Journal of
Scientific Research and Studies Vol. 4(11), pp. 304317,
November, 2017
ISSN 23758791
Copyright © 2017
Author(s) retain the
copyright of this article
http://www.modernrespub.org/jsrs/index.htm

Full Length Research Paper

On the 4th Clay
millennium problem: Proof of the regularity of the solutions of
the Euler and NavierStokes equations, based on the conservation
of particles
Constantine E.
Kyritsis
Department of Accounting/Finance,
University of Applied Sciences (TEI) of Epirus, Psathaki Preveza 48100, Greece.
Email: ckiritsi@teiep.gr,
C_kyrisis@yahoo.com
Accepted 30 October, 2017


Abstract 

The majority of the applications of fluid dynamics refer to
fluids that during the flow the particles are conserved. It is
natural to have in mind such physical applications when
examining the 4th Clay Millennium Problem. The assumptions of
the standard formulation of the above problem, although
reflecting the finiteness and conservation of the momentum and
energy, as well as the smoothness of incompressible physical
flows, do not reflect the conservation of particles of the fluid
as local structure. By formulating the later conservation law
and adding it to the hypotheses, it becomes possible to prove
the regularity both for the Euler and NavierStokes equations.
From the physical point of view this may mean that: if a) the
particles like neutrons, electrons and protons remain such
particle during the flow or if atoms exist in the fluid that if
b) the atoms remain atoms of the same atomic number during the
flow or if there are molecules in the fluid, if c) the molecules
remain molecules of the same chemical type during the flow, then
the regularity (smoothness of flow at all times) for the 4th
Clay Millennium problem is provable and holds. The methodology
for such a proof is based on proving that if a Blowup would
exist then at least for a particle range, the total energy would
also converge to infinite (see Propositions 5.1, 5.2) which is a
contradiction to the hypothesis of finite initial energy of the
standard formulation of the 4th Clay Millennium problem.
Key words: Incompressible flows, regularity, NavierStokes
equations, 4th Clay millennium problem. 

Other Journal
■ Modern Research Journal of
Agriculture
