Theory of Partial Differential Equations, Applied and Numerical Analysis
Diverse phenomena such as the motion of fluids or elastic objects, the evolution of interfaces, or the physics of quantum-mechanical particles are described accurately by partial differential equations. The Fischer group works on the mathematical analysis of partial differential equations that arise in the sciences, connecting also to areas like numerical analysis or probability.
Partial differential equations are a fundamental tool for the description of many phenomena in the sciences, ranging from the physics of continua like fluids or elastic solids over quantum mechanics to population biology. Julian Fischer and his group work on the mathematical aspects of partial differential equations. One of the group’s main themes is the mathematical justification of model simplifications: For example, an elastic material with a highly heterogeneous small-scale structure may in many cases be approximated as a homogeneous material. Likewise, a fluid with low compressibility may in many cases be approximated as ideally incompressible. To justify such approximations, the group derives rigorous estimates for the approximation error. The techniques they employ connect the analysis of PDEs with adjacent mathematical areas like numerical analysis and probability.
On this site:
Effective behavior of random materials | Evolution of interfaces in fluid mechanics and solids | Structure of fluctuations in stochastic homogenization | Entropy-dissipative PDEs
Fischer JL. 2019. The choice of representative volumes in the approximation of effective properties of random materials. Archive for Rational Mechanics and Analysis. 234(2), 635–726. View
Friesecke G, Kniely M. 2019. New optimal control problems in density functional theory motivated by photovoltaics. Multiscale Modeling and Simulation. 17(3), 926–947. View
Fischer JL, Kneuss O. 2019. Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity. Journal of Differential Equations. 266(1), 257–311. View
Duerinckx M, Fischer JL. 2018. Well-posedness for mean-field evolutions arising in superconductivity. Annales de l’Institut Henri Poincare (C) Non Linear Analysis. 35(5), 1267–1319. View
Fischer JL, Grün G. 2018. Existence of positive solutions to stochastic thin-film equations. SIAM Journal on Mathematical Analysis. 50(1), 411–455. View
since 2017 Assistant Professor, IST Austria
2014 – 2016 Postdoc, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
2013 – 2014 Postdoc, University of Zurich, Switzerland
2013 PhD, University of Erlangen-Nürnberg, Germany
2015 Dr. Körper Prize, PhD Award of the GAMM