Optimality of the specific phase-2 Γ-decay schedule

Establish whether letting the analyticity radius bound Γ(t) decay during phase 2 according to the decay schedule used to derive the phase-2 duration (i.e., the schedule tied to the enstrophy bound v_2(t) that underpins inequality (12) in the paper) is an optimal strategy within the two-phase iterative procedure for guaranteeing space analyticity of Navier–Stokes solutions.

Background

In the proposed two-phase procedure, phase 2 decreases Γ to zero while the enstrophy decreases. The authors analyze a particular decay schedule for Γ that leads to an exponential-type bound on the enstrophy (inequality (12)) and use it to estimate the phase-2 duration.

They explicitly question whether this particular choice of Γ(t) during phase 2 is optimal, noting that faster decay shortens phase 2 but may enhance enstrophy decrease via the negative right-hand side of inequality (8), which becomes more influential as Γ decreases.