On approximate solutions to the Euler–Poisson system with boundary layers

Abstract

In this article, we construct the approximate solutions to the Euler–Poisson system in an annular domain, that arises in the study of dynamics of plasmas. Due to a small parameter (proportional to the square of the Debye length) multiplied to the Laplacian operator, together with unmatched boundary conditions, we find that the solutions exhibit sharp transition layers near the boundaries, which makes the associated limit problem singular. To investigate this singular behavior, we explicitly construct the approximate solutions composed of the outer and inner solutions by the method of asymptotic expansions in appropriate order of the small parameter, turned out to be the Debye length. The equations to single out the boundary layers are determined by the inner expansions, for which we effectively treat nonlinear terms using the Taylor polynomial expansions with multinomials. We can obtain estimates showing that the approximate solutions are close enough to the original ones. We also provide numerical evidences demonstrating that the approximate solutions converge to those of the Euler–Poisson system as the parameter goes to zero.