Abstract
In this talk, I will discuss a curvature-dependent motion of plane curves in a
two-dimensional infinite cylinder with spatially undulating boundaries. The two
ends of the curve slide freely along the boundary of the domain while keeping the
constant contact angle of .
The question is whether the curve continues to travel to infinity (propagation)
or remains in a bounded area (blocking). The same problem was studied in my
earlier work under the assumption that the maximum opening angle and closing
angle of the boundaries are less than (2006 and 2013, joint work with K.-I.
Nakamura and B. Lou). Under this condition, one can show that no singularity
develops and the solution remains classical. If the opening and closing
angles are larger than , the middle part of the curve may bump into the
boundary of the domain, thereby creating singularities. In this talk, I will
discuss the long-time behavior of curves that propagates while creating
singularities. Among other things we show that if the bumps are arrayed
more densely on the boundary, then the speed of propagation increases.
In other words, denser obstacles can facilitate propagation, which may
sound somewhat paradoxical. This is joint work with Ryunosuke Mori.