  This paper addresses a question concerning the behaviour of a sequence of
global solutions to the Navier-Stokes equations, with the corresponding
sequence of smooth initial data being bounded in the (non-energy class) weak
Lebesgue space $L^{3,\infty}$. It is closely related to the question of what
would be a reasonable definition of global weak solutions with a non-energy
class of initial data, including the aforementioned Lorentz space. This paper
can be regarded as an extension of a similar problem regarding the Lebesgue
space $L_3$ to the weak Lebesgue space $L^{3,\infty}$, whose norms are both
scale invariant with the respect to the Navier-Stokes scaling.
