Global Regularity Solution of the 4th Clay Millennium Problem for the Periodic Navier Stokes Equations (original) (raw)

A rigorous proof of no finite time blowup of the 3D Incompressible Navier Stokes equations in R 3 /Z 3 has been shown by corresponding author of the present work [1]. Smooth solutions for the z−component momentum equation u z assuming the x and y component equations have vortex smooth solutions have been proven to exist, however the Clay Institute Millennium problem on the Navier Stokes equations was not proven for a general enough vorticity form and [1], [3] and references therein do not prove this as previously thought. The idea was to show that Geometric Algebra can be applied to all three momentum equations by adding any two of the three equations and thus combinatorially producing either u x , u y or u z as smooth solutions at a time. It was shown that using the Gagliardo-Nirenberg and Prékopa-Leindler inequalities together with Debreu's theorem and some auxiliary theorems proven in [1] that there is no finite time blowup for 3D Navier Stokes equations for a constant vorticity in the z direction. In part I of the present work it is shown that using Hardy's inequality for u 2 z term in the Navier Stokes Equations that a resulting PDE emerges which can be coupled to auxiliary pde's which give us wave equations in each of the three principal directions of flow. The present work is extended to all spatial directions of flow for the most general flow conditions. In Part II it is shown for the first time that the full system of 3D Incompressible Navier Stokes equations without the above mentioned coupling consists of non-smooth solutions. In particular if u x , u y satisfy a non-constant zvorticity for 3D vorticity ω, then higher order derivatives blowup in finite time but u z remains regular. So a counterexample of the Navier Stokes equations having smooth solutions is shown. A specific time dependent vorticity is also considered.