Non-univalent solutions of the Polubarinova-Galin equation

We study non-univalent solutions of the Polubarinova-Galin equation, describing the time evolution of the conformal map from the unit disk onto a Hele-Shaw blob of fluid subject to injection at one point. In particular, we tackle the difficulties arising when the map is not even locally univalent, in which case one has to pass to weak solutions developing on a branched covering surface of the complex plane. One major concern is the construction of this Riemann surface, which is not given in advance but has to be constantly up-dated along with the solution. Once the Riemann surface is constructed the weak solution is automatically global in time, but we have had to leave open the question whether the weak solution can be kept simply connected all the time (as is necessary to connect to the Polubarinova-Galin equation). A certain crucial statement, a kind of stability statement for free boundaries, has therefore been left as a conjecture only. Another major part of the paper concerns the structure of rational solutions (as for the derivative of the mapping function). Here we have fairly complete results on the dynamics. Several examples are given.