scholarly journals The Characteristic Initial Value Problem for the Einstein-Yang-Mills-Higgs System in Weighted Sobolev Spaces

2010 ◽  
Author(s):  
M. Dossa ◽  
C. Tadmon
Keyword(s):  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Weighted Sobolev Spaces ◽  
Initial Value ◽  
Yang Mills ◽  
Characteristic Initial Value Problem

scholarly journals Rapidly decreasing behaviour of solutions in nonlinear 3-D-thermoelasticity

1991 ◽  
Vol 43 (1) ◽  
pp. 89-99 ◽  
Author(s):  
Song Jiang
Keyword(s):  
Boundary Value Problem ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Weighted Sobolev Spaces ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Exterior Domains ◽  
Initial Value ◽  
Initial Boundary Value

In this paper we study the asymptotic behaviour, as |x| → ∞, of solutions to the initial value problem in nonlinear three-dimensional thermoelasticity in some weighted Sobolev spaces. We show that under some conditions, solutions decrease fast for each t as x tends to infinity. We also consider the possible extension of the method presented in this paper to the initial boundary value problem in exterior domains.

scholarly journals On some semilinear equations of Schrödinger type

2002 ◽  
Vol 90 (1) ◽  
pp. 139
Author(s):  
Rossella Agliardi ◽  
Daniela Mari
Keyword(s):  
Differential Equations ◽  
Existence Theorem ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Local Existence ◽  
Weighted Sobolev Spaces ◽  
Initial Value ◽  
Semilinear Equations ◽  
Local Existence Theorem

We study the initial value problem for some semilinear pseudo-differential equations of the form $\partial _tu + i\ H(x,D_x)u = F(u,\nabla u)$. The assumptions we make on $H$ are trivially satisfied by $\Delta$, thus our equations generalize Schrdinger type equations. A local existence theorem is proved in some weighted Sobolev spaces.

Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations

1990 ◽  
Vol 427 (1872) ◽  
pp. 221-239 ◽  
Keyword(s):  
Cauchy Problem ◽  
Perfect Fluid ◽  
Initial Value Problem ◽  
Einstein Equations ◽  
Initial Value ◽  
Fluid Source ◽  
Null Hypersurfaces ◽  
The Cauchy Problem ◽  
Well Posed ◽  
Characteristic Initial Value Problem

A method is described by means of which the characteristic initial value problem can be reduced to the Cauchy problem and examples are given of how it can be used in practice. As an application it is shown that the characteristic initial value problem for the Einstein equations in vacuum or with perfect fluid source is well posed when data are given on two transversely intersecting null hypersurfaces. A new discussion is given of the freely specifiable data for this problem.

Numerical relativity. II. Numerical methods for the characteristic initial value problem and the evolution of the vacuum field equations for space‒times with two Killing vectors

1983 ◽  
Vol 386 (1791) ◽  
pp. 373-391 ◽  
Keyword(s):  
General Relativity ◽  
Numerical Methods ◽  
Plane Wave ◽  
Numerical Solution ◽  
Initial Value Problem ◽  
Vacuum Field ◽  
Field Equations ◽  
Initial Value ◽  
Wave Space ◽  
Characteristic Initial Value Problem

This is the second of a sequence of papers on the numerical solution of the characteristic initial value problem in general relativity. Although the equations to be integrated have regular coefficients, the nonlinearity leads to the occurrence of singularities after a finite evolution time. In this paper we first discuss some novel techniques for integrating the equations right up to the singularities. The second half of the paper presents as examples the numerical evolution of the Schwarzschild and certain colliding plane wave space‒times.

An oscillation theorem for characteristic initial value problems in linear hyperbolic equations

1977 ◽  
Vol 77 (3-4) ◽  
pp. 265-271 ◽  
Author(s):  
Gordon Pagan
Keyword(s):  
Initial Value Problem ◽  
Initial Value Problems ◽  
Hyperbolic Equations ◽  
Initial Value ◽  
Oscillation Theorem ◽  
Characteristic Initial Value Problem ◽  
Linear Hyperbolic Equations

SynopsisIt is established that under certain restrictions the solution u of the characteristic initial value problem uxy+g(x, y)u = 0, u(x, 0) = p(x) and u(0, y) = q(y), where p(x) > 0 and q(y) > 0, in [0, ∞) x [0, ∞) changes sign along a monotonic decreasing curve which is asymptotic to the axes.

scholarly journals Revisiting the characteristic initial value problem for the vacuum Einstein field equations

2020 ◽  
Vol 52 (10) ◽  
Author(s):  
David Hilditch ◽  
Juan A. Valiente Kroon ◽  
Peng Zhao
Keyword(s):  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Initial Value ◽  
Bootstrap Argument ◽  
Null Hypersurfaces ◽  
Characteristic Initial Value Problem ◽  
Local Existence Of Solutions

AbstractUsing the Newman–Penrose formalism we study the characteristic initial value problem in vacuum General Relativity. We work in a gauge suggested by Stewart, and following the strategy taken in the work of Luk, demonstrate local existence of solutions in a neighbourhood of the set on which data are given. These data are given on intersecting null hypersurfaces. Existence near their intersection is achieved by combining the observation that the field equations are symmetric hyperbolic in this gauge with the results of Rendall. To obtain existence all the way along the null-hypersurfaces themselves, a bootstrap argument involving the Newman–Penrose variables is performed.

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