Some identities related to the Navier-Stokes equations

This paper consists of two parts. The first one studies integral relations to which the solutions of the Navier-Stokes equations or Euler equations satisfy in the case of fluids filling the entire three-dimensional space. The existence of these relations is due to a rapid decrease of the velocity field at infinity (but not too rapid in order that the required asymptotic forms are reproduced with time). Of special interest are the integrals of motion whose density depends quadratically on the velocities or their derivative with respect to the coordinates. Such integrals (conservation laws) for the Navier-Stokes equations were recently found by Dobrokhotov and Shafarevich. In the present paper, new conservation laws are obtained, which are quadratic in the derivatives of the velocity and lead to identities that link the averaged and pulsation characteristics of free turbulent flows.

The second part is devoted to equations of rotationally symmetric motion of a viscous incompressible liquid. There is proposed a new approach to investigation of such kind of motions. It is shown that the projection of momentum equation on the axis of cylindrical coordinate system has a form of conservation law. This gives possibility to introduce a new unknown function instead of the pressure. The resulting system consists of an elliptic equation and two second-order parabolic equations for the stream function and peripheral component of velocity, which is weakly connected with the elliptic equation. Moreover, in comparison with traditional approach, boundary conditions for all sought functions are separated completely. The obtained system has a very rich group of symmetries. On the base of this group, the new exact solutions of the Navier-Stokes equations are constructed.

References:
V.V.Pukhnachov. Integrals of motion of an incompressible fluid occupying the entire space. Journal of Applied Mechanics and Technical Physics, Vol. 45, No. 2, pp. 167-171, 2004.
S.N.Aristov, V.V.Pukhnachov. On equations of the rotationally symmetric motion of a viscous incompressible liquid. Doklady Akademii Nauk, Vol. 394, No.5, pp. 611-614, 2004.