Stellar Collapse in the Time Dependent Hartree-Fock Approximation

We prove blow-up in finite time for radially symmetric solutions to the pseudo-relativistic Hartree-Fock equation with negative energy. The non-linear Hartree-Fock equation is commonly used in the physics literature to describe the dynamics of white dwarfs. We extend thereby recent results by Fröhlich and Lenzmann, who established in [3,4] blow-up for solutions to the pseudo-relativistic Hartree equation. As key ingredient for handling the exchange term we use the conservation of the expectation of the square of the angular momentum operator.     

Nonlinear Dynamical Stability of Newtonian Rotating and Non-rotating White Dwarfs and Rotating Supermassive Stars

We prove general nonlinear stability and existence theorems for rotating star solutions which are axi-symmetric steady- state solutions of the compressible isentropic Euler-Poisson equations in 3 spatial dimensions. We apply our results to rotating white dwarf and high density supermassive (extreme relativistic) stars, stars which are in convective equilibrium and have uniform chemical composition. Also, we prove nonlinear dynamical stability of non-rotating white dwarfs with general perturbation without any symmetry restrictions. This paper is a continuation of our earlier work ([26]).     

Study of bifurcations in a model Hartree-Fock calculation

The Hartree-Fock procedure is used to study the behaviour of the ground state of a system ofM spinless electrons distributed overN equivalent and equidistant sites (M ⩽N) as a function of the strength of the mutual repulsion between the electrons. Below a critical strength, all initial configurations are seen, after repeated iterations, to converge to a unique solution. Above this critical strength, in addition to the initial configurations which lead to a unique solution, there exist configurations which on repeated iterations give rise to stable two-period solutions. Although the number of independent stable two-period solutions depends on the coupling strength, for no value of the coupling are stable solutions of periodicity higher than two seen.     

Re-evaluation of the ^23Mg（p,γ） ^24Al reaction rate

Based on a new screening Coulomb model, this paper discusses the effect of electron screening on proton capture reaction of 23Mg. The derived result shows that, in some considerable range of stellar temperatures, the effect of electron screening on resonant reaction is prominent; on the non-resonant reaction the effect is obvious only in the low stellar temperatures. The reaction rates of ^23Mg（p,γ） ^24Al would increase 15%-25% due to the fact that the electron screening are considered in typical temperature range of massive mass white dwarfs, and the results undoubtedly affect the nucleosynthesis of some heavier nuclei in massive mass white dwarfs.     

Blow-up,Zero <Emphasis Type="Italic">α</Emphasis> Limit and the Liouville Type Theorem for the Euler-Poincaré Equations

In this paper we study the Euler-Poincaré equations in . We prove local existence of weak solutions in , and local existence of unique classical solutions in , k > N/2 + 3, as well as a blow-up criterion. For the zero dispersion equation (α = 0) we prove a finite time blow-up of the classical solution. We also prove that as the dispersion parameter vanishes, the weak solution converges to a solution of the zero dispersion equation with sharp rate as α → 0, provided that the limiting solution belongs to with k > N/2 + 3. For the stationary weak solutions of the Euler-Poincaré equations we prove a Liouville type theorem. Namely, for α > 0 any weak solution is u=0; for α= 0 any weak solution is u=0.     

Finite-Time Blow-up of Solutions of an Aggregation Equation in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We consider the aggregation equation in R ⁿ , n ≥ 2, where K is a rotationally symmetric, nonnegative decaying kernel with a Lipschitz point at the origin, e.g. K(x) = e ^−|x|. We prove finite-time blow-up of solutions from specific smooth initial data, for which the problem is known to have short time existence of smooth solutions.     

The astrophysics of cool white dwarfs

Old, cool white dwarfs convey valuable information about the early history of our Galaxy. They have been used to determine the age of the Galactic disk, several open clusters and a globular cluster. We review the current understanding of the physics of cool white dwarfs, including their mass distribution, chemical evolution, and cooling. We also examine the role of white dwarfs as tracers of various stellar populations, both in terms of observational searches and theoretical models.     

Hartree-fock equations for atoms with two open shells

We have formulated the Hartree-Fock equations for multielectron systems with two open shells (the Huzinaga method) in terms of the density matrix in the LCAO approximation. In order to solve the Hartree-Fock equations, in the algebraic approximation we have obtained expressions for the derivatives of the energy with respect to the density matrix elements and the nonlinear atomic orbital parameters (the orbital exponents). We discuss the question of calculating the open shell parameters (the vector coupling coefficients) in the configurations s¹p^N and s¹d^N within the Huzinaga method. We have calculated the energy for a series of atoms with two open shells in these configurations. Using rather narrow basis sets of Slater-type atomic orbitals), we have obtained energy values close to the results of a numerical solution of the Hartree-Fock equations with sufficiently high accuracy of the virial ratio. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 74, No. 2, pp. 145–152, March–April, 2007     

Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions

We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct “peakon” and “multi-peakon” solutions for all λ ≠ 0, 1, and “shock-peakons” for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold’s approach to Euler equations on Lie groups.     

The stability of relativistic stars and the role of the adiabatic index

We study the stability of three analytical solutions of the Einstein’s field equations for spheres of fluid. These solutions are suitable to describe compact objects including white dwarfs, neutron stars and supermassive stars and they have been extensively employed in the literature. We re-examine the range of stability of the Tolman VII solution, we focus on the stability of the Buchdahl solution which is under contradiction in the literature and we examine the stability of the Nariai IV solution. We found that all the mentioned solutions are stable in an extensive range of the compactness parameter. We also concentrate on the effect of the adiabatic index on the instability condition. We found that the critical adiabatic index, depends linearly on the ratio of central pressure over central energy density \(P_c/{\mathcal{E}}_c\), up to high values of the compactness. Finally, we examine the possibility to impose constraints, via the adiabatic index, on realistic equations of state in order to ensure stable configurations of compact objects.     

General relativistic effects in the structure of massive white dwarfs

In this work we investigate the structure of white dwarfs using the Tolman–Oppenheimer–Volkoff equations and compare our results with those obtained from Newtonian equations of gravitation in order to put in evidence the importance of general relativity (GR) for the structure of such stars. We consider in this work for the matter inside white dwarfs two equations of state, frequently found in the literature, namely, the Chandrasekhar and Salpeter equations of state. We find that using Newtonian equilibrium equations, the radii of massive white dwarfs (\(M>1.3M_{\odot }\)) are overestimated in comparison with GR outcomes. For a mass of \(1.415M_{\odot }\) the white dwarf radius predicted by GR is about 33% smaller than the Newtonian one. Hence, in this case, for the surface gravity the difference between the general relativistic and Newtonian outcomes is about 65%. We depict the general relativistic mass–radius diagrams as \(M/M_{\odot }=R/(a+bR+cR^2+dR^3+kR^4)\), where a, b, c and d are parameters obtained from a fitting procedure of the numerical results and \(k=(2.08\times 10^{-6}R_{\odot })^{-1}\), being \(R_{\odot }\) the radius of the Sun in km. Lastly, we point out that GR plays an important role to determine any physical quantity that depends, simultaneously, on the mass and radius of massive white dwarfs.     

The origin of the strongest magnetic fields in dwarfs

White dwarfs have frozen in magnetic fields ranging from below the measurable limit of about 3×10³ to 10⁹ G. White dwarfs with surface magnetic fields in excess of 1 MG are found as isolated single stars and relatively more often in magnetic cataclysmic variables. Some 1253 white dwarfs with a detached low-mass main-sequence companion have been identified in the Sloan Digital Sky Survey (SDSS) but none of these shows sufficient evidence for Zeeman splitting of hydrogen lines for a magnetic field in excess of 1 MG. If such high magnetic fields in white dwarfs result from the isolated evolution of a single star then there should be the same fraction of high field white dwarfs among this SDSS binary sample as among single stars. Thus, we deduce that the origin of such high magnetic fields must be intimately tied to the formation of cataclysmic variables (CVs). The formation of a CV must involve orbital shrinkage from giant star to main-sequence star dimensions. It is believed that this shrinkage occurs as the low-mass companion and the white dwarf spiral together inside a common envelope. CVs emerge as very close but detached binary stars that are then brought together by magnetic braking or gravitational radiation. We propose that the smaller the orbital separation at the end of the common envelope phase, the stronger the magnetic field. The magnetic cataclysmic variables (MCVs) originate from those common envelope systems that almost merge. Those common envelope systems that do merge are the progenitors of the single high field white dwarfs. Thus all highly magnetic white dwarfs, be they single stars or the components of MCVs, have a binary origin. This accounts for the relative dearth of single white dwarfs with fields of 10⁴–10⁶ G. Such intermediate-field white dwarfs are found preferentially in cataclysmic variables. The bias towards higher masses for highly magnetic white dwarfs is expected if a fraction of these form when two degenerate cores merge in a common envelope. From the space density of single highly magnetic white dwarfs we estimate that about three times as many common envelope events lead to a merged core as to a cataclysmic variable.     

Anisotropic relativistic stellar models

We present a class of exact solutions of Einstein's gravitational field equations describing spherically symmetric and static anisotropic stellar type configurations. The solutions are obtained by assuming a particular form of the anisotropy factor. The energy density and both radial and tangential pressures are finite and positive inside the anisotropic star. Numerical results show that the basic physical parameters (mass and radius) of the model can describe realistic astrophysical objects like neutron stars.     

On Depletion of the Vortex-Stretching Term in the 3D Navier-Stokes Equations

Certain new cancellation properties in the vortex-stretching term are detected leading to new geometric criteria for preventing finite-time blow-up in the 3D Navier-Stokes equations.     

Probing white dwarf interiors with LISA: periastron precession in eccentric double white dwarfs

In globular clusters, dynamical interactions give rise to a population of eccentric double white dwarfs detectable by the Laser Interferometer Space Antenna (LISA) up to the Large Magellanic Cloud. In this Letter, we explore the detectability of periastron precession in these systems with LISA. Unlike previous investigations, we consider contributions due to tidal and rotational distortions of the binary components in addition to general relativistic contributions to the periastron precession. At orbital frequencies above a few mHz, we find that tides and stellar rotation dominate, opening up a possibly unique window to the study of the interior and structure of white dwarfs.     

The Beale-Kato-Majda Criterion for the 3D Magneto-Hydrodynamics Equations

We study the blow-up criterion of smooth solutions to the 3D MHD equations. By means of the Littlewood-Paley decomposition, we prove a Beale-Kato-Majda type blow-up criterion of smooth solutions via the vorticity of velocity only, namely

Modeling Type Ia supernova explosions

Despite their astrophysical significance-as a major contributor to cosmic nucleosynthesis and as distance indicators in observational cosmology-Type Ia supernovae lack theoretical explanation. Not only is the explosion mechanism complex due to the interaction of (potentially turbulent) hydrodynamics and nuclear reactions, but even the initial conditions for the explosion are unknown. Various progenitor scenarios have been proposed. After summarizing some general aspects of Type Ia supernova modeling, recent simulations of our group are discussed. With a sequence of modeling starting (in some cases) from the progenitor evolution and following the explosion hydrodynamics and nucleosynthesis we connect to the formation of the observables through radiation transport in the ejecta cloud. This allows us to analyze several models and to compare their outcomes with observations. While pure deflagrations of Chandrasekhar-mass white dwarfs and violent mergers of two white dwarfs lead to peculiar events (that may, however, find their correspondence in the observed sample of SNe Ia), only delayed detonations in Chandrasekhar-mass white dwarfs or sub-Chandrasekhar-mass explosions remain promising candidates for explaining normal Type Ia supernovae.     

Remarks on the Blow-up of the Euler Equations and the Related Equations

We consider the Euler system for inviscid incompressible fluid flows, and its perturbations in ⁿ, n2. We prove global well-posedness of this perturbed Euler system in the Triebel-Lizorkin spaces for initial vorticity which is small in the critical Besov norms. Comparison type theorems about the blow-up of solutions are proved between the Euler system and its perturbations. We also study the possiblity of the self-similar type of blow-up of solutions to the equations.     

GLOBAL EXISTENCE AND BLOW-UP PHENOMENA FOR THE PERIODIC HUNTER–SAXTON EQUATION WITH WEAK DISSIPATION

In this paper, we study the periodic Hunter–Saxton equation with weak dissipation. We first establish the local existence of strong solutions, blow-up scenario and blow-up criteria of the equation. Then, we investigate the blow-up rate for the blowing-up solutions to the equation. Finally, we prove that the equation has global solutions.     

Stability of solitary waves in nonlinear composite media

The system of equations for planar waves in elastic composite media in the presence of anisotropy is considered. In anisotropic case two two-parametric families of solitary waves are found in an explicit form. In case of the absence of anisotropy these two families coalesce into the unique three parametric family. The solitary wave solutions are found to be orbitally stable in a certain range of their phase speeds (range of stability) both in an anisotropic as well as in an isotropic materials. It is also shown that the initial value problem for the governing equations is locally well posed which is needed to prove the stability result. The local well-posedness of the initial value problem along with stability of solitary waves implies global existence result provided the initial data lie in a neighbourhood of a stable solitary wave. This complements the previous results of blow-up for this type of equations.