Existence of a superior strategy to alternating Γ growth and decay

Investigate whether there exists a strategy superior to the alternating regime that grows Γ from 0 in phase 1 and decays Γ to 0 in phase 2 across iterations, in terms of achieving a longer guaranteed time of space analyticity T∗∗ for Navier–Stokes solutions.

Background

The two-phase procedure in the paper alternates between Γ growth from 0 (phase 1) and Γ decay to 0 (phase 2). While this scheme expands the guaranteed analyticity interval compared to using only phase 1, the authors ask whether a different control or scheduling strategy for Γ could yield an even better guaranteed existence time.

This question invites exploration of alternative control paradigms beyond the simple alternation, potentially including non-monotone or adaptive Γ schedules.