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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

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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 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 Navier-Stokes 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 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 smooth and globally extended, called the regularity of the Navier-Stokes equations in 3 dimensions. We are concerned of course with solutions of the Navier-Stokes equations as in the initial Schwartz data in Fefferman (2006) that are smooth at least in a small time interval [0,t) otherwise the well-known proof of uniqueness of the solutions for the Navier-Stokes equations would not hold and the millennium problem would be considered ill-posed . The corresponding case of regularity in 2 dimensions has long ago been shown to hold, but the 3-dimensionality 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 3-dimensional 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 Navier-Stokes 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 3-ball, 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 Navier-Stokes equations.

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

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Vol. 9 Issue 1

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