On Hele-shaw Problems Arising as Scaling Limits

This work is motivated by the paper [LP]. In this paper, the authors consider the scaling limits of three discrete aggregation models with multiple sources the internal DLA (diffusion-limited aggregation), the rotor-router, and the divisible sandpile model. They show that for all three models, the scaling limit is the same, and it is a solution of the Hele-Shaw injection problem with multiple sources. In two dimensions, the latter problem can be solved explicitly using the theory of quadrature domains, which gives an explicit formula for the scaling limit. They also consider the same aggregation models in two dimensions with just one source, but with an additional condition on the positive (horizontal) half-axis (namely, the condition that a particle hitting the positive half-axis is killed, or the condition that it is directed downward). In these cases, the existence of the scaling limit remains unknown, although it is expected to exist, and computer-generated pictures for its shape are given in Fig. 4 of [LP]. The goal of this paper is to study the Hele-Shaw problems which are expected to be the scaling limits of the above models with a condition. Namely, we provide an explicit solution for the Hele-Shaw problem corresponding to the killing condition, which is a close fit with the left shape at Fig. 4 of [LP]. We also describe moment properties of the solution of the Hele-Shaw problem corresponding to the downward condition (the right shape at Fig. 4 of [LP]), although we are unable to compute this shape explicitly. Acknowledgements. This paper is dedicated to the memory of my teacher Vladimir Markovich Entov, who introduced me to the subject of Hele-Shaw flows. He was an extraordinary person and scientist, and interaction with him was one of my best experiences. I am very grateful to Lionel Levine for introducing me to the problem, and for checking that my formulas fit the results of computer