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.
Highlights
This paper studies the relaxation time limit for the quantum Navier–Stokes–Poisson (QNSP) system with linear damping, toward the quantum drift–diffusion equation
In the context of semiconductor devices for instance, quantum transport of electrons can be effectively described by the quantum drift–diffusion (QDD) equation (Roosbroeck 1950), given by
The main purpose of our paper is to rigorously prove the above limit, that is to prove that scaled finite energy weak solutions to (1.1) converge to finite energy weak solutions to (1.2)
Summary
In the three-dimensional torus T3, we consider a compressible, viscous fluid, whose dynamics is prescribed by. The system arises in the macroscopic description of electron transport in nanoscale semiconductor devices (Jüngel 2009), where quantum mechanical effects must be taken into account In this context the dissipative term −ξρu describes collisions between electrons and the semiconductor crystal lattice (see, for instance, Baccarani and Wordeman 1985), and τ = 1/ξ is the relaxation time. The main purpose of our paper is to rigorously prove the above limit, that is to prove that scaled finite energy weak solutions to (1.1) converge to finite energy weak solutions to (1.2) To this aim, in the following we shall refer to (1.4) with initial datum (ρ0, u0) and doping profile g possibly depending in a suitable way on the relaxation parameter as well.
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