Mathematical Challenges of Quantum Transport in Nano-Optoelectronic Systems - Abstract
Jüngel, Ansgar
Quantum fluid models have been recently derived by Degond, Mehats, and Ringhofer from the Wigner-BGK equation by a moment method with a quantum Maxwellian closure. In the $O(epsilon^4)$ approximation, where $epsilon$ is the scaled Planck constant, this leads to local quantum diffusion or quantum hydrodynamic equations. In this talk, we present recent results on the modeling, analysis, and numerical approximation of these models. First, we consider quantum diffusion models containing highly nonlinear fourth-order or sixth-order differential operators. The existence results are obtained from a priori estimates using entropy dissipation methods. Second, a quantum Navier-Stokes model, derived by Brull and Mehats, will be analyzed. This system contains nonlinear third-order derivatives and a density-depending viscosity. The key idea of the mathematical analysis is the reformulation of the system in terms of a new “osmotic velocity” variable, leading to a viscous quantum hydrodynamic model. Surprisingly, this variable has been also successfully employed by Bresch and Desjardins in (non-quantum) viscous Korteweg models.