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

Ludmila Petrova


It is known that the Euler and Navier-Stokes equations, which describe flows of ideal and viscid gases, are the set of equations that are modelling 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 in the set of Euler and Navier-Stokes equations 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 system 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 so on), which are described by generalized solutions.
This discloses the mechanism of such processes as emergence  vorticity and turbulence.

These results were obtained using the relation that is deduced from the Euler and Navier-Stokes equations and contains the entropy, which specifies the state of gas-dynamic system.

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Journal of Mathematics Research   ISSN 1916-9795 (Print)   ISSN 1916-9809 (Online)

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