In the talk, we consider a gradient flow of the total variation in the negative Sobolev space H−s, s∈[0,1], under the periodic boundary condition. We derive a dual formulation of a convex variational problem associated with a semi-implicit time discretization of this flow. Based on the forward-backward scheme, we construct a minimizing sequence of a given functional 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 joint work with Y. Giga (University of Tokyo) and P. Rybka (University of Warsaw).