# The initial value problem for the Euler equations of incompressible fluids viewed as a concave maximization problem

## TL;DR

This paper reformulates the initial value problem for incompressible Euler equations as a concave maximization problem, revealing connections to optimal transport and convex integration, and demonstrating existence of relaxed solutions.

## Contribution

It introduces a novel variational formulation of the Euler equations as a concave maximization problem, linking it to optimal transport and convex integration theories.

## Key findings

- Relaxed solutions always exist via the maximization problem.
- Smooth solutions can be recovered from the maximization formulation for short times.
- The approach connects Euler equations to optimal transport and convex analysis.

## Abstract

We consider the Euler equations of incompressible fluids and attempt to solve the initial value problem with the help of a concave maximization problem.We show that this problem, which shares a similar structure with the optimal transport problemwith quadratic cost, in its "Benamou-Brenier" formulation,always admits a relaxed solution that can be interpretedin terms of $sub-solution$ of the Euler equations in the sense of convex integration theory.Moreover, any smooth solution of the Euler equations can be recovered from this maximization problem, at least for short times.