KIT

Abstract

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(eps^4) approximation, where eps is the scaled Planck constant, this leads to local quantum diffusion or quantum hydrodynamic equations. In this talk, we present recent results on the global existence and long-time decay of solutions 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.