Preface -- Introduction -- Brief History of the Navier-Stokes Problem -- Statement of the Navier-Stokes Problem -- Theory of Some Hyper-Singular Integral Equations -- A Priori Estimates of the Solution to the NSP -- Uniqueness of the Solution to the NSP -- The Paradox and its Consequences -- Logical Analysis of Our Proof -- Appendix 1 - Theory of Distributions and Hyper-Singular Integrals -- Appendix 2 - Gamma and Beta Functions -- Appendix 3 - The Laplace Transform -- Bibliography -- Author's Biography.

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 [greater than or equal to] 0 and v(x,t) = 0). It is shown that if the initial data v0(x) [does not equal] 0, f(x,t) = 0 and the solution to the NSP exists for all t [epsilon] 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) x C(R+) is proved, W21(R3) is the Sobolev space, R+ = [0,[infinity]). 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.