A Solution of the 4th Clay Millennium Problem about the Navier-Stokes Equations

Konstantinos E. Kyritsis

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

In this paper it is solved the 4^th Clay Millennium problem about the Navier-Stokes equations, in the direction of regularity (no blow-up). This is proved for the Navier-Stokes equations for the non-periodic formulation and without external forcing (homogeneous case).The proof is based on discovering a new invariant as a 2D surface density of (rotatory) momentum, derived from the well-known Helmholtz-Kelvin-Stokes velocity circulation invariant. This invariant is indispensable, besides to the ordinary momentum conservation, to prove that there cannot be a blow-up in finite time, of the point vorticities, thus regularity..It is proved that not only there is no Blow-up in finite time but not even at  the  time T=+∞.

https://doi.org/10.31871/WJRR.13.2.8