## Research

**Research Area:** Analysis, Applied Mathematics

**Research Profile:***Nonlinear partial differential equations, interface dynamics.*

Interface dynamics is a study of the qualitative behaviour of solutions to certain nonlinear partial differential equations known as singular reaction-diffusion systems. These systems are usually models developed by material scientists and mathematicians in order to understand the properties of interfaces or phase boundaries in alloys, polymers or glasses. Under stress, the alloy divides into several regions where the orientation of the molecules differs. The boundary between these regions evolve in time in order to decrease surface tension. The mathematical models developed to study this evolution lead to geometrical problems for the interfaces. An important example is mean curvature evolution, where the normal velocity of the interface is given by its mean curvature. A formal asymptotic tool called the method of matched asymptotic expansion is often used to predict the behaviour for the evolution of the interfaces. Once this formal study is done, the problem is then to develop analytical and geometrical tools to verify rigorously the expected behaviour. Since the models involve nonlinear partial differential equations, new tools are often needed to successfully complete the work.

Nonlinear partial differential equations, mathematical physics