Type O pure radiation metrics with a cosmological constant

In this paper we complete the integration of the conformally flat pure radiation spacetimes with a non-zero cosmological constant Λ, and , by considering the case . This is a further demonstration of the power and suitability of the generalised invariant formalism (GIF) for spacetimes where only one null direction is picked out by the Riemann tensor. For these spacetimes, the GIF picks out a second null direction (from the second derivative of the Riemann tensor) and once this spinor has been identified the calculations are transferred to the simpler GHP formalism, where the tetrad and metric are determined. The whole class of conformally flat pure radiation spacetimes with a non-zero cosmological constant (those found in this paper, together with those found earlier for the case ) have a rich variety of subclasses with zero, one, two, three, four or five Killing vectors.     

Stationarity and large ω Brans–Dicke solutions versus general relativity

It is often claimed that the asymptotic behaviour of the Brans–Dicke solutions versus general relativity, when , is related to the trace of the stress tensor. Considering the standard Euclidean cosmological model, we argue that this claim is not correct. On the other hand, we argue that this behaviour depends on the property of the considered solutions versus stationarity and asymptotical flatness.     

The Relation of Generalized Scalar-Tensor Theory with the 5D Space-Time-Matter Theory

The relationship between cosmological solutionsof five-dimensional Space-Time-Matter (STM) theory anda Generalized Scalar-Tensor (GST) theory is investigatedin which the cosmological term Lambda depends not only on a scalar field but also onits time derivative .Identification of these solutions allows us to solve forthe functional form of the cosmological term, and mayhave relevance for the early Universe.     

Inhomogeneous dark energy and cosmological acceleration

We study the evolution of an inhomogeneous fluid with self-similarity of the second kind and anisotropic pressure. We found a class of solution to the Einstein field equations by assuming an equation of state where the radial pressure of the fluid is proportional to its energy density () and that the fluid moves along time-like geodesics. The equation of state combined with the self-similarity of second kind implies ω = −1. The energy conditions, geometrical and physical properties of the solutions are studied. We have found that, for the self-similar parameter , the solution represents an accelerated cosmological model ending in a Big Rip stage.     

Power Series for Solutions of the 3 {\mathcal D}-Navier-Stokes System on R 3

In this paper we study the Fourier transform of the -Navier-Stokes System without external forcing on the whole space R ³. The properties of solutions depend very much on the space in which the system is considered. In this paper we deal with the space of functions where and c (k) is bounded, . We construct the power series which converges for small t and gives solutions of the system for bounded intervals of time. These solutions can be estimated at infinity (in k-space) by .     

Conformal properties of nonpeeling vacuum space-times

In a previous paper we investigated a class ofnonpeeling asymptotic vacuum solutions which were shown to admit finite expressions for the Winicour-Tamburino energy-momentum and angular momentum integrals. These solutions have the property that $$\psi _0 = O(r^{ - 3 - \in _0 } ), \in _0 \leqslant 2$$ and $$\psi _1 = O(r^{ - 3 - \in _1 } ), \in _1< \in _0 and \in _1< 1$$ withψ ₂,ψ ₃, andψ ₄ having the same asymptotic behavior as they do for peeling solutions. The above investigation was carried out in the physical space-time. In this paper we examine the conformal properties of these solutions, as well as the more general Couch-Torrence solutions, which include them as a subclass. For the Couch-Torrence solutions $$\begin{gathered} \psi _0 = O(r^{ - 2 - \in _0 } ) \hfill \\ \psi _1 = O(r^{ - 2 - \in _1 } ), \in _1< \in _0 {\text{ }}and \in _1 \leqslant 2 \hfill \\ \end{gathered} $$ and , $$\psi _2 = O(r^{ - 2 - \in _2 } ),{\text{ }} \in _2< \in _1 {\text{ }}and \in _2 \leqslant 1$$ withψ ₃ andψ ₄ behaving as they do for peeling solutions. It is our purpose to determine how much of the structure generally associated with peeling space-times is preserved by the nonpeeling solutions. We find that, in general, a three-dimensional null boundary (?⁺) can be defined and that the BMS group remains the asymptotic symmetry group. For the general Couch-Torrence solutions several physically and/or geometrically interesting quantities     

A Criterion for Uniqueness of Lagrangian Trajectories for Weak Solutions of the 3D Navier-Stokes Equations

Foias, Guillopé, & Temam showed in 1985 that for a given weak solution of the three-dimensional Navier-Stokes equations on a domain Ω, one can define a ‘trajectory mapping’ that gives a consistent choice of trajectory through each initial condition , and that respects the volume-preserving property one would expect for smooth flows. The uniqueness of this mapping is guaranteed by the theory of renormalised solutions of non-smooth ODEs due to DiPerna & Lions. However, this is a distinct question from the uniqueness of individual particle trajectories. We show here that if one assumes a little more regularity for u than is known to be the case, namely that , then the particle trajectories are unique and C ¹ in time for almost every choice of initial condition in Ω. This degree of regularity is more than can currently be guaranteed for weak solutions () but significantly less than that known to ensure that u is regular ( . We rely heavily on partial regularity results due to Caffarelli, Kohn, & Nirenberg and Ladyzhenskaya & Seregin.     

Twistor Actions for Self-Dual Supergravities

We give holomorphic Chern-Simons-like action functionals on supertwistor space for self-dual supergravity theories in four dimensions, dealing with supersymmetries, the cases where different parts of the R-symmetry are gauged, and with or without a cosmological constant. The gauge group is formally the group of holomorphic Poisson transformations of supertwistor space where the form of the Poisson structure determines the amount of R-symmetry gauged and the value of the cosmological constant. We give a formulation in terms of a finite deformation of an integrable -operator on a supertwistor space, i.e., on regions in . For , we also give a formulation that does not require the choice of a background.     

On the Conserved Quantities for the Weak Solutions of the Euler Equations and the Quasi-geostrophic Equations

In this paper we obtain sufficient conditions on the regularity of the weak solutions to guarantee conservation of the energy and the helicity for the incompressible Euler equations. The regularity of the weak solutions are measured in terms of the Triebel-Lizorkin type of norms, and the Besov norms, . In particular, in the Besov space case, our results refine the previous ones due to Constantin-E-Titi (energy) and the author of this paper (helicity), where the regularity is measured by a special class of the Besov space norm , which is the Nikolskii space. We also obtain a sufficient regularity condition for the conservation of the L ^p -norm of the temperature function in the weak solutions of the quasi-geostrophic equation.     

Remarks on the Balance Relations for the Two-Dimensional Navier–Stokes Equation with Random Forcing

We use the balance relations for the stationary in time solutions of the randomly forced 2D Navier-Stokes equations, found in [10], to study these solutions further. We show that the vorticity ξ(t,x) of a stationary solution has a finite exponential moment, and that for any the expectation of the integral of over the level-set , up to a constant factor equals the expectation of the integral of over the same set.     

A Counterexample to Dispersive Estimates for Schrödinger Operators in Higher Dimensions

In dimension n > 3 we show the existence of a compactly supported potential in the differentiability class , for which the solutions to the linear Schrödinger equation in,

Stability of Two Soliton Collision for Nonintegrable gKdV Equations

We continue our study of the collision of two solitons for the subcritical generalized KdV equations

Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzschild Space

We provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzschild background. Our estimate implies that such solutions have asymptotic behavior as long as the source term is bounded in the norm . In particular this gives scattering at small amplitudes for non-linear scalar fields of the form for all 2 < p. This paper is dedicated to the memory of Hope Machedon The second author would like thank MSRI and Princeton University, where a portion of this research was conducted during the Fall of 2005. The second author was also supported by a NSF postdoctoral fellowship.     

Bianchi type-I,V and VIo models in modified generalized scalar-tensor theory

In modified generalized scalar-tensor (GST) theory, the cosmological term Λ is a function of the scalar field ϕ and its derivatives . We obtain exact solutions of the field equations in Bianchi Type-I, V and VIo space-times. The evolution of the scale factor, the scalar field and the cosmological term has been discussed. The Bianchi Type-I model has been discussed in detail. Further, Bianchi Type-V and VIo models can be studied on the lines similar to Bianchi Type-I model.      

Local Resolvent Estimates for <Emphasis Type="Italic">N</Emphasis>-body Stark Hamiltonians

For an N-body Stark Hamiltonian , the resolvent estimate holds uniformly in with Re and Im , where , and is a compact interval. This estimate is well known as the limiting absorption principle. In this paper, we report that by introducing the localization in the configuration space, a refined resolvent estimate holds uniformly in with Re and Im . Dedicated to Professor Hideo Tamura on the occasion of his 60th birthday     

Transition from decelerated to accelerated cosmic expansion in braneworld universes

Braneworld theory provides a natural setting to treat, at a classical level, the cosmological effects of vacuum energy. Non-static extra dimensions can generally lead to a variable vacuum energy, which in turn may explain the present accelerated cosmic expansion. We concentrate our attention in models where the vacuum energy decreases as an inverse power law of the scale factor. These models agree with the observed accelerating universe, while fitting simultaneously the observational data for the density and deceleration parameter. The redshift at which the vacuum energy can start to dominate depends on the mass density of ordinary matter. For _m = 0.3, the transition from decelerated to accelerated cosmic expansion occurs at z _T ≈ 0.48 ± 0.20, which is compatible with SNe data. We set a lower bound on the deceleration parameter today, namely > − 1 + 3 _m /2, i.e., > − 0.55 for _m = 0.3. The future evolution of the universe crucially depends on the time when vacuum starts to dominate over ordinary matter. If it dominates only recently, at an epoch z < 0.64, then the universe is accelerating today and will continue that way forever. If vacuum dominates earlier, at z > 0.64, then the deceleration comes back and the universe recollapses at some point in the distant future. In the first case, quintessence and Cardassian expansion can be formally interpreted as the low energy limit of our model, although they are entirely different in philosophy. In the second case there is no correspondence between these models and ours.     

Nonexistence of Self-Similar Singularities for the 3D Incompressible Euler Equations

We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations if the vorticity decays sufficiently fast near infinity in . By a similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in . This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions. The work was supported partially by the KOSEF Grant no. R01-2005-000-10077-0, and KRF Grant (MOEHRD, Basic Research Promotion Fund).     

Dilatonic Dark Energy Model with Late Time de Sitter Attractor

Based on Weyl-scaled induced gravitational theory, we regard dilaton field in this theory as a candidate of dark energy. We construct a dilatonic dark energy model and its phantom model, that admit late time de Sitter attractor solution. When we take the potential of dilaton field as the form which has been studied in supergravity model and the famous Mexican hat potential , we show mathematically that these attractor solutions correspond to an equation of state ω = −1 and a cosmic density parameter Ω_σ = 1, which are important features for a dark energy model that can meet the current observations.