Theory of hyper-singular integrals and its application to the Navier-Stokes problem

Abstract

In this paper, the convolution integrals (cid:82) t 0 ( t − s ) λ − 1 b ( s ) ds with hyper-singular kernels are considered, where λ ≤ 0 and either b is a smooth function or b belongs to L 1 ( R + ) . For such λ , these integrals diverge classically even for smooth b . These convolution integrals are defined in this paper for negative non-integer values of λ . Integral equations and inequalities are considered with the hyper-singular kernels ( t − s ) λ − 1 + for λ ≤ 0 , where t λ + := 0 for t < 0 . In particular, one is interested in the value λ = − 14 because it is important for the Navier-Stokes problem (NSP). Integral equations of the type b ( t ) = b 0 ( t ) + (cid:82) t 0 ( t − s ) λ − 1 b ( s ) ds , λ ≤ 0 , are also studied. The solution of these equations is investigated, and the existence and uniqueness of the solution is proved for λ = − 14 . The obtained results are applied to the analysis of the NSP in the space R 3 without boundaries. It is proved that the NSP is contradictory in the following sense: even if one assumes that v ( x, 0) > 0 , one proves that the solution v ( x, t ) to the NSP has the property v ( x, 0) = 0 , in general. This paradox shows that the NSP is not a correct description of the fluid mechanics problem and it proves that the NSP does not have a solution, in general.