The combined quasineutral and incompressible limit for the Navier–Stokes–Poisson system in critical spaces

Abstract

In this paper, we study the quasineutral and incompressible limit for the Navier–Stokes–Poisson system with ill-prepared initial data in whole space. Based on Strichartz estimates for the Klein–Gordon equation and the pseudo-scaling property of modified Besov norms, we prove that the density tends to a constant, the compressible part of velocity vanishes, and the incompressible part of velocity converges to the solution of the incompressible Navier–Stokes equation. We get a relation of the lifespan of the solution between the Navier–Stokes–Poisson and the incompressible Navier–Stokes equations for small Debye length and low Mach number.