Relaxed solutions for incompressible inviscid flows: A variational and gravitational approximation to the initial value problem

TL;DR

This paper explores a variational approach to modeling incompressible inviscid flows, proposing a gravitational approximation to improve the well-posedness of relaxed Euler equations for initial value problems.

Contribution

It introduces a gravitational Euler-Poisson system as a new relaxed model, enabling the formulation of the initial value problem as a concave maximization for short-term solutions.

Findings

The gravitational approximation yields well-posed initial value problems.

Short-time smooth solutions can be recovered using the proposed model.

The approach connects geometric interpretations with variational methods.

Abstract

Following Arnold's geometric interpretation, the Euler equations of an incompressible fluid moving in a domain D are known to be the optimality equation of the minimizing geodesic problem along the group of orientation and volume preserving diffeomorphisms of D. This problem admits a well-established convex relaxation which generates a set of "relaxed", "multi-stream", version of the Euler equations. However, it is unclear that such relaxed equations are appropriate for the initial value problem and the theory of turbulence, due to their lack of well-posedness for most initial data. As an attempt to get a more relevant set of relaxed Euler equations, we address the multi-stream pressure-less gravitational Euler-Poisson system as an approximate model, for which we show that the initial value problem can be stated as a concave maximization problem from which we can at least recover a large class of smooth solutions for short enough times.