Kozachok A.A., Kiev, Ukraine

Formulated by Clay Mathematics Institute the sixth Millennium Problems about existence and smoothness of solutions of the Navier – Stokes equations periodically was discussed at numerous forums (http://grani.ru/Society/Science/m.112524.html). On recognition of some commentators the complete presentation of problem’s solution can demand about thousand pages for mathematical formulas (http://lib.mexmat.ru/forum/viewtopic.php?t=4289). The author of Official Problem Description–Charles Fefferman has set the task about demonstration of existence and smoothness of the solution, instead of solution’s obtaining. However, the Navier-Stokes equations can be reduced correctly to more simple classical equations of mathematical physics . The problem of an existence proof of solutions of such equations is not so actual.
It is known, divu and divv are identical to infinitesimal magnitude and velocity of the relative modification of a volume element of the strained medium. Therefore divergency of acceleration divw, probably, there is a magnitude, identical to a acceleration of the relative modification of the same volume. In that case for incompressible liquid alongside with requirements divu=0, divv=0 it is necessary to accept divw=0.
The requirement divw=0 for incompressible liquid is formulated by analogy and proved. Operation div will convert the Navier – Stokes equations to the three-dimensional Laplace equation for pressure p=p(x,y,z,t) at some limitations of a mass force vector. The time enters into Laplace equation as parameter.
Laplacian of the Navier – Stokes equations (pressure p=p(x,y,z,t) – harmonic function) and a change of a variable (velocity on acceleration) allow to gain system of conventionally independent integro-differential equations with acceleration’s components w. In that case components of the acceleration for ideal incompressible liquid are harmonic functions too. The change of a variable allows to use boundary conditions of an adhesion of a fluid absolutely correctly. According to this requirement vectors of acceleration on firm immobile boundary line are equal to null.
Conversion of the Navier-Stokes equations to more simple equations has actually removed a problem of an existence proof and smooth finish of their solution on a background. Such demonstration in view of that one of required variables is a harmonic function, it is possible to not fulfil. It is possible to use known effects about properties of harmonic functions or representation of the common decision of the Laplace equation( http://continuum-paradoxes.narod.ru the link « Manual, p.1 », p. 58). More in detail on a site http://continuum-paradoxes.narod.ru the link “Russian pages”, “Sixth Millennium Problems (NAVIER-STOKES equations) is solvable by classical methods (in Russian)”.
Yours faithfully, Alexandr Kozachok