Asymptotics of Solutions of the Einstein Equations with Positive Cosmological Constant

A positive cosmological constant simplifies the asymptotics of forever expanding cosmological solutions of the Einstein equations. In this paper a general mathematical analysis on the level of formal power series is carried out for vacuum spacetimes of any dimension and perfect fluid spacetimes with linear equation of state in spacetime dimension four. For equations of state stiffer than radiation evidence for development of large gradients, analogous to spikes in Gowdy spacetimes, is found. It is shown that any vacuum solution satisfying minimal asymptotic conditions has a full asymptotic expansion given by the formal series. In four spacetime dimensions, and for spatially homogeneous spacetimes of any dimension, these minimal conditions can be derived for appropriate initial data. Using Fuchsian methods the existence of vacuum spacetimes with the given formal asymptotics depending on the maximal number of free functions is shown without symmetry assumptions.Communicated by Sergiu Klainermansubmitted 08/12/03, accepted 30/04/04     

Periodic Solutions of Non-linear Anisotropic Partial Differential Equations

Periodic solutions of arbitrary period to semilinear partial differential equations of Zabusky or Boussinesq type are obtained. More generally, for a linear differential operator A(y,∂), the equation A(y,∂)u = (−1)^∣γ∣∂^γf(y,∂^γu), y = (t,x)∈ℝ^k×G is studied, where homogeneous boundary conditions on ∂G and periodicity conditions on t are imposed. The solutions are obtained by variational methods in anisotropic Sobolev spaces.     

Asymptotic Gluing of Asymptotically Hyperbolic Solutions to the Einstein Constraint Equations

We show that asymptotically hyperbolic solutions of the Einstein constraint equations with constant mean curvature can be glued in such a way that their asymptotic regions are connected.     

On Localization of Solutions of Elliptic Equations with Nonhomogeneous Anisotropic Degeneracy

The work deals with the Dirichlet problem for elliptic equations with nonhomogeneous anisotropic degeneracy in a possibly unbounded domain of multidimensional Euclidean space. The existence of weak solutions is proved. Some conditions are established connecting the character of nonlinearity of the equation and the geometric characteristics of the domain which guarantee the one-dimensional localization (vanishing) of weak solutions. The equation with anisotropic degeneracy is shown to admit localized solutions even in the absence of absorption.     

Einstein Equations on a 5-Manifold with a Causal Structure in the Absence of Matter Fields

A 5-manifold with a restricted smooth structure and an appropriate group of coordinate transformations including general relativity and gauge transformations is considered. An explicit expression for the Riemannian curvature of a 4-dimensional vector distribution is obtained, which implies the classical Einstein and Maxwell equations. Bibliography: 14 titles.     

Plane Wave Solutions to the Equations of Electrodynamics in an Anisotropic Medium

Siberian Mathematical Journal - Under examination is the system of equations of electrodynamics for a nonconducting nonmagnetic medium with the simplest anisotropy of permittivity. We assume that...     

On the Regularity of Solutions to the 2D Boussinesq Equations Satisfying Type I Conditions

Journal of Nonlinear Science - We prove continuation in time of the local smooth solutions satisfying various Type I conditions for the 2D inviscid Boussinesq equations.     

一类交替型脉冲微分系统的解

主要讨论一类超前型与滞后型交替的脉冲微分系统.首先给出具常系数的脉冲微分系统解存在的充分条件以及解唯一的表达形式;对于变系数的微分系统也作了相应的讨论.     

Multiple Solutions for p-Laplacian Type Equations

We establish the existence and multiplicity of weak solutions for equations which involve a uniformly convex elliptic operator in divergence form（in particular, a p-Laplacian operator）, while the nonlinearity has a （p- 1）-superlinear growth at infinity. Our result completes and extends the relevant results of recent papers. Tile argument in the proof of our main result relies on the Z2-symmetric version of mountain pass lemma.     

Dynamics Equations for an Anisotropic Plastic Subsystem of a Fractal Medium

Model equations for the dynamics of an anisotropic plastic subsystem of a fractal medium are obtained in explicit form using fractional calculus. The slow dynamics of a number of parameters are analyzed and the results are represented graphically.     

Remark on the Lifespan of Solutions to 3-D Anisotropic Navier Stokes Equations

The goal of this article is to provide a lower bound for the lifespan of smooth solutions to 3-D anisotropic incompressible Navier-Stokes system,which in particular extends a similar type of result for the classical 3-D incompressible Navier-S tokes system.     

一类具多个偏差变元混合型p-Laplacian周期解(英文)

Based on Mansevich-Mawhin continuation theorem and some analysis skill,some sufficient conditions for the existence of periodic solutions for mixed type p-Laplacian equation with deviating arguments are established,which are complement of previously known results.     

Very Weak Solutions of p-Laplacian Type Equations with VMO Coefficients

In this note we obtain a new a priori estimate for the very weak solutions of p-Laplacian type equations with VMO coefficients when p is close to 2, and then prove that the very weak solutions of such equations are the usual weak solutions. Our approath is based on the Hodge decomposition and the L^p-estimate for the corresponding linear equations. And this also provides a simpler proof for the results in [1].     

Dynamics of Solutions and Approximation for Partial Functional Differential Equations with Delay

This work aims to investigate the existence and uniqueness of almost periodic solution for partial functional differential equations with delay. Here, we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille–Yosida condition, the delayed parts are assumed to be almost periodic with respect to the first argument and Lipschitz continuous with respect to the second argument. Using the exponential dichotomy and the contraction mapping principle, some sufficient conditions are obtained for the existence and uniqueness of almost periodic solution.     

Future Non-Linear Stability for Reflection Symmetric Solutions of the Einstein–Vlasov System of Bianchi Types II and VI0

Using the methods developed for the Bianchi I case we have shown that a boostrap argument is also suitable to treat the future non-linear stability for reflection symmetric solutions of the Einstein–Vlasov system of Bianchi types II and VI₀. These solutions are asymptotic to the Collins–Stewart solution with dust and the Ellis–MacCallum solution, respectively. We have thus generalized the results obtained by Rendall and Uggla in the case of locally rotationally symmetric Bianchi II spacetimes to the reflection symmetric case. However, we needed to assume small data. For Bianchi VI₀ there is no analogous previous result.     

Elliptic-Hyperbolic Systems and the Einstein Equations

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 quasi-linear elliptic-hyperbolic system of evolution equations. We prove that the Cauchy problem is locally strongly well posed and that a continuation principle holds.¶For initial data satisfying the Einstein constraint and gauge conditions, the solutions to the elliptic-hyperbolic system defined by the gauge fixed Einstein evolution equations are shown to give vacuum space-times.