Relaxation limit from the Quantum-Navier-Stokes equations to the Quantum Drift Diffusion equation

TL;DR

This paper rigorously derives the quantum drift-diffusion equation as a relaxation limit of the Quantum-Navier-Stokes system, using energy and entropy estimates without assuming prior properties of the limit.

Contribution

It provides a novel mathematical proof of the relaxation limit from Quantum-Navier-Stokes to quantum drift-diffusion equations without restrictive assumptions.

Findings

Established the relaxation limit rigorously.

Revealed how energy and entropy estimates behave in the limit.

Provided an alternative proof for existence of solutions to the quantum drift-diffusion equation.

Abstract

The relaxation-time limit from the Quantum-Navier-Stokes-Poisson system to the quantum drift-diffusion equation is performed in the framework of finite energy weak solutions. No assumptions on the limiting solution are made. The proof exploits the suitably scaled a priori bounds inferred by the energy and BD entropy estimates. Moreover, it is shown how from those estimates the Fisher entropy and free energy estimates associated to the diffusive evolution are recovered in the limit. As a byproduct, our main result also provides an alternative proof for the existence of finite energy weak solutions to the quantum drift-diffusion equation.