Uniform stability of expanding simple waves for Euler--Poisson--Boltzmann: a singular compensation approach

We rigorously justify the quasineutral limit for the warm-ion Euler--Poisson system with Maxwell--Boltzmann electrons on the cylinder $\R\times\T$. The reference dynamics are governed by a globally smooth expanding planar simple wave of the effective Euler system, connecting distinct neutral far-field states. For every finite order $M$, we construct an even Debye expansion with residual $O(\eps^{2M+2})$ and establish uniform nonlinear stability on any fixed time interval $[t_0,T]$, with a lifespan and energy estimates entirely independent of the Debye length $0<\eps\le\eps_0$. The central analytical contribution is a singular compensation mechanism: by integrating the fluid transport by parts and coupling the warm-ion symmetrizer directly to the time-differentiated nonlinear Poisson constraint, we extract an $\eps$-uniform relative energy topology. This bounds the potential in $H^s$ and its gradient in the scaled norm $\eps H^s$, bypassing the fatal $\eps^{-1}$ penalty in the momentum equation. Consequently, the framework accommodates genuinely two-dimensional transverse perturbations, including the transport of specific vorticity. This yields a quantitative, arbitrary-order quasineutral expansion for well-prepared data, strictly isolating the stability of the smooth expansion wave from the geometric singularities of the centered Riemann fan.