Isolated singularity for the stationary Navier-Stokes system
The main aim of the paper is to find growth conditions near a singular point for a solution of the stationary Navier-Stokes system, in order to remove the singularity. The method is analogous to the one that is used to remove the isolated singularities of harmonic functions. In the first part of the paper a classical solution to the stationary Stokes system is given, near an isolated singular point, as a power series expansion in terms of homogeneous biharmonic polynomials. By means of this expansion, the authors find a growth condition on the solution for a singularity to be removable. Also, they obtain a new proof of a well-known Liouville-type theorem. In the second part they use the previous results to represent a solution to the stationary Navier-Stokes system as a power series expansion. The nonlinear part is treated as a non-homogeneous term. They are able then to deduce the following removable isolated singularity theorem for solutions to the stationary Navier-Stokes equations (\(d\geq 3\)): If \((u,p)\) is a classical solution in \(B_R-\{0\}\) such that \(|u(x)|=o(|x|^{-\frac{d-1}{2}})\) as \(|x|\to 0\) or \(u\in L^{\frac{2d}{d-1}}(B_R)\), then \((u,p)\) is a distributional solution in \(B_R\). Moreover if \(u\in L^q(B_R)\), with \(q>d\), then \((u,p)\) is smooth in \(B_R\). (A similar statement holds also when \(d=2\)).
scientific article; zbMATH DE number 1484690
scientific article