On the well-posedness of the vacuum Einstein��s equations

The Cauchy problem of the vacuum Einstein’s equations aims to find a semi-metric g _αβ of a spacetime with vanishing Ricci curvature R _α,β and prescribed initial data. Under the harmonic gauge condition, the equations R _α,β  = 0 are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein’s equations are a proper Riemannian metric h _ab and a second fundamental form K _ab . A necessary condition for the reduced Einstein equation to satisfy the vacuum equations is that the initial data satisfy Einstein constraint equations. Hence the data (h _ab , K _ab ) cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The goal of the present article is to resolve this incompatibility and to show that under the harmonic gauge the vacuum Einstein equations are well-posed in one type of Sobolev spaces.     

Complete description of invariant Einstein metrics on the generalized flag manifold SO(2n)/U(p)?× U(n ? p)

We find the precise number of non-K?hler SO(2n)-invariant Einstein metrics on the generalized flag manifold M = SO(2n)/U(p)×U(n−p) with n ≥ 4 and 2 ≤ p ≤ n−2. We use an analysis on parametric systems of polynomial equations and we give some insight towards the study of such systems. We also examine the isometric problem for these Einstein metrics.     

Non-relativistic global limits of entropy solutions to the extremely relativistic Euler equations

We are concerned in this paper with the non-relativistic global limits of the entropy solutions to the Cauchy problem of 3 × 3 system of relativistic Euler equations modeling the conservation of baryon numbers, momentum, and energy respectively. Based on the detailed geometric properties of nonlinear wave curves in the phase space and the Glimm’s method, we obtain, for the isothermal flow, the convergence of the entropy solutions to the solutions of the corresponding classical non-relativistic Euler equations as the speed of light c → +∞.     

On global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities

We consider the Cauchy problem for the equations of selfgravitating motions of a barotropic gas with density-dependent viscosities μ(ρ), and λ(ρ) satisfying the Bresch–Desjardins condition, when the pressure P(ρ) is not necessarily a monotone function of the density. We prove that this problem admits a global weak solution provided that the adiabatic exponent γ associated with P(ρ) satisfies ${\gamma > \frac{4}{3}}${\gamma > \frac{4}{3}}.     

On the Cauchy problem for nonlinear parabolic equations with variable density

We investigate the well-posedness of the Cauchy problem for a class of nonlinear parabolic equations with variable density. Sufficient conditions for uniqueness or nonuniqueness in L ^∞(IR ^N  × (0, T)) (N ≥ 3) are established in dependence of the behavior of the density at infinity. We deal with conditions at infinity of Dirichlet type, and possibly inhomogeneous.     

On the Riemann matrices of the hyperbolic system dual to the system of equations of one-dimensional gas dynamics

We consider the Cauchy problem for the quasilinear hyperbolic system describing a one-dimensional flow of a gas with the equation of state p = p(ϱ), p′(ϱ) > 0, and with initial data satisfying a monotonicity condition. We suggest an approach to solving it by reduction to the Cauchy problem for the linear hyperbolic system obtained from the original system by the hodograph transformation. These constructions are extended to a system of elasticity equations describing nonlinear vibrations of a one-dimensional medium. The main result is illustrated by two examples.     

Stability properties for solution operators

We study stability properties of certain evolution equations including the fractional Cauchy problem. Under some spectral assumptions these equations are governed either by a resolvent or a regularized resolvent or a k-convoluted semigroup. We investigate the long time behavior for bounded solutions by a direct application of the ergodic theorems for regularized resolvents of Lizama and Prado (J. Approx. Theory 122:42–61, 2003), Prado (Semigroup Forum 73:243–252, 2006). We apply our results to the qualitative study of the fractional diffusion-wave equation on L ^p (ℝ). The author is partially supported under FONDECYT Grant n^o 1070127.     

Asymptotic analysis of some classes of ordinary differential equations with large high-frequency terms

A complete asymptotic expansion is constructed for solutions of the Cauchy problem for nth order linear ordinary differential equations with rapidly oscillating coefficients, some of which may be proportional to ω ^n/2, where ω is oscillation frequency. A similar problem is solved for a class of systems of n linear first-order ordinary differential equations with coefficients of the same type. Attention is also given to some classes of first-order nonlinear equations with rapidly oscillating terms proportional to powers ω ^d . For such equations with d ∈ (1/2, 1], conditions are found that allow for the construction (and strict justification) of the leading asymptotic term and, in some cases, a complete asymptotic expansion of the solution of the Cauchy problem.     

Generalized Solutions of Nonlinear Parabolic Systems and the Vanishing Viscosity Method

We show in this paper that stochastic processes associated with nonlinear parabolic equations and systems allow one to construct a probabilistic representation of a generalized solution to the Cauchy problem. We also show that in some cases the derived representation can be used to construct a solution to the Cauchy problem for a hyperbolic system via the vanishing viscosity method. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 7–39.     

A degenerate hyperbolic equation under Levi conditions

Abstract We consider the Cauchy problem for a second order equation of hyperbolic type. This equation degenerates in two different ways. On one hand, the coefficients have a bad behavior with respect to time: there is a blow-up phenomenon in the first time derivative of the principal part’s coefficients, that is the derivative vanishes at the time t=0. On the other hand, the equation is weakly hyperbolic and the multiplicity of the roots is not constant, but zeroes are of finite order. Here we overcome the blow-up problem and, moreover, the finitely degeneration of the Cauchy problem allows us to give an appropriate Levi condition on the lower order terms in order to get C^∞ well posedness of the Cauchy problem. Keywords: Cauchy problem, Hyperbolic equations, Levi conditions     

The formation of trapped surfaces in spherically-symmetric Einstein–Euler spacetimes with bounded variation

We study the evolution of a self-gravitating compressible fluid in spherical symmetry and we prove the existence of weak solutions with bounded variation for the Einstein–Euler equations of general relativity. We formulate the initial value problem in Eddington–Finkelstein coordinates and prescribe spherically symmetric data on a characteristic initial hypersurface. We introduce here a broad class of initial data which contain no trapped surfaces, and we then prove that their Cauchy development contains trapped surfaces. We therefore establish the formation of trapped surfaces in weak solutions to the Einstein equations. This result generalizes a theorem by Christodoulou for regular vacuum spacetimes (but without symmetry restriction). Our method of proof relies on a generalization of the “random choice” method for nonlinear hyperbolic systems and on a detailed analysis of the nonlinear coupling between the Einstein equations and the relativistic Euler equations in spherical symmetry.     

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.     

Multidimensional ultrametric pseudodifferential equations

We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions and the space D′₀(X) of generalized functions on a multidimensional ultrametric space. We also consider some family of pseudodifferential operators on multidimensional ultrametric spaces. The notions of Cauchy problem for ultrametric pseudodifferential equations and of ultrametric characteristics are introduced. We prove an existence theorem and describe all solutions for the Cauchy problem (an analog of the Kovalevskaya theorem).     

Global solutions of the evolutionary Faddeev model with small initial data

We consider the Cauchy problem for evolutionary Faddeev model corresponding to maps from the Minkowski space ℝ¹⁺ⁿ to the unit sphere $ \mathbb{S} $ \mathbb{S} ², which obey a system of non-linear wave equations. The nonlinearity enjoys the null structure and contains semi-linear terms, quasi-linear terms and unknowns themselves. We prove that the Cauchy problem is globally well-posed for sufficiently small initial data in Sobolev space.     

Solvability of semilinear differential equations with singularity

We prove local theorems on the existence of solutions of the Cauchy problem for singular equations of the form {fx262-01} in Banach spaces. Solvability conditions depend on the type of the singularity of the pencil λA + B of closed linear operators A and B. Examples and applications to finite-dimensional differential-algebraic equations, infinite systems of differential equations, and partial differential equations of the non-Kovalevskaya type are presented. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 225–239, February, 2008.     

All Vacuum Near Horizon Geometries in D-dimensions with (D − 3) Commuting Rotational Symmetries

We explicitly construct all stationary, non-static, extremal near horizon geometries in D dimensions that satisfy the vacuum Einstein equations, and that have D−3 commuting rotational symmetries. Our work generalizes [arXiv:0806.2051] by Kunduri and Lucietti, where such a classification had been given in D = 4,5. But our method is different from theirs and relies on a matrix formulation of the Einstein equations. Unlike their method, this matrix formulation works for any dimension. The metrics that we find come in three families, with horizon topology S ² × T ^D-4, or S ³ × T ^D-5, or quotients thereof. Our metrics depend on two discrete parameters specifying the topology type, as well as (D − 2)(D − 3)/2 continuous parameters. Not all of our metrics in D ≥ 6 seem to arise as the near-horizon limits of known black hole solutions.     

On the Maxwell problem

We study the large-time behavior of global smooth solutions to the Cauchy problem for hyperbolic regularization of conservation laws. An attracting manifold of special smooth global solutions is determined by the Chapman–Enskog projection onto the phase space of consolidated variables. For small initial data we construct the Chapman–Enskog projection and describe its properties in the case of the Cauchy problem for moment approximations of kinetic equations. The existence conditions for the Chapman–Enskog projection are expressed in terms of the solvability of the Riccati matrix equations with parameter. Bibliography: 21 titles. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 27–63.     

On an operational method of solving initial-value problems for partial differential equations induced by generalized separation of variables

We propose an operational method of solving the Cauchy problem for partial differential equations and systems of partial differential equations. We demonstrate its superiority to the known methods. We give a number of illustrative examples of applications of the method. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1 1998, pp. 136–145.     

Vibrations of a fractal elastic string

We construct the solution of the fractional space-time equations that describe the vibrations of a quasi-one-dimensional fractal elastic string. We give the solution of the Cauchy problem for fractional differential equations with initial conditions. We carry out a numerical analysis and construct the graphic variation of the displacement function of a fractal elastic string. Three figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 142–147     

On existence of a harmonic variational flow subjected to two-sided conditions

We construct weak solutions to parabolic systems of the variational flow type related to a quadratic functional with the initial and boundary data from a suitable Sobolev space and subjected to two-sided conditions. We present an approach based on Rothe’s method. It is applicable to solving the Cauchy problem and initial boundary-value problem for many types of equations. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov, POMI, Vol. 243, 1997, pp. 324–337.