The Navier-Stokes problem. Solution of a millennium problem related to the Navier-Stokes equations

Abstract

The goal of this paper is to present the author's results concerning the Navier-Stokes problem (NSP) in \(\mathbb{R}^3\) without boundaries. It is proved that the NSP is contradictory in the following sense:

Assume (for simplicity only) that the exterior force \(f=f(x,t)=0\). If one assumes that the initial data \(v(x,0)\not\equiv 0\), \(v(x,0)\) is a smooth and rapidly decaying at infinity vector function, \(\nabla \cdot v(x,0)=0\), and the solution to the NSP exists for all \(t\ge 0\), then one proves that the solution \(v(x,t)\) to the NSP has the property \(v(x,0)=0\).

This paradox (the NSP paradox) shows that the NSP is not a correct description of the fluid mechanics problem and the NSP does not have a solution defined for all times \(t>0\). This solves the millennium problem concerning the Navier-Stokes equations: the solution does not exist for all \(t>0\) if \(v(x,0)\not\equiv 0\), \(v(x,0)\) is a smooth and rapidly decaying at infinity vector function, \(\nabla \cdot v(x,0)=0\). In the exceptional case, when the data are equal to zero, the solution \(v(x,t)\) to the NSP exists for all \(t\ge 0\) and is equal to zero, \(v(x,t)\equiv 0\).