Determination of Time Evolution Equations for Hydrodynamic Variables with Arbitrary Initial Data

Using a time evolution equation of the single-particle distribution, we propose an alternate method for obtaining time evolution equations for hydrodynamic local variables such as density, velocity fields, and kinetic energy. We venture to suggest that our time evolution equations cover both problems of the Boltzmann equation and the Navier-Stokes equation. Our time evolution equations may well be a useful alternative to these two formulations. Indeed it would be very productive if numerous applications of the old Boltzmann equation and Navier-Stokes equation are reformulated with our time evolution approach. It is proposed that the prescription will have numerous applications in hydrodynamics, including solutions to the Navier-Stokes equation, when applied to various pairing potentials between monoatoms, geometry, and initial data.