scholarly journals A minimization problem for the lapse and the initial-boundary value problem for Einstein's field equations

2006 ◽  
Vol 23 (16) ◽  
pp. S477-S504 ◽  
Author(s):  
Gabriel Nagy ◽  
Olivier Sarbach
Keyword(s):  
Boundary Value Problem ◽  
Minimization Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Einstein’S Field Equations ◽  
Field Equations ◽  
Einstein's Field Equations ◽  
Initial Boundary Value

scholarly journals A MODEL PROBLEM FOR THE INITIAL-BOUNDARY VALUE FORMULATION OF EINSTEIN'S FIELD EQUATIONS

2005 ◽  
Vol 02 (02) ◽  
pp. 397-435 ◽  
Author(s):  
OSCAR REULA ◽  
OLIVIER SARBACH
Keyword(s):  
General Relativity ◽  
Boundary Conditions ◽  
Model Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Einstein’S Field Equations ◽  
Field Equations ◽  
Einstein's Field Equations ◽  
Initial Boundary Value ◽  
Well Posed

In many numerical implementations of the Cauchy formulation of Einstein's field equations one encounters artificial boundaries which raises the issue of specifying boundary conditions. Such conditions have to be chosen carefully. In particular, they should be compatible with the constraints, yield a well posed initial-boundary value formulation and incorporate some physically desirable properties like, for instance, minimizing reflections of gravitational radiation. Motivated by the problem in General Relativity, we analyze a model problem, consisting of a formulation of Maxwell's equations on a spatially compact region of space–time with timelike boundaries. The form in which the equations are written is such that their structure is very similar to the Einstein–Christoffel symmetric hyperbolic formulations of Einstein's field equations. For this model problem, we specify a family of Sommerfeld-type constraint-preserving boundary conditions and show that the resulting initial-boundary value formulations are well posed. We expect that these results can be generalized to the Einstein–Christoffel formulations of General Relativity, at least in the case of linearizations about a stationary background.

GLOBAL WEAK SOLUTIONS OF AN INITIAL BOUNDARY VALUE PROBLEM FOR SCREW PINCHES IN PLASMA PHYSICS

2009 ◽  
Vol 19 (06) ◽  
pp. 833-875 ◽  
Author(s):  
JIANWEN ZHANG ◽  
SONG JIANG ◽  
FENG XIE
Keyword(s):  
Magnetic Field ◽  
Boundary Value Problem ◽  
Plasma Physics ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Field Equations ◽  
Initial Boundary Value

This paper is concerned with an initial-boundary value problem for screw pinches arisen from plasma physics. We prove the global existence of weak solutions to this physically very important problem. The main difficulties in the proof lie in the presence of 1/x-singularity in the equations at the origin and the additional nonlinear terms induced by the magnetic field. Solutions will be obtained as the limit of the approximate solutions in annular regions between two cylinders. Under certain growth assumption on the heat conductivity, we first derive a number of regularities of the approximate physical quantities in the fluid region, as well as a lot of uniform integrability in the entire spacetime domain. By virtue of these estimates we then argue in a similar manner as that in Ref. 20 to take the limit and show that the limiting functions are indeed a weak solution which satisfies the mass, momentum and magnetic field equations in the entire spacetime domain in the sense of distributions, but satisfies the energy equation only in the compact subsets of the fluid region. The analysis in this paper allows the possibility that energy is absorbed into the origin, i.e. the total energy be possibly lost in the limit as the inner radius goes to zero.

RESEARCH OF THE IMPACT DEVICE MODEL OF THE ROD TYPE BY DIFFERENCE METHOD

2020 ◽  
pp. 73-80
Author(s):  
А.М. Слиденко ◽  
В.М. Слиденко
Keyword(s):  
Boundary Value Problem ◽  
Difference Method ◽  
Model Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cross Sectional ◽  
Impact Device ◽  
Initial Boundary Value ◽  
The Impact

Приводится анализ механических колебаний элементов ударного устройства с помощью модели стержневого типа. Ударник и инструмент связаны упругими и диссипативными элементами, которые имитируют их взаимодействие. Аналогично моделируется взаимодействие инструмента с рабочей средой. Сформулирована начально-краевая задача для системы двух волновых уравнений с учетом переменных поперечных сечений стержней. Площади поперечных сечений определяются параметрическими формулами при сохранении объемов стержней. Параметрические формулы позволяют получать различного вида зависимости площади поперечного сечения стержня от его длины. Начальные условия отражают физическую картину взаимодействия инструмента с ударником и рабочей средой. Краевые условия описывают контактные взаимодействия ударника с инструментом и последнего с рабочей средой. В качестве модельной задачи рассматривается соударение ударника и инструмента через элемент большой жесткости. Начально-краевая задача исследуется разностным методом. Проводится сравнение решений задачи, полученных с помощью двухслойной и трехслойной разностных схем. Такие схемы реализованы в общей компьютерной программе в системе Mathcad. Показано, что при вычислениях распределения нормальных напряжений по длине стержня лучшими свойствами относительно устойчивости обладает двухслойная схема The article gives the analysis of mechanical vibrations of the impact device elements using the model of the rod type. The hammer and the tool are connected by elastic and dissipative elements that simulate their interaction. The interaction of the tool with the processing medium is simulated in a similar way. An initial boundary-value problem is formulated for a system of two wave equations taking into account the variable cross sections of the rods. Cross-sectional areas are determined by parametric formulas maintaining the volume of the rods. Parametric formulas allow one to obtain various dependence types of the cross-sectional area of the rod on its length. The initial and boundary conditions reflect the physical phenomenon of the tool interaction with the processing medium, and also describe the contact interactions of the hammer with the tool. The impacting of the hammer and the tool through an element of high rigidity is considered as a model problem. To control the limiting values, the solution of the model problem by the Fourier method is used. The initial-boundary-value problem is investigated by the difference method. A comparison of solutions obtained for the two-layer and three-layer difference schemes is given. Such schemes are realized in a common computer program in the Mathcad. It is shown that the two-layer scheme has the best properties in relation to stability while calculating the distribution of normal voltage along the length of the rod

scholarly journals Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Liming Xiao ◽  
Mingkun Li
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Evolution Problem ◽  
Positive Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

scholarly journals An extension of Gronwall inequality in the theory of bodies with voids

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1161-1167
Author(s):  
Marin Marin ◽  
Praveen Ailawalia ◽  
Ioan Tuns
Keyword(s):  
Boundary Value Problem ◽  
Internal State ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Result ◽  
Internal Variables ◽  
State Variables ◽  
Gronwall’S Inequality ◽  
Initial Boundary Value

Abstract In this paper, we obtain a generalization of the Gronwall’s inequality to cover the study of porous elastic media considering their internal state variables. Based on some estimations obtained in three auxiliary results, we use this form of the Gronwall’s inequality to prove the uniqueness of solution for the mixed initial-boundary value problem considered in this context. Thus, we can conclude that even if we take into account the internal variables, this fact does not affect the uniqueness result regarding the solution of the mixed initial-boundary value problem in this context.

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