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Optimal control of Γ during phase 2

Identify the optimal time-dependent control law for the analyticity radius bound Γ(t) during phase 2 of the two-phase iterative procedure for Navier–Stokes solutions that best balances the trade-off between shorter phase-2 duration and enhanced enstrophy decrease (given the negative right-hand side of inequality (8) involving v_{3/2} and Γ^{-1/2}), thereby optimizing the guaranteed analyticity interval T∗∗.

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Background

The authors observe that a faster decay of Γ shortens phase 2 but can amplify the enstrophy decrease due to the structure of the nonlinear terms (as captured by inequality (8)), which includes factors v_{3/2} and Γ{-1/2} that are large near the end of phase 2.

This creates a nontrivial control problem: choosing Γ(t) to optimally trade off phase duration against enstrophy reduction. The paper leaves this optimization question explicitly open.

References

If Γ decreases faster, then phase 2 is shorter, resulting in the larger factor e−2νΛ(2)j, but the decrease in the enstrophy is enhanced by the negative r.h.s. of (8), involving two factors: v3/2, which can be large and Γ−1/2, which is large towards the ends of phases 2. So, what is the optimal control of Γ in phase 2?

— Depletion of nonlinearity in space-analytic space-periodic solutions to equations of diffusive magnetohydrodynamics (2404.14429 - Zheligovsky, 18 Apr 2024) in Section 4 (Conclusions)