On the convergence of the forward-backward scheme for the minimizing total variation flow in $H^{-s}$

In the talk, we consider a gradient flow of the total variation in the negative Sobolev space $H^{-s}$, $s\in[0,1]$, under the periodic boundary condition. We explain how to construct a minimizing sequence based on a dual formulation and discuss issues concerning its convergence. We also show and compare results of numerical experiments for simple initial data and different values of the index s.
This is a join work with Y. Giga (University of Tokyo) and P. Rybka (Warsaw University).