# Neumann heat flow and gradient flow for the entropy on non-convex domains

## TL;DR

This paper proves that for many non-convex domains in various geometric settings, the gradient flow of the Boltzmann entropy exists and matches the heat flow with Neumann boundary conditions, despite the entropy's lack of semiconvexity.

## Contribution

It establishes the existence and equivalence of the entropy gradient flow and heat flow with Neumann boundary conditions on non-convex domains in diverse geometric contexts.

## Key findings

- Gradient flow for entropy exists on non-convex domains.
- The entropy gradient flow coincides with the heat flow with Neumann boundary conditions.
- Results apply to subsets in Euclidean spaces, Riemannian manifolds, and RCD-spaces.

## Abstract

For large classes of non-convex subsets $Y$ in ${\mathbb R}^n$ or in Riemannian manifolds $(M,g)$ or in RCD-spaces $(X,d,m)$ we prove that the gradient flow for the Boltzmann entropy on the restricted metric measure space $(Y,d_Y,m_Y)$ exists - despite the fact that the entropy is not semiconvex - and coincides with the heat flow on $Y$ with Neumann boundary conditions.