Communications in Mathematical Sciences

Volume 16 (2018)

Number 2

Quasineutral limit for the compressible quantum Navier–Stokes–Maxwell equations

Pages: 363 – 391

DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n2.a3

Authors

Min Li (School of Mathematics and Statistics, Chongqing University, Chongqing, China)

Xueke Pu (School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, China)

Shu Wang (School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, China)

Abstract

In this paper, we study the quasi-neutral limit of the full quantum Navier–Stokes–Maxwell equation as the Debye length tends to zero. We justify rigorously the quasi-neutral limit by establishing rigorous uniform estimates on the error functions with respect to the Debye length and by using the formal asymptotic expansion and singular perturbation methods combined with $\textrm{curl-div}$ decomposition of the gradient. The key difficulty is to deal with the quantum effects, which do play important roles in establishing a priori estimates.

Keywords

quantum Navier–Stokes–Maxwell system, quasineutral limit, uniform energy estimates, curl-div decomposition, singular perturbation methods

2010 Mathematics Subject Classification

35B40, 35B45, 35C20, 76Y05

Received 28 April 2017

Accepted 22 November 2017

Published 14 May 2018