Removable singularities of weak solutions to the Navier-Stokes equations

Consider the Navier-Stokes equations in Ω × (0,T), where Ω is a domain in R³. We show that there is an absolute constant ε₀ such that every weak solution u with the property that sup_t∈(a,b) ∥u(t)∥_L3W(D) ≤ ε₀ is necessarily of class C^∞ in the space-time variables on any compact subset of D x (a, b), where D ⊂⊂ Ω and 0 < a < b < T. As an application, we prove that if the weak solution u behaves around (x₀, t₀) ∈ Ω × (0,T) like u(x,t) = 0(|x - x₀l^-1) as x → x₀ uniformly in t in some neighbourhood of t₀, then (x₀,t₀) is actually a removable singularity of u.