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MILLENNIUM PRIZE PROBLEM (NAVIER– STOKES EQUATIONS) IS SOLVABLE BY CLASSICAL METHODS

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continuum...@narod.ru

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Aug 12, 2008, 8:36:29 AM8/12/08
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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://
dxdy.ru/topic4289.html). 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 smoothness of
their solution. It is possible to use known effects about properties
of harmonic functions or representation of the common decision of the
Laplace equation. Read more http://continuum-paradoxes.narod.ru,
the link “Russian pages”, “Sixth Millennium Problems (NAVIER-STOKES
equations) is solvable by classical methods (in Russian)”. Discussion
- http://dxdy.ru/topic12373.html .
Yours faithfully, Alexandr
Kozachok

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