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A. Jüngel:
"Global weak solutions to compressible Navier-Stokes equations for quantum fluids";
SIAM Journal on Mathematical Analysis, 42 (2010), 3; 1025 - 1045.

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
third-order differential operator, with the quantum Bohm potential, and a density-dependent viscosity.
The system has been derived by Brull and M´ehats [Derivation of viscous correction terms for the
isothermal quantum Euler model, 2009, submitted] 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.

Siehe englisches Abstract.

Navier-Stokes equations; Bohm potential;quantum hydrodynamic equations