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A. Jüngel:
"Global weak solutions to compressible Navier-Stokes equations for quantum fluids";
in: "ASC Report 41/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

The global-in-time existence of weak solutions to the barotropic compressible quantum
Navier-Stokes equations in a three-dimensional torus for large data is proved. The model consists
of the mass conservation equation and a momentum balance equation, including a nonlinear thirdorder
differential operator, with the quantum Bohm potential, and a density-dependent viscosity.
The system has been derived by Brull and M´ehats [10] from a Wigner equation using a moment
method and a Chapman-Enskog expansion around the quantum equilibrium. The main idea of the
existence analysis is to reformulate the quantum Navier-Stokes equations by means of a so-called
effective velocity involving a density gradient, leading to a viscous quantum Euler system. The advantage
of the new formulation is that there exists a new energy estimate which implies bounds on
the second derivative of the particle density. The global existence of weak solutions to the viscous
quantum Euler model is shown by using the Faedo-Galerkin method and weak compactness techniques.
As a consequence, we deduce the existence of solutions to the quantum Navier-Stokes system
if the viscosity constant is smaller than the scaled Planck constant.

Compressible Navier-Stokes equations, quantum Bohm potential, density-dependent

http://www.asc.tuwien.ac.at/preprint/2009/asc41x2009.pdf