📝 SummarXiv

Abstract

Our objective is to demonstrate the global existence of classical solutions for the nonlinear irrotational Euler-Nordstroem 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 Nordstroem 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 Nordstroem 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.