⍀ t be the bounded open domain such that Ѩ ⍀ t ϭ ⌫ t . Then for each x ʦ ⌫ t , V(x, t) is positive if ⌫ t moves inwards to
We use a boundary integral technique to study the two space dimensional Mullins-Sekerka free boundary problem which origi-⍀ t , and (x, t) is positive if the center of the osculating nates from a study of solidification and liquidation of materials circle is on the side of ⍀ t .
of negligible specific heat. This is an area preserving and curve Note that (1.1) is a geometric motion problem in the shortening motion. Evolution equations for the free boundaries are sense that ⌫ t depends only on the initial position ⌫ 0 . In derived in terms of the tangent angle and total arclength, which fact, (1.1) can be written in a short form as makes a small scale decomposition possible and the Fourier transform a powerful tool in numerical calculations. With this formulation, implicit schemes can be implemented to avoid the difficult
numerical stiffness associated with explicit schemes. We can compute solutions up to the time when there is a topological change, where K represents the harmonic extension of the curvai.e., when particles touch or break up. Our numerical results for ture of ⌫ t over ޒ 2 . This motion is an area preserving and systems of a single particle or multi-particles provide some valuable information in the particle dynamics, such as the circularization of curve shortening motion. To see this, let us denote by each individual particle, and the mass transfer between different A(t) the area of ⍀ t and L(t) the arclength of ⌫ t . Then we particles during particle interactions.