The steady Euler-Poisson system couples fluid flow with electrostatic self-interaction — charged particles flowing through a domain while generating and responding to their own electric field. Subsonic solutions exist, and near one-dimensional background states they were known to be locally unique: if you perturb slightly, there’s only one subsonic solution nearby.

This paper proves global uniqueness. Without any smallness assumption on the perturbation, within the full class of subsonic solutions satisfying the same boundary data, the solution is unique. The previous contraction mapping argument worked in a neighborhood; this result covers the entire subsonic regime.

The jump from local to global uniqueness is not incremental. Local uniqueness says “nearby, there’s only one.” Global uniqueness says “everywhere in the subsonic class, there’s only one.” The gap between them contains the possibility of bifurcations, disconnected solution branches, and nonlinear resonances — none of which occur. The subsonic constraint is strong enough to pin down the solution completely, regardless of how far it is from the one-dimensional background. The boundary data determines the flow.