Michael Kniely

Current Projects

Electro–Energy–Reaction–Diffusion Systems

The first goal of this project is the construction of global solutions to a thermodynamically consistent reaction–diffusion model for a system of charged constituents. While the underlying PDE system is formulated in Onsager form employing the internal energy as the primal thermodynamical variable, the electrostatic potential enters via Poisson's equation and a subtle coupling to the diffusion operator. A similar result has been obtained recently together with Julian Fischer, Katharina Hopf, and Alexander Mielke in the absence of charge carriers (see the project below).
Equilibrium states are found by maximizing the total entropy over all admissible states subject to given values for the total charge and the total energy. This notion of equilibria originates from the monotonicity of the entropy and the conservation of charge and energy as a function of time. Employing the Lagrange multiplier approach, one can prove the existence and uniqueness as well as regularity properties of the equilibrium.
Finally, we aim to establish exponential equilibration of our main model in a quantitative manner using entropy methods. This strategy was already sucessfully applied to models omitting either the temperature dependence or the electrostatic potential.

Renormalized solutions to temperature-dependent PDE systems

The existence of global renormalized solutions to temperature-dependent reaction–diffusion equations has been established in a joint work with Julian Fischer, Katharina Hopf, and Alexander Mielke. We focused on a PDE system in Onsager form for the concentrations of the involved species and the internal energy. Employing the internal energy (and not the temperature) as the main thermodynamic variable, facilitates some modeling issues; for example, reactions typically influence the temperature but leave the internal energy unchanged. Moreover, realistic models also have to account for cross-diffusive behavior between the concentrations and the internal energy to describe, for example, the Soret and the Dufour effect.
Global weak solutions are shown to exist for reactions with limited growth, while global renormalized solutions are constructed in the case that the reactions are not subject to any growth conditions.

Large-time behavior of reaction–diffusion-type equations

Since my PhD studies under the supervision of Klemens Fellner, I have investigated the large-time behavior of various classes of reaction–diffusion equations, which arise in the modeling of semiconductors. By using an appropriate entropy functional and by deriving an inequality between the entropy and the entropy production functional, exponential convergence to equilibrium can be proven.
One of the investigated models considers a two-level system for electrons and holes with an intermediate level for immobile electrons in so-called trapped states. These additional electron states inside the bandgap of the semiconductor are due to impurities within the crystal of the semiconductor and typically facilitate the excitation of an electron from the valence band to the conduction band. Another model takes into account that electrons and holes are electrically charged and, therefore, influencing their motion in a self-consistent manner. This additional coupling is described by Poisson's equation for the electrostatic potential leading to an additional drift term for electrons and holes. Furthermore, exponential equilibration for the self-consistent version of the trapped states model has been shown.

Past Projects

Regularity of random elliptic operators with degenerate coefficients

During my time at TU Dortmund University, I studied regularity properties of degenerate random elliptic operators together with Peter Bella. A central quantity in this context is the minimal radius, which corresponds to the minimal scale at which a certain large-scale regularity holds for functions in the kernel of the degenerate random elliptic operator. The size of the minimal radius can be estimated in terms of stretched exponential moments, which was achieved by Gloria, Neukamm, and Otto in the uniformly elliptic case.
Generalizing the above approach, we were able to derive a stretched exponental moment bound on the minimal radius in the degenerate elliptic situation. Moreover, we deduced corresponding estimates on the corrector and a quantitative two-scale expansion for degenerate random elliptic operators.

Variance reduction for effective energies of random atomic lattices

At IST Austria, I have been working with Julian Fischer on a variance reduction technique for effective energies of random atomic lattices. A frequently employed approach for calculating material properties of random alloys is the method of special quasirandom structures, which aims at constructing a periodic lattice with finite periodicity cell—the representative volume—which resembles the original lattice as closely as possible concerning certain statistics. The desired material properties are then calculated by solving the Thomas–Fermi–von Weizsäcker (TFW) equations for the electron density and by subsequently evaluating the TFW energy functional on the representative volume.
In our work, we proved that this strategy indeed leads to a variance reduction of the TFW energy as compared to calculating the electron density and the TFW energy on a randomly chosen finite volume. We thereby derived the first rigorous variance reduction result related to an application of this selection approach to a nonlinear PDE model.

Material design for photovoltaics within Kohn–Sham DFT

Together with Gero Friesecke, I also investigated a novel optimal control problem in the context of density functional theory (DFT) with possible applications to photovoltaics. Given a nuclear charge distribution as our control, one can calculate—by solving the Kohn–Sham equations—the separation of the generated electron–hole pair after the system has been excited by absorbing a photon. Finding the optimal control then corresponds to designing a material, which gives rise to a maximal separation of positive and negative charge after a light-induced excitation.
We proved the existence of an optimal nuclear charge density in a 3D setting, and we observed a significant transfer of charge in a 1D numerical experiment when the system is excited from the ground state to the first excited state.