Communications in Mathematical Sciences

Volume 19 (2021)

Number 8

Existence and uniqueness for a stationary hybrid quantum hydrodynamical model with general pressure functional

Pages: 2049 – 2079

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n8.a1

Authors

Federica Di Michele (Gran Sasso Science Institute (GSSI), L’Aquila, Italy)

Ming Mei (Department of Mathematics, Champlain College Saint-Lambert, Quebec, Canada; and Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada)

Bruno Rubino (Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, Italy)

Rosella Sampalmieri (Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, Italy)

Abstract

In this paper we generalize the results obtained in [F. Di Michele, M. Mei, B. Rubino, and R. Sampalmieri, Int. J. Numer. Anal. Model., 13:898–925, 2016], where a hybrid model for semiconductor devices has been presented. In particular we consider a more general pressure function, which allows us to account also for the isotropic case. General Dirichlet boundary conditions are also included. In this case we need a different and more restrictive subsonic condition which directly involves the first derivative of the quantum function $Q(x)$. The existence of solutions is obtained by regularizing the problem and performing a suitable vanishing viscosity limit. Also the zero-charge-space limit is discussed and our results are tested on a simple toy model.

Keywords

hybrid quantum hydrodynamic model, isotropic pressure, stationary solutions, existence, uniqueness, classical limit

2010 Mathematics Subject Classification

35L50, 35L60, 35L65, 76R50

Full Text (PDF format)

Received 1 March 2020

Accepted 20 April 2021

Published 7 October 2021