages the knowledge of the background solution to subtract…
This paper addresses the inverse problem of simultaneously recovering multiple unknown parameters for semilinear wave equations from boundary measurements. We consider an initial-boundary value problem for a wave equation with a general…
The dynamics of gravitational waves is investigated in full 3+1 dimensional numerical relativity, emphasizing the difficulties that one might encounter in numerical evolutions, particularly those arising from non-linearities and gauge…
The goal of this monograph is to prove that any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension, with periodic, even in space, initial data of small size $\epsilon$, is almost…
We present a numerical method to solve the equations of general relativistic hydrodynamics in a given external gravitational field. The method is based on a generalization of Roe's approximate Riemann solver for the non relativistic Euler…
We construct new static, spherically symmetric non-extremal black hole solutions of four-dimensional ${\cal N}=2$ supergravity, using a systematic technique based on dimensional reduction over time (the c-map) and the real formulation of…
We present a new formulation of the Einstein equations that casts them in an explicitly first order, flux-conservative, hyperbolic form. We show that this now can be done for a wide class of time slicing conditions, including maximal…
HyperCR Einstein--Weyl equations in 2+1 dimensions reduce to a pair of quasi-linear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can be in principle constructed from a twistor correspondence, thus establishing the…
We find new classes of exact solutions to the Einstein-Maxwell system of equations for a charged sphere with a particular choice of the electric field intensity and one of the gravitational potentials. The condition of pressure isotropy is…
The linear Einstein-Boltzmann equations describe the evolution of perturbations in the universe and its numerical solutions play a central role in cosmology. We revisit this system of differential equations and present a detailed…
The Einstein evolution equations are studied in a gauge given by a combination of the constant mean curvature and spatial harmonic coordinate conditions. This leads to a coupled quasilinear elliptic--hyperbolic system of evolution…
We establish an optimal \emph{Widder theory} for a weighted porous medium equation with rough and inhomogeneous density that may be singular at a point and tends to zero at spatial infinity. Specifically, for this equation, we identify a…
In a previous work [I. Rodnianski and Y. Shlapentokh-Rothman, Naked Singularities for the Einstein Vacuum Equations: The Exterior Solution, arXiv:1912.08478] we constructed solutions to the Einstein vacuum equations in 3+1 dimensions which…
We show how to apply ideas from the theory of rough paths to the analysis of low-regularity solutions to non-linear dispersive equations. Our basic example will be the one dimensional Korteweg--de Vries (KdV) equation on a periodic domain…
The fundamental quasinormal modes of black holes in higher-derivative gravity given by the Einstein-Weyl action are known to be moderately corrected by the Weyl term. Here we will show that the first several overtones are highly sensitive…