Projects per year
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.
- 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.
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Projects
- 5 Finished
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Generalised and Low-Regularity Solutions of Nonlinear Partial Differential Equations
Moser, R. (PI) & Matthies, K. (CoI)
Engineering and Physical Sciences Research Council
1/07/21 → 30/06/24
Project: Research council
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Derivation of kinetic equation: From Newton to Boltzmann via trees
Matthies, K. (PI)
1/10/20 → 31/03/24
Project: UK charity
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ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS
Matthies, K. (PI) & Burstall, F. (CoI)
Engineering and Physical Sciences Research Council
1/09/05 → 31/08/10
Project: Research council
Research output
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Derivation of kinetic and diffusion equations from a hard-sphere Rayleigh gas using collision trees and semigroups
Matthies, K. & Syntaka, T., 3 May 2024, (Acceptance date) London Mathematical Society, (London Mathematical Society Lecture Note Series).Research output: Working paper / Preprint › Preprint
Open AccessFile34 Downloads (Pure) -
Fractional diffusion as the limit of a short range potential Rayleigh gas
Matthies, K. & Syntaka, T., 29 May 2024, arXiv.Research output: Working paper / Preprint › Preprint
File18 Downloads (Pure) -
Solitary solutions to the steady Euler equations with piecewise constant vorticity in a channel
Matthies, K., Sewell, J. & Wheeler, M. H., 15 Aug 2024, In: Journal of Differential Equations. 400, p. 376-422 47 p.Research output: Contribution to journal › Article › peer-review
Open AccessFile1 Citation (SciVal)95 Downloads (Pure) -
SU(2)^2xU(1)-invariant G_2-instantons on the AC limit of the C7 family
Matthies, K., Nordström, J. & Turner, M., 11 Dec 2024, In: Communications in Analysis & Geometry. 32, 9, p. 2505-2549Research output: Contribution to journal › Article › peer-review
Open AccessFile39 Downloads (Pure) -
Second-order asymptotic expansion and thermodynamic interpretation of a fast-slow Hamiltonian system
Klar, M., Matthies, K. & Zimmer, J., 31 Dec 2022, In: Letters in Mathematical Physics. 112, 6, 32 p., 119.Research output: Contribution to journal › Article › peer-review
Open AccessFile67 Downloads (Pure)