Related papers: A New Generalized Harmonic Evolution System
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 is a short review of a series of papers which, in collaboration with Yue Ma, establish several novel existence results for systems of coupled wave-Klein-Gordon equation. Our method, the Hyperbolic Hyperboloidal Method, has allowed us…
A well-posed initial-boundary value problem is formulated for the model problem of the vector wave equation subject to the divergence-free constraint. Existence, uniqueness and stability of the solution is proved by reduction to a system…
We introduce a new wave formulation for the relativistic Euler equations with vacuum boundary conditions that consists of a system of non-linear wave equations in divergence form with a combination of acoustic and Dirichlet boundary…
Techniques are developed for projecting the solutions of symmetric hyperbolic evolution systems onto the constraint submanifold (the constraint-satisfying subset of the dynamical field space). These optimal projections map a field…
Rapid growth of constraints is often observed in free evolutions of highly gravitating systems. To alleviate this problem we investigate the effect of adding spatial derivatives of the constraints to the right hand side of the evolution…
We present a symmetric hyperbolic formulation of the Einstein equations in affine-null coordinates. Giannakopoulos et. al. (arXiv:2007.06419) recently showed that the most commonly numerically implemented formulations of the Einstein…
This paper is concerned with the initial-boundary value problem for an evolutionary variational inequality complying with three intrinsic properties: complete irreversibility, unilateral equilibrium of an energy and an energy conservation…
In this paper we focus on the Cahn-Hilliard equation with dynamic boundary conditions, by adding two hyperbolic relaxation terms to the system. We verify that the energy of the total system is decreasing with time. By adding two…
We describe in detail how to eliminate nonphysical degrees of freedom in the Lagrangian and Hamiltonian formulations of a constrained system. Two important and distinct steps in our method are the fixing of ambiguities in the dynamics and…
Numerical simulation of the spherically symmetric Einstein--Euler (EE) system faces severe challenges due to the stringent physical admissibility constraints of relativistic fluids and the geometric singularities inherent in metric…
We survey some of our recent results on inverse problems for evolution equations. The goal is to provide a unified approach to solve various types of evolution equations. The inverse problems we consider consist in determining unknown…
Singular solutions of the harmonic Einstein evolution equation are constructed which are related to spatially global and time-local solutions for a certain class of quasilinear hyperbolic systems of second order. The constructed…
We establish a variant, which has the advantage of introducing only physical characteristics, of the symmetric quasi linear first order system given by H.\ Friedrich for the evolution equations of gravitating fluid bodies in General…
A general form of the fifth-order nonlinear evolution equation is considered. Helmholtz solution of the inverse variational problem is used to derive conditions under which this equation admits an analytic representation. A Lennard type…
A class of parabolic cross-diffusion systems modeling the interaction of an arbitrary number of population species is analyzed in a bounded domain with no-flux boundary conditions. The equations are formally derived from a random-walk…
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 find a one-parameter family of variables which recast the 3+1 Einstein equations into first-order symmetric-hyperbolic form for any fixed choice of gauge. Hyperbolicity considerations lead us to a redefinition of the lapse in terms of an…
Inflationary spatially homogeneous cosmological models within an Einstein-Aether gravitational framework are investigated. The matter source is assumed to be a scalar field which is coupled to the aether field expansion and shear scalars…
We prove global existence for Einstein's equations with a charged scalar field for initial conditions sufficiently close to the Minkowski spacetime without matter. The proof relies on generalized wave coordinates adapted to the outgoing…