Existence of a solution of the TV Wasserstein gradient flow
Kexin Lin, Filippo Santambrogio
TL;DR
The paper proves the existence of a TV-Wasserstein gradient flow on the flat torus for densities bounded above and below, extending prior 1D results to higher dimensions by introducing a lower-bound regularization in a TV-JKO scheme. It establishes a minimum principle for the approximated flow, demonstrates convergence of the approximated scheme to a weak solution, and derives explicit BV-norm decay rates: $J(\rho(t)) \le \min\{A t^{-1}, B t^{-1/3}, J(\rho_0)\}$ with $A=\kappa/(4\pi^2)$, $B=3\kappa^{1/3}$ and $\kappa=\beta/\alpha$, where $\alpha\le\rho_0\le\beta$. These results yield existence for BV initial data and extend to general densities via BV-approximation, preserving density bounds and providing decay in BV that is robust to initial regularity. The methods blend a detailed BV/optimal-transport framework with a regularized JKO scheme to overcome convexity obstacles in higher dimensions.
Abstract
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 $ρ_0$ 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 $ρ_0\notin BV$, 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 $ρ_0\notin BV$. 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.