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

Abstract

In this paper the convolution integrals $\int_0^t(t-s)^{λ-1}b(s)ds$ with hyper-singular kernels are considered, where $λ\le 0$ and $b$ is a smooth or $b$ is in $L^1(\mathbb{R}_+)$. For such $λ$ these integrals diverge classically even for smooth $b$. These convolution integrals are defined in this paper for $λ\le 0$, $λ\neq 0,-1,-2,...$. Integral equations and inequalities are considered with the hyper-singular kernels $(t-s)^{λ-1}_+$ for $λ\le 0$, where $t^λ_+:=0$ for $t<0$. In particular, one is interested in the value $λ=-\frac 14$ because it is important for the Navier-Stokes problem (NSP). Integral equations of the type $b(t)=b_0(t)+ \int_0^t(t-s)^{λ-1}b(s)ds$, $λ\le 0$, are studied. The solution of these equations is investigated, existence and uniqueness of the solution is proved for $λ=-\frac 1 4$. This special value of $λ$ is of basic importance for a study of the Navier-Stokes problem (NSP). The above results are applied to the analysis of the NSP in the space $\mathbb{R}^3$ without boundaries. It is proved that the NSP is contradictory in the following sense: even if one assumes that the initial data $v_0(x):=v(x,0)\not\equiv 0$, $\nabla \cdot v_0(x)=0$ one proves that the solution $v(x,t)$ to the NSP has the property $v(x,0)=0$. 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.