# On the total variation Wasserstein gradient flow and the TV-JKO scheme

## TL;DR

This paper investigates the JKO scheme for total variation, characterizes its optimizers, and proves convergence to a nonlinear fourth-order PDE under certain boundedness conditions, with results specific to one-dimensional and radially symmetric cases.

## Contribution

It provides a detailed analysis of the TV-JKO scheme, including optimizer properties and convergence results, extending understanding of total variation gradient flows.

## Key findings

- Characterization of optimizers for the TV-JKO scheme
- Proof of maximum and minimum principles in certain cases
- Convergence to a fourth-order nonlinear PDE with bounded density assumptions

## Abstract

We study the JKO scheme for the total variation, characterize the optimizers, prove some of their qualitative properties (in particular a form of maximum principle and in some cases, a minimum principle as well). Finally, we establish a convergence result as the time step goes to zero to a solution of a fourth-order nonlinear evolution equation, under the additional assumption that the density remains bounded away from zero. This lower bound is shown in dimension one and in the radially symmetric case.