Let 
be a facet which is shrinking toward zero length due to its
motion and the motion of the neighboring facets.
There are two cases to consider: 
and 
.
We claim that a facet with never approaches zero
length, since
if it did, equations 1 and 6
would imply that 
becomes
unbounded on with the same sign as 
as 
. If the neighboring facets are not also all having
their potentials becoming unbounded at the same rate (and with the
same sign) then that would also produce infinite
gradients in potential and provide an infinite driving
force to lengthen the edge. And because there is volume conservation,
it is not possible for all edges in connected component to go to zero
length simultaneously. (Proofs will appear in [8].)
If , then as the length ,
along since 
. If the change in
flux across the facet is
also going to zero, then the facet can disappear ``in place'' as its
neighbors become collinear;
in this case we merge the neighbors, as stated above.
Otherwise if the change in flux is larger than 
,
by equation 10
its velocity will become very large.
The small facet will either: rapidly move to a position, if it
exists, where the flux entering matches the exiting flux; or,
if such a position does not exist, it will move rapidly
until one of its adjacent segments also becomes
very short, with most of its length having been effectively
transferred to the other adjacent segment, Fig. 2.
In the second case, as this adjacent segment
becomes short, with nonzero change of flux across it,
it too begins to move rapidly (see Eq. 10).
That makes the original short segment rapidly become longer, and as
it grows longer it rapidly slows its motion, squeezing the new short
facet very much but not down to zero length.
The new short facet in its rapid
motion again effectively transfers the
length of the neighbor ahead of it back onto the neighbor behind it
(the one that used to be short), until the neighbor ahead of the
moving facet itself becomes short (See Figure 2).
This process of successive short, increasingly
rapidly moving facets continues until the fluxes across the short
facets
match; often this occurs when there is another rapidly moving
short facet coming from the opposite direction and several facets
annihilate at the same instant.
The motion of the short facets
occurs on a time-scale which is rapid compared to the rest of the
polygon:
a facet becomes short at one
position, the short step zigs along the curve, apparently around its
corners, disappearing somewhere far away.
Since the larger facets are nearly stationary during this short period
of time, the interface looks much
like it did before the small facets settled or annihilated
but with differently
named segments occupying the positions of the original segments.