# Inverse problem on conservation laws

## TL;DR

This paper introduces the first explicit formulation of the inverse problem on conservation laws, aiming to determine differential equations that admit specific conservation laws, with applications to fluid dynamics and parameterization schemes.

## Contribution

It provides a novel explicit formulation of the inverse problem on conservation laws and applies it to classify conservative parameterization schemes for Euler equations.

## Key findings

- Classified conservative first-order parameterization schemes for eddy-vorticity flux.
- Solved inverse problem for equations with infinite-dimensional conservation law spaces.
- Applied results to derive closed, averaged Euler equations with conserved quantities.

## Abstract

The explicit formulation of the general inverse problem on conservation laws is presented for the first time. In this problem one aims to derive the general form of systems of differential equations that admit a prescribed set of conservation laws. The particular cases of the inverse problem on first integrals of ordinary differential equations and on conservation laws for evolution equations are studied. We also solve the inverse problem on conservation laws for differential equations admitting an infinite dimensional space of zeroth-order conservation-law characteristics. This particular case is further studied in the context of conservative first-order parameterization schemes for the two-dimensional incompressible Euler equations. We exhaustively classify conservative first-order parameterization schemes for the eddy-vorticity flux that lead to a class of closed, averaged Euler equations possessing generalized circulation, generalized momentum and energy conservation.