Solution of the Millennium Problem Related to the Navier-Stokes Equations
DOI:
https://doi.org/10.65904/3083-1733.2026.2.11Keywords:
Global Solutions, Integral Equations, Millennium Problem, Hypersingular Integrals, Navier–Stokes EquationsAbstract
This paper contains a solution to the millennium problem related to the Navier-Stokes problem (NSP) in R3 without boundaries. In the first part of this paper the author’s results on hypersingular integrals are presented. By hypersingular integrals we understand the integrals diverging from the classical analysis point of view. We discuss integral equations with such integrals, the Cauchy integrals with densities that are distributions, and the boundary values of analytic functions with distributional density of the Cauchy integrals. Solving integral equations with hypersingular integrals is used in the second part. The second part of the paper is based on the first, and contains an analysis of the Navier-Stokes problem (NSP) in R3 without boundaries. It is shown that a nontrivial solution to NSP cannot exist for all time t > 0. The only global solution is trivial. The NSP paradox is proved. This paradox shows that there cannot exist a solution to the NSP in R3 without boundaries for all t > 0.
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