Projects and Publications

Projects

  • Interfaces, complex structures, and singular limits in continuum mechanics


    (Third Party Funds Group – Overall project)
    Term: 01-04-2018 - 30-09-2022
    Funding source: DFG / Graduiertenkolleg (GRK)

  • Free boundary propagation and noise: analysis and numerics of stochastic degenerate parabolic equations


    (Third Party Funds Single)
    Term: 01-04-2018 - 31-03-2020
    Funding source: Deutsche Forschungsgemeinschaft (DFG), DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
    URL: https://www1.am.uni-erlangen.de/~gruen/
    Abstract

    The porous-medium equation and the thin-film equation are prominent examples of nonnegativity preserving degenerate parabolic equations which give rise to free boundary problems with the free boundary at time t > 0 defined as the boundary of the solution’s support at that time.
    As they are supposed to describe the spreading of gas in a porous-medium or the spreading of a viscous droplet on a horizontal surface, respectively, mathematical results on the propagation of free boundaries become relevant in applications. In contrast to, e.g., the heat equation, where solutions to initial value problems with compactly supported nonnegative initial data
    instantaneously become globally positive, finite propagation and waiting time phenomena are characteristic features of degenerate parabolic equations.
    In this project, stochastic partial differential equations shall be studied which arise from the aforementioned degenerate parabolic equations by adding multiplicative noise in form of source terms or of convective terms. The scope is to investigate the impact of noise on the propagation of free boundaries, including in particular necessary and sufficient conditions for the occurrence
    of waiting time phenomena and results on the size of waiting times. Technically, the project relies both on rigorous mathematical analysis and on numerical simulation.

  • Molecular Communication Systems


    (FAU Funds)
    Term: 01-01-2017 - 31-12-2019
    URL: https://www.idc.tf.fau.de/efi-mcs/
    Abstract

    Neuentstehende Anwendungen in der Biologie, Nanotechnologie und Medizin machen die Vernetzung von Objekten und Maschinen mit Abmessungen im Nano- und Mikrometerbereich erforderlich. Traditionelle elektromagnetische Ansätze für den Entwurf entsprechender Kommunikationssysteme sind für solch kleine Größenordnungen nicht geeignet. In der Natur jedoch ist die Kommunikation zwischen Nano- und Mikro-Objekten, wie z.B. Bakterien und anderen Zellen, weit verbreitet. Dabei kommen oft Signalmoleküle als Informationsträger zum Einsatz, so dass ein natürliches molekulares Kommunikationssystem entsteht. Das Projekt bündelt die an der FAU vorhandene Expertise auf den Gebieten Elektrotechnik, Biologie, Materialwissenschaften, Mathematik und Nanomedizin, um – ausgehend von in der Natur vorkommenden Mechanismen und Prozessen – synthetische molekulare Kommunikationssysteme zu entwerfen und zu implementieren.

  • Diffuse interface models for transport processes at fluidic interfaces


    (Third Party Funds Group – Sub project)
    Overall project: SPP 1506: Fluide Grenzflächen
    Term: 01-06-2013 - 31-10-2017
    Funding source: DFG / Schwerpunktprogramm (SPP)
    Abstract

    In recent years, diffuse interface models turned out to be a promising approach to describe fundamental features of two-phase flow like droplet break-up or coalescence. In the second funding period, novel thermodynamical consistent phase-field models for species transport in two-phase 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 sharp-interface models shall be derived by formal asymptotic analysis.
    For selected diffuse-interface models, existence of solutions and stability of fluidic interfaces will be investigated by rigorous mathematical analysis. Stable numerical schemes shall be formulated and implemented in two and three space dimensions. By numerical simulations, partially guided by the "Leitmassnahme" Taylor-flow, the models shall be validated and further improved. By numerical analysis, convergence shall be established for the prototypical problem of species transport in two-phase flow with general mass densities.

  • Diffuse interface models for transport processes at fluidic interfaces


    (Third Party Funds Group – Sub project)
    Overall project: SPP 1506: Fluide Grenzflächen
    Term: 01-04-2010 - 30-04-2013
    Funding source: DFG / Schwerpunktprogramm (SPP)
    Abstract

    Topological transitions like droplet coalescence or droplet break-up are fundamental features of two-phase flows. In recent years, diffuse interface models turned out to be a promising approach to describe such phenomena. Species transport across fluidic interfaces and the effects exerted by soluble and insoluble surfactants are additional issues of still increasing technological importance. For those phenomena, novel thermodynamically consistent diffuse interface models shall be developed taking in particular general densities into account. Based on rigorous mathematical analysis, existence and qualitative behaviour of solutions will be investigated, this way enhancing the understanding of the fundamental model properties. Starting from energy and entropy inequalities, stable and convergent numerical schemes shall be formulated and implemented in two and three spatial dimensions. By numerical simulations, the models shall be validated and further improved.

  • Fronts and Interfaces in Science and Technology


    (Third Party Funds Group – Sub project)
    Overall project: Fronts and Interfaces in Science and Technology
    Term: 01-02-2010 - 31-12-2013
    Funding source: EU - 7. RP / People / Initial Training Networks (ITN)
    Abstract

    With this network, the universities of Bath, Eindhoven, Erlangen, Haifa (Technion), Madrid (Complutense), Paris (Orsay), Rome (La Sapienza), Zürich and the industrial partners EGIS and SIEMENS AG foster a joint training platform for PhD-students working on analysis and control of interfacial phenomena. Applications range from image processing over reaction-diffusion systems to complex multi-phase flow.
    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 two-phase flow with electrolyte solutions. The goal is to derive thermodynamically consistent diffuse-interface 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 multi-scale 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 road-systems. 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.

Publications