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New quantum hydrodynamic equations are derived from a Wigner-Boltzmann model, using the quantum entropy minimization method recently developed by Degond and Ringhofer. The model consists of conservation equations for the carrier, momentum, and energy densities. The derivation is based on a careful expansion of the quantum Maxwellian in powers of the Planck constant. In difference to the standard quantum hydrodynamic equations derived by Gardner, the new model includes vorticity terms and a dispersive term for the velocity. Numerical current-voltage characteristics of a one-dimensional resonant tunneling diode for both the new quantum hydrodynamic equations and Gardner's model are presented. The numerical results indicate that the dispersive velocity term regularizes the solution of the system.

Abstract. Viscous stabilizations of the quantum hydrodynamic equations are studied. The quantum hydrodynamic model consists of the conservation laws for the particle density, momentum, and energy density, including quantum corrections from the Bohm potential. Two different stabilizations are analyzed. First, viscous terms are derived using a Fokker-Planck collision operator in the Wigner equation. The existence of solutions (with strictly positive particle density) to the isothermal, stationary, one-dimensional viscous model for general data and nonhomogeneous boundary conditions is shown. The estimates depend on the viscosity and do not allow to perform the inviscid limit. Second, the numerical viscosity of the second upwind finite-difference discretization of the inviscid quantum hydrodynamic model is computed. Finally, numerical simulations using the non-isothermal, stationary, one-dimensional model of a resonant tunnelling diode show the influence of the viscosity on the solution.Key words. quantum hydrodynamics, viscous quantum hydrodynamics, existence of stationary solutions, numerical dispersion, numerical viscosity, resonant tunnelling diode, semiconductors AMS subject classifications. 76Y05, 35J40, 65N05 IntroductionThe fluiddynamical formulation of the Schrödinger equation, which has been known since the early years of quantum mechanics [32], has been used in many applications for quantum plasmas [34] or quantum semiconductor devices [12,14]. Quantum effects appear through terms involving the Bohm potential and containing third-order derivatives which express the dispersive structure of the equations. These terms are mathematically difficult to handle, and in fact there are only partial analytical and numerical results. This work is devoted to an analytical and numerical study of these equations when they contain additional viscosity terms originating either from particle collisions or from an upwind numerical discretization of the equations.More specifically, we consider the scaled stationary quantum hydrodynamic (QHD) equations for the particle density n, the current density J, and the energy density ne,

For two-phase flow in porous media, the natural medium heterogeneity necessarily gives rise to capillary nonequilibrium effects. The relaxation to the equilibrium is a slow process which should be introduced in macroscopic flow models. Many nonequilibrium models are based on a phenomenological approach. At the same time there exists a rigorous mathematical way to develop the nonequilibrium equations. Its formalism, developed by Bourgeat and Panfilov [Computational Geosciences 2, 191 (1998)], is based on the homogenization of the microscale flow equations over medium heterogeneities. In contrast with the mentioned paper, in which the case of a sufficiently fast relaxation was analyzed, we consider the case of long relaxation, which leads to the appearance of long-term memory on the macroscale. Due to coupling between the nonlinearity and nonlocality in time, the macroscopic model remains, however, incompletely homogenized, in the general case. At the same time, frequently only the relationship for the nonequilibrium capillary pressure is of interest for applications. In the present paper, we obtain such an exact relationship in two different independent forms for the case of long-term memory. This relationship is more general than that obtained by Bourgeat and Panfilov. In addition, we prove the comparison theorem which determines the upper and lower bounds for the macroscopic model. These bounds represent linear flow models, which are completely homogenized. The results obtained are illustrated by numerical simulations.

Abstract. The global-in-time existence of weak nonnegative solutions to a sixth-order nonlinear parabolic equation in one space dimension with periodic boundary conditions is proved. The equation arises from an approximation of the quantum drift-diffusion model for semiconductors and describes the evolution of the electron density in the semiconductor crystal. The existence result is based on two techniques. First, the equation is reformulated in terms of exponential and power variables, which allows for the proof of nonnegativity of solutions. The existence of solutions to an approximate equation is shown by fixed-point arguments. Second, a priori bounds uniformly in the approximation parameters are derived from the algorithmic entropy construction method which translates systematic integration by parts into polynomial decision problems. The a priori estimates are employed to show the exponential time decay of the solution to the constant steady state in the L 1 norm with an explicit decay rate. Furthermore, some numerical examples are presented.Key words. Sixth-order parabolic equation, quantum semiconductors, global existence of solutions, algorithmic entropy construction, convergence of transport plans, long-time behavior of the solutions.