Optimal choice of enstrophy amplification factors in phase 1

Determine the sequence of enstrophy amplification factors a_j > 1 used at the end of phase 1 in the two-phase iterative procedure for space-analytic solutions to the three-dimensional incompressible Navier–Stokes equations on the periodic torus (with modified solution v defined via V_n e^{-Γ|n|}) that maximizes T∗∗, the total guaranteed time of space analyticity equal to the sum over iterations of the durations of phase 1 (with Γ(t) = ν^{-1} t and enstrophy v_1 increased by a factor a_j) and phase 2 (with Γ decreased to zero while the enstrophy decreases).

Background

The paper introduces a two-phase iterative procedure to estimate a guaranteed interval of space analyticity for solutions to the incompressible Navier–Stokes equation (and analogously for MHD). The procedure alternates between: (i) phase 1, where the analyticity radius bound Γ grows linearly in time as Γ(t) = ν^{-1} t and the enstrophy v_1 of the modified solution increases by a selected factor a_j > 1; and (ii) phase 2, where Γ is reduced to zero while the enstrophy decreases.

The total guaranteed analyticity time T∗∗ is defined as the sum of the durations of all phases across iterations. The authors raise the problem of choosing the sequence {a_j} to maximize T∗∗, highlighting that this design choice directly affects the length of the guaranteed analyticity interval.