Stability of a one-dimensional full viscous quantum hydrodynamic system

TL;DR

This paper investigates the existence, uniqueness, and exponential stability of steady-state solutions in a one-dimensional viscous quantum hydrodynamic system, employing regularization and entropy methods.

Contribution

It establishes the first rigorous results on steady-state existence, uniqueness, and stability for this complex quantum hydrodynamic model.

Findings

Existence and uniqueness of steady-state solutions are proven.

Local-in-time existence of solutions is demonstrated using viscous regularization.

Steady-state solutions are shown to be exponentially stable.

Abstract

A full viscous quantum hydrodynamic system for particle density, current density, energy density and electrostatic potential coupled with a Poisson equation in one dimensional bounded intervals is studied. First, the existence and uniqueness of a steady-state solution to the quantum hydrodynamic system is established. Then, utilizing the fact that the third order perturbation term has an appropriate sign, the local-in-time existence of the solution is investigated by introducing a fourth order viscous regularization and using the entropy dissipation method. In the end, the exponential stability of the steady-state solution is shown by constructing a uniform a-priori estimate.