On classical solutions of the Nordstr\"om-Vlasov system

TL;DR

This paper proves existence, uniqueness, and conditions for global solutions of the Nordstr"om-Vlasov system in three dimensions, and analyzes finite-time blow-up in the repulsive case, also providing results for the one-dimensional system.

Contribution

It establishes the first rigorous results on classical solutions for the 3D Nordstr"om-Vlasov system, including global existence criteria and blow-up conditions, and extends analysis to the 1D case.

Findings

Existence and uniqueness of classical solutions in 3D

Global solutions under certain conditions

Finite-time blow-up for the repulsive case

Abstract

The Nordstr\"om-Vlasov system describes the dynamics of a self-gravitating ensemble of collisionless particles in the framework of the Nordstr\"om scalar theory of gravitation. We prove existence and uniqueness of classical solutions of the Cauchy problem in three dimensions and establish a condition which guarantees that the solution is global in time. Moreover, we show that if one changes the sign of the source term in the field equation, which heuristically corresponds to the case of a repulsive gravitational force, then solutions blow up in finite time for a large class of initial data. Finally, we prove global existence of classical solutions for the one dimensional version of the system with the correct sign in the field equation.