Projects per year

## Personal profile

### Willing to supervise PhD

Projects available in

Averaging and Homogenisation for PDE;

Infinite-dimensional dynamics: PDEs and lattice ODEs

Many particle dynamics and derivation of kinetic equations

### Research interests

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.

- 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.
- 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.
- Deriving continuum equations from atomistic equations: This research aims at the basic question how equations on different scales can have fundamentally different behaviour.

## Fingerprint Dive into the research topics where Karsten Matthies is active. These topic labels come from the works of this person. Together they form a unique fingerprint.

## Projects 2005 2015

- 3 Finished

### ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS

Engineering and Physical Sciences Research Council

1/09/05 → 31/08/10

Project: Research council

## Research Output 1999 2019

### Rescaled Objective Solutions of Fokker-Planck and Boltzmann equations

Matthies, K. & Theil, F., 18 Apr 2019, In : SIAM Journal on Mathematical Analysis (SIMA). 51, 2, p. 1321–1348 28 p.Research output: Contribution to journal › Article

### Solitary waves in atomic chains and peridynamical media

Herrmann, M. & Matthies, K., 7 Mar 2019, In : Mathematics in Engineering. 1, 2, p. 281-308 28 p.Research output: Contribution to journal › Article

### Stability of high-energy solitary waves in Fermi-Pasta-Ulam-Tsingou chains

Herrmann, M. & Matthies, K., 1 Sep 2019, In : Transactions of the American Mathematical Society. 372, 5, p. 3425–3486 62 p.Research output: Contribution to journal › Article

### Asymptotic properties of high-speed waves in atomic chains

Herrmann, M. & Matthies, K., 17 Dec 2018, In : PAMM - Proceedings in Applied Mathematics and Mechanics. 18, 1, 2 p., e201800305.Research output: Contribution to journal › Conference article

### Derivation of a Nonautonomous Linear Boltzmann Equation from a Heterogeneous Rayleigh Gas

Matthies, K. & Stone, G., 1 Jul 2018, In : Discrete and Continuous Dynamical Systems - Series A. 38, 7, p. 3299-3355 57 p.Research output: Contribution to journal › Article

## Thesis

## Derivation of Kinetic Equations from Particle Models

Author: Stone, G. R., 23 Oct 2017Supervisor: Matthies, K. (Supervisor)

Student thesis: Doctoral Thesis › PhD

## On the phase transition in certain percolation models.

Author: Daniels, C., 22 Feb 2016Supervisor: Penrose, M. (Supervisor) & Matthies, K. (Supervisor)

Student thesis: Doctoral Thesis › PhD

## Powerfully Nilpotent p-groups

Author: Williams, J., 3 Apr 2019Supervisor: Matthies, K. (Supervisor) & Traustason, G. (Supervisor)

Student thesis: Doctoral Thesis › PhD

## Travelling waves in heterogeneous media

Author: Boden, A., 30 Apr 2013Supervisor: Matthies, K. (Supervisor)

Student thesis: Doctoral Thesis › PhD