Title: Global existence of the irrotational Euler-Norstrom equations with a positive cosmological constant: The gravitational field equation Show Chinese title

Title: 无旋欧拉-诺特隆方程在正宇宙学常数下的整体存在性：引力场方程

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. Show Chinese abstract

Abstract: 我们的目标是证明非线性无旋欧拉-诺德斯特罗姆系统的经典解的全局存在性，该系统包含一个线性状态方程和一个宇宙学常数。 在此框架中，引力场由一个满足特定半线性波动方程的标量函数表示。 我们关注背景度规的空间周期性偏差，因此我们在三维环面$\mathbb{T}^3$上研究半线性波动方程，并在 Sobolev 空间$H^m(\mathbb{T}^3)$中进行研究。 这项工作分为两部分。 首先，我们研究由一个由线性状态方程控制的无旋流体产生的源项的诺德斯特罗姆方程。 在第二部分中，我们分析完整的耦合系统。 这样划分的原因之一是，具有线性状态方程的无旋流体会在诺德斯特罗姆方程中引入一个包含分数阶非线性项的源项。 这种非线性使得不能直接应用我们在先前工作中使用的技术\cite{Brauer_Karp_23}，其中我们依赖对称双曲系统、能量估计和齐次 Sobolev 空间。 相反，我们开发了一个适当的能量泛函，并建立了针对所考虑的波动方程的相应能量估计。