Optimal stability estimate at the two-thirds threshold

Determine whether the optimal inviscid-damping estimate represented by equation (est-id) for the three-dimensional stratified Boussinesq equations around Couette flow on \(\mathbb{T}\times\mathbb{R}^2\) can be achieved under an initial-data condition with stability exponent \(\gamma\leq\frac{2}{3}+\delta\), or identify the sharp stability threshold.

Background

The paper proves global stability for unequal but comparable viscosity and thermal diffusivity under an initial-data size of order approximately ν3/4+δ\nu^{3/4+\delta}, while its main theorem establishes a threshold of order ν2/3+\nu^{2/3+} with weaker nonlinear decay estimates. In the stable regime β2>1/4\beta^2>1/4, the authors explain that the optimal linear decay estimates (est-id) can be recovered only with the smaller-data assumption used in the proposition.

The unresolved issue is whether the optimal estimate can persist at the larger data size associated with the $2/3+$ threshold, and, more broadly, whether the sharp nonlinear stability threshold can be identified.

References

It remains open whether the optimal estimate est-id can be achieved under the condition \gamma\leq\frac{2}{3}+\delta, or whether the sharp stability threshold can be further identified.

— The stability threshold for Boussinesq equations around stratified Couette flow with unequal viscosity and thermal diffusivity on $\mathbb{T}\times\mathbb{R}^2$  (2609.29916 - Chen et al., 24 Sep 2026) in Remark following Proposition, immediately after Proposition (the proposition establishing (est-id))