The main result of this book is a proof of the contradictory nature of the Navier‒Stokes problem (NSP). It is proved that the NSP is physically wrong, and the solution to the NSP does not exist on ℝ] (except for the case when the initial velocity and the exterior force are both equal to zero; in this case, the solution 𝑣(𝑥, 𝑡) to the NSP exists for all 𝑡 >= 0 and 𝑣(𝑥, 𝑡) = 0).

It is shown that if the initial data 𝑣0(𝑥) ≢ 0, 𝑓(𝑥,𝑡) = 0 and the solution to the NSP exists for all 𝑡 ϵ ℝ+, then 𝑣0(𝑥): = 𝑣(𝑥, 0) = 0.

This Paradox proves that the NSP is physically incorrect and mathematically unsolvable, in general. Uniqueness of the solution to the NSP in the space 𝑊21(ℝ3) C(ℝ+) is proved, 𝑊21(ℝ3) is the Sobolev space, ℝ+ = 0, ∞).

Theory of integral equations and inequalities with hyper-singular kernels is developed. The NSP is reduced to an integral inequality with a hyper-singular kernel.