Research at the Department for Analysis of Partial Differential Equations at the Johann Radon Institute for Computational and Applied Mathematics (RICAM)

Extract from the research site of RICAM

This research group will work mainly on modelling with and analysis of partial differential equations. It will provide a direct interface to the groups of the Radon Institute in financial mathematics (Schachermayer) and of Engl/Langer in inverse problems/numerical analysis and simulation. The interaction will be very tight since modern pde modelling/analysis and numerics/simulation cannot survive without one another in the near future. Although we fully appreciate the fact that research guidelines have to be extremely flexible (also pushed by an unprecedented hardware development which nowadays makes simulations possible which nobody dreamt about only ten years ago), some main new topics, which will be developed within the Radon Institute, have emerged.

1) The 'next' generation of pde-analysts will to a large extent be concerned with non-linear system of pdes. It will be a main challenge to overcome already well-developed tools for scalar problems (like maximum principles) pushing the so far underdeveloped theory of pde systems.
2)Continuing research on (fully) non-linear pdes: Monge-Ampere equation, Monge-Kantorovich mass transportation problems, Wasserstein metric, entropies for diffusive systems, convex Sobolev inequalities, Hamilton-Jacobi equations, free boundary problems (typical technique: viscosity solutions), connections to control theory, geometric motions (mean curvature flow, pattern formation), Hamiltonian-Lagrangian (quantum) mechanics, homogenisation of non-linear pdes. There are close links to level set methods as mentioned in the description of the Inverse Problems Group.
3) stochastic pdes with space-time stochasticity (random Schrödinger equations, Brownian motions on manifolds, random diffusion systems, stochastic non-linear pdes). Typical applications are in quantum mechanics (impurity scattering), geophysics, finance mathematics, mathematical biology.
Already ongoing research of the group of Peter Markowich on models for the dynamics of large particle ensembles will continue. This covers the range from basic quantum mechanical models (e.g., (non-linear) Schrödinger and Dirac equations) to classical macroscopic fluid models, including (semi-)classical and quantum kinetic models (e.g., Boltzmann, Fokker-Planck, Wigner equations). A particular emphasis will be on scaling limits and the analysis of large time behaviour. Specific topics are:
4) Kinetic models in population dynamics and mathematical biology: the dynamic interaction of genotype/phenotype and spatial (geography) effects will be studied by means of kinetic modelling (the "genotype/phenotype" variables play the role of the momentum variable, dual to the spatial variable). Chemotaxis will be studied on the kinetic level, with particular emphasis ondiffusion limits and microscopic modelling of the motion of bacteria. It is conceivable that finite-time blow-up can be avoided on the kinetic level by careful modelling of reorientation processes.
5) Macroscopic limits of kinetic models: Applications include charge transport in semiconductors, ionisation of gases, and kinetic models for wave-particle scattering. An understanding of fluid limits of the latter might lead to a new approach for understanding macroscopic models of gas dynamics.
6) Modelling and Simulation of Bose-Einstein condensation: Bose-Einstein condensation is one of the extremely hot topics in low-temperature atomic physics (cf. the Nobel Prize in physics 2001). The typical model after condensation is the Gross-Pitaevskii equation (cubically non-linear Schrödinger equation with harmonic confinement), while the bosonic Boltzmann equation is used to describe the condensation process. We expect to continue numerical simulations and the analysis of the condensation process (transition regime of the bosonic Boltzmann equation to the Gross-Pitaevskii equation).
7) Derivation, analysis and numerics of quantum Boltzmann equations by means of Wigner transforms: This work is supposed to shed new light on the origin of irreversibility in semi-classical and other scaling limits. New insights into open quantum systems are expected by employing quantum entropy and quantum entropy-dissipation techniques similarly to the classical-mechanics case.
8) inverse problems in fluid-type pdes: applications in semiconductors, nano-technology, geophysics. In this area there will be significant interaction with semiconductor device engineers outside the Radon Institute and with the in-house groups of Engl and Langer. Main topics will be doping profile identification and device performance optimisation.