Asymptotic behavior of solutions to a crystalline flow
Alina Stancu · Hokkaido Mathematical Journal · 1998
The paper extends an author's result that, under certain technical assump- tions, presents the self-similar solutions to a crystalline flow defined independently by M. Gurtin and J.E .Taylor as attractors for any other solutions.The existing result con- cerns the crystalline flow defined on the space of closed, convex polygons with respect to a reference convex body.On the same space of polygons, the present work shows that even for an arbitrary weight function \gamma defined on a certain set of normal directions, the self-similar solutions are attractors in the following sense: Let the system of equations defining the flow bewhere h_{i}(t)=h(\theta_{i}, t) is the distance from the origin to the i-th side of the evolving convex polygon, while l_{i}(t)=l(\theta_{i}, t) is the length of the i-th side at the moment t , and \gamma i=\gamma(\theta_{i}) is a strictly positive function on the set of normal directions to the sides of the polygon.Our main result says that if the family of convex polygonal curves which evolve by (*) is normalized to enclose constant area, then for any sequence of times diverging to infinity, there is a convergent subsequence of polygons which converges to the shape of a self-similar solution.Moreover, for a \pi -periodic weight function, there is a unique self- similar solution of the flow which is a global attractor for the family of evolving polygons.