Quasi-neutral limit for Euler-Poisson system in the presence of boundary layers in an annular domain

Abstract

We investigate the quasi-neutral limit (the zero Debye length limit) for the Euler-Poisson system with radial symmetry in an annular domain. Under physically relevant conditions at the boundary, referred to as the Bohm criterion, we first construct the approximate solutions by the method of asymptotic expansion in the limit parameter, the square of the rescaled Debye length, whose detailed derivation and analysis are carried out in our companion paper [8]. By establishing Hm-norm, (m≥2), estimate of the difference between the original and approximation solutions, provided that the well-prepared initial data is given, we show that the local-in-time solution exists in the time interval, uniform in the quasi-neutral limit, and we prove the difference converges to zero with a certain convergence rate validating the formal expansion order. Our results mathematically justify the quasi-neutrality of a plasma in the regime of plasma sheath, indicating that a plasma is electrically neutral in bulk, whereas the neutrality may break down in a scale of the Debye length.