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Journal of Scientific Research and  Studies

Journal of Scientific Research and Studies Vol. 4(11), pp. 304-317, November, 2017

ISSN 2375-8791

Copyright 2017

Author(s) retain the copyright of this article


Full Length Research Paper

On the 4th Clay millennium problem: Proof of the regularity of the solutions of the Euler and Navier-Stokes 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. E-mail:,


Accepted 30 October, 2017




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 Navier-Stokes 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 Blow-up 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, Navier-Stokes equations, 4th Clay millennium problem.

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