Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics

The conservation laws for second order scalar partial differential equations and systems of partial differential equations which occur in fluid mechanics are constructed using different approaches. The direct method, Noether’s theorem, the characteristic method, the variational approach (multiplier approach) for arbitrary functions as well as on the solution space, symmetry conditions on the conserved quantities, the direct construction formula approach, the partial Noether approach and the Noether approach for the equation and its adjoint are discussed and explained with the help of an illustrative example on a non-linear field equation describing the relaxation to a Maxwellian distribution. The conservation laws for the non-linear diffusion equation for the spreading of an axisymmetric thin liquid drop, the system of two partial differential equations governing flow in a laminar two-dimensional jet and the system of two partial differential equations governing flow in a laminar radial jet are discussed via these approaches.