# Current Research topics of the Research group for Analysis of Partial Differential Equations

Our research area is partial differential equations. In particular, we are interested in free boundary problems, free surface flows, traveling waves, singular limits, the calculus of variations as well as regularity. Presently, we are working on the following problems:

### Mathematical problems in free surface flows

(for example water waves, jets and cavities)

Water waves have been intriguing objects to men since the early cultures. Many famous scientists and mathematicians such as Newton, Laplace, Lagrange, Euler, Cauchy, Poisson and Stokes worked on the mathematical analysis of fluid flow and water waves. Many questions (even simple ones) in this research area turn out to be hard mathematical problems of which some remain unsolved to this date. In higher dimensions even existence questions are largely open. In our group we proved some generalized Stokes-conjectures concerning water waves of maximal amplitude [Varvaruca-Weiss 2011, Varvaruca-Weiss 2012]. We also considered singularities of capillary gravity water waves [Weiss-Zhang 2012].

### ElektroHydroDynamic Equations

Currently we are working on several fascinating problems caused by coupling of fluid flow and electric field in which Sir G. Taylor did pioneering research. The combination of surface tension and Neumann-type free boundary conditions as well as important phenomena (even in industrial applications) such as cusp-formed jets and singular cones creates new and interesting problems for the analysis of partial differential equations [Smit Vega Garcia-Varvaruca-Weiss 2016].

### Mathematical problems in combustion theory

Experiments in combustion of gases and porous media as well as heterogeneous catalysis show a rich pattern formation. We have investigated for premixed gaseous combustion the regime of high activation energy and reduced the respective reaction-diffusion system to one partial differential equation [Weiss 2003], [Weiss-Zhang 2010].

For combustion of porous media, which has applications in the synthesis by combustion, we have rigorously shown that the case of high activation energy is linked to mathematical models for freezing of supercooled water [Monneau-Weiss 2009] .