Asymptotic limit of nonlinear Schrödinger-Poisson system with general initial data

Abstract

The asymptotic limit of the nonlinearSchrödinger-Poisson system with general WKB initial data isstudied in this paper. It is proved that the current, defined bythe smooth solution of the nonlinear Schrödinger-Poisson system,converges to the strong solution of the incompressible Eulerequations plus a term of fast singular oscillating gradient vectorfields when both the Planck constant $\hbar$ and the Debyelength $\lambda$ tend to zero. The proof involves homogenizationtechniques, theories of symmetric quasilinear hyperbolic systemand elliptic estimates, and the key point is to establish theuniformly bounded estimates with respect to both the Planckconstant and the Debye length.