Abstract. 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 Méhats [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.Key words. Compressible Navier-Stokes equations, quantum Bohm potential, density-dependent viscosity, global existence of solutions, viscous quantum hydrodynamic equations, third-order derivative, energy estimates. Gardner [20] from the Wigner equation by a moment method. More recently, dissipative quantum fluid models have been proposed. For instance, the moment method applied to the Wigner-Fokker-Planck equation leads to viscous quantum Euler models [21], and a Chapman-Enskog expansion in the Wigner equation leads under certain assumptions to quantum Navier-Stokes equations [10]. In this paper, we will reveal a connection between these two models by introducing an effective velocity variable, first used in capillary Korteweg-type models [5], and we will prove the global existence of weak solutions to the multidimensional initial-value problems for any finite-energy initial data. AMSIn the following, we describe the two dissipative quantum systems studied in this paper. The barotropic quantum Navier-Stokes equations for the particle density n and the particle velocity u read as

Abstract. A hierarchy of diffusive partial differential equations is derived by a moment method and a Chapman-Enskog expansion from the semiconductor Boltzmann equation assuming dominant elastic collisions. The moment equations are closed by employing the entropy maximization principle of Levermore. The new hierarchy contains the well-known drift-diffusion model, the energy-transport equations, and the six-moments model of Grasser et al. It is shown that the diffusive models are of parabolic type. Two different formulations of the models are derived: a drift-diffusion formulation, allowing for a numerical decoupling, and a symmetric formulation in generalized dual entropy variables, inspired by nonequilibrium thermodynamics. An entropy inequality (or H-theorem) follows from the latter formulation.