No Finite Time Blowup for Incompressible Navier Stokes Equations via Scaling Invariance (original) (raw)

A closely related problem to The Clay Math Institute "Navier-Stokes, breakdown of smooth solutions here on an arbitrary cube subset of three dimensional space with periodic boundary conditions is examined. The incompressible Navier-Stokes Equations are presented in a new and conventionally different way here, by naturally reducing them to an operator form which is then further analyzed. It is shown that a reduction to a general 2D N-S system decoupled from a 1D non-linear partial differential equation is possible to obtain. This is executed using integration over n-dimensional compact intervals which allows decoupling. Here we extract the measure-zero points in the domain where singularities may occur and are left with a pde that exhibits finite time singularity. The operator form is considered in a physical geometric vorticity case, and a more general case. In the general case, the solution is revealed to have smooth solutions which exhibit finite-time blowup on a fine measure zero set and using the Gagliardo-Nirenberg inequalities it is shown that for any non zero and sufficiently large measure set in the form of cube subset of 3D there is finite time blowup for the non-starred parametrized velocity in the z-direction of flow. It is proposed that for the non-dimensional quantity δ used in the reparametrization of the Navier-Stokes equations, the starred velocity in the z * direction does not blowup and this corresponds to δ approaching infinity.