Related papers: Well posed constraint-preserving boundary conditio…
We present two families of first-order in time and second-order in space formulations of the Einstein equations (variants of the Arnowitt-Deser-Misner formulation) that admit a complete set of characteristic variables and a conserved energy…
We establish the existence of Lipschitz continuous solutions to the Cauchy Dirichlet problem for a class of evolutionary partial differential equations of the form $$ \partial_tu-\text{div}_x \nabla_\xi f(\nabla u)=0 $$ in a space-time…
We derive conditions for well-posedness of semilinear evolution equations with unbounded input operators. Based on this, we provide sufficient conditions for such properties of the flow map as Lipschitz continuity,…
A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the…
We construct and parametrize solutions to the constraint equations of general relativity in a neighborhood of Minkowski spacetime with arbitrary prescribed decay properties at infinity. We thus provide a large class of initial data for the…
The present paper is devoted to the study of the well-posedness issue for the density-dependent Euler equations in the whole space. We establish local-in-time results for the Cauchy problem pertaining to data in the Besov spaces embedded in…
Two problems concerning asymptotically hyperbolic manifolds with an inner boundary are studied. First, we study scalar curvature presciption with either Dirichlet or mean curvature prescription interior boundary condition. Then we apply…
The formulation of the initial value problem for the Einstein equations is at the heart of obtaining interesting new solutions using numerical relativity and still very much under theoretical and applied scrutiny. We develop a specialised…
We revisit the problem of solving the Einstein constraint equations in vacuum by a new method, which allows us to prescribe four scalar quantities, representing the full dynamical degrees of freedom of the constraint system. We show that…
The global-in-time existence of bounded weak solutions to general cross-diffusion systems describing the evolution of $n$ population species is proved. The equations are considered in a bounded domain with no-flux boundary conditions. The…
We provide a formulation of the initial boundary value problem for Friedrich's extended conformal Einstein field equations in which boundary data is prescribed on a timelike hypersurface located at a finite position in the spacetime. Our…
Answering a question left open in \cite{MZ2}, we show for general symmetric hyperbolic boundary problems with constant coefficients, including in particular systems with characteristics of variable multiplicity, that the uniform Lopatinski…
In this paper we provide a complete local well-posedness theory for the free boundary relativistic Euler equations with a physical vacuum boundary on a Minkowski background. Specifically, we establish the following results: (i) local…
We study the ``hyperboloidal Cauchy problem'' for linear and semi-linear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behaviour at the boundary, or with polyhomogeneous initial data.…
We investigate how the accuracy and stability of numerical relativity simulations of 1D colliding plane waves depends on choices of equation formulations, gauge conditions, boundary conditions, and numerical methods, all in the context of a…
By employing the Bianchi identities for the Riemann tensor in conjunction with the Einstein equations, we construct a first order symmetric hyperbolic system for the evolution part of the Cauchy problem of general relativity. In this…
We investigate the relevance of the conformal method by investigating stability issues for the Einstein-Lichnerowicz conformal constraint system in a nonlinear scalar-field setting. We prove the stability of the system with respect to…
In this paper, we delve into the study of evolution equations that exhibit white-noise boundary conditions. Our primary focus is to establish a necessary and sufficient condition for the existence of solutions, by utilizing the concept of…
We give a short proof of the existence of a small piece of null infinity for $(3+1)$-dimensional spacetimes evolving from asymptotically flat initial data as solutions of the Einstein vacuum equations. We introduce a modification of the…
We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to…