# Regularizing effect of homogeneous evolution equations: case homogeneous order zero

## TL;DR

This paper develops a functional analytical framework showing that solutions to certain homogeneous evolution equations of order zero are regular, with applications to total variational flow and fractional 1-Laplace operators.

## Contribution

It introduces a new theory linking mild and strong solutions for first-order homogeneous operators of order zero, with specific applications to variational and nonlocal operators.

## Key findings

- Mild solutions are shown to be strong solutions with regularity estimates.
- Results apply to total variational flow and fractional 1-Laplace operators.
- Global regularity depends only on initial data and time.

## Abstract

In this paper, we develop a functional analytical theory for establishing that mild solutions of first-order Cauchy problems involving homogeneous operators of order zero are strong solutions; in particular, the first-order time derivative satisfies a global regularity estimate depending only on the initial value and the positive time. We apply those results to the Cauchy problem associated with the total variational flow operator and the nonlocal fractional 1-Laplace operator.