Global existence of the irrotational Euler-Nordström equations with a positive cosmological constant: The gravitational field equation

TL;DR

This work establishes global existence and asymptotic stability for the irrotational Euler–Nordström system with a positive cosmological constant, recast as a semilinear damped wave equation on the three‑torus. A novel energy functional is developed to control a fractional-power nonlinear source arising from an irrotational fluid with a linear equation of state, enabling uniform-in-time energy bounds and global existence for small data. The authors prove that the solution exists for all time and that its mean converges to a constant, with the full solution approaching this constant in the long-time limit. The analysis combines Sobolev space techniques on the torus, Moser-type estimates for nonlinearities, and a bootstrap energy method to handle the dissipative and fractional nonlinear terms, yielding a rigorous description of the asymptotic behaviour.

Abstract

Our objective is to demonstrate the global existence of classical solutions for the nonlinear irrotational Euler-Nordström system, which includes a linear equation of state and a cosmological constant. In this framework, the gravitational field is represented by a single scalar function that satisfies a specific semi-linear wave equation. We focus on spatially periodic deviations from the background metric, which is why we study the semi-linear wave equation on the three-dimensional torus $\mathbb{T}^3$ within the Sobolev spaces $H^m(\mathbb{T}^3)$. This work is divided into two parts. First, we examine the Nordström equation with a source term generated by an irrotational fluid governed by a linear equation of state. In the second part, we analyze the full coupled system. One reason for this separation is that an irrotational fluid with a linear equation of state introduces a source term for the Nordström equation containing a nonlinear term of fractional order. This nonlinearity precludes the direct application of the techniques used in our earlier work \cite{Brauer_Karp_23}, where we relied on symmetric hyperbolic systems, energy estimates, and homogeneous Sobolev spaces. Instead, we develop an appropriate energy functional and establish the corresponding energy estimates tailored to the wave equation under consideration.

Theorems & Definitions (30)

• Definition 1.1: Irrotational fluid flow
• Proposition 1.1: The source term of an irrotational fluid with an linear equation of state
• Remark 1.1: On the character of the source term in \ref{['eq:nordstroem-field:25']}
• Remark 1.2: On the notation for the source term
• Remark 1.3: On the smallness assumption of $a(t,x)$
• Definition 2.1: Sobolev spaces over the torus
• Theorem 2.1: Local existence of solutions to a nonlinear wave equation
• Theorem 2.2: Local existence of solutions to the Nordström equation
• Proposition 2.1: Sobolev inequality
• Proposition 2.2: Multiplications of derivatives in $L^2({\mathord{\mathbb T}}^n)$