Isolated Singularities for Solutions of the Nonlinear Stationary Navier-Stokes Equations

Abstract

ABSTRACT. The notion for (u, p)

Isolated Singularity for the Stationary Navier—Stokes System

“…In [4] Shapiro showed that if n ≥ 3, u ∈ L β for some β > n, and |u(x)| = o (|x| − n− 1 2 ) and |p(x)| = o (|x| −(n−1) ) as |x| → 0, then {0} is removable. Also he was able to remove the growth condition on the pressure p (see [5]).…”

A removable isolated singularity theorem for the stationary Navier–Stokes equations

“…Hence a regularity result due to Shapiro [13] enables us to conclude that if u satisfies the additional condition…”

“…Then it follows from a removable singularity theorem in [5] that if u ∈ L n+ε (B R ) and p ∈ L n+ε (B R ) for some ε > 0, then (u, p) can be defined at 0 so that it is a smooth solution of (NS) in the whole ball B R , that is, {0} is removable. Next, using the theory of multiple trigonometric series to investigate fine behaviors of solutions near isolated singular points, Shapiro [13,14] showed that the only condition on the velocity u is sufficient to remove the isolated singularities. A different proof of this result may be based on the theories of hydrodynamic potentials and homogeneous harmonic polynomials.…”

Point Singularities of 3D Stationary Navier–Stokes Flows

“…There are a lot of studies on this problem [4,5,8,15,16]. A typical result is to show that, under some conditions, the solution is a very weak solution across the origin without singular forcing supported at the origin (removable singularity), and is regular, i.e., locally bounded, under possibly more assumptions (regularity).…”

“…Dyer-Edmunds [5] proved removable singularity and regularity assuming both u, p ∈ L 3+ε (B 2 ) for some ε > 0. Shapiro [15,16] proved removable singularity and regularity assuming u ∈ L 3+ε (B 2 ) for some ε > 0 and u(x) = o(|x| −1 ) as x → 0, without assumption on p. Choe and Kim [4] proved removable singularity assuming u ∈ L 3 (B 2 ) or u(x) = o(|x| −1 ) as x → 0, and regularity assuming u ∈ L 3+ε (B 2 ) for some ε > 0. Kim and Kozono [8] recently proved removable singularity under the same assumptions as [4], and regularity assuming u ∈ L 3 (B 2 ) or u is small in weak L 3 .…”