Karsten Matthies

Projects available in
Averaging and Homogenisation for PDE;
Infinite-dimensional dynamics: PDEs and lattice ODEs
Many particle dynamics and derivation of kinetic equations

The main goal of my research is to develop rigorous mathematical methods to understand and describe the dynamical (temporal) behaviour of solutions of partial differential equations and other infinite-dimensional dynamical systems. The studied equations are motivated by models in the physical sciences, where the aim is a mathematically rigorous analysis of model problems to achieve a proper and lasting understanding of structure and effects.

In particular I am interested in equations with additional properties like dependence on fast scales or broken symmetries. A key question is to identify some limiting description, when e.g. the period of the fast scale tending to zero in averaging or homogenisation. Then qualitative differences (e.g. pinning, splitting of separatrices) between the various systems are studied. The final aim is to give a quantitative description of effects causing the differences through rigorous error bounds.

There are three related areas of my research.

1.     Averaging and homogenisation aims at the description of partial differential equations with fast spatial and/or fast temporal scales, these break the symmetry in autonomous or homogeneous equations.
2.     Dynamics of waves: The understanding of existence, stability and behaviour of travelling waves is a prime example of dynamical behaviour in partial differential equations and discrete lattice equations.
3.     Deriving continuum equations from atomistic equations: This research aims at the basic question how equations on different scales can have fundamentally different behaviour.