Existence of a solution of the TV Wasserstein gradient flow
On the flat torus in any dimension we prove existence of a solution to the TV Wasserstein gradient flow equation, only assuming that the initial density $\rho_0$ is bounded from below and above by strictly positive constants. This solution preserves upper and lower bounds of the densities, and shows a certain decay of the BV norm (of the order of $t^{-1/3}$ for $t\to 0$ -- if $\rho_0\notin BV$, otherwise the BV norm is of course bounded -- and of the order of $t^{-1}$ as $t\to\infty$). This generalizes a previous result by Carlier and Poon, who only gave a full proof in one dimension of space and did not consider the case $\rho_0\notin BV$. The main tool consists in considering an approximated TV-JKO scheme which artificially imposes a lower bound on the density and allows to find a continuous-in-time solution regular enough to prove that the lower bounds of the initial datum propagates in time, and study on this approximated equation the decay of the BV norm.