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, $\mathrm{\nabla }\cdot v(x,0)=0$, 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, $\mathrm{\nabla }\cdot v(x,0)=0$. 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, $v(x,t)\equiv 0$.