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 +, then 0( ) := ( , 0) = 0.
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 ���� ϵ 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, ∞). 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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