Related papers: On the initial boundary value problem for the Eins…
In axial symmetry, there is a gauge for Einstein equations such that the total mass of the spacetime can be written as a conserved, positive definite, integral on the spacelike slices. This property is expected to play an important role in…
We discuss an initial-boundary value problem for a fractional diffusion equation with Caputo time-fractional derivative where the coefficients are dependent on spatial and time variables and the zero Dirichlet boundary condition is…
We investigate the existence and the global causal structure of plane symmetric spacetimes with weak regularity when the matter consists of an irrotational perfect fluid with pressure equal to its mass-energy density. Our theory encompasses…
The Einstein initial-value equations in the extrinsic curvature (Hamiltonian) representation and conformal thin sandwich (Lagrangian) representation are brought into complete conformity by the use of a decomposition of symmetric tensors…
Four-dimensional spacetimes foliated by a two-parameter family of homologous two-surfaces are considered in Einstein's theory of gravity. By combining a 1+(1+2) decomposition, the canonical form of the spacetime metric and a suitable…
We find new classes of exact solutions of the initial momentum constraint for vacuum Einstein's equations. Considered data are either invariant under a continuous symmetry or they are assumed to have the exterior curvature tensor of a…
In this paper, we consider the initial-boundary value problem for the time-dependent Maxwell--Schr\"{o}dinger equations in the Coulomb gauge. We first prove the global existence of weak solutions to the equations. Next we propose an…
A common approach for the numerical simulation of wave propagation on a spatially unbounded domain is to truncate the domain via an artificial boundary, thus forming a finite computational domain with an outer boundary. Absorbing boundary…
An outstanding issue in the treatment of boundaries in general relativity is the lack of a local geometric interpretation of the necessary boundary data. For the Cauchy problem, the initial data is supplied by the 3-metric and extrinsic…
In certain models of conformal gravity, the propagation of gravitational waves is governed by a fourth order scalar partial differential equation. We study the initial value problem for a generalization of this equation, and derive a…
When Einstein's equations for an asymptotically flat, vacuum spacetime are reexpressed in terms of an appropriate conformal metric that is regular at (future) null infinity, they develop apparently singular terms in the associated conformal…
We examine initial-boundary value problems for diffusion equations with distributed order time-fractional derivatives. We prove existence and uniqueness results for the weak solution to these systems, together with its continuous dependency…
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in $\R^n$ with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat…
The classical boundary-value problem of the Einstein field equations is studied with an arbitrary cosmological constant, in the case of a compact ($S^{3}$) boundary given a biaxial Bianchi-IX positive-definite three-metric, specified by two…
This paper is concerned with the Dirichlet initial-boundary value problem of a 2-D parabolic-elliptic system proposed to model the formation of biological transport networks. Even if global weak solutions for this system are known to exist,…
In the first of two papers, we study the initial boundary-value problem that underlies the theory of the Boltzmann equation for general non-spherical hard particles. In this work, for two congruent ellipses and for a large class of…
Spacetime is foliated by spatial hypersurfaces in the 3+1 split of General Relativity. The initial value problem then consists of specifying initial data for all relevant fields on one such a spatial hypersurface. These fields are the…
We initiate a systematic study of Einstein-Gauss-Bonnet gravity in the presence of boundaries subject to conformal boundary conditions, in which the conformal class of the boundary metric is kept fixed. In Einstein gravity, the trace of the…
One of the central difficulties of settling the $L^2$-bounded curvature conjecture for the Einstein -Vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to…
The main objective of this paper is analysis of the initial-boundary value problems for the linear and semilinear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo…