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Personal profile

Willing to supervise doctoral students

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.

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Projects

Research Output

  • 27 Article
  • 2 Chapter
  • 1 Conference contribution
  • 1 Conference article

Nonlinear and Nonlocal Eigenvalue Problems: variational existence, decay properties, approximation, and universal scaling-

Herrmann, M. & Matthies, K., 25 Mar 2020, (Acceptance date) In : Nonlinearity.

Research output: Contribution to journalArticle

Rescaled Objective Solutions of Fokker-Planck and Boltzmann equations

Matthies, K. & Theil, F., 31 Dec 2019, In : SIAM Journal on Mathematical Analysis (SIMA). 51, 2, p. 1321–1348 28 p.

Research output: Contribution to journalArticle

Open Access
File
  • 76 Downloads (Pure)

    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
  • 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
  • 34 Downloads (Pure)

    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

  • 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