Regularity criterion on weak solutions to the navier-stokes equations

In a domain ω ⊂ Rⁿ, consider a weak solution u of the Navier-Stokes equations in the class u ε L∞(0, T;Lⁿ(ω)). If lim sup_t→t*-0 ||u(t)||ⁿ_n-||u(t_*)||ⁿ_n is small at each point of t_* ε (0, T), then u is regular on ω̄ × (0, T). As an application, we give a precise characterization of the singular time; i.e., we show that if a solution u of the Navier-Stokes equations is initially smooth and loses its regularity at some later time T_* < T, then either lim sup_t→T*-0 ||u(t)||Lⁿ(ω) = +∞, or u(t) oscillates in Lⁿ(ω) around the weak limit wlim_{t→ T*-0} u(t) with sufficiently large amplitude. Furthermore, we prove that every weak solution u of bounded variation on (0, T) with values in Lⁿ(ω) becomes regular.