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In fluid mechanics, a lot of authors have been executing their researches to obtain the analytical solutions of Navier-Stokes equations, even for 3D case of compressible gas flow or 3D case of non-stationary flow of incompressible fluid. But there is an essential deficiency of non-stationary solutions indeed. We explore the ansatz of derivation of non-stationary solution for the Navier-Stokes equations in the case of incompressible flow, which was suggested earlier. In general case, such a solution should be obtained from the mixed system of 2 Riccati ordinary differential equations (in regard to the time-parameter t). But we find an elegant way to simplify it to the proper analytical presentation of exact solution (such a solution is exponentially decreasing to zero for t going to infinity). Also it has to be specified that the solutions that are constructed can be considered as a class of perturbation absorbed exponentially as t going to infinity by the null solution., 21 pages, 1 figure; this article is accepted for publication in "Rendiconti del Circolo Matematico di Palermo". Keywords: Navier-Stokes equations, non-stationary incompressible flow, Riccati ODE

Here a generalization of photogravitational restricted 3-body problem to the case of influence of Yarkovsky effect is presented, which is known as the reason of additional infinitesimal acceleration of small bodies in the space (due to anisotropic re-emission of absorbed energy from the sun, and other stellar sources). Asteroid is supposed to move under the influence of gravitational forces from 2 massive bodies (which are rotating around their common center of masses on Kepler's trajectories), as well as under the influence of pressure of light from both the primaries. Analyzing the ODE system of motion, we explore the existence of equilibrium points for a small body (asteroid) in the case when the 2-nd primary is non-oblate spheroid. In such a case, the existence of maximally 256 different non-planar libration points in generalized photogravitational restricted 3-body problem when we take into consideration even a small Yarkovsky effect is proved.

A new presentation of general solution of Navier-Stokes equations is considered here. We consider equations of motion for 3-dimensional non-stationary incompressible flow. The field of flow velocity as well as the equation of momentum should be split to the sum of two components: an irrotational (curl-free) one, and a solenoidal (divergence-free) one. The obviously irrotational (curl-free) part of equation of momentum used for obtaining of the components of pressure gradient. As a term of such an equation, we used the irrotational (curl-free) vector field of flow velocity, which is given by the proper potential (besides, the continuity equation determines such a potential as a harmonic function). The other part of equation of momentum could also be split to the sum of 2 equations: - with zero curl for the field of flow velocity (viscous-free), and the proper Eq. with viscous effects but variable curl. A solenoidal Eq. with viscous effects is represented by the proper Heat equation for each component of flow velocity with variable curl. Non-viscous case is presented by the PDE-system of 3 linear differential equations (in regard to the time-parameter), depending on the components of solution of the above Heat Eq. for the components of flow velocity with variable curl. So, the existence of the general solution of Navier-Stokes equations is proved to be the question of existence of the proper solution for such a PDE-system of linear equations. Final solution is proved to be the sum of 2 components: - an irrotational (curl-free) one and a solenoidal (variable curl) components., 13 pages; Keywords: Keywords: Navier-Stokes equations, incompressible flow, Riccati equations. Appears in International Journal of Fluid Mechanics Research 06/2015

A new quasi-periodic solution of Einstein-Friedman equations is presented here. The equation for law of energy saving is proved to be transformed to the proper Abel ordinary differential equation. The equation for evolution of the density of inter-stellar matter is reduced to linear ODE in the case of arbitrary equation of state. The equation of state in a form of linear connexion between the density and pressure of inter-stellar matter is used for the obtaining of such a solution. Besides, the component of solution for the density of inter-stellar matter is proved to be expressed in term of a proper elliptical integral. The component of solution for the radius of space curvature is expressed depending on the density of inter-stellar matter.