No Finite Time Blowup for 3D Incompressible Navier Stokes Equations via Scaling Invariance

Abstract

The 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.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 Prékopa-Leindler and Gagliardo-Nirenberg inequalities it is shown that for any non zero measure set in the form of cube subset of 3D there is no finite time blowup for the starred velocity for large dimension of cube and small δ .In particular vortices are shown to exist and it is shown that zero is in the attractor of the 3D Navier-Stokes equations.

Locations

• Mathematics and Statistics - View - PDF