Convergence of the three-dimensional compressible Navier-Stokes-Poisson-Korteweg equation to the incompressible Euler equation

Abstract

We study the combination of quasi-neutral limit and viscosity limit of smooth solution for the three-dimensional compressible viscous Navier-Stokes-Poisson-Korteweg equation for plasmas and semiconductors. When the Debye length and viscosity coefficients are sufficiently small, the initial value problem of the model has a unique smooth solution in the time interval where the corresponding incompressible Euler equation has a smooth solution. We also establish a sharp convergence rate of smooth solutions for three-dimensional compressible viscous Navier-Stokes-Poisson-Kortewe equation towards those for the incompressible Euler equation in combining quasi-neutral limit and viscosity limit. Moreover, if the incompressible Euler equation has a global smooth solution, the maximal existence time of three-dimensional compressible Navier-Stokes-Poisson-Korteweg equation tends to infinity as the Debye length and viscosity coefficients goes to zero.