Global existence of the irrotational Euler-Nordstr\"om equations with a positive cosmological constant: The gravitational field equation

Abstract

Our aim is to establish the global existence of classical solutions to the nonlinear irrotational Euler--Nordstr\"om system, which incorporates a linear equation of state and a cosmological constant. In this setting, gravitation is described by a single scalar field satisfying a specific semilinear wave equation. We restrict attention to spatially periodic perturbations of the background metric and therefore study this equation on the three-dimensional torus $\mathbb{T}^3$, working within the Sobolev spaces $H^m(\mathbb{T}^3)$. We begin by analysing the Nordstr\"om equation in isolation, with a source term generated by an irrotational fluid obeying a linear equation of state. This separation is motivated by the fact that such a fluid produces a source term containing a nonlinear contribution of fractional order. To obtain a global solution for the gravitational field, the fractional-order nonlinearity $(1+u)^\mu$, with $\mu\in\mathbb{R}$, must remain smooth throughout the evolution. This condition, in turn, requires that $u$ remain small for all time. We ensure this by introducing a suitably chosen energy functional. We also prove that, asymptotically, the solutions tend to a constant.