## Theory of Partial Differential Equations, Applied and Numerical Analysis Julian Fischer is a mathematician working in the fields of applied analysis and partial differential equations arising in the applied sciences. His research frequently also features connections of mathematical analysis with numerical analysis and stochastics.

Partial differential equations (PDEs) are a fundamental tool in the description of many physical phenomena. To name a few examples, the Navier-Stokes equation provides a description of the flow of a fluid, while the behavior of electromagnetic fields is determined by Maxwell's equations. Associated with partial differential equations, there are many mathematical questions: Does a given PDE have a solution? Is the solution unique or does the equation admit additional unphysical solutions? How may one approximate the solution for practical purposes? Is it possible to use a simpler equation in a given physical situation?

In its current focus topics, Fischer's group addresses in particular the last of these questions: For most physical situations, many PDE-based models with different degrees of accuracy are available. One would then like to select the PDE that may be solved with least computational effort while providing a sufficiently accurate description of physical reality. To facilitate the selection of the model, it is highly desirable to estimate the modeling error associated with a given model. One focus topic of Fischer's group are a posteriori modeling error estimates, a mathematical concept that uses an (exact or numerical) solution of a PDE model as an input and provides a bound on the modeling error, as compared to a given more accurate model.

The second focus topic of Fischer's group is stochastic homogenization: Many real-world materials feature random heterogeneities on small scales and nevertheless behave on large scales like a homogeneous material. The subject of the theory of homogenization is the derivation of effective models for the macroscopic behavior of a material as a limit of material models which include the microstructure of the material. The homogeneous effective model is typically easier to solve computationally, but provides a less accurate description of the material behavior ("homogenization error"). Of particular interest to Fischer's research group is the quantitative theory of stochastic homogenization, which for example is concerned with quantitative estimates for the homogenization error.

Contact
Julian Fischer

Institute of Science and Technology Austria (IST Austria)
Am Campus 1
A – 3400 Klosterneuburg

E-mail: julian.fischer@ist.ac.at

Assistant
Sandra Widdmann
Phone: +43 (0)2243 9000-1178
E-mail:  sandra.widdmann@ist.ac.at

Team

• Sebastian Hensel, PhD Student
• Michael Kniely, Postdoc

Selected Publications

• Fischer J. 2018. The choice of representative volumes in the approximation of effective properties of random materials. arXiv:1807.00834, 84 pp.
• Fischer J, Otto F. 2016. A higher-order large-scale regularity theory for random elliptic operators. Comm. Partial Differential Equations 41 (7), 1108-1148.
• Fischer J. 2015. A posteriori modeling error estimates for the assumption of perfect incompressibility in the Navier-Stokes equation. SIAM J. Numer. Anal. 53 (5), 2178-2205.
• Fischer J. 2015. Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems. Arch. Ration. Mech. Anal. 218 (1), 553-587.
• Fischer J. 2014. Upper bounds on waiting times for the thin-film equation: the case of weak slippage. Arch. Ration. Mech. Anal. 211 (3), 771-818.

Career
As of 2017 Assistant Professor, IST Austria
2014-2016 Postdoc, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
2013-2014 Postdoc, University of Zürich, Switzerland
2011-2013 PhD in Mathematics, University of Erlangen-Nürnberg, Germany

Selected Distinctions
2015 Dr. Körper Prize, PhD Award of the GAMM

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