Well-Posedness for the Euler-Nordstrom System with Cosmological Constant

TL;DR

This paper proves the well-posedness of the Euler-Nordstrom system with a cosmological constant, using adapted energy currents despite the system's non-symmetric hyperbolic nature.

Contribution

It introduces a method to establish well-posedness for a non-symmetric hyperbolic PDE system using energy currents derived from a Lagrangian framework.

Findings

Proves well-posedness in Sobolev spaces for the Euler-Nordstrom system with cosmological constant.

Adapts Christodoulou's energy current method to a non-symmetric hyperbolic system.

Demonstrates existence of solutions for initial data close to constant solutions.

Abstract

In this paper the author considers the motion of a relativistic perfect fluid with self-interaction mediated by Nordstrom's scalar theory of gravity. The evolution of the fluid is determined by a quasilinear hyperbolic system of PDEs, and a cosmological constant is introduced in order to ensure the existence of non-zero constant solutions. Accordingly, the initial value problem for a compact perturbation of an infinitely extended quiet fluid is studied. Although the system is neither symmetric hyperbolic nor strictly hyperbolic, Christodoulou's constructive results on the existence of energy currents for equations derivable from a Lagrangian can be adapted to provide energy currents that can be used in place of the standard energy principle available for first-order symmetric hyperbolic systems. After providing such energy currents, the author uses them to prove that the Euler-Nordstrom system with a cosmological constant is well-posed in a suitable Sobolev space.