Inverse problem on conservation laws

Roman O. Popovych, Alexander Bihlo

Published 2017-05-09Version 1

The first concise formulation of the inverse problem on conservation laws is presented. 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 considered. We also solve the inverse problem on conservation laws for differential equations admitting an infinite dimensional space of zero-order characteristics. This particular case is further studied in the context of conservative parameterization schemes for the two-dimensional incompressible Euler equations. We exhaustively classify conservative 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.