Integrability and the Properties of Solutions to Euler and Navier- Stokes Equations

It is known that the Euler and Navier-Stokes equations, which describe flows of ideal and viscid gases, are the
set of equations of the conservation laws for energy, linear momentum and mass. As it will be shown, the
integrability and properties of the solutions to the Euler and Navier-Stokes equations depend, firstly, on the
consistency of equations of the conservation laws and, secondly, on the properties of conservation laws.
It was found that the Euler and Navier-Stokes equations have solutions of two types, namely, the solutions that
are not functions (depend not only on coordinates) and generalized solutions that are functions but realized
discretely and hence, functions or their derivatives have discontinuities. A transition from the solutions of first
type to generalized solutions describes the process of transition of gas-dynamic medium from non-equilibrium
state to the locally-equilibrium one. Such a process is accompanied by the emergence of any observable
formations (such as waves, vortices, turbulent pulsations and soon). This discloses the mechanism of such
processes as emergence vorticity and turbulence.
Such results were obtained when studying the equations the conservation laws for energy and linear momentum,
which turned out to be inconsistent, due to the non-commutativity of the conservation laws.