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 ( , ) to the NSP exists for all = 0 and ( , ) = 0). It is shown that if the initial data 0( ) 0, ( , ) = 0 and the solution to the NSP exists for all R+, then 0( ) := ( , 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 21(R3) × C(R+) is proved, 21(R3) is the Sobolev space, R+ = [0, 8). 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.

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Alexander G. Ramm, Ph.D., was born in Russia, immigrated to the U.S. in 1979, and is a U.S. citizen. He is Professor of Mathematics with broad interests in analysis, scattering theory, inverse problems, theoretical physics, engineering, signal estimation, tomography, theoretical numerical analysis, and applied mathematics. He is an author of 690 research papers, 16 monographs, and an editor of 3 books. He has lectured in many universities throughout the world, presented approximately 150 invited and plenary talks at various conferences, and has supervised 11 Ph.D. students. He was Fulbright Research Professor in Israel and in Ukraine, distinguished visiting professor in Mexico and Egypt, Mercator professor, invited plenary speaker at the 7th PACOM, won the Khwarizmi international award, and received other honors. Recently he solved inverse scattering problems with non-over-determined data and the many-body wave-scattering problem when the scatterers are small particles of an arbitraryshape; Dr. Ramm used this theory to give a recipe for creating materials with a desired refraction coefficient, gave a solution to the refined Pompeiu problem and proved the refined Schiffers conjecture.