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Research

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

Journal articles

  • R. Schulz, S. Gärttner, N. Ray: Investigations of effective dispersion models for electroosmotic flow with rigid and free boundaries in a thin strip, Math. Meth. Appl. Sci., 1 - 23, DOI10.1002/mma.9651, (2023).

  • S. Zech, N. Ray, R. Schulz: Numerical investigations of degenerate equations for fluid flow and reactive transport in clogging porous media, J. Math. Anal. Appl. 512 (2), DOI: 10.1016/j.jmaa.2022.126384, (2022).

  • R. Schulz: Uniqueness of Degenerating Solutions to a Diffusion-Precipitation Model for Clogging Porous Media,  Math. Model. Anal. 27 (3), 471 - 491, DOI: 10.3846/mma.2022.15132, (2022).

  • N. Ray, R. Schulz: Existence and uniqueness of solutions to a flow and transport problem with degenerating coefficients, Eur. J. Appl. Math. 34 (1), 1 - 22, DOI: 10.1017/S0956792522000018, (2022)

  • Schnepf A., Carminati A., Ahmed M., Ami M., Benard P., Bentz J., Bonkowski M., Brax M., Diehl D., Duddek P., Kröner E., Javaux M., Landl M., Lehndorff E., Lippold E., Lieu A., Müller CW., Oburger E., Otten W., Portell-Canal X., Phalempin M., Prechtel A., Schulz R., Vanderborght J., Vetterlein D.: Linking rhizosphere processes across scales: Opinion, Plant and Soil 478, 5 - 42, DOI: 10.1007/s11104-022-05306-7, (2022)
  • R. Schulz: Degenerate equations in a diffusion-precipitation model for clogging porous media, Eur. J. Appl. Math. 31 (6), 1050 - 1069, DOI: 10.1017/S0956792519000391, (2020).
  • R. Schulz: Degenerate equations for flow and transport in clogging porous media, J. Math. Anal. Appl. 483 (2), DOI: 10.1016/j.jmaa.2019.123613, (2020).
  • R. Schulz, N. Ray, S. Zech, A. Rupp, P. Knabner: Beyond Kozeny-Carman: Predicting the permeability in porous media, Transp. Porous Med. 130 (2), 487 - 512, DOI: 10.1007/s11242-019-01321-y, (2019).
  • N. Ray, R. Schulz: Derivation of an effective dispersion model for electroosmotic flow involving free boundaries in a thin strip, J. Eng. Math. 119, 167 - 197, (2019).
  • R. Schulz: Biofilm modeling in evolving porous media with Beavers-Joseph condition, Z. Angew. Math. Mech. 99 (3), DOI: 10.1002/zamm.201800123, (2019).
  • R. Schulz: Crystal precipitation and dissolution in a porous medium: Evolving microstructure and perforated solid matrix, Special Topics Rev. Porous Media 10 (4), 305 - 321, DOI: 10.1615/SpecialTopicsRevPorousMedia.2019029274, (2019).
  • N. Ray, A. Rupp, R. Schulz, P. Knabner: Old and New Approaches Predicting the Diffusion in Porous Media, Transp. Porous Med. 124 (3), 803 - 824, DOI: 10.1007/s11242-018-1099-x, (2018)
  • R. Farwig, R. Schulz, Taniuchi Y.: Spatial asymptotic profiles of solutions to the Navier-Stokes system in a rotating frame with fast decaying data, Journal of the Hokkaido University, Faculty of Science, Series VII: Geophysics 47, 501 - 529, DOI: 10.14492/hokmj/1537948828, (2018)
  • R. Schulz: Boundedness in a biofilm-chemotaxis model in evolving porous media, Math. Model. Anal. 22 (6), 852 - 869, DOI: 10.3846/13926292.2017.1389772, (2017).
  • R. Schulz, Knabner P.: An effective model for biofilm growth made by chemotactical bacteria in evolving porous media, SIAM Journal on Applied Mathematics 77, 1653 - 1677, DOI: 10.1137/16M108817X, (2017)
  • R. Schulz, P. Knabner: Derivation and analysis of an effective model for biofilm growth in evolving porous media, Mathematical Methods in the Applied Sciences 40, 2930 - 2948, DOI: 10.1002/mma.4211, (2017)
  • R. Schulz, N. Ray, F. Frank, H. Mahato, P. Knabner: Strong solvability up to clogging of an effective diffusion-precipitation model in an evolving porous medium, European Journal of Applied Mathematics, 1 - 29, DOI: 10.1017/S0956792516000164, (2016)
  • R. Farwig, R. Schulz, M. Yamazaki: Concentration-diffusion phenomena of heat convection in an incompressible fluid, Asymptotic Analysis 88, 17 - 41, (2014).

Preprints & Proceedings

  • A. Prechtel, A. Lieu, M. Phalempin, R. Schulz: Evaluating the effective diffusion in soil columns including biopores by homogenization, FAU Erlangen-Nürnberg, Preprint Series Angewandte Mathematik 416, (2022)
  • R. Schulz: Analysis of chemotactical biofilm growth in evolving microstructures, 88th GAMM Annual Meeting, Proceedings in Applied Mathematics and Mechanics PAMM, DOI: 10.1002/pamm.201710326, (2017)
  • N. Ray, R. Schulz: Derivation of an effective dispersion model for electroosmotic flow involving free boundaries in a thin strip, FAU Erlangen-Nürnberg, Preprint Series Angewandte Mathematik 398, (2017)
  • N. Ray, R. Schulz, A. Rupp, P. Knabner: Past and present approaches to calculate hydrodynamic parameters in evolving porous media, FAU Erlangen-Nürnberg, Preprint Series Angewandte Mathematik 395, (2016).

Textbooks

  • P. Knabner, B. Reuter, R. Schulz: Mit Mathe richtig anfangen Eine Einführung mit integrierter Anwendung der Programmiersprache Python, Berlin Heidelberg: Springer Spektrum, ISBN: 978-3-662-59229-8, DOI: 10.1007/978-3-662-59230-4, (2019)

Theses

  • R. Schulz: Mathematical modeling and analysis of processes in evolving microstructures, habilitation thesis, (2019)
  • R. Schulz: Spatial asymptotic profile in geophysical fluid dynamics, dissertation thesis, (2012)
  • R. Schulz: Global solvability of two-dimensional Boussinesq-equations with non-decaying initial data, diploma thesis, (2008).

Projects

Fluid flow and reactive transport in coupled porous and free-flow domains

BayIntAn - Promoting Cooperation

(Funding Programme for the Initiation of International Projects)

Laufzeit: 28-02-2024 - 31-12-2024
Mittelgeber: Bavarian Research Alliance

Partner: Department of Mathematics, University of Bergen, Norway

Mathematical modeling and analysis of transport phenomena in porous media with variable hydrodynamic properties

Special reasearch fund

(Individual funding for research stay)

Term: 01-02-2020 - 31-12-2020
Funding source: Hasselt University, Belgium

Partner: Computational Mathematics, Hasselt University, Belgium

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

DFG Priority Programme 2089

(Third Party Funds Group – Sub project)

Overall project: 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=4649

Upscaling Evolving Microstructures and its Applications

German-Norwegian collaborative research support scheme

(Third Party Funds Single)

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

Partner: Department of Mathematics, University of Bergen, Norway

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