Journal of
Scientific Research and Studies Vol. 9(1), pp. 816, February, 2022
ISSN 23758791
Copyright © 2022
Author(s) retain the
copyright of this article
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A shorter
solution to the Clay millennium problem about regularity of the
NavierStokes equations
Konstantinos
E. Kyritsis
Department of Accounting and Finance,
University of
Ioannina,
Psathaki Preveza, 48100, Greece.
Email:
ckiritsi@uoi.gr,
C_kyrisis@yahoo.com
Accepted 15 February, 2022

The Clay millennium problem regarding the
NavierStokes equations is one of the seven famous mathematical
problems for which the Clay Mathematics Institute has set a high
monetary award for its solution. It is considered a difficult
problem because it has refused to solve it for almost a whole
century. The NavierStokes equations, which are the equations
that govern the flow of fluids, were formulated long ago in
mathematical physics, before matter was known to be composed of
atoms. So in effect they formulated the old infinitely divisible
material fluids. Although it is known that the set of NavierStokes
equations has a unique smooth local time solution under the
assumptions of the millennium problem, it is not known whether
this solution can always be smooth and globally extended, called
the regularity of the NavierStokes equations in 3 dimensions.
We are concerned of course with solutions of the NavierStokes
equations as in the initial Schwartz data in Fefferman (2006)
that are smooth at least in a small time interval [0,t)
otherwise the wellknown proof of uniqueness of the solutions
for the NavierStokes equations would not hold and the
millennium problem would be considered illposed . The
corresponding case of regularity in 2 dimensions has long ago
been shown to hold, but the 3dimensionality refuses to prove
it. Of course, the natural outcome would be that the regularity
also holds for 3 dimensions. Many people feel that this
difficulty hides our lack of understanding of the 3dimensional
flow laws of incompressible fluids. Compared to the older
solution proposed by Kyritsis (2021a, 2013), this paper presents
a shorter solution to the Clay Millennium problem about the
NavierStokes equations. The longer solution is based on the
equivalence of smooth Schwartz initial data in the original
formulation of the problem with simply connected compact and
smooth boundary initial data (e.g., on a 3ball, see Kyritsis,
2017a). The current short solution is in the context of smooth
Schwartz initial data and is an independent solution logically
different from the previous one. The next strategy is as
follows: (1) from the finite initial energy and energy
conservation, and due to the incompressibility as well as the
conservative field of the pressure forces, we obtain the
regularity in the pressures; (2) from the regularity in the
pressures, we obtain the regularity of the material velocities,
which leads to the regularity of the NavierStokes equations.
Key words: Incompressible flows, regularity, NavierStokes
equations, 4^{th} Clay millennium problem. 