Isolated Singularity for the Stationary Navier—Stokes System

Abstract

In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in $ B_R \setminus \{0\} $ , $ n \geq 3 $ and $ |u(x)| = o\,(|x|^{-(n - 1)/2}) $ as $ |x| \to 0 $ or $ u \in L^{2n/(n - 1)}(B_R) $ , then (u, p) is a distribution solution and if in addition, $ u \in L^{\beta}(B_R) $ for some $ \beta > n $ then ( u, p) is smooth in B R .