PDE seminar: An example of Non-finite time stopping solution for 1-harmonic map flow equation

2014-4-21 17:00 - 2014-4-21 18:00
Faculty of Science Building #3, Room 309
Hirotoshi Kuroda (Hokkaido University)
1-harmonic flow is the gradient system of the total variation in the L^2-topology. This system has a very strong diffusivity, so it is well-known that the solution tends to a steady solution in finite time under some boundary conditions.

In this talk we consider 1-harmonic map flow with values in a unit sphere S^2 when initial data is a piecewise constant. Then there is a unique global solution under the Dirichlet condition in the class of piecewise constant functions. Moreover we construct a example of the solution which does not stop in finite time. This fact shows that there is a direction where the diffusion is not very strong for higher dimensional cases.