On the existence of a scalar pressure field in the Br\"odinger problem

TL;DR

This paper proves the existence of a scalar pressure field in the entropic regularization of the Brenier problem for incompressible fluids, linking it to a Lagrange multiplier via PDE reformulation.

Contribution

It demonstrates the existence of a pressure field in the regularized Brenier problem, extending the understanding of pressure in fluid dynamics with entropic regularization.

Findings

Existence of a scalar pressure field proven.

Pressure interpreted as a Lagrange multiplier.

Reformulation of the problem in PDE terms.

Abstract

This work deals with the entropic regularization of the Brenier problem for perfect incompressible fluids introduced by Arnaudon, Cruzeiro, L\'eonard and Zambrini. We show that as in the original setting, there exists a scalar pressure field which is interpreted as the Lagrange multiplier associated to the incompressibility constraint. The proof goes through a reformulation of the problem in PDE terms.