REU 1988: Time Evolution of Planar Domains (M. Peterson)

The REU project for 1988 directed by Mark Peterson investigated the evolution of a planar domain D whose boundary moves with velocity proportional to the gradient of its external Green's function. This growth law arises in many physical situations, including aggregation of diffusing particles, electrodeposition, and viscous fingering at a fluid-fluid boundary. This strangely unstable growth law produces fractal-like shapes, like the one below from the poster for STATPHYS 86:

[Image of Diffusion Limited Aggregation] Image of Diffusion Limited Aggregation

We defined a singular case of this dynamics, for degenerate domains, and found analytic solutions for the simplest cases, which are slit domains.



One can think of the evolution as the growth of needle crystals by deposition of material. The needles can simply grow along their length -- this is one possible consequence of the dynamics. The singular nature of the Green's function means that all growth is at the tip, so that slit domains stay slit domains. A non-obvious (but plausible) consequence of the dynamics is that longer needles can "shadow" short needles and grow at their expense. We were able to describe the evolution exactly of certain slit domains like the one shown. Remarkably, the dynamics is not deterministic! At any time, a needle tip may bifurcate into two tips growing at an angle to each other. The allowed angles are restricted only by the "shadowing" mentioned above. This means that solutions to the governing equations, an infinite system of ODE's, are "maximally non-unique." Every solution has infinitely many other solutions tangent to it at every point.

This research project was continued by the 1992 REU group. It led to two publications:

The student participants were:
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