Related papers: Theorems on existence and global dynamics for the …
This paper studies the Cauchy problem for systems of semi-linear wave equations on $\mathbb{R}^{3+1}$ with nonlinear terms satisfying the null conditions. We construct future global-in-time classical solutions with arbitrarily large initial…
The Cauchy problem for the inelastic Boltzmann equation is studied for small data. Existence and uniqueness of mild and weak solutions is obtained for sufficiently small data that lies in the space of functions bounded by Maxwellians. The…
This paper studies a Boltzmann-Nordheim equation in a slab with two-dimensional velocity space and pseudo-Maxwellian forces. Strong solutions are obtained for the Cauchy problem with large initial data in an $ L^1 \cap L^{\infty} $ setting.…
This paper studies the global existence and uniqueness of strong solutions and its large-time behavior for the compressible isothermal Euler equations with a nonlocal dissipation. The system is rigorously derived from the kinetic…
Our interest itself of this paper is strongly inspired from an open problem in the paper [1] published by D'Abbicco. In this article, we would like to study the Cauchy problem for a weakly coupled system of semi-linear structurally damped…
The Cauchy problem of the vacuum Einstein's equations aims to find a semi-metric $g_{\alpha\beta}$ of a spacetime with vanishing Ricci curvature $R_{\alpha,\beta}$ and prescribed initial data. Under the harmonic gauge condition, the…
We investigate the Cauchy problem for the nonlinear Schr\"odinger equation with a time-dependent linear damping term. Under non standard assumptions on the loss dissipation, we prove the blow-up in the inter-critical regime, and the global…
We review selected recent results concerning the global structure of solutions of the vacuum Einstein equations. The topics covered include quasi-local mass, strong cosmic censorship, non-linear stability, new constructions of solutions of…
We prove, for the relativistic Boltzmann equation on a Bianchi type I space-time, a global existence and uniqueness theorem, for arbitrarily large initial data.
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 consider a system representing self-gravitating balls of dust in an expanding Universe. It is demonstrated that one can prescribe data for such a system at infinity and evolve it backward in time without the development of shocks or…
This article begins with a brief introduction to numerical relativity aimed at readers who have a background in applied mathematics but not necessarily in general relativity. I then introduce and summarise my work on the problem of treating…
We consider gravitational field equations which are Einstein equations written in terms of embedding coordinates in some higher dimensional Minkowski space. Our main focus is to address some tricky issues relating to the Cauchy problem and…
Our goal is to prove existence results for classical solutions to some general nondegenerate Cauchy problems which are natural generalizations of Isaacs equations. For the latter we are able to extend our results by admitting local…
The existence of a simple spherically symmetric and static solution of the Einstein equations in the presence of a cosmological constant vanishing outside a definite value of the radial distance is investigated. A particular succession of…
We investigate a class of cosmological solutions of Einstein's field equations in higher dimensions with a cosmological constant and an ideal fluid matter distribution as a source. We discuss the dynamical evolution of the universe subject…
We prove a global in time existence theorem for the initial value problem for the Einstein-Boltzmann system, with positive cosmological constant and arbitrarily large initial data, in the spatially homogeneous case, in a Robertson-Walker…
The stability properties of the Einstein Static solution of General Relativity are altered when corrective terms arising from modification of the underlying gravitational theory appear in the cosmological equations. In this paper the…
We consider static cosmological solutions along with their stability properties in the framework of a recently proposed theory of massive gravity. We show that the modifcation introduced in the cosmological equations leads to several new…
In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately…