The main result of this book is a proof of the contradictory nature of the NavierStokes 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 solution𝑣(𝑥,𝑡) to the NSP exists for all𝑡 0 and𝑣(𝑥,𝑡) = 0).

It is shown that if the initial data𝑣0(𝑥) 0,𝑓(𝑥,𝑡) = 0 and the solution to the NSP exists for all𝑡 +, then𝑣0(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 space𝑊21(3) × 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.

PrefaceIntroductionBrief History of the NavierStokes ProblemStatement of the NavierStokes ProblemTheory of Some Hyper-Singular Integral EquationsA Priori Estimates of the Solution to the NSPUniqueness of the Solution to the NSPThe Paradox and its ConsequencesLogical Analysis of Our ProofAppendix 1 Theory of Distributions and Hyper-Singular IntegralsAppendix 2 Gamma and Beta FunctionsAppendix 3 The Laplace TransformBibliographyAuthor's Biography