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

Our research area is nonlinear 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:

### Singularities of obstacle-type equations

The obstacle problem arguably is the most extensively studied free boundary problem. It may be derived from a simple model for spanning an elastic membrane over some given (concave) obstacle. Alternatively it can be derived from a certain setting in the Stefan problem, the simplest model for the melting of ice, or from the Hele-Shaw problem.

The obstacle problem may be expressed in the single nonlinear partial differential equation

$$\Delta u = c(x) \chi_{\{ u>0\}}, u \ge 0.$$

The set $$u=0$$ is called coincidence set and the interface $$\partial \{ u>0\}$$ is called the free boundary.
It is known that even for $$C^\infty$$-coefficients $$c(x)$$, the behaviour of the free boundary may be very complicated,
including singular sets which are generalised Cantor sets.

In [Eberle-Shahgholian-Weiss, 2021. 54] we obtained a result on the behavior of the regular part of the free boundary
of the obstacle problem close to singularities. Extending the result to the critical dimension $$N=3$$ is work in
progress with Simon Eberle and Alessio Figalli.

### Convergence of Algorithms in Machine Learning

With Simon Eberle and the group of Arnulf Jentzen (Münster) we are investigating
gradient flows and related algorithms in the training of artificial neural networks with $$ReLU$$ activation
[Eberle-Jentzen-Riekert-Weiss, 2021. 55]. This project does not involve partial differential equations.

### Models of Neutron Stars

With Sagun Chanillo (Rutgers) we investigated the free surface of a neutron star in a classical model
[Chanillo-Weiss, 2012. 40] and we are currently working on a relativistic model.

Research areas we are still interested in include:

### 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. 36, Varvaruca-Weiss, 2012. 39].We also considered singularities of capillary gravity water waves [Weiss-Zhang 2012. 38].

### 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. 47].

### 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. 18], [Weiss-Zhang, 2010. 33].
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. 31] .