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In this paper we present new results on the preservation of polynomial stability of damped wave equations under addition of perturbing terms. We in particular introduce sufficient conditions for the stability of perturbed two-dimensional…
We present a new formulation of the multipolar expansion of an exact boundary condition for the wave equation, which is truncated at the quadrupolar order. Using an auxiliary function, that is the solution of a wave equation on the sphere…
We present three-dimensional simulations of Einstein equations implementing a symmetric hyperbolic system of equations with dynamical lapse. The numerical implementation makes use of techniques that guarantee linear numerical stability for…
In this paper, we introduce the notion of boundary delay equations, establishing a unified framework for analyzing linear time-invariant systems with pure time-delayed boundary conditions. We establish mild sufficient conditions for the…
The Einstein equations have proven surprisingly difficult to solve numerically. A standard diagnostic of the problems which plague the field is the failure of computational schemes to satisfy the constraints, which are known to be…
In the Bondi-Sachs gauge, the Einstein equations with a cosmological constant coupled to a scalar field in spherical symmetry are cast into a first order strongly hyperbolic formulation in which the lapse and shift are the fundamental…
In the middle of last century, Bondi and his coworkers proposed an out going boundary condition for the Einstein equations. Based on such boundary condition the authors theoretically solved the puzzle of the existence problem of…
We prove well posedness and stability in $\mathbf{L}^1$ for a class of mixed hyperbolic-parabolic non linear and non local equations in a bounded domain with no flow along the boundary. While the treatment of boundary conditions for the…
We define an abstract setting to treat wave equations equipped with time-dependent acoustic boundary conditions on bounded domains of ${\bf R}^n$. We prove a well-posedness result and develop a spectral theory which also allows to prove a…
In a recent article, we constructed a hierarchy B_L of outer boundary conditions for Einstein's field equations with the property that, for a spherical outer boundary, it is perfectly absorbing for linearized gravitational radiation up to a…
We derive a class of energy preserving boundary conditions for incompressible Newtonian flows and prove local-in-time well-posedness of the resulting initial boundary value problems, i.e. the Navier-Stokes equations complemented by one of…
This brief note wants to bring to attention that the formulation of physically reasonable initial-boundary value problems for wave equations in Lorentzian space-times is not unique, i.e., that there are inequivalent such formulations that…
Motivated by the partial differential equations of mixed type that arise in the reduction of the Einstein equations by a helical Killing vector field, we consider a boundary value problem for the helically-reduced wave equation with an…
In this work, we study of the algebraic-hyperbolic formulation of the Einstein constraint equations for numerically constructing initial data sets for inhomogeneous cosmological space-times with $\mathbb{T}^3$ topology. We implement a…
This paper considers the time-harmonic Maxwell equations with impedance boundary condition.We present $H^2$-norm bound and other high-order norm bounds for strong solutions. The $H^2$-estimate have been derived in [M. Dauge, M. Costabel and…
In this paper, we investigate the stability of the linear wave equation where one part of the boundary, which is seen as a lower-dimensional Riemannian manifold, is governed by a coupled wave equation, while the other part is subject to a…
We present Helmholtz or Helmholtz like equations for the approximation of the time-harmonic wave propagation in gases with small viscosity, which are completed with local boundary conditions on rigid walls. We derived approximative models…
We study the linearization stability of the Einstein constraint equations on an asymptotically hyperbolic manifold. In particular we prove that these equations are linearization stable in the neighborhood of vacuum solutions for a…
This article solves the initial boundary value problem for the vacuum Einstein equations with a negative cosmological constant in dimension 4, giving rise to asymptotically Anti-de Sitter spaces. We introduce a new family of geometric…
Well-posedness of the initial (boundary) value problem is an essential property, both of meaningful physical models and of numerical applications. To prove well-posedness of wave-type equations their level of hyperbolicity is an essential…