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A. Jüngel:
"Modeling and analysis of diffusive quantum fluid equations";
Talk: Workshop Theory and numerics for nonlinear Schroedinger equations, Nizza, Universität Nizza (invited); 2015-01-12 - 2015-01-14.

Quantum fluid equations are an interesting alternative to more sophisticated but computationally very expensive quantum transport models. In this talk, two classes of quantum fluid models are analyzed: quantum drift-diffusion and quantum Navier-Stokes equations. These models can be formally derived from a collisional Wigner equation in the diffusion limit. The quantum Navier-Stokes equations can be also obtained from a dissipative Euler-Lagrange equation with a quantum action functional on the space of probability measures, which allows one to relate quantum fluid models with the (one-particle) Schroedinger equation. The main mathematical challenge of the diffusive quantum models is the treatment of the highly nonlinear higher-order differential operators. The existence of global weak solutions is proved by using new analytical tools like entropy methods, systematic integration of parts, and osmotic velocity variable transformation.