On a degenerate elliptic problem arising in the least action principle for Rayleigh-Taylor subsolutions

TL;DR

This paper investigates a degenerate elliptic variational problem linked to the least action principle for averaged solutions of inhomogeneous Euler equations in a Rayleigh-Taylor setup, establishing existence and regularity of minimizers.

Contribution

It derives a functional from differential inclusions related to Euler equations, proves the existence of regular minimizers, and explores the least action principle as a selection criterion for solutions.

Findings

Existence of a minimizer with partial regularity.

Connection between the variational problem and convex integration.

Discussion on the least action principle as a solution selector.

Abstract

We address a degenerate elliptic variational problem arising in the application of the least action principle to averaged solutions of the inhomogeneous Euler equations in Boussinesq approximation emanating from the horizontally flat Rayleigh-Taylor configuration. We give a detailed derivation of the functional starting from the differential inclusion associated with the Euler equations, i.e. the notion of an averaged solution is the one of a subsolution in the context of convex integration, and illustrate how it is linked to the generalized least action principle introduced by Brenier in \cite{Brenier89,Brenier18}. Concerning the investigation of the functional itself, we use a regular approximation in order to show the existence of a minimzer enjoying partial regularity, as well as other properties important for the construction of actual Euler solutions induced by the minimizer. Furthermore, we discuss to what extent such an application of the least action principle to subsolutions can serve as a selection criterion.