[Paper Review] Well-Posedness of the Einstein-Euler System in Asymptotically Flat Spacetimes

Uwe Brauer, Lavi Karp|ArXiv.org|Oct 28, 2008

Navier-Stokes equation solutions31 references3 citations

TL;DR

This paper establishes the local well-posedness of the Einstein-Euler system in asymptotically flat spacetimes by introducing a novel framework of weighted Sobolev spaces of fractional order. It proves existence and uniqueness of classical solutions under physically motivated conditions—vanishing or decaying energy density at infinity and a polytropic equation of state—overcoming limitations of prior works that required time symmetry or integer-order regularity.

ABSTRACT

We prove a local in time existence and uniqueness theorem of classical solutions of the coupled Einstein--Euler system, and therefore establish the well posedness of this system. We use the condition that the energy density might vanish or tends to zero at infinity and that the pressure is a certain function of the energy density, conditions which are used to describe simplified stellar models. In order to achieve our goals we are enforced, by the complexity of the problem, to deal with these equations in a new type of weighted Sobolev spaces of fractional order. Beside their construction, we develop tools for PDEs and techniques for elliptic and hyperbolic equations in these spaces. The well posedness is obtained in these spaces. The results obtained are related to and generalize earlier works of Rendall for the Euler-Einstein system under the restriction of time symmetry and of Gamblin for the simpler Euler--Poisson system.

Motivation & Objective

• To establish local existence and uniqueness of classical solutions for the coupled Einstein-Euler system in asymptotically flat spacetimes.
• To generalize prior results by Rendall and Gamblin by removing the restriction of time symmetry and allowing non-integer regularity.
• To develop a functional framework based on weighted Sobolev spaces of fractional order to handle the singular behavior of the fluid equations when energy density vanishes.
• To construct initial data for both the Einstein constraint equations and the fluid equations within the same functional space, ensuring consistency across the system.
• To generalize earlier results on the Euler-Poisson and Einstein-Euler systems by allowing a broader class of equations of state with non-integer $γ$.

Proposed method

• Formulate the Einstein-Euler system using a polytropic equation of state $p = K\epsilon^\gamma$ with $1 < \gamma$, modeling relativistic self-gravitating fluids.
• Introduce a new variable $w = \epsilon^{(\gamma-1)/2}$ (Makino variable) to regularize the fluid equations and handle the singularity at $\epsilon = 0$.
• Define and analyze a new class of weighted Sobolev spaces $H_{s,\delta}^{k}$ with fractional order $s$, decay rate $\delta$, and weight $(1+|x|)^\delta$, tailored for asymptotically flat spacetimes.
• Establish embedding and density theorems for these spaces, including the density of $C_0^\infty$ functions in $H_{s,\delta}$ and control of weighted norms via dyadic decomposition.
• Use dyadic partition of unity and rescaling techniques to control pointwise decay and regularity, proving $\|u\|_{C_{\beta}} \leq C\|u\|_{H_{s,\delta}}$ under $s > \frac{3}{2}+\delta$, $\delta + \frac{3}{2} \geq \beta$.
• Prove that the evolution equations form a first-order symmetric hyperbolic system in the unknowns $g_{\alpha\beta}, u^\alpha, w$, with lower-order terms depending on $\epsilon$, which is controlled via $w$ in the norm.

Experimental results

Research questions

• RQ1 Can the Einstein-Euler system be proven well-posed in asymptotically flat spacetimes without assuming time symmetry?
• RQ2 What functional framework is necessary to handle the singular behavior of the fluid equations when energy density vanishes at spatial infinity?
• RQ3 How does the choice of equation of state $p = K\epsilon^\gamma$ constrain the required regularity of solutions in weighted Sobolev spaces?
• RQ4 Can the Makino variable $w = \epsilon^{(\gamma-1)/2}$ be used to regularize the system and allow existence theorems in fractional-order weighted spaces?
• RQ5 What embedding and density properties must weighted Sobolev spaces of fractional order satisfy to support the analysis of hyperbolic and elliptic systems in this context?

Key findings

• The authors prove local existence and uniqueness of classical solutions to the Einstein-Euler system in weighted Sobolev spaces of fractional order, establishing well-posedness in asymptotically flat spacetimes.
• The solution framework allows for energy density $\epsilon$ to vanish or decay at infinity, consistent with stellar models, without requiring compact support.
• The equation of state $p = K\epsilon^\gamma$ is admitted for $1 < \gamma \leq \frac{2+k}{k}$, where $k$ is the order of the Sobolev space, thus generalizing prior results that required $\gamma \in \mathbb{N}$.
• The Makino variable $w = \epsilon^{(\gamma-1)/2}$ enables regularization of the fluid equations, allowing the system to be treated as a symmetric hyperbolic system in the unknowns $g_{\alpha\beta}, u^\alpha, w$.
• A new density theorem is proven: $C_0^\infty(\mathbb{R}^3)$ is dense in $H_{s,\delta}$, and for $s' > s$, approximations in $H_{s',\delta}$ can be bounded in terms of the original $H_{s,\delta}$ norm.
• The pointwise decay estimate $\sup_x (1+|x|)^\beta |u(x)| \leq C \|u\|_{H_{s,\delta}}$ holds for $\beta \leq \delta + \frac{3}{2}$ and $s > \frac{3}{2} + \delta$, ensuring decay control in the solution space.