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 + (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 +, 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( 3) x C( +) is proved, 𝑊21( 3) is the Sobolev space, + = [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.