Seminar topics

In the classical wave equation information travels with a certain speed intrinsic to the wave phenomenon described. This might be the speed of sound in the context of acoustic waves or the speed of light in the context of electro magnetic waves; thus, leading to the notion of wave fronts, forward/backward light cones or domains of incfluence etc. In this project the finiteness of the propagation speed of information is analysed in the context of certain models combining the thermodynamic effects with elastic responses, that is, thermoelasticity. A primary source can be found at https://doi.org/10.1080/01495739.2011.601257. 

One of the main objects of study of solutions to partial differential equations depending on time is their asymptotic behaviour as time tends to infinity. Most prominently, one considers the exponential decay of the energy (well-defined in the given context) of solutions. This project concentrates on showing exponential decay for solutions of certain models describing thermoelastic materials. The strategy of proof is parallel to the concepts developed in https://link.springer.com/chapter/10.1007/978-3-030-89397-2_11

The (one+one-dimensional) model under consideration can be found in https://ems.press/content/serial-article-files/24931

It is a classical topic in the theory of partial differential equations to analyse the asymptotic of divergence form problems with highly oscialltory coefficients. In many cases it is possible to actually compute a limit problem modelling the case of  ``infinitely large oscillations''. This limit problem is characterised by so-called `effective' or `homogenised' coefficients. It is the aim of this project to compute these coefficients by operator-theoretic means as it is being exemplified in Section 14.4 of https://link.springer.com/book/10.1007/978-3-030-89397-2#toc.

The analysis of homogenisation problems associated with equations with oscillations perpendicular to the directions of the derivative is rather complicated. In this context surprising effects can occur. For instance, it can be shown that large oscillations 'in x-direction' of ordinary differential equations with respect to time lead to memory effects in the fictitous limit of infinitely large oscillations. It is the aim of this project to understand this process and exemplarily compute the effective equations for prototype examples. The main source is Chapter 13 of https://link.springer.com/book/10.1007/978-3-030-89397-2#toc.