##### Title : Solution of the millennium problem concerning the Navier Stokes equations

#### Abstract:

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

v0 + (v, ∇)v = −∇p + ν ?v + f, x ∈ R3 , t ≥ 0, ∇ · v = 0, v(x, 0) = v0(x),

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

The aim of this talk is to analyse the Navier-Stokes problem (NSP) in R3 without boundaries. It is proved that the NSP is contradictory in the following sense:

I**f one assumes that the initial data** v(x, 0) 6≡ 0, ∇ · v(x, 0) = 0 **and the solution to the NSP exists for all** t ≥ 0, **then one proves that the solution** v(x, t) t**o the NSP has the property **v(x, 0) = 0.

This paradox (**the NSP paradox**) shows that the NSP is not a correct de- scription of the fluid mechanics problem for incompressible viscous fluid and the NSP does not have a solution defined for all t > 0. In the exceptional case, when the data are equal to zero, the solution v(x, t) to the NSP does exist for all t ≥ 0 and is equal to zero, v(x, t) ≡ 0.

These results are proved in [1]–[4].

These results solve one of the millennium problems.