Admissible weak solutions for the Muskat problem
- Clemens Förster (MPI MiS, Leipzig)
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.