Continuous dependence of the pressure field with respect to endpoints for ideal incompressible fluids

TL;DR

This paper proves that in a variational model for ideal incompressible fluids, the optimal action and pressure field depend continuously on the initial and final particle positions, measured via Monge-Kantorovich distance.

Contribution

It establishes the H"older continuity of the optimal action and pressure field with respect to the initial-final position law in the Brenier model.

Findings

Optimal action is H"older continuous with respect to the data.

Pressure field varies continuously with initial-final position law.

Results apply to the Brenier variational model for perfect fluids.

Abstract

In the Brenier variational model for perfect fluids, the datum is the joint law of the initial and final positions of the particles. In this paper, we show that both the optimal action and the pressure field are H\"older continuous with respect to this datum metrized in Monge-Kantorovic distance.