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Research

Interests

  • Nonlinear (multi-scale) partial differential systems (arising in fluid dynamics and evolving microstructures): Weak solvability, regularity, boundedness, etc.
  • Degenerating parabolic equations: Existence of weak solutions by regularization
  • Mathematical modeling of biological processes (biofilms, chemotaxis, etc.) in evolving microstructures: Homogenization in a level-set framework
  • Numerical analysis of nonlinear (multi-scale) PDE systems: (Upwind, mixed) finite element methods

Publications

Projects

  • Multiscale modeling with evolving microstructure: An approach to
    emergence in the rhizosphere via effective soil functions

    (Third Party Funds Group – Sub project)

    Overall project: DFG Priority Programme 2089 “Rhizosphere Spatiotemporal Organisation – a Key to Rhizosphere Functions”
    Term: 01-02-2019 - 31-01-2022
    Funding source: DFG / Schwerpunktprogramm (SPP)
    URL: https://www.ufz.de/spp-rhizosphere/index.php?en=46495

    The self-organization of the aggregates in the rhizosphere by various
    attracting forces influenced by geochemistry, and microbiology shall
    be studied by a novel, comprehensive model. This model should
    account for processes on the microscale (single roots, pore scale),
    and then be upscaled to the root system scale (macroscale) by
    mathematical homogenization. This goal exceeds the functional range
    of existing models for aggregation and needs the introduction of an
    explicit phase of mucilage, and attachment properties of root hairs in
    the rhizosheath. The project aims at the development of a mechanistic modeling approach that allows for dynamic structural reorganization of the rhizosphere at the single root scale and couples this evolving microscale model to the root system scale including the inference of soil functions. This means that we do not assume a static rhizosphere but develop a tool that is capable to dynamically track this zone on the basis of the underlying spatiotemporal aggregegate formation and geochemical patterns. The collaboration with experimental groups – analyzing CT images in various moisture and growth conditions - the Central Experiment will allow to derive the properties of the mucilage phase, the pore structure and thus the
    influence of root hairs on aggregation mechanisms.

  • German-Norwegian collaborative research support scheme

    (Third Party Funds Single)

    Term: 01-01-2016 - 31-12-2017
    Funding source: Deutscher Akademischer Austauschdienst (DAAD)

    Homogenisierung reaktiven Transports in variablen Mikrostrukturen

Articel

  • Schulz R.: “Clogging and Degenerate Equations for Flow and Transport in Evolving Porous Media”, in preparation.
  • Schulz R., Ray N., Zech S., Rupp A., Knabner P.: “Beyond Kozeny-Carman: Predicting the permeability in porous media”, submitted
  • Schulz R.: “Crystal precipitation and dissolution in a porous medium: Evolving microstructure and perforated solid matrix”, Special Topics Rev. Porous Media, accept
  • Schulz R. (2019): “Biofilm modeling in evolving porous media with Beavers-Joseph condition”, Z. Angew. Math. Mech., doi 10.1002/zamm.201800123.
  • Ray N., Rupp A., Schulz R., Knabner P. (2018): “Old and New Approaches Predicting the Diffusion in Porous Media”, Transp. Porous Med., doi 10.1007/s11242-018-1099-x.
  • Farwig R., Schulz R., Taniuchi Y. (2018): “Spatial asymptotic profiles of solutions to the Navier-Stokes system in a rotating frame with fast decaying data”, Hokkaido Math. J. 47 (3), 501-529.
  • Schulz R. (2017): “Boundedness in a biofilm-chemotaxis model in evolving porous media”, Math. Model. Anal. 22 (6), 852-869, doi 10.3846/13926292.2017.1389772.
  • Schulz R., Knabner P. (2017): “An effective model for biofilm growth made by chemotactical bacteria in evolving porous media”, SIAM J. Appl. Math. 77 (5), 1653-1677, doi 10.1137/16M108817X.
  • Schulz R., Knabner P. (2017 published online 2016): “Derivation and analysis of an effective model for biofilm growth in evolving porous media”, Math. Method Appl. Sci. 40 (8), 2930-2948, doi 10.1002/mma.4211.
  • Schulz R., Ray N., Frank F., Mahato H., Knabner P. (2017 published online 2016): “Strong solvability up to clogging of an effective diffusion-precipitation model in an evolving porous medium”, Eur. J. Appl. Math. 28 (2), 179-207, doi 10.1017/S0956792516000164.
  • Farwig R., Schulz R., Yamazaki M. (2014): “Concentration-diffusion phenomena of heat convection in an incompressible fluid”, Asymptotic Anal. 88, 17 – 41, doi 10.3233/ASY-131211.

Preprints

  • Schulz R. (2017): “Analysis of chemotactical biofilm growth in evolving microstructures”, Proceedings in Applied Mathematics and Mechanics PAMM 17, 715-716.
  • Ray N., Schulz R. (2017): “Derivation of an effective model for electroosmotic flow involving free boundaries in a thin strip”, FAU Erlangen, Preprint Series Angewandte Mathematik 398.
  • Ray N., Schulz R., Rupp A., Knabner P. (2016): “Past and present approaches to calculate hydrodynamic parameters in evolving porous media”, FAU Erlangen, Preprint Series Angewandte Mathematik 395.

Thesis

  • Diploma Thesis: Global Solvability of two-dimensional Boussinesq Equations with Non-decaying Initial Data, TU Darmstadt, 2008. Advisor: Prof. Dr. Reinhard Farwig