Abstract for the talk on 07.04.2017 (11:00 h)Arbeitsgemeinschaft ANGEWANDTE ANALYSIS
Clemens Förster (MPI MIS, Leipzig)
Admissible weak solutions for the Muskat problem
In this talk we consider the Muskat problem for the incompressible porous media equation. By this we mean a special initial value problem, where the density is given by two different constants separated by a curve. Recent results of Castro, Cordoba and Faraco showed the existence of infinitely many weak solutions, whenever the initial curve is in H^5(R). In particular, these weak solutions show a mixing behavior in a zone around the curve. I will show an alternative proof of this result. The main difference is to use a piecewise constant density. Therefore, instead of solving a nonlinear equation for the curve, it will be sufficient to make a power series ansatz up to order two. Moreover I will show that the necessary regularity of the initial curve is the same as before. If time allows, I give some remarks about an analogous ansatz for the Euler equations.