Ingenuin Gasser

Quantum hydrodynamics, Wigner transforms, the classical limit

We analyse the classical limit of the quantum hydrodynamic equations as the Planck constant tends to zero. The equations have the form of an Euler system with a constant pressure and a dispersive regularisation term, which (formally) tends to zero in the classical limit. The main tool of the analysis is the exploitation of a kinetic equation, which lies behind the quantum hydrodynamic system. The presented analysis can also be interpreted as an alternative approach to the geometrical optics (WKB)-analysis of the Schrequation.

Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors

Abstract The asymptotic behavior of classical solutions of the bipolar hydrodynamical model for semiconductors is considered in the present paper. This system takes the form of Euler–Poisson with electric field and frictional damping added to the momentum equation. The global existence of classical solutions is proven, and the nonlinear diffusive phenomena is observed in large time in the sense that both densities of electron and hole tend to the same unique nonlinear diffusive wave.

The initial time layer problem and the quasineutral limit in the semiconductor drift-diffusion model

The classical time-dependent drift-diffusion model for semiconductors is considered for small scaled Debye length (which is a singular perturbation parameter). The corresponding limit is carried out on both the dielectric relaxation time scale and the diffusion time scale. The latter is a quasineutral limit, and the former can be interpreted as an initial time layer problem. The main mathematical tool for the analytically rigorous singular perturbation theory of this paper is the (physical) entropy of the system.

Closure conditions for classical and quantum moment hierarchies in the small-temperature limit

Abstract We analyze closure conditions in the small initial temperature limit for classical and quantum mechanical moment hierarchies of corresponding collisionless kinetic equations. Euler equations with a nondiagonal pressure tensor are obtained in the classical case. In the quantum case we consider the cases of fixed and small (scaled) Planck constant and derive Quantum Hydrodynamic equations.

A geometric singular perturbation analysis of detonation and deflagration waves

The existence of steady plane wave solutions of the Navier-Stokes equations for a reacting gas is analyzed. Under the assumption of an ignition temperature the existence of detonation and deflagration waves close to the corresponding waves of the ZND-model is proved in the limit of small viscosity, heat conductivity, and diffusion. The method is constructive, since the classical solutions of the ZND-model serve as singular solutions in the context of geometric singular perturbation theory. The singular solutions consist of orbits on which the dynamics are slow-driven by chemical reaction and of orbits on which the dynamics are fast-driven by gasdynamic shocks. The approach is geometric and leads to a clear, complete picture of the existence, structure, and asymptotic behavior of detonation and deflagration waves.

Gas Pipeline Models Revisited: Model Hierarchies, Nonisothermal Models, and Simulations of Networks

By using asymptotic analysis we derive most of the known and also new nonisothermal pipeline models starting from transient gas equations. We introduce proper scalings to identify valid regimes for the derived models and extend them to networks. Finally, we perform numerical simulations on a single pipe as well-posed as on a small network. We compare both isothermal and nonisothermal flow and pressure predictions with results obtained from the literature.

The quantum hydrodynamic model for semiconductors in thermal equilibrium

Abstract. The thermal equilibrium states of the quantum hydrodynamic model are analyzed. We show existence of solutions, perform the classical limit and investigate the uniqueness of the solutions. Numerical examples show the difference to the classical thermal equilibrium states.

The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors

The combined relaxation and vanishing Debye length limit for the hydrodynamic model for semiconductors is considered in both the unipolar and the bipolar case. The resulting limit problems are non-linear drift driven hyperbolic equations. We make use of non-standard entropy functions and the related entropy productions in order to obtain uniform estimates. In the bipolar case additional time-dependent L∞-type estimates, available from the existence theory, are needed in order to control the entropy production terms. Finally, strong convergence of the electric field allows the limit towards the limiting problem. Copyright

Radiation models for thermal flows at low Mach number

Simplified approximate models for radiation are proposed to study thermal effects in low Mach flow in open tunnels. The governing equations for fluid dynamics are derived by applying a low Mach asymptotic in the compressible Navier-Stokes problem. Based on an asymptotic analysis we show that the integro-differential equation for radiative transfer can be replaced by a set of differential equations which are independent of angle variable and easy to solve using standard numerical discretizations. As an application we consider a simplified fire model in vehicular tunnels. The results presented in this paper show that the proposed models are able to predict temperature in the tunnels accurately with low computational cost.

Regularity and propagation of moments in some nonlinear Vlasov systems

We introduce a new approach to prove the regularity of solutions to transport equations of the Vlasov type. Our approach is mainly based on the proof of propagation of velocity moments, as in a previous paper by Lions and Perthame [16]. We combine it with Moment Lemmas which assert that, locally in space, velocity moments can be gained from the kinetic equation itself. We apply our theory to two cases. First, to the Vlasov-Poisson system, and we solve a long standing conjecture, namely the propagation of any moment larger than two. Next, to the Vlasov-Stokes system where we prove the same result for fairly singular kernels.