Title: Inverse problem on conservation laws,
Abstract: 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.