The Navier-Stokes problem in ℝ³ consists of solving the equations:

where v = v(x, t) is the velocity of the incompressible viscous fluid, p = p(x, t) is the pressure, the density ρ = 1, f = f(x, t) is the force, v₀ = v₀(x) is the initial velocity.

The aim of this talk is to explain and prove the author’s result concerning the Navier-Stokes problem (NSP) in ℝ³ without boundaries.

It is proved that the NSP is contradictory in the following sense:

If one assumes that the initial data and the solution to the NSP exists for all t ≥ 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 on all t ≥ 0.

In the exceptional case, when the data are equal to zero, the solution v(x, t) to the NSP exists for all t ≥ 0 and is equal to zero, v(x, t) ≡ 0.

The results, mentioned above, are proved in the author’s monographs [1], [5] and paper [3].

Our results solve the millennium problem concerning the Navier-Stokes equations, see [5].

These results are based on the author’s theory of integral equations with hyper-singular kernels, see [2], [4].

In paper [6], p.472, Theorem 2, there is a statement that, for f(x, t) = 0 and u₀(x) sufficiently small, the solution to the NSP exists for all t ≥ 0 if m ≤ q, where m is the dimension of the space and the solution is in L^q. In our case m = 3 and q = 2, so the condition m ≤ q does not hold. Therefore, the claim in [6], p. 472, is not applicable in our case.