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- dissipative Schroedinger-type operators (4)
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Project
We regard drift-diffusion equations for semiconductor devices in Lebesgue spaces. To that end we reformulate the (generalized) van Roosbroeck system as an evolution equation for the potentials to the driving forces of the currents of electrons and holes. This evolution equation falls into a class of quasi-linear parabolic systems which allow unique, local in time solution in certain Lebesgue spaces. In particular, it turns out that the divergence of the electron and hole current is an integrable function. Hence, Gauss' theorem applies, and gives the foundation for space discretization of the equations by means of finite volume schemes. Moreover, the strong differentiability of the electron and hole density in time is constitutive for the implicit time discretization scheme. Finite volume discretization of space, and implicit time discretization are accepted custom in engineering and scientific computing. ---This investigation puts special emphasis on non-smooth spatial domains, mixed boundary conditions, and heterogeneous material compositions, as required in electronic device simulation.
An one-dimensional Kohn-Sham system for spin particles is considered which effectively describes semiconductor nanostructures and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schrödinger operator with effective Kohn-Sham potential and obtain $W^1,2$-bounds of the associated particle density operator. Afterwards, compactness and continuity results allow to apply Schauder's fixed point theorem. In case of vanishing exchange-correlation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero.
Building on a transformation formula that we previously found in connection with Landau-Lifshitz-Gilbert equations, we present a strategy for utilizing compensated compactness methods in the context of Ginzburg-Landau approximation for harmonic maps and related problems in magnetism and superconductivity. Its applicability is illustrated by a number of new examples.
The LSW model with encounters has been suggested by Lifshitz and
Slyozov as a regularization of their classical mean-field model for
domain coarsening to obtain universal self-similar long-time
behavior. We rigorously establish that an exponentially decaying
self-similar solution to this model exist, and show that this
solutions is isolated in a certain function space. Our proof relies
on setting up a suitable fixed-point problem in an appropriate
function space and careful asymptotic estimates of the solution to a
corresponding homogeneous problem.
Propagation of short optical pulses in a nonlinear dispersive medium is considered without the use of slow envelope and
unidirectional propagation approximations. The existence of uniformly moving solitary solutions is predicted in the anomalous
dispersion domain. A four-parametric family of such solutions is found that contains the classical envelope soliton in the limit of
large pulse durations. In the opposite limit we get another family member, which in contrast to the envelope soliton strongly depends on nonlinearity model and represents the shortest and the most intense pulse which can propagate in a stationary manner.
The nonlinear Schrödinger equation based on the Taylor approximation of the material dispersion can become invalid for ultrashort and few-cycle optical pulses. Instead, we use a rational fit to the dispersion function such that the resonances are naturally accounted for. This approach allows us to derive a simple non-envelope model for short pulses propagating in one spatial dimension. This model is further investigated numerically and analytically.
In ferromagnetic materials, the gyrotropic nature of Landau-Lifshitz-Gilbert dynamics and anisotropic effects from stray-field interaction lead in certain regimes effectively to a wave-type dynamic equation. In the case of soft thin films and small Gilbert damping, we investigate the motion of Néel walls and prove the existence of traveling wave solutions under a small constant forcing.