Vanishing viscosity limit to rarefaction wave with vacuum for an ionized plasma

Abstract

In this paper, we consider the vanishing viscosity limit to rarefaction wave with vacuum for an ionized plasma whose equations of motion are described by the one-dimensional compressible Navier–Stokes–Poisson system for ions with [Formula: see text]-law pressure. In plasma physics, it is very often assumed that the plasma is quasineutral. The quasineutrality assumption can be obtained formally from the limit of the Debye length [Formula: see text] in the Poisson equation. For the Navier–Stokes–Poisson system for ions, the Debye length [Formula: see text] is much smaller than the ion viscosity coefficient [Formula: see text], which implies when [Formula: see text], it must have [Formula: see text]. Thus, by letting [Formula: see text], we can obtain the corresponding quasineutral Euler system whose Riemann solutions include vacuum state. Then given a rarefaction wave with one-side vacuum state to the quasineutral Euler system, we construct a sequence of solutions to the Navier–Stokes–Poisson system for ions which converge to the above rarefaction wave with vacuum as the viscosity tends to zero. Moreover, the uniform convergence rate is obtained.