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

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