PIMS/IAM Distinguished Lecture: Martin Oberlack

It has long been known that 3D time-dependent Navier-Stokes equations for incompressible fluids admit the classical conservation laws (CL) of mass, momentum, angular momentum and centre-of-mass theorem. For inviscid flows, i.e. Eulers equation, this is extended by the conservation of helicity and energy. Employing the “direct method” (DM) by Anco, Bluman (1997) it has been shown that this set of conservation laws is complete for primitive variables. The DM is a substantial generalization of Noethers theorem and does not rely on a variational principle, and, further, is directly applicable to any type of differential equation, even dissipative ones. With this an additional infinite set of CL for Navier-Stokes equations in vorticity formulation are derived. Various examples are shown. Interesting enough, even more CLs exist for Euler and Navier-Stokes equations in spatially reduced coordinate systems such as for plane, axisymmetric and helically symmetric flows. E.g. an infinite set of CLs for the generalization of helicity has been derived, and, surprisingly, even new CL for plane flows haven been identified.