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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.

  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.

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.

Boltzmann equation Engineering & Materials Science
Homogenization Mathematics
Solitary Waves Mathematics
Ludwig Boltzmann Mathematics
Traveling Wave Mathematics
Linear Boltzmann Equation Mathematics
Periodic Media Mathematics
Averaging Mathematics

Projects 2005 2015

Research Output 1999 2019

  • 26 Article
  • 2 Chapter
  • 1 Conference contribution
  • 1 Conference article
32 Downloads (Pure)

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 journalArticle

Open Access
File
Fokker-Planck equation
shear flow
energy
symmetry

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 journalArticle

Open Access
norms
traveling waves
differential equations
solitary waves
potential energy
24 Downloads (Pure)

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 journalArticle

Open Access
File
Solitary Waves
Solitons
High Energy
Unstable
Eigenvalues and eigenfunctions

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 journalConference article

Asymptotic Properties
High Speed
10 Downloads (Pure)

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 journalArticle

Open Access
File
Linear Boltzmann Equation
Boltzmann equation
Rayleigh
Tagged Particle
Collision

Thesis

Derivation of Kinetic Equations from Particle Models

Author: Stone, G. R., 23 Oct 2017

Supervisor: Matthies, K. (Supervisor)

Student thesis: Doctoral ThesisPhD

File

On the phase transition in certain percolation models.

Author: Daniels, C., 22 Feb 2016

Supervisor: Penrose, M. (Supervisor) & Matthies, K. (Supervisor)

Student thesis: Doctoral ThesisPhD

File

Powerfully Nilpotent p-groups

Author: Williams, J., 3 Apr 2019

Supervisor: Matthies, K. (Supervisor) & Traustason, G. (Supervisor)

Student thesis: Doctoral ThesisPhD

File

Travelling waves in heterogeneous media

Author: Boden, A., 30 Apr 2013

Supervisor: Matthies, K. (Supervisor)

Student thesis: Doctoral ThesisPhD

File