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Gradient Existence and Energy Finiteness of Local Minimizers in the Wasserstein $L^\infty$ Topology for Binary-Star Systems

Hangsheng Chen

TL;DR

This work strengthens McCann's variational treatments of binary-star equilibria in the Euler–Poisson framework by exploiting a Wasserstein $L^{\infty}$ topology. It proves gradient existence for local minimizers, enabling a rigorous passage from Euler–Lagrange equations to the Euler–Poisson system, and shows that $L^{\infty}$ functions populate every $W^{\infty}$-neighborhood of a density, ensuring meaningful perturbations. The paper also analyzes the energy functional's variational structure, providing an explicit derivative, a locally constant (and negative) Lagrange multiplier, and a comparison of minimizer existence across topologies, including the finite-energy constraint and the role of uniform rotation. Collectively, these results establish a solid mathematical foundation for uniformly rotating binary-star configurations and their local energy stability in the Wasserstein framework, with implications for star–planet systems and related astrophysical models.

Abstract

In this paper, we refine and complement McCann's results on binary-star systems \cite{McC06}, a compressible fluid model governed by the Euler-Poisson equations. We consider a general form of the equation of state that includes polytropic gaseous stars indexed by $γ$ as a special case. Beyond revisiting McCann's framework and conclusions -- where solutions to the Euler-Poisson equations are obtained as local energy minimizers via variational methods under the topology induced by the Wasserstein $L^\infty$ distance -- we focus on a detailed exploration of the properties of local energy minimizers in this topology, addressing three key aspects: (1) the feasibility of transitioning from the Euler-Lagrange equation to the Euler-Poisson equation by demonstrating gradient existence; (2) the existence of $L^\infty$ functions within neighborhoods in this topology; and (3) the finiteness of the energy of local minimizers in this topology, contrasted with the non-existence of finite-energy local minimizers and the existence of infinite-energy weak local minimizers in the topology inherited from topological vector spaces.

Gradient Existence and Energy Finiteness of Local Minimizers in the Wasserstein $L^\infty$ Topology for Binary-Star Systems

TL;DR

This work strengthens McCann's variational treatments of binary-star equilibria in the Euler–Poisson framework by exploiting a Wasserstein topology. It proves gradient existence for local minimizers, enabling a rigorous passage from Euler–Lagrange equations to the Euler–Poisson system, and shows that functions populate every -neighborhood of a density, ensuring meaningful perturbations. The paper also analyzes the energy functional's variational structure, providing an explicit derivative, a locally constant (and negative) Lagrange multiplier, and a comparison of minimizer existence across topologies, including the finite-energy constraint and the role of uniform rotation. Collectively, these results establish a solid mathematical foundation for uniformly rotating binary-star configurations and their local energy stability in the Wasserstein framework, with implications for star–planet systems and related astrophysical models.

Abstract

In this paper, we refine and complement McCann's results on binary-star systems \cite{McC06}, a compressible fluid model governed by the Euler-Poisson equations. We consider a general form of the equation of state that includes polytropic gaseous stars indexed by as a special case. Beyond revisiting McCann's framework and conclusions -- where solutions to the Euler-Poisson equations are obtained as local energy minimizers via variational methods under the topology induced by the Wasserstein distance -- we focus on a detailed exploration of the properties of local energy minimizers in this topology, addressing three key aspects: (1) the feasibility of transitioning from the Euler-Lagrange equation to the Euler-Poisson equation by demonstrating gradient existence; (2) the existence of functions within neighborhoods in this topology; and (3) the finiteness of the energy of local minimizers in this topology, contrasted with the non-existence of finite-energy local minimizers and the existence of infinite-energy weak local minimizers in the topology inherited from topological vector spaces.
Paper Structure (14 sections, 25 theorems, 101 equations)

This paper contains 14 sections, 25 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. McCann's Construction and Results of 2-Body Systems
  3. Notations and problem setting
  4. Uniform rotation minimizes kinetic energy
  5. McCann's results for 2-body systems
  6. Detailed Analysis of Theorem \ref{['Properties of LEM']}
  7. Wasserstein $L^{\infty}$ Metric on ${R}\left(\mathbb{R}^3\right)$
  8. Analysis of Lagrange Multiplier and Properties of Local Minimizers
  9. Variational derivative of $E_J$ and finite energy of local minimizer
  10. Locally constant Lagrange multiplier
  11. Minimizers' existence and nonexistence
  12. Appendix
  13. Calculus of Variations in Vector Spaces
  14. Properties of Sobolev Spaces

Key Result

Lemma 2.6

If $P(\rho)$ satisfies assumptions F1F2F3, then $A'$ is continuous and strictly increasing, where $A$ is given in (A). Moreover, $A$ is convex.

Theorems & Definitions (113)

  • Definition 2.1: Notations
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6: Strictly Increasing of $A^{\prime}$
  • proof
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • ...and 103 more