TY - JOUR AB - The global existence of renormalised solutions and convergence to equilibrium for reaction-diffusion systems with non-linear diffusion are investigated. The system is assumed to have quasi-positive non-linearities and to satisfy an entropy inequality. The difficulties in establishing global renormalised solutions caused by possibly degenerate diffusion are overcome by introducing a new class of weighted truncation functions. By means of the obtained global renormalised solutions, we study the large-time behaviour of complex balanced systems arising from chemical reaction network theory with non-linear diffusion. When the reaction network does not admit boundary equilibria, the complex balanced equilibrium is shown, by using the entropy method, to exponentially attract all renormalised solutions in the same compatibility class. This convergence extends even to a range of non-linear diffusion, where global existence is an open problem, yet we are able to show that solutions to approximate systems converge exponentially to equilibrium uniformly in the regularisation parameter. AU - Fellner, Klemens AU - Fischer, Julian L AU - Kniely, Michael AU - Tang, Bao Quoc ID - 10550 JF - Journal of Nonlinear Science SN - 0938-8974 TI - Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion VL - 33 ER - TY - JOUR AB - We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear. AU - Fischer, Julian L AU - Hopf, Katharina AU - Kniely, Michael AU - Mielke, Alexander ID - 10547 IS - 1 JF - SIAM Journal on Mathematical Analysis KW - Energy-Reaction-Diffusion Systems KW - Cross Diffusion KW - Global-In-Time Existence of Weak/Renormalised Solutions KW - Entropy Method KW - Onsager System KW - Soret/Dufour Effect SN - 0036-1410 TI - Global existence analysis of energy-reaction-diffusion systems VL - 54 ER - TY - JOUR AB - In this paper, we establish convergence to equilibrium for a drift–diffusion–recombination system modelling the charge transport within certain semiconductor devices. More precisely, we consider a two-level system for electrons and holes which is augmented by an intermediate energy level for electrons in so-called trapped states. The recombination dynamics use the mass action principle by taking into account this additional trap level. The main part of the paper is concerned with the derivation of an entropy–entropy production inequality, which entails exponential convergence to the equilibrium via the so-called entropy method. The novelty of our approach lies in the fact that the entropy method is applied uniformly in a fast-reaction parameter which governs the lifetime of electrons on the trap level. Thus, the resulting decay estimate for the densities of electrons and holes extends to the corresponding quasi-steady-state approximation. AU - Fellner, Klemens AU - Kniely, Michael ID - 7866 JF - Journal of Elliptic and Parabolic Equations SN - 22969020 TI - Uniform convergence to equilibrium for a family of drift–diffusion models with trap-assisted recombination and the limiting Shockley–Read–Hall model VL - 6 ER - TY - JOUR AB - In the computation of the material properties of random alloys, the method of 'special quasirandom structures' attempts to approximate the properties of the alloy on a finite volume with higher accuracy by replicating certain statistics of the random atomic lattice in the finite volume as accurately as possible. In the present work, we provide a rigorous justification for a variant of this method in the framework of the Thomas–Fermi–von Weizsäcker (TFW) model. Our approach is based on a recent analysis of a related variance reduction method in stochastic homogenization of linear elliptic PDEs and the locality properties of the TFW model. Concerning the latter, we extend an exponential locality result by Nazar and Ortner to include point charges, a result that may be of independent interest. AU - Fischer, Julian L AU - Kniely, Michael ID - 8697 IS - 11 JF - Nonlinearity SN - 09517715 TI - Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizsäcker model VL - 33 ER - TY - JOUR AB - We present and study novel optimal control problems motivated by the search for photovoltaic materials with high power-conversion efficiency. The material must perform the first step: convert light (photons) into electronic excitations. We formulate various desirable properties of the excitations as mathematical control goals at the Kohn-Sham-DFT level of theory, with the control being given by the nuclear charge distribution. We prove that nuclear distributions exist which give rise to optimal HOMO-LUMO excitations, and present illustrative numerical simulations for 1D finite nanocrystals. We observe pronounced goal-dependent features such as large electron-hole separation, and a hierarchy of length scales: internal HOMO and LUMO wavelengths < atomic spacings < (irregular) fluctuations of the doping profiles < system size. AU - Friesecke, Gero AU - Kniely, Michael ID - 6762 IS - 3 JF - Multiscale Modeling and Simulation SN - 15403459 TI - New optimal control problems in density functional theory motivated by photovoltaics VL - 17 ER -