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

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.