**Ansgar Jüngel**

*Dissipative quantum fluid models*

**Pages:** 217-290

**Received:** 29 November 2010

**Accepted in revised form:** 11 February 2011

**Mathematics Subject Classification (2000):** 35A05, 35B40, 35B45, 35G25, 35K30, 35M10, 35Q30, 35Q40, 76Y05, 81S30.

**Keywords:** Quantum transport, Schrödinger equation, Wigner equation, Wigner-Fokker-Planck equation,
quantum drift-diffusion equation, DLSS equation, quantum hydrodynamic equations,
viscous quantum hydrodynamic equations, quantum Navier-Stokes equations.

**Abstract:**
Some macroscopic models for diffusive quantum systems are reviewed.
The model equations are derived from Wigner-Boltzmann equations by a moment method.
Depending on the properties of the collision operator and the scaling of the Wigner equation,
three model classes are presented: the quantum drift-diffusion equations,
inviscid and viscous quantum hydrodynamic equations, and quantum Navier-Stokes equations.
For each of these models, the derivation is sketched and analytical results are reviewed.

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