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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 Fingerprint is based on mining the text of the person's scientific documents to create an index of weighted terms, which defines the key subjects of each individual researcher.

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

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Projects 2005 2015

Spectrum of Lattice Equations

Matthies, K.

1/07/1430/11/14

Project: UK charity

Homogenisation and Modulation Equations

Matthies, K.

1/03/1431/10/15

Project: Research council

ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS

Matthies, K. & Burstall, F.

1/09/0531/08/10

Project: Research council

Research Output 1999 2019

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

Rescaled Objective Solutions of Fokker-Planck and Boltzmann equations

Matthies, K. & Theil, F., 31 Jan 2019, (Accepted/In press) In : Siam Journal on Mathematical Analysis.

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
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

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

Linear Boltzmann Equation
Boltzmann equation
Rayleigh
Tagged Particle
Collision