| Abstract: |
| Our research aim is to construct a mathematical model of stretching and shrinking motions for compressible
elastic curves, for instance, rubber bands. We consider initial and boundary value problems for the beam
equation known as a mathematical model for elastic material. The unknown function represents a position on
the plane. In our research, we discuss two problems. The first problem has no viscosity term, and the second
problem has viscosity term. The feature of these problems is that the strain is nonlinear and non-smooth.
To overcome this difficulty, we suppose that the stress function has a singularity point. Thanks to this, we
can obtain the lower boundedness of the strain. By this estimate we can prove existence and uniqueness of
not only weak solutions but also strong solutions. For the problem with the viscosity, we can also show the
large time behavior of solutions. It means that the omega - limit set of the solution orbit is not empty and
included to the set of steady solutions. The key in the proof is a priori estimate for center of mass globally
in time. This is a joint work with Toyohiko AIKI (Japan Women`s University), Japan. |
|