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A. Jüngel:
"Higher-order nonlinear PDEs for quantum fluids";
Talk: Mathematisches Kolloquium Universidad Granada, Granada (invited); 2011-02-24.

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.

Siehe englisches Abstract.

Quantum diffusion; quantum Navier-Stokes