[Paper Review] The fast Newtonian limit for perfect fluids

Todd A. Oliynyk|arXiv (Cornell University)|Aug 31, 2009

Navier-Stokes equation solutions10 references4 citations

TL;DR

This paper establishes the existence of a large class of dynamical solutions to the Einstein-Euler equations for perfect fluids that converge to solutions of the Newtonian Poisson-Euler equations in the limit $\epsilon \searrow 0$, where $\epsilon = v_T/c$ is a small parameter. Using a non-local symmetric hyperbolic formulation with weighted Sobolev spaces and energy-dispersive estimates, the author proves $\epsilon$-independent bounds and shows convergence of fluid density, velocity, and gravitational potential at rate $O(\epsilon)$, generalizing prior results to weaker initial data conditions.

ABSTRACT

We prove the existence of a large class of dynamical solutions to the Einstein-Euler equations for which the fluid density and spatial three-velocity converge to a solution of the Poisson-Euler equations of Newtonian gravity. The results presented here generalize those of \cite{Oli06} to allow for a larger class of initial data. As in \cite{Oli06}, the proof is based on a non-local symmetric hyperbolic formulation of the Einstein-Euler equations which contain a singular parameter $\ep=v_T/c$ with $v_T$ a characteristic speed associated to the fluid and $c$ the speed of light. Energy and dispersive estimates on weighted Sobolev spaces are the main technical tools used to analyze the solutions in the singular limit $\ep\searrow 0$.

Motivation & Objective

• To generalize previous results on the Newtonian limit of the Einstein-Euler equations by allowing a broader class of initial data.
• To establish rigorous convergence of relativistic fluid solutions to Newtonian solutions in the limit $\epsilon \searrow 0$, where $\epsilon = v_T/c$.
• To develop a framework using non-local symmetric hyperbolic systems that enables $\epsilon$-independent energy and dispersive estimates for the singular limit.
• To support future improvements in post-Newtonian expansions, particularly to second post-Newtonian order.
• To provide a rigorous foundation for Newtonian gravity as an approximation to general relativity under weak-field, slow-motion conditions.

Proposed method

• Formulate the Einstein-Euler equations in a dimensionless, singularly perturbed form using a small parameter $\epsilon = v_T/c$, where $v_T$ is a typical fluid speed.
• Introduce new variables: $\bar{\mathfrak{u}}^{ij}$ for the gravitational field and $w^i$ for the fluid velocity, which remain regular as $\epsilon \to 0$.
• Use an isentropic equation of state $p = K\rho^{(n+1)/n}$ to enable regularization via the density variable $\alpha$, defined by $\rho = (4Kn(n+1))^{-n} \alpha^{2n}$.
• Reformulate the system as a non-local symmetric hyperbolic system of the form $b^0(\epsilon W)\partial_t W = \frac{1}{\epsilon}c^I\partial_I W + b^I(\epsilon,W)\partial_I W + F(\epsilon,W)$, suitable for singular limit analysis.
• Apply weighted Sobolev norms $H^{k-2}_{\delta-1,\epsilon}$ and derive $\epsilon$-independent energy and dispersive estimates to control the solution behavior as $\epsilon \to 0$.
• Use the weighted multiplication lemma and embedding results to compare relativistic and Newtonian solutions in $H^{k-2}$-type norms, establishing convergence rates.

Experimental results

Research questions

• RQ1 Does the Newtonian limit of the Einstein-Euler equations hold for a larger class of initial data than previously established?
• RQ2 Can the convergence of relativistic fluid solutions to Newtonian solutions be proven using a non-local symmetric hyperbolic formulation with $\epsilon$-independent estimates?
• RQ3 What is the rate of convergence of the fluid density, velocity, and gravitational potential in the $\epsilon \to 0$ limit?
• RQ4 Can the techniques developed here be used to improve post-Newtonian expansions beyond the first order?
• RQ5 How do the new variables $\bar{\mathfrak{u}}^{ij}$ and $w^i$ ensure regularity and compatibility with the Newtonian limit?

Key findings

• The fluid density $\rho_\epsilon$ and spatial velocity $w^I_\epsilon$ converge to the Newtonian solutions $\tilde{\rho}$ and $\tilde{w}^I$ at rate $O(\epsilon)$ in $H^{k-2}$-norm.
• The temporal component of the fluid velocity $w^4_\epsilon$ converges to zero at rate $O(\epsilon)$ in $H^{k-2}_{\delta-1,\epsilon}$-norm.
• The gravitational potential $\Phi_\epsilon$ and its derivatives converge to the Newtonian potential $\tilde{\Phi}$ at rate $O(\epsilon)$ in $H^{k-2}_{\delta-1,\epsilon}$-norm.
• The metric-related variable $\bar{\mathfrak{u}}^{ij}_\epsilon$ converges to its Newtonian counterpart $\tilde{\mathfrak{u}}^{ij}_\epsilon$ at rate $O(\epsilon)$ in $L^6 \cap H^{k-2}$-type norms.
• The $\epsilon$-independent energy estimates ensure uniform boundedness of solutions in weighted Sobolev spaces for all $\epsilon \in (0, \epsilon_0]$, enabling the singular limit analysis.
• The convergence results are established for compact fluid bodies (stars) with isentropic equations of state, extending prior results to less restrictive initial data.