Quasineutral limit for the quantum Navier-Stokes-Poisson equation

TL;DR

This paper rigorously analyzes the quasineutral limit of the quantum Navier-Stokes-Poisson equation, showing how quantum effects influence the transition to the neutral incompressible Navier-Stokes regime.

Contribution

It provides a formal expansion and uniform estimates for the quantum Navier-Stokes-Poisson system, highlighting the role of quantum effects in the quasineutral limit.

Findings

Derivation of the neutral incompressible Navier-Stokes equation from the quantum system.

Uniform estimates that depend on the Planck constant.

Quantum effects significantly influence the asymptotic behavior.

Abstract

In this paper, we study the quasineutral limit and asymptotic behaviors for the quantum Navier-Stokes-Possion equation. We apply a formal expansion according to Debye length and derive the neutral incompressible Navier-Stokes equation. To establish this limit mathematically rigorously, we derive uniform (in Debye length) estimates for the remainders, for well-prepared initial data. It is demonstrated that the quantum effect do play important roles in the estimates and the norm introduced depends on the Planck constant $\hbar>0$.