Abstract
The goal of this paper is to present the author’s solution to the millennium problem related to the Navier–Stokes problem (NSP) in \(\mathbb{R}^{3}\) without boundaries. It is proved that the NSP is contradictory in the following sense: Assume that the exterior force \(f=f(x,t)=0\) . If 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 \(t\geq 0\) , then one proves that the solution \(v(x,t)\) to the NSP has the property \(v(x,0)=0\) . This paradox (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 \(t>0\) , because the assumption \(v(x,0)\not\equiv 0\) and the conclusion \(v(x,0)=0\) are not compatible. The NSP paradox solves the millennium problem concerning the Navier–Stokes equations. It proves that the solution to the NSP problem 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 \(t\geq 0\) and is equal to zero, \(v(x,t)\equiv 0\) .
