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We show that Quantum Energy-Transport and Quantum Drift-Diffusion models can be derived through diffusion limits of a collisional Wigner equation. The collision operator relaxes to an equilibrium defined through the entropy minimization principle. Both models are shown to be entropic and exhibit fluxes which are related with the state variables through… (More)

- Naoufel Ben Abdallah, Florian Méhats, Christian Schmeiser, Rada-Maria Weishäupl
- SIAM J. Math. Analysis
- 2005

The nonlinear Schrödinger equation with general nonlinearity of polynomial growth and harmonic confining potential is considered. More precisely, the confining potential is strongly anisotropic; i.e., the trap frequencies in different directions are of different orders of magnitude. The limit as the ratio of trap frequencies tends to zero is carried out. A… (More)

- Naoufel Ben Abdallah, Florian Méhats, Olivier Pinaud
- SIAM J. Math. Analysis
- 2005

Asymptotic quantum transport models of a two-dimensional electron gas are presented. The starting point is a singular perturbation of the three-dimensional Schrödinger-Poisson system. The small parameter ε is the scaled width of the electron gas and appears as the lengthscale on which a one dimensional confining potential varies. The rigorous ε → 0 limit is… (More)

- Naoufel Ben Abdallah, François Castella, Fanny Delebecque-Fendt, Florian Méhats
- SIAM Journal of Applied Mathematics
- 2009

We study the limit of the three-dimensional Schrödinger-Poisson system with a singular perturbation, to model a quantum electron gas that is strongly confined near an axis. For well-prepared data, which are polarized on the ground space of the transversal Hamiltonian, the resulting model is the cubic defocusing nonlinear Schrödinger equation. Our main tool… (More)

Weak solutions of a Vlasov-Schrödinger-Poisson system are shown to exist in the stationary and time-dependent situations. This system models the transport and interaction of electrons in a bidimensional electron gas. The particles are assumed to have a wave behaviour in the confinement directions (z) and to behave like point particles in the directions… (More)

We consider the three dimensional gravitational Vlasov-Poisson (GVP) system in both classical and relativistic cases. The classical problem is subcritical in the natural energy space and the stability of a large class of ground states has been derived by various authors. The relativistic problem is critical and displays finite time blow up solutions. Using… (More)

In this paper, we consider two recently derived models: the Quantum Hydrodynamic model (QHD) and the Quantum Energy Transport model (QET). We propose different equivalent formulations of these models and we use a commutator formula for stating new properties of the models. A gauge invariance lemma permits to simplify the QHD model for irrotational flows. We… (More)

In this work, we give an overview of recently derived quantum hydrodynamic and diffusion models. A quantum local equilibrium is defined as a minimizer of the quantum entropy subject to local moment constraints (such as given local mass, momentum and energy densities). These equilibria relate the thermodynamic parameters (such as the temperature or chemical… (More)

- Mohammed Lemou, Florian Méhats
- SIAM J. Scientific Computing
- 2012

We introduce a new micro-macro decomposition of collisional kinetic equations in the specific case of the diffusion limit, which naturally incorporates the incoming boundary conditions. The idea is to write the distribution function f in all its domain as the sum of an equilibrium adapted to the boundary (which is not the usual equilibrium associated with… (More)

We study the gravitational Vlasov Poisson system f t + v · ∇ x f − E · ∇ v f = 0 where E(x) = ∇ x φ(x), ∆ x φ = ρ(x), ρ(x) = R N f (x, v)dxdv, in dimension N = 3, 4. In dimension N = 3 where the problem is subcritical, we prove using concentration com-pactness techniques that every minimizing sequence to a large class of minimization problems attained on… (More)