The Euler and Navier-Stokes equations revisited

Abstract: The present paper is motivated by recent mathematical work on the incompressible Euler and Navier-Stokes equations, partly having physically problematic results and unrealistic expectations. The Euler and Navier-Stokes equations are rederived here from the roots, starting at the kinetic equation for the distribution function in phase space. The derivation shows that the Euler and Navier-Stokes equations are valid only if the fluid under consideration is an ideal gas, and if deviations from equilibrium are small in a defined sense, thereby excluding fully nonlinear solutions. Furthermore, the derivation shows that the Euler and Navier-Stokes equations are unseparably coupled with an appertaining equation for the temperature, whereby, in conjunction with the continuity equation, a closed system of transport equations is set up which leaves no room for any additional equation, with the consequence that the frequently used incompressibility condition $\nabla\cdot{\bf v}=0$ can, at best, be applied to simplify these transport equations, but not to supersede any of them.