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Journal of Scientific Research and  Studies

Journal of Scientific Research and Studies Vol. 9(1), pp. 8-16, February, 2022

ISSN 2375-8791

Copyright 2022

Author(s) retain the copyright of this article


Full Length Research Paper

A shorter solution to the Clay millennium problem about regularity of the Navier-Stokes equations


Konstantinos E. Kyritsis

Department of Accounting and Finance, University of Ioannina, Psathaki Preveza, 48100, Greece. E-mail:,


Accepted 15 February, 2022




The Clay millennium problem regarding the Navier-Stokes equations is one of the seven famous difficult and significant mathematical problems. Although it is known that the set of Navier-Stokes 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 extended for all times smoothly, which is called the regularity (no blow-up) of the Navier-Stokes equations in 3 dimensions. Of course, the natural outcome would be that the regularity also holds for 3 dimensions since we know that it holds in 2 dimensions. Compared to the older solution proposed by Kyritsis (2021a) for the non-periodic setting without external forcing, this paper solves it also for the case with the periodic setting without external forcing. The strategy is based again in discovering new momentum density invariants derived from the well-known Helmholtz-Kelvin-Stokes theorem of the velocity circulation.

Key words: Incompressible flows, regularity, Navier-Stokes equations, 4th Clay millennium problem.

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Vol. 9 Issue 1

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