The steady Euler-Poisson system describes a charged fluid flowing through a bounded domain under its own electric field — the kind of problem that arises in semiconductor device modeling, where electrons (the fluid) flow through a device (the domain) driven by an applied voltage (the boundary conditions). The system couples fluid dynamics (conservation of mass, momentum, energy) with electrostatics (Poisson’s equation for the potential).

Previous existence proofs for multidimensional subsonic solutions constructed them as small perturbations of one-dimensional background states. In the perturbative regime, a contraction mapping argument works: the solution is the unique fixed point of a map that is contractive when the perturbation is small enough. This gives local uniqueness — the solution is unique among solutions close to the one-dimensional background — but says nothing about solutions far from the background.

Global uniqueness removes the smallness assumption. The subsonic solution is unique within the entire class of subsonic solutions satisfying the same boundary data, not just within a neighborhood of a particular background. The proof cannot use contraction mapping (which requires smallness) and instead relies on structural properties of the Euler-Poisson system that hold for all subsonic flows.

The subsonic condition is essential. For supersonic flows, uniqueness fails — multiple solutions with shocks can satisfy the same boundary data, and the selection among them requires additional criteria (entropy conditions, viscous regularization). The subsonic regime is special because the equations are elliptic (information propagates in all directions), which provides enough structure for uniqueness. In the supersonic regime, the equations are hyperbolic (information propagates along characteristics), and the characteristics can focus to create shocks where uniqueness is lost.

One set of boundary conditions, one subsonic solution. Not approximately (near a background) but exactly (among all subsonic flows). The flow that fits the boundaries is the only one that can.