Sharpness of partial anomalous regularization in supercritical regimes

Determine whether the partial anomalous regularity exponent obtained for the supercritical parameter regimes of the two-dimensional active scalar equations is merely an artifact of the proof or is sharp.

Background

The paper establishes anomalous regularization for vanishing-viscosity approximations of two-dimensional active scalar equations, including the Euler, SQG, and IPM systems. In some scaling-supercritical parameter regimes, the authors obtain the full regularity exponent $\beta=1-\alpha$, while in other supercritical regimes they prove only a partial gain with $\beta<1-\alpha$.

The authors do not determine whether this lower exponent reflects a genuine limitation of the solutions or only a limitation of their estimates and proof strategy. Resolving this issue would characterize the optimal regularization behavior in the remaining supercritical regimes.