Methods for deriving conservation laws

Abstract

Systematic methods are used to find all possible conservation laws of a given type for certain systems of partial differential equations, including some from fluid mechanics. The necessary and sufficient conditions for a vector to be divergence-free are found in the form of a system of first order, linear, homogeneous partial differential equations, usually overdetermined. Incompressible, inviscid fluid flow is treated in the unsteady two-dimensional and steady three-dimensional cases. A theorem about the degrees of freedom of partial differential equations, needed for finding conservation laws, is proven. Derivatives of the dependent variables are then included in the divergence-free vectors. Conservation laws for Laplace's equation are found with the aid of complex variables, used also to treat the two-dimensional steady flow case when first derivatives are included in the vectors. Conservation laws, depending on an arbitrary number of derivatives, are found for a general first order quasi-linear equation in two independent variables, using two differential operators, which are associated with the derivatives with respect to the independent variables. Linear totally hyperbolic systems are then treated using an obvious generalization of the above operators.