Removability of time-dependent singularities of solutions to the Navier-Stokes equations

Abstract

Let Ω be a bounded domain in RN and ξ∈Cα([0,T];Ω) for 1/N<α≤1. Suppose that u is a smooth solution of the Navier-Stokes equations in ⋃0<t<T(Ω∖{ξ(t)})×{t}, namely, ⋃0<t<T{ξ(t)}×{t} is supposed to be the set of moving singularities of u in Ω×[0,T]. We prove that ifu(x,t)=o(|x−ξ(t)|−N+β)locally uniformly in t∈(0,T) as x→ξ(t) for β≡max⁡{1/α,N−1}, then u is, in fact, smooth in the whole region Ω×(0,T). Our result may be regarded as a theorem on removable time-dependent singularities of solutions to the Navier-Stokes equations.