Attainment of the double-logarithmic separation-time lower bound

Determine whether the lower bound of order \(\log\!\log(1/\nu)\) for the Euler–Navier–Stokes separation time can be attained by a nonstationary two-dimensional Euler flow with exponentially growing velocity gradient.

Background

The appendix derives a lower bound for the separation time of nonstationary Euler and Navier–Stokes flows that grows like a double logarithm in the inverse viscosity. The bound is related to the possible exponential growth of the Euler velocity gradient through the accumulated stretching rate.

The authors explicitly leave unresolved whether this rate is sharp. Achieving it would require constructing a nonstationary Euler flow whose velocity gradient grows exponentially, a flow with the specific behavior that the authors state is currently unknown.

References

Whether this lower bound can be achieved is left for future investigation. Indeed, to achieve this rate, one would need a nonstationary Euler flow with exponentially growing velocity gradient, which is currently unknown.

— Fast separation of Euler and Navier--Stokes flows in two dimensions  (2610.20540 - He et al., 14 Oct 2026) in Remark, Appendix A, Section “Lower bound on the separation time”