One-parameter Family Of Initial Data Research Articles

The stability of diverging traveling fronts and threshold phenomenon for the buffered bistable system

This paper is devoted to a bistable system for calcium buffering. We first prove two kinds of stability of diverging traveling fronts of the degenerate system (All buffers do not diffuse): the local C0-norm stability and the asymptotic stability. In particular, as for asymptotic stability, we prove a Liouville-type result by the sliding method, and then use the truncation technique to study the long-time behavior of diverging wave like solution. These stabilities imply that, under suitable conditions on the initial data, the solution locally uniformly approaches to the high equilibrium state (1,b2). Then, by examining the behavior of solutions with one-parameter family of initial data, we show that the parameter-dependent solutions can be divided into three categories: convergence to the basal equilibrium state (0,b0) for small parameter values, convergence to the high equilibrium state (1,b2) for large parameter values, whereas neither of these behaviors occurs for intermediate parameter values. We refer to such phenomenon as threshold phenomenon. These intermediate parameter values are called threshold values, and the corresponding solutions are called threshold solutions. At the end of the paper, we present some important properties of the threshold solution and provide some numerical simulations.

Small deformations of extreme five dimensional Myers–Perry black hole initial data

We demonstrate the existence of a one-parameter family of initial data for the vacuum Einstein equations in five dimensions representing small deformations of the extreme Myers–Perry black hole. This initial data set has ‘\(t-\phi ^i\)’ symmetry and preserves the angular momenta and horizon geometry of the extreme solution. Our proof is based upon an earlier result of Dain and Gabach-Clement concerning the existence of \(U(1)\)-invariant initial data sets which preserve the geometry of extreme Kerr (at least for short times). In addition, we construct a general class of transverse, traceless symmetric rank 2 tensors in these geometries.

Black hole universe: Construction and analysis of initial data

We numerically construct an one-parameter family of initial data of an expanding inhomogeneous universe model which is composed of regularly aligned black holes with an identical mass. They are initial data for vacuum solutions of the Einstein equations. We call this universe model the "black hole universe" and analyze the structure of these initial data. We study the relation between the mean expansion rate of the 3-space, which corresponds to the Hubble parameter, and the mass density of black holes. The result implies that the same relation as that of the Einstein-de Sitter universe is realized in the limit of the large separation between neighboring black holes. The applicability of the cosmological Newtonian $N$-body simulation to the dark matter composed of black holes is also discussed. The deviation of the spatial metric of the cosmological Newtonian $N$-body system from that of the black hole universe is found to be smaller than about 1% in a region distant from the particles, if the separation length between neighboring particles is 20 times larger than their gravitational radius. By contrast, the deviation of the square of the Hubble parameter of the cosmological Newtonian $N$-body system from that of the black hole universe is about 20% for the same separation length.

Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$

This paper is devoted to a class of nonautonomous parabolic equations of the form ut = Δu + f(t, u) on \({\mathbb{R}^N}\) . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric in the sense that all its limit profiles, as t → ∞, are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic equations.

A Gluing Construction Regarding Point Particles in General Relativity

We develop a gluing construction which adds scaled and truncated asymptotically Euclidean solutions of the Einstein constraint equations to compact solutions with potentially non-trivial cosmological constants. The result is a one-parameter family of initial data which has ordinary and scaled "point-particle" limits analogous to those of Gralla and Wald ("A rigorous derivation of gravitational self-force," Class. Quantum Grav. 2008). In particular, we produce examples of initial data which generalize Schwarzschild - de Sitter initial data and gluing theorems of IMP-type.

Critical behavior of collapsing surfaces

We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling that of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the existence of a critical initial surface that develops a degenerate neckpinch. The limiting flow of the type II singularity is accurately modeled by the rotationally symmetric translating soliton.

Boson stars driven to the brink of black hole formation

We present a study of black hole threshold phenomena for a self-gravitating, massive complex scalar field in spherical symmetry. We construct Type I critical solutions dynamically by tuning a one-parameter family of initial data composed of a boson star and a massless real scalar field. The real field is used to perturb the boson star via a gravitational interaction which results in a {\em significant} transfer of energy. The resulting critical solutions, which show great similarity with unstable boson stars, persist for a finite time before dispersing or forming a black hole. We extend the stability analysis of Gleiser and Watkins [Nucl. Phys. B319, 733 (1989)], providing a method for calculating the radial dependence of boson star modes of nonzero frequency. We find good agreement between our critical solutions and boson star modes. For critical solutions less than 90% of the maximum boson star mass $M_{\rm max} \simeq 0.633 M_{Pl}^2/m$, a small halo of matter appears in the tail of the solution. This halo appears to be an artifact of the collision between the original boson star and the real field, and does not belong to the true critical solution. It seems that unstable boson stars are unstable to dispersal in addition to black hole formation. Given the similarity in macroscopic stability between boson and neutron stars, we suggest that neutron stars at or beyond the point of instability may also be unstable to explosion.

Critical behavior and scaling in vacuum axisymmetric gravitational collapse.

We report a second example of critical behavior in gravitational collapse. Collapse of axisymmetric gravitational wave packets is computed numerically for a one-parameter family of initial data. A black hole first appears along the sequence at a critical parameter value ${\mathit{p}}^{\mathrm{*}}$. As with spherical scalar-field collapse, a power law is found to relate black-hole mass (the order parameter) and critical separation: ${\mathit{M}}_{\mathrm{BH}}$\ensuremath{\propto}\ensuremath{\Vert}p-${\mathit{p}}^{\mathrm{*}}$${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{\beta}}}$. The critical exponent is \ensuremath{\beta}\ensuremath{\simeq}0.37, remarkably close to that observed by Choptuik. Near-critical evolutions produce echoes from the strong-field region which appear to exhibit scaling.