# On global stability of optimal rearrangement maps

## TL;DR

This paper proves that solutions to a nonlocal vectorial transport equation on bounded domains converge exponentially fast to the optimal rearrangement of initial maps close to convex potentials, establishing global stability.

## Contribution

It provides a rigorous proof of global stability and exponential convergence of solutions to the optimal rearrangement map for initial data near convex potentials.

## Key findings

- Solutions are global in time for initial maps close to convex potentials.
- Solutions converge exponentially fast to the optimal rearrangement.
- The stability result is rigorously justified for the nonlocal transport equation.

## Abstract

We study the nonlocal vectorial transport equation $\partial_ty+ (\mathbb{P} y \cdot \nabla) y=0$ on bounded domains of $\mathbb{R}^d$ where $\mathbb{P}$ denotes the Leray projector. This equation was introduced to obtain the unique optimal rearrangement of a given map $y_0$ as the infinite time limit of the solution with initial data $y_0$ (\cite{AHT, Macthesis, Brenier09}). We rigorously justify this expectation by proving that for initial maps $y_0$ sufficiently close to maps with strictly convex potential, the solutions $y$ are global in time and converge exponentially fast to the optimal rearrangement of $y_0$ as time tends to infinity.