Related papers: The Cauchy Problem for the Einstein Equations

In this paper, we investigate the Cauchy problem for both linear and semi-linear elliptic equations. In general, the equations have the form \[ \frac{\partial^{2}}{\partial…

We deal with the existence of weak solutions for a mixed Neumann-Robin-Cauchy problem. The existence results are based on global-in-time estimates of approximating solutions, and the passage to the limit exploits compactness techniques. We…

In this paper, we considered the problem of analytical continuation of the solution of the system equations of the moment theory of elasticity in spacious bounded domain from its values and values of its strains on part of the boundary of…

Degenerate hyperbolic equations are dealing with many important issues for applied nature. While a variety of degenerate equations and boundary conditions, successfully matched to these differential equation, most in the characteristic…

We obtain a system for the spatial metric and extrinsic curvature of a spacelike slice that is hyperbolic non-strict in the sense of Leray and Ohya and is equivalent to the Einstein equations. Its characteristics are the light cone and the…

This paper studies a space-inhomogeneous Boltzmann-Nordheim equation with pseudo-Maxwellian forces. Strong solutions are obtained for the Cauchy problem in a setting with large bounded L1 initial data. The main results are existence,…

We prove local existence and uniqueness of the Cauchy problem for a large class of tensorial second order linear hyperbolic partial differential equations with coefficients of low regularity in a suitable class of generalized functions.

We present a general method of solving the Cauchy problem for a linear parabolic partial differential equation of evolution type with variable coefficients and demonstrate it on the equation with derivatives of orders two, one and zero. The…

In this work we study constant-coefficient first order systems of partial differential equations and give necessary and sufficient conditions for those systems to have a well posed Cauchy Problem. In many physical applications, due to the…

The causal structure of Einstein's evolution equations is considered. We show that in general they can be written as a first order system of balance laws for any choice of slicing or shift. We also show how certain terms in the evolution…

In this paper we consider the problem of analytical continuation of solutions to the system of thermoelasticity in a bounded domain from their values and values of their strains on a part of the boundary of this domain, i.e., the Cauchy…

In the Cauchy problem of general relativity one considers initial data that satisfies certain constraints. The evolution equations guarantee that the evolved variables will satisfy the constraints at later instants of time. This is only…

We consider the Cauchy problem for strictly hyperbolic $m$-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that the problem is $L^2$ well-posed in the case of Lipschitz…

In this paper we consider some new classes of integral equations that arise from Lie symmetry analysis. Specifically, we consider the task of obtaining solutions of a Cauchy problem for some classes of second order hyperbolic partial…

We study the Cauchy problem for the parabolic infinity Laplace equation. We prove a new comparison principle and obtain uniqueness of viscosity solutions in the class of functions with a polinomial growth at infinity, improving previous…

An introduction to solutions of the Einstein equations defining cosmological models with accelerated expansion is given. Connections between mathematical and physical issues are explored. Theorems which have been proved for solutions with…

In this paper, we study the solvability of a Cauchy- Dirichlet problem for nonlinear parabolic equation with non standard growths and nonlocal terms. We show the existence of weak solutions of the considered problem under more general…

The Einstein evolution equations may be written in a variety of equivalent analytical forms, but numerical solutions of these different formulations display a wide range of growth rates for constraint violations. For symmetric hyperbolic…

In the present article, a modified Cauchy problem (problem C) for the hyperbolic equation of the third order with the data on the equation's coefficients singularity plane is solved by Riemann method. The special class in which the solution…

This paper develops a method for solving Einstein's equation numerically on multi-cube representations of manifolds with arbitrary spatial topologies. This method is designed to provide a set of flexible, easy to use computational…