Ansgar Jüngel

Transport Equations for Semiconductors

Basic Semiconductor Physics.- Microscopic Semi-Classical Models.- Derivation of Macroscopic Equations.- Collisionless Models.- Scattering Models.- Macroscopic Semi-Classical Models.- Drift-Diffusion Equations.- Energy-Transport Equations.- Spherical Harmonics Expansion Equations.- Diffusive Higher-Order Moment Equations.- Hydrodynamic Equations.- Microscopic Quantum Models.- The Schr#x00F6 dinger Equation.- The Wigner Equation.- Macroscopic Quantum Models.- Quantum Drift-Diffusion Equations.- Quantum Diffusive Higher-Order Moment Equations.- Quantum Hydrodynamic Equations.

GLOBAL WEAK SOLUTIONS TO COMPRESSIBLE NAVIER-STOKES EQUATIONS FOR QUANTUM FLUIDS

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 Mehats [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 s...

A system of parabolic equations in nonequilibrium thermodynamics including thermal and electrical effects

Abstract The time-dependent equations for a charged gas or fluid consisting of several components, exposed to an electric field, are considered. These equations form a system of strongly coupled, quasilinear parabolic equations which in some situations can be derived from the Boltzmann equation. The model uses the duality between the thermodynamic fluxes and the thermodynamic forces. Physically motivated mixed Dirichlet-Neumann boundary conditions and initial conditions are prescribed. The existence of weak solutions is proven. The key of the proof is ( i ) a transformation of the problem by using the entropic variables, or electro-chemical potentials, which symmetrizes the equations, and ( ii ) a priori estimates obtained by using the entropy function. Finally, the entropy inequality is employed to show the convergence of the solutions to the thermal equilibrium state as the time tends to infinity.

Global Nonnegative Solutions of a Nonlinear Fourth-Order Parabolic Equation for Quantum Systems

The existence of nonnegative weak solutions globally in time of a nonlinear fourth-order parabolic equation in one space dimension is shown. This equation arises in the study of interface fluctuations in spin systems and in quantum semiconductor modeling. The problem is considered on a bounded interval subject to initial and Dirichlet and Neumann boundary conditions. Further, the initial datum is assumed only to be nonnegative and to satisfy a weak integrability condition. The main difficulty of the existence proof is to ensure that the solutions stay nonnegative and exist globally in time. The first property is obtained by an exponential transformation of variables. Moreover, entropy-type estimates allow for the proof of the second property. Results concerning the regularity and long-time behavior are given. Finally, numerical experiments underlining the preservation of positivity are presented.

THE DERRIDA-LEBOWITZ-SPEER-SPOHN EQUATION : EXISTENCE, NONUNIQUENESS, AND DECAY RATES OF THE SOLUTIONS

The logarithmic fourth-order equation

Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model

\partial_t u + \frac12\sum_{i,j=1}^d\partial_{ij}^2(u\partial_{ij}^2\log u) = 0,

Numerical Discretization of Energy-Transport Models for Semiconductors with Nonparabolic Band Structure

called the Derrida–Lebowitz–Speer–Spohn equation, with periodic boundary conditions is analyzed. The global-in-time existence of weak nonnegative solutions in space dimensions

Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas

d\leq 3

An algorithmic construction of entropies in higher-order nonlinear PDEs

is shown. Furthermore, a family of entropy-entropy dissipation inequalities is derived in arbitrary space dimensions, and rates of the exponential decay of the weak solutions to the homogeneous steady state are estimated. The proofs are based on the algorithmic entropy construction method developed by the authors and on an exponential variable transformation. Finally, an example for nonuniqueness of the solution is provided.

A Positivity-Preserving Numerical Scheme for a Nonlinear Fourth Order Parabolic System

Summary. A positivity-preserving numerical scheme for a strongly coupled cross-diffusion model for two competing species is presented, based on a semi-discretization in time. The variables are the population densities of the species. Existence of strictly positive weak solutions to the semidiscrete problem is proved. Moreover, it is shown that the semidiscrete solutions converge to a non-negative solution of the continuous system in one space dimension. The proofs are based on a symmetrization of the problem via an exponential transformation of variables and the use of an entropy functional.