Quasi-neutral and zero-viscosity limit to planar rarefaction waves of two-dimensional Navier–Stokes–Poisson equations

In this paper we are concerned with the quasi-neutral and zero-viscosity limit of the two-dimensional compressible Navier–Stokes–Poisson equations in the domain Ω=R×T. We prove that if the solution of the quasi-neutral Euler system is piecewise constant with planar rarefaction waves, there exists a family of smooth solutions to the compressible Navier–Stokes–Poisson equations which converges to the planar rarefaction waves at a rate of ε29|ln⁡ε| as the scaled Debye length constant ɛ tends to zero, provided that the shear viscosity and the bulk viscosity are both the same order as ɛ. The results in this paper demonstrate that the introduction of the hyperbolic wave can not only recover the physical viscosity of the inviscid approximate rarefaction wave profile but also effectively suppress the singularities induced by the degeneration of elliptic constraint on electrostatic potential.