### Non-univalent solutions of the Polubarinova-Galin equation

Gustafsson, B *(KTH - Royal Institute of Technology)*

Wednesday 09 April 2014, 14:00-15:00

Seminar Room 1, Newton Institute

#### Abstract

The Polubarinova-Galin equation describes the time evolution of the conformal map onto a Hele-Shaw blob of fluid, which expands by injection of fluid at one point. As the evolution goes on different parts of the fluid region may collide with each other. This can be handled by turning to a weak formulation of the problem, but in the talk I will discuss another way, namely by keeping the fluid region simply connected by letting it go up onto a Riemann surface. In this way one can let the solution go on forever as a simply connected solution, but the construction of the Riemann surface is by no means trivial, because it cannot constructed in advance. It has to be updated all the time, as new collisions occurs
in the fluid region.
The talk describes work in progress, in collaboration with Yu-Lin Lin (KTH).

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