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

Let Omega be a bounded domain in R-N and xi is an element of C-alpha([0, T]; Omega) for 1/N< alpha <= 1. Suppose that u is a smooth solution of the Navier-Stokes equations in boolean OR(0<t<T) (Omega \ {xi(t)}) x {t}, namely, boolean OR(0< t<T){xi(t)} x{t} is supposed to be the set of moving singularities of u in Omega x [0, T]. We prove that if u(x, t) = o(vertical bar x - xi(t)|(-N+beta)) locally uniformly in t is an element of (0, T ) as x -> xi(t) for beta equivalent to max{1/alpha, N - 1}, then u is, in fact, smooth in the whole region Omega x (0, T ). Our result may be regarded as a theorem on removable time-dependent singularities of solutions to the Navier-Stokes equations. (c) 2023 Elsevier Inc. All rights reserved.