Removable time-dependent singularities of solutions to the Stokes equations

Abstract

Let Ω⊂RN and let ξ∈Cα([0,T];Ω) for 0<α≤12. We consider the situation that u=u(x,t) is a classical solution of the Stokes equations in ⋃0<t<T(Ω∖{ξ(t)})×{t}, that is, {ξ(t)}0<t<T is regarded as the time-dependent singularities of u in Ω×(0,T). If u behaves around ξ(t) like |u(x,t)|=o(|x−ξ(t)|2−N+(1/α−2)) as x→ξ(t) uniformly in t∈(0,T), then {ξ(t)}0<t<T is a family of removable singularities of u, which implies that u can be extended as a smooth solution in the whole space and time Ω×(0,T).