Current projects
Free boundary propagation and noise: analysis and numerics of stochastic degenerate parabolic equationsDFG Research Grant Dates: 20182020Participants: Prof. Dr. Günther Grün, Hubertus Grillmeier, M.Sc.

Interfaces, ComplexStructures, and Singular Limits in Continuum MechanicsDFG Research Training Group GRK 2339Dates: 20182022 (first funding period), Participants at FAU: Proff. Bänsch, Burger, Grün (CoSpokesperson), Knabner, Pratelli, Neuss Radu, and Dr. Ray. The DFG Research Training Group IntComSin is a joint doctorate program of FAU and University of Regensburg in Applied Mathematics. The research focuses on many aspects of advanced mathematical modeling, analysis and numerics with the perspective to understand complex phenomena observed in fluid mechanics, material engineering, biological systems and other applied sciences. Typically interfaces, multiple scales/fields and small parameters (singular limits) are the core features of the problems to be studied. The doctoral program offers a structured course program in partial differential equations, calculus of variations, numerical analysis, scientific computing, mathematical modeling and professional skills. The topics explored in the doctoral projects address: • interfaces (twophase flows, transport processes at interfaces, fluidic and elastic effects in membranes, shape optimisation, fluidstructure interactions), • complex structures ( multiple scales, homogenisation of porous media, micromacro models for complex fluids, microstructures generated by nonconvex variational problems), • singular limits and dimension reduction (thin film limits, plates/shells and beams, asymptotic limits in phase field models). For further information, see www.uniregensburg.de/IntComSin  
Completed projects
Diffusive interface models for transport processes at fluidic interfaces (Part 2)DFG Priority Programme SPP 1056 "Transport processes at fluidic interfaces"  
In recent years, diffuse interface models turned out to be a promising approach to describe fundamental features of twophase flow like droplet breakup or coalescence. In the second funding period, novel thermodynamical consistent phasefield models for species transport in twophase flow shall be derived with an emphasis on soluble surfactants. Additional phenomena  ranging from microscale effects like molecule orientation over thermal effects to electrostatic interactions  shall be included as well. On this basis, new sharpinterface models shall be derived by formal asymptotic analysis. 
Fronts and Interfaces in Science and Technology (FIRST) / Marie Curie Initial Training Networks  
FAU is involved in three projects, guided by Proff. Grün, Knabner, and Leugering. The first one is concerned with the effects electric fields have on twophase flow with electrolyte solutions. The goal is to derive thermodynamically consistent diffuseinterface models for general mass densities and ion distributions and to prove existence and regularity of solutions. The second one is a tandem project with Prof. Peletier (TU Eindhoven) devoted to contaminant flow in porous media. There is experimental evidence that attachment to colloids strongly enhances contaminant transport. Derivation and analysis of appropriate multiscale models are in the focus of this project. Prof. Leugering's project  jointly with Prof. Coron (University Pierre et Marie Curie, Paris)  is devoted to optimal control and stabilization of flow of gas, water, and traffic in networked pipe and roadsystems. It focusses on reachability and stabilizability properties under constraints both in states and controls and on the derivation of appropriate sensitivities for a numerical treatment of optimal controls for systems of realistic size.

Diffusive interface models for transport processes at fluidic interfaces (Part 1)DFG Priority Programme SPP 1056 "Transport processes at fluid interfaces"  

Mathematical Analysis of Models Describing the Evolution of Liquid Patterns on Material Interfaces (Part 2 and Part 3)DFG Priority Programme SPP 1052 "Benetzung und Strukturbildung an Grenzflächen""  
The goal of the project is to analyse, to evaluate, and to improve mathematical models for the dewetting of thin liquid films on homogeneous surfaces and the formation of fluid structures on inhomogeneous substrates. During the second period of the Schwerpunktprogramm, investigations on evaporation and condensation processes will be included. The proposed project consists of three main parts which can be summarized as follows:Modelling: Based on lubrication approximation, evolution equations for the height of condensing fluids on inhomogeneous surfaces shall be derived.Analysis: Methods from the calculus of variations and the theory of partial differential equations shall be used to obtain results on existence and qualitative behaviour of solutions to the corresponding evolution equations for film height h and pressure p. Besides their obvious importance for a better understanding of the asymptotic behaviour of solutions, these theoretical results are also the key ingredient to formulate fast and reliable algorithms for numerical sumulations.Numerical simulations: During the last two years, G. Grün and M. Rumpf succeded in developing and analysing a finiteelement/finitevolume scheme which drastically reduces the computation time for the simulation of spreading phenomena. Based on this scheme, a general finiteelement solver shall be designed to enable numerical simulations of all the phenomena mentioned above. The evaluation will be performed by comparison with experimental data; it strongly depends on an intense cooperation with experimentalists inside the Schwerpunktprogramm. Wetting Kaleidoscope>> 
DAAD project "Mathematical Analysis and Numerical Simulation of Dynamic Electrowetting" ("projektbezogener Personenaustausch Spanien")  
Electric fields may influence the wetting behaviour of charged droplets. This effect may be used to manipulate droplet motion and to enforce droplet coalescence or breakup. It is summarized under the notion of electrowetting. First studied in the late 19th century by Lippmann and others, very recently this effect gained the interest of physicists and engineers on account of a variety of applications in microfluidics ("labonachip"). However, mathematical modeling of this effect is still in its childhood  the dependency of contact angles on the applied voltage is modeled by the so called Lippmann formula which may hold true at most in a small voltage regime. In this project, new thermodynamically consistent models shall be derived to describe the evolution of droplet, voltage and thereby to shed light on the mechanisms underlying electrowetting. Existence of solutions shall be proven and efficient numerical schemes shall be developed as well. Poster>>

Complex rheologiesProject in SFB 611 "Singular phenomena and scaling in mathematical models", University of Bonn (together with Christiane Helzel and Felix Otto)  
In this project, models for complex flow phenomena are investigated, for instance pattern formation in sheared suspensions, moving contact lines or RayleighBénard convection. By means of rigorous analysis and numerical simulations, properties of the models will be predicted. The aforementioned specific models may be used to develop new methods for PDE in general.

Scaling laws and their crossovers: global analysis of rheological processesProject in SFB 611 "Singular phenomena and scaling in mathematical models", University of Bonn (together with Felix Otto)  
The goal of this project is to investigate scaling laws in rheological processes. More precisely, analytical tools to rigorously infer such scaling laws shall be developed. A common feature of the friction dominated processes to be investigated is their gradient flow structure. This structure translates the physical free energy and the underlying dissipation mechanism into the mathematical terms of functional and metric tensor, respectively. The idea is to use global PDE methods, rather than local methods, like matched asymptotic expansions, to analyze the scaling laws. Phenomena to be studied include spreading of viscous films, caseII diffusion of solvents in polymeric solids, and the demixing of polymer solutions and spongelike pattern formation.

DFG project "Mathematical Analysis of Models Describing the Evolution of Liquid Patterns on Material Interfaces ")Project in SFB 611 DFG Priority Programme SPP 1052 "Benetzung und Strukturbildung an Grenzflächen"  
This project is concerned with the analytical description of the qualitative and asymptotic behaviour of solutions to certain higher order parabolic differential equations arising in modelling phenomena of wetting and film rupture. It is planned to develop efficient numerical tools  in particular algorithms based on FiniteVolumeMethods  to enable computer simulations of film rupture in space dimension N = 3.
