Radial Transonic Shock Solutions to Euler-Poisson System with Varying Background Charge in an Annulus

Abstract

This paper concerns both the structural and dynamical stabilities of radially symmetric transonic shock solutions for two-dimensional Euler-Poisson system in an annulus. The density of fixed, positively charged background ions is allowed to be different constants in supersonic and subsonic regimes. First, the existence and structural stability of a steady transonic shock solution are obtained by the monotonicity between the shock location and the density on the outer circle. Second, any radially symmetric transonic shock solution with respect to small perturbations of the initial data is shown to be dynamically stable. The proof relies on the decay estimates and coupled effects from electric field and geometry of the annulus, together with the methods from [18]. These results generalize previous stability results on transonic shock solutions for constant background charge.

Keywords

Euler-Poisson equations / radial symmetry / transonic shock / varying background charge / stability

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